Robust Control

2
Robust Control
Ad Damen and Siep Weiland
(tekst bij het college:
Robuuste Regelingen
5P430, najaarstrimester)
Measurement and Control Group
Department of Electrical Engineering
Eindhoven University of Technology
P.O.Box 513
5600 MB Eindhoven
Draft version of August 23, 2001

4
Indien een mondelinge nabespreking door de grootte van de groep niet mogelijk is
kan gedurende de tentamenperiode dit college worden afgesloten met een `take home'
tentamen.
Computers
Preface
Om praktische ervaring op te doen met het ontwerp van robuuste regelsystemen zal voor
enkele opgaven in de eerste helft van het trimester, alsmede voor de te presenteren computersimulaties
gebruik worden gemaakt van de Robust Control Toolbox in MATLAB.
Deze software is door de vakgroep Meten en Regelen op commerciele basis aangekocht
voor onderzoeksdoeleinden. Heel nadrukkelijk wordt er op gewezen dat het niet is toegestaan
software te kopieren. Een beknopte en elementaire handleiding over MATLAB is op
uitleenbasis beschikbaar bij het secretariaat van de vakgroep Meten en Regelen (EH-4.34).
Computerapparatuur met deze applicatie is beschikbaar op onderstaande lokaties.
Opzet
1. E-hoog zaal 6.05. (6 pc's, alleen 's morgens beschikbaar).
2. W-hoog zalen 2.141 (486) en 3A.014 (AT), buiten de voor andere colleges en praktika
gereserveerde uren.
3. Open Shop RC.
Voor enkele computersimulatieopgaven zal gebruik worden gemaakt van een menu gestuurd
pakket voor H1-regelaarontwerp. Uitleg hierover zal tijdens een van de colleges (chapter
13) worden gegeven en een handleiding voor deze applicatie is eveneens op uitleenbasis
beschikbaar.
Beoordeling
Dit college heeft het karakter van een werkgroep. Dit betekent dat u geen kant en klare
portie `wetenschap' ter bestudering krijgt aangeboden, maar dat van u een actieve participatie
zal worden verwacht indevorm van bijdragen aan discussies en presentaties. In dit
college willen we een overzicht aanbieden van moderne, deels nog in ontwikkeling zijnde,
technieken voor het ontwerpen van robuuste regelaars voor dynamische systemen.
In de eerste helft van het trimester zal de theorie over robuust regelaarontwerp aan
de orde komen in reguliere hoorcolleges. Als voorkennis is vereist de basis klassieke regeltechniek
en wordt aanbevolen kennis omtrent LQG-control en matrixrekening/functionaal
analyse. In het college zal de nadruk liggen op het toegankelijk maken van robuust regelaarontwerp
voor regeltechnici en niet op een uitputtende analyse van de benodigde mathematiek.
Gedurende deze periode zullen zes keer oefenopgaven worden verstrekt die bedoeld
zijn om u ervaring op te laten doen met zowel theoretische als praktische aspecten
m.b.t. dit onderwerp. De instruktieopgaven zullen worden gecorrigeerd en zullen voor de
eindbeoordeling meetellen volgens een bonussysteem (zie `beoordeling' hieronder).
De bruikbaarheid en de beperkingen van de theorie zullen vervolgens worden getoetst
aan diverse toepassingen die in de tweede helft van het college door u en uw collegastudenten
worden gepresenteerd en besproken. De opdrachten zijn deels opgezet voor
individuele oplossing en deels voor uitwerking in koppels. Ukunthierbij een keuze maken
uit :
Het eindcijfer E = P + B, waarin P 2 [1 10] een gewogen gemiddelde is van de beoordeling
van uw presentatie, uw discussiebijdrage bij andere presentaties uw verslag en
het eindgesprek. De bonus bestaat uit .1 punt per ingeleverde en voldoende beoordeelde
hoofdstukoefening.
het kritisch evalueren van een artikel uit de toegepast wetenschappelijke literatuur
een regelaarontwerp voor een computersimulatie
Cursusmateriaal
een regelaarontwerp voor een laboratoriumproces.
Naast het collegediktaat is het volgende een beknopt overzicht van aanbevolen literatuur:
[1]
Zeer bruikbaar naslagwerk voorradig in de TUE boekhandel
[2]
Geeft zeker een goed inzicht in de problematiek met methoden om voor SISOsystemen
zelf oplossingen te creeren. Mist evenwel de toestandsruimte aanpak voor
MIMO-systemen.
[3]
Zeer praktijk gericht voor procesindustrie. Mist behoorlijk overzicht.
Meer informatie hierover zal op het eerste college worden gegeven alwaar intekenlijsten
gereed liggen.
Iedere presentatie duurt 45 minuten, inclusief discussietijd. De uren en de verroostering
van de presentaties zullen nader bekend worden gemaakt. Benodigd materiaal voor
de presentaties (sheets, pennen, e.d.) zullen ter beschikking worden gesteld en zijn verkrijgbaar
bij het secretariaat van de vakgroep Meten en Regelen (E-hoog 4.32 's ochtends
geopend). Er wordt verwacht dat u bij tenminste 13 presentaties aanwezig bent en dat u
aktief deelneemt aan de discussies. Een presentielijst zal hiervoor worden bijgehouden.
Over uw bevindingen t.a.v. het door ugekozen onderwerp schrijft ueenkort verslag
(maximaal 4 Aviertjes) dat u dient in te leveren voor aanvang van de tentamenperiode.
Gedurende de tentamenperiode zal dit college worden beeindigd met een bespreking over
de inhoud van dit college en de inhoud van uw verslag.
3

6
5
[4]
Dankzij stormachtige ontwikkelingen in het onderzoeksgebied van H1 regeltheorie,
was dit boek reeds verouderd op het moment van publikatie. Desalniettemin een
goed geschreven inleiding over H1 regelproblemen.
[5]
Goed leesbaar standaardwerk voor vervolgstudie
[6]
Van de uitvinders zelf ::: Aanbevolen referentie voor ;analyse.
[7]
Een boek vol formules voor de liefhebbers van `harde' bewijzen.
[8]
Een korte introductie, die wellicht zonder al te veel details de hoofdlijnen verduidelijkt.
[9]
Robuuste regelingen vanuit een wat ander gezichtspunt.
[12]
Dit boek omvat een groot deel van het materiaal van deze cursus. Goed geschreven,
mathematisch georienteerd, met echter iets te weinig aandacht voor de praktische
aspecten aangaande regelaarontwerp.
[13]
Uitgebreide verhandeling vanuit een wat andere invalshoek: de parametrische benadering.
[14]
Doorwrocht boek geschreven door degenen, die aan de mathematische wieg van
robuust regelen hebben gestaan. Wiskundig georienteerd.
[15]
Dit boek is geschreven in de stijl van het dictaat. Helaas kwam het te laat uit.
Uitstekende voorbeelden, die ook in ons college gebruikt worden.

8 CONTENTS
6 Weighting lters 63
6.1 The use of weighting lters . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.1.2 Singular value loop shaping . . . . . . . . . . . . . . . . . . . . . . . 63
6.1.3 Implications for control design . . . . . . . . . . . . . . . . . . . . . 69
6.2 Robust stabilization of uncertain systems . . . . . . . . . . . . . . . . . . . 70
6.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.2.2 Modeling model errors . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.2.3 The robust stabilization problem . . . . . . . . . . . . . . . . . . . . 75
6.2.4 Robust stabilization: main results . . . . . . . . . . . . . . . . . . . 78
6.2.5 Robust stabilization in practice . . . . . . . . . . . . . . . . . . . . . 80
6.2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Contents
7 General problem. 83
7.1 Augmented plant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.2 Combining control aims. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.3 Mixed sensitivity problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.4 A simple example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
7.5 The typical compromise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
7.6 An aggregated example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.7 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
1 Introduction 11
1.1 What's robust control? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 H1 in a nutshell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 What about LQG? 17
2.1 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
8 Performance robustness and -analysis/synthesis. 97
8.1 Robust performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
8.2 -analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
8.3 Computation of the -norm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
8.3.1 Maximizing the lower bound. . . . . . . . . . . . . . . . . . . . . . . 105
8.3.2 Minimising the upper bound. . . . . . . . . . . . . . . . . . . . . . . 106
8.4 -analysis/synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
8.5 A simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3 Control goals 23
3.1 Stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Disturbance reduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Tracking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 Sensor noise avoidance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.5 Actuator saturation avoidance. . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.6 Robust stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.7 Performance robustness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4 Internal model control 33
4.1 Maximum Modulus Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
9 Filter Selection and Limitations. 115
9.1 A zero frequency set-up. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
9.1.1 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
9.1.2 Actuator saturation, parsimony and model error. . . . . . . . . . . . 116
9.1.3 Bounds for tracking and disturbance reduction. . . . . . . . . . . . . 118
9.2 Frequency dependent weights. . . . . . . . . . . . . . . . . . . . . . . . . . 120
9.2.1 Weight selection by scaling per frequency. . . . . . . . . . . . . . . . 120
9.2.2 Actuator saturation: Wu . . . . . . . . . . . . . . . . . . . . . . . . . 122
9.2.3 Model errors and parsimony. . . . . . . . . . . . . . . . . . . . . . . 123
9.2.4 We bounded by fundamental constraint: S + T = I . . . . . . . . . 126
9.3 Limitations due to plant characteristics. . . . . . . . . . . . . . . . . . . . . 128
9.3.1 Plant gain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
9.3.2 RHP-zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
9.3.3 Bode integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
9.3.4 RHP-poles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
9.3.5 RHP-poles and RHP-zeros . . . . . . . . . . . . . . . . . . . . . . . . 138
9.3.6 MIMO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
9.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5 Signal spaces and norms 39
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2 Signals and signal norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2.1 Periodic and a-periodic signals . . . . . . . . . . . . . . . . . . . . . 40
5.2.2 Continuous time signals . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.2.3 Discrete time signals . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.2.4 Stochastic signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.3 Systems and system norms . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.3.1 The H1 norm of a system . . . . . . . . . . . . . . . . . . . . . . . . 48
5.3.2 The H2 norm of a system . . . . . . . . . . . . . . . . . . . . . . . . 53
5.4 Multivariable generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.4.1 The singular value decomposition . . . . . . . . . . . . . . . . . . . . 56
5.4.2 The H1 norm for multivariable systems . . . . . . . . . . . . . . . . 59
5.4.3 The H2 norm for multivariable systems . . . . . . . . . . . . . . . . 59
5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
7

10 CONTENTS
CONTENTS 9
10 Design example 147
10.1 Plant denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
10.2 Classic control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
10.3 Augmented plant andweight lter selection . . . . . . . . . . . . . . . . . . 154
10.4 Robust control toolbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
10.5 H1 design in mutools. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
10.6 LMI toolbox. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
10.7 designinmutools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
11 Basic solution of the general problem 171
11.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
12 Solution to the general H1 control problem 177
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
12.2 The computation of system norms . . . . . . . . . . . . . . . . . . . . . . . 178
12.2.1 The computation of the H2 norm . . . . . . . . . . . . . . . . . . . . 178
12.2.2 The computation of the H1 norm . . . . . . . . . . . . . . . . . . . 180
12.3 The computation of H2 optimal controllers . . . . . . . . . . . . . . . . . . 182
12.4 The computation of H1 optimal controllers . . . . . . . . . . . . . . . . . . 186
12.5 The state feedback H1 control problem . . . . . . . . . . . . . . . . . . . . 190
12.6 The H1 ltering problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
12.7 Computational aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
12.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
13 Solution to the general H1 control problem 197
13.1 Dissipative dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . 197
13.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
13.1.2 Dissipativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
13.1.3 A rstcharacterization of dissipativity . . . . . . . . . . . . . . . . . 200
13.2 Dissipative systems with quadratic supply functions . . . . . . . . . . . . . 202
13.2.1 Quadratic supply functions . . . . . . . . . . . . . . . . . . . . . . . 202
13.2.2 Complete characterizations of dissipativity . . . . . . . . . . . . . . . 203
13.2.3 The positive real lemma . . . . . . . . . . . . . . . . . . . . . . . . . 205
13.2.4 The bounded real lemma . . . . . . . . . . . . . . . . . . . . . . . . 205
13.3 Dissipativity andH1 performance . . . . . . . . . . . . . . . . . . . . . . . 206
13.4 Synthesis of H1 controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
13.5 H1 controller synthesis in Matlab . . . . . . . . . . . . . . . . . . . . . . . 210

12 CHAPTER 1. INTRODUCTION
Chapter 1
Figure 1.1: Simple block scheme of a controlled system
Introduction
avoidence of actuator saturation The actuator, not explicitly drawn here but taken
as the rst part of process P , should not become saturated but has to operate as a
linear transfer.
1.1 What's robust control?
robustness If the real dynamics of the process change by an amount P , the performance
of the system, i.e. all previous desiderata, should not deteriorate to an
unacceptable level. (In speci c cases it may be that only stability is considered.)
It will be clear that all above desiderata can only be ful lled to some extent. It will be
explained how some constraints put similar demands on the controller C, while others
require contradictory actions, and as a result the nal controller can only be a kind of
compromise. To that purpose it is important that we can quantify the various aims
and consequently weight each claim against the others. As an example, emphasis on the
robustness requirement weakens the other achievable constraints, because a performance
should not only hold for a very speci c process P , where the control action can be tuned
very speci cally, but also for deviating dynamics. The true process dynamics are then
given by:
In previous courses the processes to be controlled were represented by rather simple transfer
functions or state space representations. These dynamics were analysed and controllers
were designed such that the closed loop system was at least stable and showed some desired
performance. In particular, the Nyquist criterion used to be very popular in testing
the closed loop stability and some margins were generally taken into account to stay `far
enough' from instability. It was readily observed that as soon as the Nyquist curve passes
the point {1 too close, the closed loop system becomes `nervous'. It is then in a kind of
transition phase towards actual instability. And, if the dynamics of the controlled process
deviate somewhat from the nominal model, the shift may cause the encirclement of the
point {1 resulting in an unstable system. So, with these margins, stability was e ectively
made robust against small perturbations in the process dynamics. The proposed margins
were really rules of thumb: the allowed perturbations in dynamics were not quantised and
only stability of the closed loop is guarded, not the performance. Moreover, the method
does not work for multivariable systems. In this course we will try to overcome these four
de ciencies i.e. provide very strict and well de ned criteria, de ne clear descriptions and
bounds for the allowed perturbations and not only guarantee robustness for stability but
also for the total performance of the closed loop system even in the case of multivariable
systems. Consequently a de nition of robust control could be stated as:
Ptrue = P + P (1.1)
where now P takes the role of the nominal model while P represents the additive
model perturbation. Their is no way to avoid P considering the causes behind it:
unmodelled dynamics The nominal model P will generally be taken linear, time-invariant
and of low order. As a consequence the real behaviour is necessarily approximated,
since real processes cannot be caught in those simple representations.
Design a controller such that some level of performance ofthecontrolled system
is guaranteed irrespective of changes in the plant dynamics within a prede ned
class.
time variance Inevitably the real dynamics of physical processes change in time. They
are susceptable to wear during aging (e.g. steel rollers), will be a ected by pollution
(e.g. catalysts) or undergo the in uence of temperature (or pressure, humidity ::: )
changes (e.g. day and night uctuations in glass furnaces).
For facilitating the discussion consider a simple representation of a controlled system
in Fig. 1.1.
The control block C is to be designed such that the following goals and constraints
can be realised in some optimal form:
varying loads Dynamics can substantially change, if the load is altered: the mass and
the inertial moment of a robot arm is determined considerably by the load unless
you are willing to pay foravery heavy robot that is very costly in operation.
stability The closed loop system should be stable.
tracking The real output y should follow the reference signal ref.
manufacturing variance A prototype process may be characterised very accurately.
This is of no help, if the variance over the production series is high. Alowvariance
production can turn to be immensely costly, if one thinks e.g. of a CD-player.
Basically, one can produce a drive with tolerances in the micrometer-domain but,
thanks to control, we can be satis ed with less.
disturbance rejection The output y should be free of the in uences of the disturbing
noise.
sensor noise rejection The noise introduced by the sensor should not a ect the output
y.
11

14 CHAPTER 1. INTRODUCTION
1.2. H1 IN A NUTSHELL 13
enough to tackle this problem. However, the intermediate popularity and evolution of the
LQG-design in time domain was not in vain, as we will elucidate in the next chapter 2 and
in the discussion of the nal solution in chapters 11 and 13. It will then follow thatLQG
is just one alternative inavery broad set of possible robust controllers each characterised
by their own signal and system spaces. This may appear very abstract at the moment but
these normed spaces are necessary to quantify signals and transfer functions in order to
be able to compare and weight the various control goals. The de nitions of the various
normed spaces are given in chapter 5 while the translation of the various control goals is
described in detail in chapter 3. Here we will shortly outline the whole procedure starting
with a rearrangment in Fig.1.3 of the structure of the problem in Fig.1.1.
limited identi cation Even if the real process were linear and time-invariant, we still
have to measure or identify its characteristics and this cannot be done without an
error. Measuring equipment and identi cation methods, using nite data sets of
limited sample rate, will inevitably be su ering from inaccuracies.
actuators & sensors What has been said about the process can be attributed to actuators
and sensors as well, that are part of the controlled system. One might require
a minimum level of performance (e.g. stability) of the controlled system in case of
e.g. sensor failure or actuator degradation.
In Fig. 1.2 the e ect of the robustness requirement is illustrated.
6
-
P
6
u
-
z
outputs
-
-
?
- -
?
- -
C
-
P
- -
6
-
?
6
6
+
-
+
6;
+
+
+ e y
+
;
+
?
d
r
? - 6+
inputs
Figure 1.2: Robust performance
Figure 1.3: Structure dictated by exogenous inputs and outputs to be minimised
On the left we havegathered all inputs of the nal closed loop system that we do not
know beforehand but that will live in certain bounded sets. These so called exogenous
inputs consist in this case of the reference signal r,the disturbance d and the measurement
noise . These signals will be characterised as bounded by a (mathematical) ball of radius 1
in a normed space together with lters that represent their frequency contents as discussed
in chapter 5. Next, at the right side we haveput together those output signals that have
to be minimised according to the control goals in a similar characterisation as the input
signals. We are not interested in minimising the actual output y (so this is not part of
the output) but only in the way that y follows the reference signal r. Consequently the
error z = r ; y is taken as an output to be minimised. Note also that we have taken
the di erence with the actual y and not the measured error e. As an extra output to be
minimised is shown the input u of the real process in order to avoid actuator saturation.
How strong this constraint is in comparison to the tracking aim depends on the quality
and thus price of the actuator and is going to be translated in forthcoming weightings
and lters. Another goal, i.e. the attenuation of e ects of both the disturbance d and the
measurement noiseis automatically represented by theminimisation of output z. In a
more complicated way also the e ect of perturbation P on the robustness of stability
and performance should be minimised. As is clearly observed from Fig.1.3 P is an extra
transfer between output u and input d . If we can keep the transfer from d to u small by a
proper controller, the loop closed by P won't have much e ect. Consequently robustness
is increased implicitely by keeping u small as we will analyse in chapter 3. Therefor we
In concedance to the natural inclination to consider something as being "better" if
it is "higher", optimal performance is a maximum here. This is contrary to the criteria,
to be introduced later on, where the best performance occurs in the minimum. So here
the vertical axis represents a degree of performance where higher value indicate better
performance. Positive values are representing improvements by the control action compared
to the uncontrolled situation and negative values correspond to deteriorations by
the very use of the controller. For extreme values ;1 the system is unstable and +1 is
the extreme optimist's performance. In this supersimpli ed picture we let the horizontal
axis represent all possible plant behaviours centered around the nominal plant P with a
deviation P living in the shaded slice. So this slice represents the class of possible plants.
If the controller is designed to perform well for just the nominal process, it can really be
ne-tuned to it, but for a small model error P the performance will soon deteriorate
dramatically. We can improve this e ect by robustifying the control and indeed improve
the performance for greater P but unfortunately and inevitably at the cost of the performance
for the nominal model P . One will readily recognise this e ect in manytechnical
designs (cars,bikes,tools, ::: ), but also e.g. in natural evolution (animals, organs, ::: ).
1.2 H1 in a nutshell
The techniques, to be presented in this course, are named H1-control and -analysis/synthesis.
They have beendevelopped since the beginning of the eighties and are, as a matter of fact,
a well quantised application of the classical control design methods, fully applied in the
frequency domain. It thus took about forty years to evolve a mathematical context strong

16 CHAPTER 1. INTRODUCTION
1.2. H1 IN A NUTSHELL 15
1.3 Exercise
d
?
+ y
- Ci
- P - -
+
?
l l
6;
Let the true process be a delay ofunknown value :
Pt = e ;s (1.2)
0 :01 (1.3)
have to quantify the bounds of P again by a proper ball or norm and lters.
At last we have toprovide a linear, time-invariant, nominal model P of the dynamics
of the process that may be a multivariable (MIMO Multi Input Multi Input) transfer.
In the multivariable case all single lines then represent vectors of signals. Provisionally
we will discuss the matter in s-domain so that P is representing a transfer function in
s-domain. In the multivariable case, P is a transfer matrix where each entry is a transfer
function of the corresponding input to the corresponding output. The same holds for the
controller C and consequently the signals (lines) represent vectors in s-domain so that we
can write e.g. u(s) =C(s)e(s). Having characterised the control goals in terms of outputs
to be minimised provided that the inputs remain con ned as de ned, the principle idea
behind the control design of block C now consists of three phases as presented in chapter
11:
Let the nominal model be given by unity transfer:
P =1 (1.4)
1. Compute a controller C0 that stabilises P .
2. Establish around this central controller C0 the set of all controllers that stabilise P
according to the Youla parametrisation.
Let there be some unknown disturbance d additive to the output consisting of a single sine
wave ! =25 ):
3. Search in this last set for that (robust) controller that minimises the outputs in the
proper sense.
d = sin (25 t) (1.5)
By an appropriate controller Ci the disturbance will be reduced and the output will
be:
y(t) =^y sin(25 t + ) (1.6)
De ne the performance of the controlled system in steady state by:
; ln j^yj (1.7)
a) Design a proportional controller C1 = K for the nominal model P , so completely
ignoring the model uncertainty, such thatj^yj is minimal and thus performance
is maximal. Possible actuator saturation can be ignored in this academic example.
Plot the actual performance as a function of .
This design procedure is quite unusual at rst instance so that we start to analyse it for
stable transfers P where we can apply the internal model approach in chapter 4. Afterwards
the original concept of a general solution is given in chapter 11. This historically
rst method is treated as it shows a clear analysis of the problem. In later times improved
solution algorithms have been developped by means of Riccati equations or by means of
Linear Matrix Inequalities (LMI) as explained in Chapter 13. In the next chapter 8 the
robustness concept will be revisited and improved which will yield the -analysis/synthesis.
After the theory, which is treated till here, chapter 9 is devoted to the selection of
appropriate design lters in practice, while in the last chapter 10 an example illustrates
the methods, algorithms and programs. In this chapter you will also get instructions how
to use dedicated toolboxes in MATLAB.
Hint: Analyse the Nyquist plot.
b) Design a proportional controller C2 = K by incorporating the knowledge about
the model uncertainty P = e ;s ; 1 where is unknown apart from its range.
Robust stability is required. Plot again the actual performance as a function of .
c) The same conditions as indicated sub b) but now for an integrating controller
C3 = K=s. If you have expressed the performance as a function of in the form:
; ln j ^ yj = ; ln jX( )+jY ( )j (1.8)
the following Matlab program can help you to compute the actual function and to
plot it:
>> for k=1:100
theta(k)=k/10000
perf(k)=-log(sqrt(X(theta(k)) 2 +Y(theta(k)) 2 )
end
>>plot(theta,perf)

18 CHAPTER 2. WHAT ABOUT LQG?
where u is the control input, y is the measured output, x is the state vector, v is the state
disturbance and w is the measurement noise. This multivariable process is assumed to be
completely detectable and reachable. Fig. 2.1 intends to recapitulate the set-up of the
LQG-control, where the state feedback matrix L and the Kalman gain K are obtained
from the well known citeria to be minimised:
Chapter 2
L = arg min Efx T Qx + u T Rug (2.1)
K = arg min Ef(x ; ^x) T (x ; ^x)g (2.2)
for nonnegative Q and positive de nite R. Certainly, the closed loop LQG-scheme is
nominally stable, but the crucial question is, whether stability is possibly lost, if the real
system, represented by state space matrices fAtBtCtg, does no longer correspond to the
model of the form fA B Cg. The robust stability, which is then under study, can best be
illustrated by anumerical example .
Consider a very ordinary, stable and minimum phase transfer function:
What about LQG?
v w
(2.3)
s +2
(s +1)(s +3)
P (s) =
+
+ y
- - - - -
+ ? +
Bt n ? l
x
I Ct
1
s
-
which admits the following state space representation:
6
6
+
(2.4)
u + v
0 1
_x =
x +
;3 ;4
0
1
y = ; 2 1 x + w
?
At
u
plant
where v and w are independent white noise sources of variances:
- m - 1
s I - C
- m
A
?
6
^x
+
6
+
6
?
+
+
;
e
;L
(2.5)
1225 ;2135
;2135 3721
Efw 2 g =1 Efvv T g =
B
? -
and the control criterion given by:
x + u 2 g (2.6)
Efx T 2800 80 p 35
80 p 35 80
From this last criterion we can easily obtain the state feedback matrix L by solving the
corresponding Riccati equation. If we were able to feed back the real states x, the stability
properties could easily be studied by analysing the looptransfer L(sI ;A) ;1B as indicated
in Fig. 2.2. The feedback loop is then interrupted at the cross at input u to obtain the
?
K
controller
Figure 2.1: Block scheme LQG-control
Figure 2.2: Real state feedback.
Before submerging in all details of robust control, it is worthwhile to show, why the
LQG-control, as presented in the course \modern control theory", is leading to a dead
end, when robustness enters the control goals. Later in this course, we will see, how the
accomplishments of LQG-control can be used and what LQG means in terms of robust
control. At the moment we can only show, how the classical interpretation of LQG gives
no clues to treat robustness. This short treatment is a summarised display of the article
[10], written just before the emergence of H1-control.
Given a linear, time invariant model of a plant in state space form:
loop transfer (LT). Note, that we analyse with the modelparameters fA Bg, while the real
process is supposed to have true parameters fAtBtg. This subtlety is caused by the fact,
that we only have the model parameters fA B Cg available and may assume, that the
_x = Ax + Bu + v
y = Cx + w
17

20 CHAPTER 2. WHAT ABOUT LQG?
19
decreases, the Nyquist curve shrinks to the origin and soon the point {1 is tresspassed,
causing instability.
The problem now is, how to e ect robustness. An obvious idea is to modify the
Kalman gain K in some way, such that the loop transfer resembles the previous loop
transfer, when feeding back the real states. This can indeed be accomplished in case of
stable and minimum phase processes. Without entering into many details, the procedure
is in main lines:
Put K equal to qBW, where W is a nonsingular matrix and q a positive constant. If we
let q increase in the (thus obtained) loop transfer:
L(sI ; A + qBWC
| {z } +BL);1 qBWC
| {z } (sI ; A);1B (2.8)
the underbraced term in the rst inverted matrix will dominate and thus almost completely
annihilate the same second underbraced expression and we are indeed left with the simple
looptransfer L(sI ; A) ;1B. In doing so , it appears, that some observer poles (the real
cause of the problem) shift to the zeros of P and cancel out, while the others are moved
to ;1. In Fig. 2.3 some loop transfers for increasing q have been drawn and indeed the
transfer converges to the original robust loop transfer. However, all that matters here is,
that, by doing so, we have implemented a completely nonoptimal Kalman gain as far as
disturbance reduction is concerned. We are dealing now with very extreme entries in K
which will cause a very high impact of measurement noise w. So we have sacri ced our
optimal observer for obtaining su cient robustness.
Alternatively, we could have taken the feedback matrix L as a means to e ect robustness.
Along similar lines we would then nd extreme entries in L, so that certainly the
actuator would saturate. Then this saturation would be the price for robustness. Next, we
could of course try to distribute the pain over both K and L, but we haveno clear means
to balance the increase of the robustness and the decrease of the remaining performance.
And then, we do not even talk about robustness of the complete performance. On top of
that we havecon ned ourselves implicitly by departing from LQG and thus to the limited
structure of the total controller as given in Fig. 2.1, where the only tunable parameters
are K and L. Conclusively, we thus have to admit, that we rst ought to de ne and
quantify the control aims very clearly (see next chapter) in order to be able to weight
them relatively and then come up with some machinary, that is able to design controllers
in the face of all these weighted aims. And surely, the straightforward approach of LQG
is not the proper way.
Figure 2.3: Various Nyquist curves.
real parameters fAtBtCtg are very close (in some norm). The Nyquistplot is drawn in
Fig. 2.3. You will immediately notice, that this curve is far from the endangering point {1,
so that stability robustness is guaranteed. This is all very well, but in practice we cannot
measure all states directly. We have to be satis ed with estimated states ^x, so that the
actual feedback is brought about according to Fig. 2.4. Check for yourself, that cutting
Figure 2.4: Feedback with observer.
aloopatcross (1) would lead to the same loop transfer as before under the assumption,
that the model and process parameters are exactly the same (then e=0!). Unfortunately,
thefullfeedbackcontroller is as indicated by the dashed box, so that we have tointerrupt
the true loop at e.g. cross (2), yielding the looptransfer:
(2.7)
Ct(sI ; At) ;1 Bt | {z }
process transfer
L(sI ; A + KC + BL) ;1 K
| {z }
model parametersfABCg
All we can do, is substitute the model parameters for the unknown process parameters
and study the Nyquist plot in Fig. 2.3. Amazingly, the robustness is now completely
lost and we even have to face conditional stability: If, e.g. by aging, the process gain

22 CHAPTER 2. WHAT ABOUT LQG?
2.1. EXERCISE 21
2.1 Exercise
v w
P
y
l? - - 1
?
- m -
s+1
6
u
6
C
?
;KL
s+1+K+L
Above blockscheme represents
a process P of rst order disturbed by white state noise v and independent white
measurement noise w. L is the state feedback gain. K is the Kalman observer gain based
upon the known variances of v and w.
a) If we do not penalise the control signal u, what would be the optimal L? Could
this be allowed here?
b) Suppose that for this L the actuator is not saturated. Is the resultant controller
C robust (in stability)? Is it satisfying the 450 phase margin?
c) Consider the same questions when P = 1
s(s+1) and in particular analyse what you
have to compute and how.(Do not try to actually do the computations.) What can
you do if the resultant solution is not robust?

24 CHAPTER 3. CONTROL GOALS
straints can be listed as stability, robust stability and (avoidance of) actuator saturation.
Within the freedom, left by these constraints, one wants to optimise, in a weighted balance,
aims like disturbance reduction and good tracking without introducing too much e ects of
the sensor noise and keeping this total performance on a su cient level in the face of the
system perturbations i.e. performance robustness against model errors. In detail:
Chapter 3
3.1 Stability.
Unless one is designing oscillators or systems in transition, the closed loop system is
required to be stable. This can be obtained by claimingthat, nowhere in the closed loop
system, some nite disturbance can cause other signals in the loop to grow to in nity: the
so-called BIBO-stability from Bounded Input to Bounded Output. Ergo all corresponding
transfers have tobechecked on possible unstable poles. So certainly the straight transfer
between the reference input r and the output y, given by :
Control goals
y = PC(I + PC) ;1 r (3.1)
In this chapter we will list and analyse the various goals of control in more detail. The
relevant transfer functions will be de ned and named and it will be shown, how some
groups of control aims are in con ict with each other. To start with we reconsider the
block scheme of a simple con guration in Fig. 3.1 which is only slightly di erent from
Fig.1.1 in chapter 1.
But this alone is not su cient as, in the computation of this transfer, possibly unstable
poles may vanish in a pole-zero cancellation. Another possible input position of stray
signals can be found at the actual input of the plant, additive to what is indicated as x
(think e.g. of drift of integrators). Let us de ne it by dx. Then also the transfer of dx to
say y has to be checked for stability whichtransfer is given by:
(3.2)
y =(I + PC) ;1 Pdx = P (I + CP) ;1 dx
Consequently for this simple scheme we distinguish four di erent transfers from r and dx
to y and x, because a closer look soon reveals that inputs d and are equivalent torand outputs z and u are equivalent toy.
Figure 3.1: Simple control structure
3.2 Disturbance reduction.
Without feedback the disturbance d is fully present in the real output y. By means of the
feedback the e ect of the disturbance can be in uenced and at least be reduced in some
frequency band. The closed loop e ect can be easily computed as read from:
d (3.3)
y = PC(I + PC) ;1 (r ; )+(I + PC) ;1
| {z }
S
The underbraced expression represents the Sensitivity S of the output to the disturbance
thus de ned by:
(3.4)
Notice that we havemade the sensor noise explicit in . Basically, the sensor itself has
a transfer function unequal to 1, so that this should be inserted as an extra block in the
feedback scheme just before the sensor noise addition. However, a good quality sensor has
a at frequency response for a much broader band than the process transfer. In that case
the sensor transfer may be neglected. Only in case the sensor transfer is not su ciently
broadbanded (easier to manufacture and thus cheaper), a proper block has to be inserted.
In general one will avoid this, because the ultimate control performance highly depends
on the quality of measurement: the resolution of the sensor puts an upper limit to the
accuracy of the output control as will be shown.
The process or plant (the word "system" is usually reserved for the total, controlled
structure) incorporates the actuator. The same remarks, as made for the sensor, hold for
the actuator. In general the actuator will be made su ciently broadbanded by proper
control loops and all possibly remaining defects are supposed to be represented in the
transfer P . Actuator disturbances are combined with the output disturbance d by computing
or rather estimating its e ect at the output of the plant. Therefor one should know
the real plant transfer Pt consisting of the nominal model transfer P plus the possible
additive model error P . As only the nominal model P and some upper bound for the
model error P is known, it is clear that only upper bounds for the equivalent of actuator
disturbances in the output disturbances d can be established. The e ects of model errors
(or system perturbations) is not yet made explicit in Fig. 3.1 but will be discussed later
in the analysis of robustness.
Next we will elaborate on various common control constraints and aims. The con-
S =(I + PC) ;1
If we want to decrease the e ect of the disturbance d on the output y, we thus have to
choose controller C such that the sensitivity S is small in the frequency band where d has
most of its power or where the disturbance is most \disturbing".
3.3 Tracking.
Especially for servo controllers, but in fact for all systems where a reference signal is
involved, there is the aim of letting the output track the reference signal with a small
23

26 CHAPTER 3. CONTROL GOALS
3.4. SENSOR NOISE AVOIDANCE. 25
The relevant (underbraced) transfer is named control sensitivity for obvious reasons and
symbolised by R thus:
error at least in some tracking band. Let us de ne the tracking error e in our simple
system by:
R = C(I + PC) ;1 (3.10)
(3.5)
| {z }
T
e def
= r ; y =(I + PC) ;1
(r ; d)+PC(I + PC) ;1
| {z }
S
In order to keep x small enough we have tomakesure that the control sensitivity R is small
in the bands of r, and d. Of course with proper relative weightings and \small" still to
be de ned. Notice also that R is very similar to T apart from the extra multiplication by
P in T . We willinterprete later that this P then functions as an weighting that cannot be
in uenced by C as P is xed. So R can be seen as a weighted T and as such the actuator
saturation claim opposes the other aims related to S. Also in LQG-design we have met
this contradiction inamoretwo-faced disguise:
Note that e is the real tracking error and not the measured tracking error observed as
signal u in Fig. 3.1, because the last one incorporates the e ect of the measurement
noise substantially di erently. In equation 3.5 we recognise (underbraced) the sensitivity
as relating the tracking error to both the disturbance and the reference signal r. It is
therefore also called awkwardly the \inverse return di erence operator". Whatever the
name, it is clear that we have to keep S small in both the disturbance and the tracking
band.
Actuator saturation was prevented by proper choice of the weights R and Q in the
design of the state feedback for disturbance reduction.
The e ect of the measurement noise was properly outweighted in the observer design.
3.4 Sensor noise avoidance.
Also the stability was stated in LQG, but its robustness and the robustness of the total
performance was lacking and hard to introduce. In this H1- context this comes quite
naturally as follows:
3.6 Robust stability.
Without any feedback it is clear that the sensor noise will not haveanyin uence on the real
output y. On the other hand the greater the feedback the greater its e ect in disrupting
the output. So we have towatch that in our enthousiasm to decrease the sensitivity, we are
not introducing too much sensor noise e ects. This actually reminiscences to the optimal
Kalman gain. As the reference r is a completely independent signal, just compared with
y in e, we may as well study the e ect of on the tracking error e in equation 3.5. The
coe cient (relevant transfer) of is then given by:
Robustness of the stability in the face of model errors will be treated here rather shortly
as more details will follow in chapter 5. The whole concept is based on the so-called
small gain theorem which trivially applies to the situation sketched in Fig. 3.2 . The
(3.6)
T = PC(I + PC) ;1
and denoted as the complementary sensitivity T . This name is induced by the following
simple relation that can easily be veri ed:
S + T = I (3.7)
and for SISO (Single Input Single Output) systems this turns into:
Figure 3.2: Closed loop with loop transfer H.
S + T =1 (3.8)
stable transfer H represents the total looptransfer in a closed loop. If we require that the
modulus (amplitude) of H is less than 1 for all frequencies it is clear from Fig. 3.3 that the
polar curve cannot encompass the point -1and thus we know from the Nyquist criterion
that the loop will always constitute a stable system. So stability is guaranteed as long as:
This relation has a crucial and detrimental in uence on the ultimate performance of the
total control system! If we want tochoose S very close to zero for reasons of disturbance
and tracking we are necessarily left with a T close to 1 which introduces the full sensor
noise in the output and vice versa. Ergo optimality will be some compromise and the
more because, as we will see, some aims relate to S and others to T .
k H k1 def
= sup jH(j!)j < 1 (3.11)
!
\Sup" stands for supremum which e ectively indicates the maximum. (Only in case that
the supremum is approached at within any small distance but never really reached it is
not allowed to speak of a maximum.) Notice that we haveused no information concerning
the phase angle which istypically H1. In above fomula we get the rst taste of H1 by
the simultaneous de nition of the in nity norm indicated by k : k1. More about this in
chapter 5 where we also learn that for MIMO systems the small gain condition is given
by:
3.5 Actuator saturation avoidance.
The input signal of the actuator is indicated by x in Fig. 3.1 because the actuator was
thought to be incorporated into the plant transfer P . This signal x should be restricted
to the input range of the actuator to avoid saturation. Its relation to all exogenous inputs
is simply derived as:
x =(I + CP) ;1 C(r ; ; d) =C(I + PC) ;1(r
; ; d) (3.9)
(H(j!)) < 1 (3.12)
k H k1 def
= sup
!
| {z }
R

28 CHAPTER 3. CONTROL GOALS
3.6. ROBUST STABILITY. 27
Schwartz inequality so that we maywrite: k R P k1 k R k1k P k1 (3.13)
Ergo, if we can guarantee that:
(3.14)
1
k P k1
a su cient condition for stability is:
k R k1< (3.15)
If all we require from P is stated in equation 3.13 then it is easy to prove that the
condition on R is also a necessary condition. Still this is a rather crude condition but it
can be re ned by weighting over the frequency axis as will be shown in chapter 5. Once
again from Fig. 3.5 we recognise that the robustness stability constraint e ectively limits
the feedback from the point, where both the disturbance and the output of the model
error block P enter, and the input of the plant such thatthe loop transfer is less than
one. The smaller the error bound 1= the greater the feedback can be and vice versa!
We so analysed the e ect of additive model error P . Similarly we can study the
e ect of multiplicative error which isvery easy if we take:
Figure 3.3: Small gain stability in Nyquist space
The denotes the maximum singular value (always real) of the transfer H (for the !
under consideration).
All together, these conditions may seem somewhat exaggerated, because transfers, less
than one, are not so common. The actual application is therefore somewhat \nested" and
very depictively indicated in literature as \the baby small gain theorem" illustrated in
Fig. 3.4. In the upper blockscheme all relevant elements of Fig. 3.1 have been displayed
Ptrue = P + P =(I + )P (3.16)
-
P
6 -
where obviously is the bounded multiplicative model error. (Together with P it evidently
constitutes the additive model error P .) In similar blockschemes we nowget Figs.
3.6 and 3.7. The \baby"-loop now contains explicitly and we notice that transfer P
+
?
< ;equivalent; > ;C(I + PC) ;1
- P -
6
9 +
?
- C - +
P -
6;
-
-
= ;R
?
6 -
6 -
:
+
+
?
?
< ;equivalent; > ;PC(I + PC) ;1
6
- C - P -
+
Figure 3.4: Baby small gain theorem for additive model error.
6;
= ;T
?
in case we have to deal with an additive model error P . We now consider the \baby"
loop as indicated containing P explicitly. The lower transfer between the output and
the input of P , as once again illustrated in Fig. 3.5, can be evaluated and happens to
Figure 3.6: Baby small gain theorem for multiplicative model error.
is somewhat \displaced"out of the additive perturbation block. The result is that sees
itself fed back by (minus) the complementary sensitivity T . (The P has, so to speak,
been taken out of P and adjoined to R yielding T .) If we require that:
(3.17)
1
k k1
the robust stability follows from:
k T k1 k T k1k k1 1 (3.18)
Figure 3.5: Control sensitivity guards stability robustness for additive model error.
yielding as nal condition:
k T k1< (3.19)
Again proper weighting may re ne the condition.
be equal to the control sensitivity R as shown in the lower blockscheme. (Actually we get
a minus sign that can be joined to P . Because we only consider absolute values in the
small gain theorem, this minus sign is irrelevant: it just causes a phase shift of 1800 which
leaves the conditions unaltered.) Now it is easy to apply the small gain theorem to the
total looptransfer H = R P . The in nity norm will appear to be an induced operator
norm in the mapping between identical signal spaces L2 in chapter 5 and as such itfollows

30 CHAPTER 3. CONTROL GOALS
3.7. PERFORMANCE ROBUSTNESS. 29
The second group centers around the complementary sensitivity and requires the controller
C to minimise T :
avoidance of sensor noise
avoidance of actuator saturation
stability robustness
robustness of S
If we were dealing with real numbers only, the choice would be very easy and limited.
Remembering that
Figure 3.7: Complementary sensitivity guards stability robustness for multiplicative model
error
S =(I + PC) ;1 (3.24)
T = PC(I + PC) ;1
(3.25)
3.7 Performance robustness.
a large C would imply a small S but T I while a small C would yield a small T and
S I. Besides, for no feedback, i.e. C = 0, , necessarily T ! 0 and S ! I. This
is also true for very large ! when all physical processes necessarily have a zero transfer
(PC ! 0). So ultimately for very high frequencies, the tracking error and the disturbance
e ect is inevitably 100%.
This may givesome rough ideas of the e ect of C, but the real impact is more di cult
as:
Till now, all aims could be grouped around either the sensitivity S or the complementary
sensitivity T . Once we have optimised some balanced criterion in both S and T and
thus obtained a nominal performance, we wish that this performance is kept more or less,
irrespective of the inevitable model errors. Consequently, performance robustness requires
that S and T change only slightly, if P is close to the true transfer Pt. We can analyse
the relative errors in these quantities for SISO plants:
We deal with complex numbers .
= (1 + PtC) ;1 ; (1 + PC) ;1
(1 + PtC) ;1 = (3.20)
The transfer may bemultivariable and thus we encounter matrices.
= ; T (3.21)
PC
1+PC
= ; P
P
St ; S
St
1+PC
= 1+PC; 1 ; PtC
The crucial quantities S and T involve matrix inversions (I + PC) ;1
and:
The controller C may only be chosen from the set of stabilising controllers.
= PtC(1 + PtC) ;1 ; PC(1 + PC) ;1
PtC(1 + PtC) ;1 = (3.22)
Tt ; T
Tt
It happens that we can circumvent the last two problems, in particular when we are dealing
with a stable transfer P . This can be done by means of the internal model control concept
as shown in the next chapter. We will later generalise this for also unstable nominal
processes.
S (3.23)
1
1+PC
P
Pt
P
=
P
= PtC ; PC
PtC(1 + PC)
As a result we note that in order to keep the relative change in S small we have to take
the product of and T small. The smaller the error bound is, the greater a T can we
a ord and vice versa. But what is astonishingly is that the smaller S is and consequently
the greater the complement T is (see equation 3.7), the less robust is this performance
measured in S. The same story holds for the performance measured in T where the
robustness depends on the complement S. This explains the remark in chapter 1 that
increase of performance for a particular nominal model P decreases its robustness for
model errors. So also in this respect the controller will have to be a compromise!
Summary
We can distinguish two competitive groups because S + T = I. One group centered
around the sensitivity that requires the controller C to be such thatSis \small" and can
be listed as:
disturbance rejection
tracking
robustness of T

32 CHAPTER 3. CONTROL GOALS
3.8. EXERCISES 31
3.8 Exercises
3.1:
u
?
+
+
? ?
- n - C - n - P
- -
6{
?
0
d
+
u
+ y
+
+
r
a) Derive by reasoning that in the above scheme internal stability is guaranteed if
all transfers from u0 and d to u and y are stable.
b) Analyse the stability for
(3.26)
(3.27)
P = 1
1 ; s
1 ; s
C =
1+s
3.2:
d
r +
- C1
- k - +
C2
- ? j - P
-
6{
+ y
?
C3
Which transfers in the given scheme are relevant for:
a) disturbance reduction
b) tracking

34 CHAPTER 4. INTERNAL MODEL CONTROL
Chapter 4
Figure 4.3: Equivalence of the `internal model' and the `conventional' structure.
Internal model control
and from this we get:
C ; CPQ = Q (4.2)
so that reversely:
Q =(I + CP) ;1 C = C(I + PC) ;1 = R (4.3)
In the internal model control scheme, the controller explicitly contains the nominal model
of the process and it appears that, in this structure, it is easy to denote the set of all
stabilising controllers. Furthermore, the sensitivity and the complementary sensitivity
take very simple forms, expressed in process and controller transfer, without inversions. A
severe condition for application is that the process itself is a stable one.
In Fig. 4.1 we repeat the familiar conventional structure while in Fig. 4.2 the internal
Remarkably, the Q equals the previously encountered control sensitivity R! The reason
behind this becomes clear, if we consider the situation where the nominal model P exactly
equals the true process Pt. As outlined before, we haveno other choice than taking P = Pt
for the synthesis and analysis of the controller. Re nement can only occur by using the
information about the model error P that will be done later. If then P = Pt, it is
obvious from Fig. 4.2 that only the disturbance d and the measurement noise are fed
back because the outputs of P and Pt are equal. Also the condition of stability of P
is then trivial, because there is no way to correct for ever increasing but equal outputs
of P and Pt (due to instability) by feedback. Since only d and are fed back, we may
draw the equivalent as in Fig. 4.4. So, e ectively, there seems to be no feedback in this
Figure 4.1: Conventional control structure.
model structure is shown. The di erence actually is the nominal model which isfedby the
Figure 4.4: Internal model structure equivalent for P = Pt.
structure and the complete system is stable, i (i.e. if and only if) transfer Q = R is stable,
because P was already stable by condition. This is very revealing, as we nowsimply have
the complete set of all controllers that stabilise P ! We only need to search for proper
stabilising controllers C by studying the stable transfers Q. Furthermore, as there is no
actual feedback in Fig. 4.4 the sensitivity and the complementary sensitivity contain no
inversions, but take so-called a ne expressions in the transfer Q, which are easily derived
as:
Figure 4.2: Internal model controller concept.
(4.4)
T = PR = PQ
S = I ; T = I ; PQ
Extreme designs are now immediately clear:
same input as the true process, while only the di erence of the measured and simulated
output is fed back. Of course, it is allowed to subtract the simulated output from the
feedback loop after the entrance of the reference yielding the structure of Fig. 4.3. The
similarity with the conventional structure is then obvious, where we identify the dashed
block as the conventional controller C. So it is easy to relate C and the internal model
control block Q as:
C = Q(I ; PQ) ;1
(4.1)
33

36 CHAPTER 4. INTERNAL MODEL CONTROL
35
minimal complementary sensitivity T :
T =0! S = I ! Q =0! C =0 (4.5)
Figure 4.5: Pole zero inversion of nonminimum phase,stable process.
there is obviously neither feedback nor control causing:
In fact poles and zeros in the open left half plane can easily be compensated for by
Q. Also the poles in the closed right half plane cause no real problems as the rootloci
from them in a feedback can be \drawn" over to the left plane in a feedback by putting
zeros there in the controller. The real problems are due to the nonminimum phase zeros
i.e. the zeros in the closed right half plane, as we will analyse further. But before doing
so, we haveto state that in fact all physical plants su er more or less from this negative
property.
We need some extra notion about the numbers of poles and zeros, their de nition and
considerations for realistic, physical processes. Let np denote the number of poles and
similarly nz the number of zeros in a conventional, SISO transfer function where denominator
and numerator are factorised. We can then distinguish the following categories by
the attributes:
{ no measurement in uence (T =0)
{ no actuator saturation (R=Q=0)
{ 100% disturbance in output (S=I)
{ 100% tracking error (S = I)
{ stability (Pt was stable)
{ robust stability (R=Q=0 and T =0)
{ robust S (T =0), but this \performance" can hardly be worse.
minimal sensitivity S:
S =0! T = I ! Q = P ;1 ! C = 1 (4.6)
if at least P ;1 exists and is stable, we get in nite feedback causing:
proper if np nz
biproper if np = nz
strictly proper if np > nz
nonproper if np < nz
Any physical process should be proper because nonproperness would involve:
{ all disturbance is eliminated from the output (S =0)
{ y tracks r exactly (S=0)
{ y is fully contaminated by measurement noise (T = I)
{ stability only in case Q = P ;1 is stable
{ very likely actuator saturation (Q = R will tend to in nity see later)
{ questionable robust stability (Q = R will tend to in nity see later)
{ robust T (S = 0), but this \performance" can hardly be worse too.
P (j!) =1 (4.8)
lim
!!1
so that the process would e ectively have poles at in nity, would have an in nitely
large transfer at in nity and would certainly start oscillating at frequency ! = 1. On
the other hand a real process can neither be biproper as it then should still have a nite
transfer for ! = 1 and at that frequency the transfer is necessarily zero. Consequently
any physical process is by nature strictly proper. But this implies that:
Once again it is clear that a good control should be a well designed compromise between
the indicated extremes. What is left is to analyse the possibility of the above last sketched
extreme where we neededthatPQ = I and Q is stable.
It is obvious that the solution could be Q = P ;1 if P is square and invertible and the
inverse itself is stable. If P is wide (more inputs than outputs) the pseudo inverse would
su ce under the condition of stability. If P is tall (less inputs than outputs) there is no
solution though. Nevertheless, the problem is more severe, because we can show that,
even for SISO systems, the proposed solution yielding in nite feedback is not feasible
for realistic, physical processes. For a SISO process, where P becomes a scalar transfer,
inversion of P turn poles into zeros and vice versa. Let us take a simple example:
P (j!) =0 (4.9)
lim
!!1
(4.7)
s + a
s ; b
s ; b
s + a a>0 b>0 ! P ;1 =
and thus P has e ectively (at least) one zero at in nity which is in the closed right
half space! Take for example:
P =
(4.10)
s + a
K
P = K
s + a a>0 ! P ;1 =
and consequently Q = P ;1 cannot be realised as it is nonproper.
where the corresponding pole/zero-plots are shown in Fig. 4.5.
It is clear that the original zeros of P have to live in the open (stable) left half plane,
because they turn into the poles of P ;1 that should be stable. Ergo, the given example,
where this is not true, is not allowed. Processes which havezeros in the closed right half
plane, named nonminimum phase, thus cause problems in obtaining agoodperformance
in the sense of a small S.

38 CHAPTER 4. INTERNAL MODEL CONTROL
4.1. MAXIMUM MODULUS PRINCIPLE. 37
4.3 Exercises
4.1 Maximum Modulus Principle.
u1
4.1:
The disturbing fact about nonminimum phase zeros can now be illustrated with the use
of the so-called Maximum Modulus Principle which claims:
yt
-
Pt
-
? +
k
-
6+
-
? +
l
P
? +
m
r +
- - Q
u -
6{
?-
+
j
6{
-
u2
y
-
?
a) Derive by reasoning that for IMC internal model stability is guaranteed if all
transfers from r, u1 and u2 to yt, y and u are stable. Take all signal lines to be
single.
b) To which simple a condition this boils down if P = Pt?
8H 2H1 :k H k1 jH(s)js2C + (4.11)
It says that for all stable transfers H (i.e. no poles in the right half plane denoted
+ by C ) the maximum modulus on the imaginary axis is always greater than or equal
to the maximum modulus in the right half plane. We will not prove this, but facilitate
its acceptance by the following concept. Imagine that the modulus of a stable transfer
function of s is represented by a rubber sheet above the s-plane. Zeros will then pinpoint
the sheet to the zero, bottom level, while poles will act as in nitely high spikes lifting the
sheet. Because of the strictly properness of the transfer, there is a zero at in nity, sothat,
in whatever direction we travel, ultimately the sheet will come to the bottom. Because of
stability there are no poles and thus spikes in the right halfplane.It is obvious that such
a rubber landscape with mountains exclusively in the left half plane will gets its heights
in the right half plane only because of the mountains in the left half plane. If we cut it
precisely at the imaginary axis we will notice only valleys at the right hand side. It is
always going down at the right side and this is exactly what the principle tells.
We are now in the position to apply the maximum modulus principle to the sensitivity
function S of a nonminimum phase SISO process P :
s=zn
z}|{
k S k1= sup jS(j!)j jS(s)js2C + = j1 ; PQjs2C + = 1 (4.12)
!
+ where zn ( C )isanynonminimum phase zero of P . As a consequence we haveto accept
that for some ! the sensitivity has to be greater than or equal to 1. For that frequency
the disturbance and the tracking errors will thus be minimally 100%! So for some band
we will get disturbance ampli cation if we want to decrease it by feedback in some other
(mostly lower) band. That seems to be the price. And reminding the rubber landscape,
it is clear that this band, where S > 1, is the more low frequent thecloser the troubling
zero is to the origin of the s-plane!
By proper weighting over the frequency axis we can still optimise a solution. For an
appropriate explanation of this weighting procedure we rst present the intermezzo of the
next chapter about the necessary norms.
c) What if P 6= Pt ?
4.2: For the general scheme let P = Pt =1.
Suppose that d is white noise with power density dd = 1 and similarly that is white
noise with power density = :01.
a) Design for an IMC set-up a Q such that the power density yy is minimal. (As you
are dealing with white noises all variables are constants independent of the frequency
!.) Compute yy, S, T , Q and C. What is the bound on k P k1 for guaranteed
stability ?
b) In order not to saturate the actuator we nowadd the extra constraint uu 1.
It has been shown that internal model control can greatly facilitate the design procedure of
controllers. It only holds, though, for stable processes and the generalisation to unstable
systems has to wait until chapter 11. Limitations of control are recognised in the e ects
of nonminimum phase zeros of the plant and in fact all physical plant su er from these at
least at in nity.
b) We want to obtain good tracking for a low pass band as broad as possible. At
least the ` nal error' for a step input should be zero. What can we reach byvariation
of K and ? (MATLAB can be useful)
c) The same question a) but now the zero of P is at ;1.

40 CHAPTER 5. SIGNAL SPACES AND NORMS
total of q > 0 real valued quantities, then the signal space W consists of q copies of the
set of real numbers, i.e.,
W =R | ::: {z R}
q copies
which is denoted as W =Rq . A signal s : T !Rq thus represents at each time instant
t 2 T a vector
Chapter 5
1
C
A
s1(t)
s2(t)
0
B
@
s(t) =
Signal spaces and norms
.
sq(t)
where si(t), the i-the component, is a real number for each time instant t.
The `size' of a signal is measured by norms. Suppose that the signal space is a complex
valued q-dimensional space, i.e. W =C q for some q > 0. We will attach to each vector
w =(w1w2:::wq) 0 2 W its usual `length'
q
jwj := w1 w1 + w2 w2 + :::+ wq wq
5.1 Introduction
which is the Euclidean norm of w. (Here, w denotes the complex conjugate of the complex
number w. That is, if w = x + jy with x the real part and y the imaginary part of w,
then w = x ; jy). If q =1this expresses the absolute value of w, which is the reason
for using this notation. This norm will be attached to the signal space W , and makes it
a normed space.
Signals can be classi ed in manyways. We distinguish between continuous and discrete
time signals, deterministic and stochastic signals, periodic and a-periodic signals.
In the previous chapters we de ned the concepts of sensitivity and complementary sensitivity
and we expressed the desire to keep both of these transfer functions `small' in a
frequency band of interest. In this chapter we will quantify in a more precise way what
`small' means. In this chapter we will quantify the size of a signal and the size of a system.
We will be rather formal to combine precise de nitions with good intuition. A rst section
is dedicated to signals and signal norms. We then consider input-output systems and
de ne the induced norm of an input-output mapping. The H1 norm and the H2 norm of
a system are de ned and interpreted both for single input single output systems, as well
as for multivariable systems.
5.2 Signals and signal norms
5.2.1 Periodic and a-periodic signals
De nition 5.2 Suppose that the time set T is closed under addition, that is, for any two
points t1t2 2 T also t1 + t2 2 T . A signal s : T ! W is said to be periodic with period P
(or P -periodic) if
s(t) =s(t + P ) t 2 T:
A signal that is not P -periodic for any P is a-periodic.
We will start this chapter with some system theoretic basics which willbeneeded in the
sequel. In order to formalize concepts on the level of systems, we need to rst recall some
basics on signal spaces. Manyphysical quantities (suchasvoltages, currents, temperatures,
pressures) depend on time and can be interpreted as functions of time. Such functions
quantify how information evolves over time and are called signals. It is therefore logical
to specify a time set T , indicating the time instances of interest. We will think of time as
a one dimensional entity and we therefore assume that T R. We distinguish between
continuous time signals (T a possibly in nite interval ofR) and discrete time signals (T
a countable set). Typical examples of frequently encountered time sets are nite horizon
discrete time sets T = f0 1 2:::Ng, in nite horizon discrete time sets T =Z+ or T =Z
Common time sets such asT =Zor T =R are closed under addition, nite time sets
such as intervals T = [a b] are not. Well known examples of continuous time periodic
signals are sinusoidal signals s(t) =A sin(!t + ) or harmonic signals s(t) =Aej!t . Here,
A, ! and are constants referred to as the amplitude, frequency (in rad/sec) and phase,
respectively. These signals have frequency !=2 (in Hertz) and period P = 2 =!. We
emphasize that the sum of two periodic signals does not need to be periodic. For example,
s(t) =sin(t) + sin( t) isa-periodic. The class of all periodic signals with time set T will
be denoted by P(T ).
or, for sampled signals, T = fk s j k 2Zg where s > 0 is the sampling time. Examples
of continuous time sets include T =R, T =R+ or intervals T =[a b].
The values which aphysically relevant signal assumes are usually real numbers. However,
complex valued signals, binary signals, nonnegative signals, angles and quantized
signals are very common in applications, and assume values in di erent sets. We therefore
introduce a signal space W , which is the set in which a signal takes its values.
5.2.2 Continuous time signals
De nition 5.1 A signal is a function s : T ! W where T R is the time set and W is
a set, called the signal space.
It is convenient tointroduce various signal classi cations. First, we consider signals which
have nite energy and nite power. To introduce these signal classes, suppose that I(t)
More often than not, it is necessary that at each time instant t 2 T , a number of
physical quantities are represented. If we wish a signal s to express at instant t 2 T a
39

42 CHAPTER 5. SIGNAL SPACES AND NORMS
5.2. SIGNALS AND SIGNAL NORMS 41
Note that these quantities are de ned for nite or in nite time sets T . In particular, if
T = R, ksk2 2 = Es, i.e the energy content of a signal is the same as the square of its
2-norm.
denotes the current through a resistance R producing a voltage V (t). The instantaneous
power per Ohm is p(t) =V (t)I(t)=R = I2 (t). Integrating this quantity over time, leads
to R de ning the total energy (in Joules). The per Ohm energy of the resistance is therefore
1
;1 jI(t)j2dt Joules.
Remark 5.6 To be precise, one needs to check whether these quantities indeed de ne
norms. Recall from your very rst course of linear algebra that a norm is de ned as a
real-valued function which assigns to each element s of a vector space a real number k s k,
called the norm of s, with the properties that
1. k s k 0andk s k= 0 if and only if s =0.
De nition 5.3 Let s be a signal de ned on the time set T =R. The energy content Es
of s is de ned as
Z 1
Es := js(t)j
;1
2 dt
2. k s1 + s2 k ks1 k + k s2 k for all s1 and s2.
If Es < 1 then s is said to be a ( nite) energy signal.
3. k s k= j jks k for all 2C .
The quantities de ned by k s k1, k s k2 and k s k1 indeed de ne (signal) norms and have
the properties 1,2 and 3 of a norm.
Clearly, not all signals have nite energy. Indeed, for harmonic signals s(t) = cej!t we
have that js(t)j2 = jcj2 so that Es = 1 whenever c 6= 0. In general, the energy content of
periodic signals is in nite. We therefore associate with periodic signals their power:
Example 5.7 The sinusoidal signal s(t) :=Asin(!t + ) for t 0 has nite amplitude
k s k1= A but its two-norm and one-norm are in nite.
De nition 5.4 Let s be a continuous time periodic signal with period P . The power of
s is de ned as
Z t0+P
Example 5.8 As another example, consider the signal s(t) which is described by the
di erential equations
js(t)j 2 dt (5.1)
Ps := 1
P
t0
where t0 2R. If Ps < 1 then s is said to be a ( nite) power signal.
(5.5)
dx
= Ax(t)
dt
s(t) =Cx(t)
where A and C are real matrices of dimension n n and 1 n, resp. It is clear that s is
uniquely de ned by these equations once an initial condition x(0) = x0 has been speci ed.
Then s is equal to s(t) =CeAtx0 where we take t 0. If the eigenvalues of A are in the
In case of the resistance, the power of a (periodic) current I is measured per period and
will be in Watt. It is easily seen that the power is independent of the initial time instant t0
in (5.1). A signal which is periodic with period P is also periodic with period nP , where
n is an integer. However, it is a simple exercise to verify that the right hand side of (5.1)
does not change if P is replaced by nP . It is in this sense that the power is independent of
the period of the signal. We emphasize that all nonzero nite power signals have in nite
energy.
Z 1
left-half complex plane then
x T
0 eAT t T At
C Ce x0dt = x T
0 Mx0
k s k 2 2 =
0
Example 5.5 The sinusoidal signal s(t) =A sin(!t+ ) is periodic with period P =2 =!,
has in nite energy and has power
Z =!
Z
with the obvious de nition for M. The matrix M has the same dimensions as A, is
symmetric and is called the observability gramian of the pair (A C). The observability
gramian M is a solution of the equation
sin 2 ( + ) d = A 2 =2:
;
A 2 sin 2 (!t + ) dt = A2
2
; =!
Ps = !
2
A T M + MA+ C T C =0
which is the Lyapunov equation associated with the pair (A C).
Let s : T !Rq be a continuous time signal. The most important norms associated
with s are the in nity-norm, the two-norm and the one-norm de ned either over a nite
or an in nite interval T . They are de ned as follows
The sets of signals for which the above quantities are nite will be of special interest.
De ne
jsi(t)j (5.2)
sup
t2T
k s k1 = max
i
o 1=2
nZ
L1(T ) = fs : T ! W jks k1 < 1g
L2(T ) = fs : T ! W jks k2 < 1g
L1(T ) = fs : T ! W jks k1 < 1g
P(T ) = fs : T ! W j p Ps < 1g
(5.3)
js(t)j 2 dt
k s k2 =
t2T
Z
js(t)jdt (5.4)
k s k1 =
t2T
More generally, the p-norm, with 1 p

44 CHAPTER 5. SIGNAL SPACES AND NORMS
5.2. SIGNALS AND SIGNAL NORMS 43
5.2.3 Discrete time signals
For discrete time signals s : T ! Rq a similar classi cation can be set up. The most
important norms are de ned as follows.
drop the T in the above signal spaces whenever the time set is clear from the context. As
an example, the sinusoidal signal of Example 5.7 belongs to L1[0 1) and P[0 1), but
not to L2[0 1) and neither to L1[0 1).
For either nite or in nite time sets T , the space L2(T )isa Hilbert space with inner
jsi(t)j (5.9)
Z
product de ned by
sup
t2T
k s k1 = max
i
nX
s T
2 (t)s1(t) dt:
hs1s2i =
(5.10)
js(t)j 2o 1=2
k s k2 =
t2T
t2T
js(t)j (5.11)
k s k1 = X
Two signals s1 and s2 are orthogonal if hs1s2i = 0. This is a natural extension of
orthogonality inRn .
t2T
The Fourier transforms
More generally, the p-norm, with 1 p

46 CHAPTER 5. SIGNAL SPACES AND NORMS
5.3. SYSTEMS AND SYSTEM NORMS 45
y(t)
-
5.2.4 Stochastic signals
H
-
u(t)
Figure 5.1: Input-output systems: the engineering view
Occasionally we consider stocastic signals in this course. We will not give a complete
treatise of stochastic system theory at this place but instead recall a few concepts.A
stationary stochastic process is a sequence of real random variablesu(t) where t runs over
some time set T . By de nition of stationarity,its mean, (t) := E[u(t)] is independent
of the time instant t, and the second order moment E[u(t1)u(t2)] depends only on the
di erence t1 ; t2. The covariance of such a process is de ned by
Remark 5.13 Again a philoso cal warning is in its place. If an input-output system is
mathematically represented as a function H, then to each input u 2 U, H attaches a
unique output y = H(u). However, more often than not, the memory structure of many
physical systems allows various outputs to correspond to one input signal. A capacitor C
imposes the relation C d dtV = I on voltage-current pairs VI. Taking I = 0 as input allows
the output V to be any constant signal V (t) = V0. Hence, there is no obvious mapping
I 7! V modeling this simple relationship!
Ru( ):=E (u(t + ) ; )(u(t) ; )
where = (t) = E[u(t)] is the mean. A stochastic (stationary) process u(t) is called a
white noise process if its mean = E[u(t)] = 0 and if u(t1) and u(t2) are uncorrelated for
all t1 6= t2. Stated otherwise, the covariance of a (continuous time) white noise process
is Ru( ) = 2 2 ( ). The number is called the variance. The Fourier transform of the
covariance function Ru( )is
Z 1
Of course, there are manyways to represent input-output mappings. We will be particularly
interested in (input-output) mappings de ned by convolutions and those de ned by
transfer functions. Undoubtedly, you have seen various of the following de nitions before,
but for the purpose of this course, it is of importance to understand (and fully appreciate)
the system theoretic nature of the concepts below. In order not to complicate things from
the outset, we rst consider single input single output continuous time systems with time
set T =R and turn to the multivariable case in the next section. This means that we will
focus on analog systems. We will not treat discrete time (or digital) systems explicitly, for
their de nitions will be similar and apparent from the treatment below.
In a (continuous time) convolution system, an input signal u 2U is transformed to an
Ru( )e ;j! d
u(!) :=
;1
and is usually referred to as the power spectrum, energy spectrum or just the spectrum of
the stochastic process u.
5.3 Systems and system norms
h(t ; )u( )d (5.12)
output signal y = H(u) according to the convolution
Z 1
y(t) =(Hu)(t) =(h u)(t) =
;1
where h :R !R is a function called the convolution kernel. In system theoretic language,
h is usually referred to as the impulse response of the system, as the output y is equal
to h whenever the input u is taken to be a Dirac impulse u(t) = (t). Obviously, H
de nes a linear map (as H(u1 + u2) =H(u1)+H(u2) andH( u) = H(u)) and for this
reason the corresponding input-output system is also called linear. Moreover, it de nes a
time-invariant system in the sense that H maps the time shifted input signal u(t ; t0) to
the time shifted output y(t ; t0).
No mapping is well de ned if we are lead to guess what the domain U of H should be.
There are various options:
A system is any setSof signals. In engineering we usually study systems which havequite some structure. It is common engineering practice to consider systems whose signals are
naturally decomposed in two independent sets: a set of input signals and a set of output
signals. A system then speci es the relations among the input and output signals. These
relations may be speci ed by transfer functions, state space representations, di erential
equations or whatever mathematical expression you can think of. We nd this theme
in almost all applications where lter and control design are used for the processing of
signals. Input signals are typically assumed to be unrestricted. Filters are designed so
as to change the frequency characteristics of the input signals. Output signals are the
responses of the system (or lter) after excitation with an input signal. For the purpose of
this course, we exclusively consider systems in which an input-output partitioning of the
signals has already been made. In engineering applications, it is good tradition to depict
input-output systems as `blocks' as in Figure 5.3, and you probably have a great deal
of experience in constructing complex systems by interconnecting various systems using
block diagrams. The arrows in Figure 5.3 indicate the causality direction.
One can take bounded signals, i.e., U = L1.
One can take harmonic signals, i.e., U = fce j!t j c 2C ! 2Rg.
Remark 5.12 Also a word of warning concerning the use of blocks is in its place. For
example, many electrical networks do not have a `natural' input-output partition of system
variables, neither need such a partitioning of variables be unique. Ohm's law V = RI
imposes a simple relation among the signals `voltage' V and `current' I but it is not
evident whichsignal is to be treated as input and which as output.
One can take energy signals, i.e., U = L2.
One can take periodic signals with nite power, i.e., U = P.
The mathematical analog of such a `block' is a function or an operator H mapping
inputs u taken from an input space U to output signals y belonging to an output space Y.
We write
The input class can also exist of one signal only. If we are interested in the impulse
response only, we take U = f g.
H : U ;!Y:

48 CHAPTER 5. SIGNAL SPACES AND NORMS
5.3. SYSTEMS AND SYSTEM NORMS 47
5.3.1 The H1 norm of a system
Induced norms
One can take white noise stochastic processes as inputs. In that case U consists of
all stationary zero mean signals u with nite covariance Ru( )= 2 ( ).
Let T be a continuous time set. If we assume that the impulse response h : R ! R
satis es k h k1= R 1
;1 jh(t)jdt < 1 (in other words, if we assume that h 2L1), then H is
a stable system in the sense that bounded inputs produce bounded outputs. Thus, under
this condition,
Example 5.14 For example, the response to a harmonic input signal u(t) =ej!t is given
by
Z 1
y(t) = h( )e
;1
j!(t; ) d = b j!t
h(!)e
H : L1(T ) ;!L1(T )
where b h is the Fourier transform of h as de ned in (5.7).
and we can de ne the L1-induced norm of H as
k H(u) k1
k u k1
k H k (11) := sup
u2L1
Example 5.15 A P -periodic signal with line spectrum fukg, k 2Z, canberepresented as u(t) = P1 k=;1 ukejk!t where ! =2 =P and its corresponding output is given by
Interestingly, under the same condition, H also de nes a mapping from energy signals to
energy signals, i.e.
b h(k!)uke jk!t :
1X
y(t) =
k=;1
H : L2(T ) ;!L2(T )
with the corresponding L2-induced norm
Consequently, y is also periodic with period P and the line spectrum of the output is given
by yk = bh(k!)uk, k 2Z.
k H(u) k2
k u k2
k H k (22) := sup
u2L2
Assume that both U and Y are normed linear spaces. Then we call H bounded if there
is a constant M 0such that
In view of our de nition of `energy' signals, this norm is also referred to as the induced
energy norm. The power does not de ne a norm for the class P of periodic signals.
Nevertheless, Example 5.15 shows that
k H(u) k M k u k :
H : P(T ) !P(T )
and we de ne the power-induced norm
Note that the norm on the left hand side is the norm de ned on signals in the output
space Y and the norm on the right hand side corresponds to the norm of the input signals
in U. In system theoretic terms, boundednes of H can be interpreted in the sense that H
is stable with respect to the chosen input class and the corresponding norms. If a linear
map H : U ! Y is bounded then its norm k H k can be de ned in several alternative
(and equivalent) ways:
:
p
Py
p
Pu
kHkpow := sup
Pu6=0
The following result characterizes these system norms
Theorem 5.16 Let T =R orR+ be the time set and let H be de ned by(5.12). Suppose
that h 2L1. Then
(5.13)
k H k = inffM
jkHu k M k u k for all u 2Ug
M
k Hu k
= sup
u2Uu6=0 k u k
= sup k Hu k
u2Ukuk 1
1. the L1-induced norm of H is given by
k Hu k
= sup
u2Ukuk=1
k H k (11)=k h k1
2. the L2-induced norm of H is given by
k H k (22)= max
!2R jbh(!)j (5.14)
For linear operators, all these expressions are equal and either one of them serves as
de nition for the norm of an input-output system. The norm k H k is often called the
induced norm or the operator norm of H and it has the interpretation of the maximal
`gain' of the mapping H : U !Y. A most important observation is that
3. the power-induced norm of H is given by
k H kpow= max
!2R jbh(!)j (5.15)
the norm of the input-output system de ned byH depends on the class of inputs
U and on the signal norms for elements u 2U and y 2Y. A di erent class of
inputs or di erent norms on the input and output signals results in di erent
operator norms of H.

5.3. SYSTEMS AND SYSTEM NORMS 49
We will extensively use the above characterizations of the L2-induced and powerinduced
norm. The rst characterization on the 1-induced norm is interesting, but will
not be further used in this course. The Fourier transform b h of the impulse response h is
generally referred to as the frequency response of the system (5.12). It has the property
that whenever h 2L1 and u 2L2,
y(t) =(h u)(t) () by(!) =b h(!)bu(!) (5.16)
Loosely speaking, this result states that convolution in the time domain is equivalent to
multiplication in the frequency domain.
Remark 5.17 The quantity max!2R j ^ h(!)j satis es the axioms of a norm, and is precisely
equal to the L1-norm of the frequency response, i.e, k ^ h k1= max!2R j ^ h(!)j.
Remark 5.18 The frequency response can be written as
b h(!) =jb h(!)je j (!) :
Various graphical representations of frequency responses are illustrative to investigate
system properties like bandwidth, system gains, etc. A plot of j ^ h(!)j and (!) as function
of ! 2R is called a Bode diagram. See Figure 5.2. In view of the equivalence (5.16) a
Bode diagram therefore provides information to what extent the system ampli es purely
harmonic input signals with frequency ! 2R. In order to interpret these diagrams one
usually takes logarithmic scales on the ! axis and plots 20 10 log( ^ h(!)) to get units in
dB. Theorem 5.16 states that the L2 induced norm of the system de ned by (5.12)
equals the highest gain value occuring in the Bode plot of the frequency response of the
system. In view of Example 5.14, any frequency !0 for which this maximum is attained
has the interpretation that a harmonic input signal u(t) = e j!0t results in a (harmonic)
output signal y(t) with frequency !0 and maximal amplitude j ^ h(!0)j. (Unfortunately,
sin(!0t) =2L2, sowe cannot use this insight directly in a proof of Theorem 5.16.)
To prove Theorem 5.16, we derive fromParseval's identity that
which shows that k H k (22)
k H k 2 (22)
k h u k
= sup
u2L2
2 2
k u k2 2
1=(2 ) k[h u k 2 2
= sup
^u2L2 1=(2 ) k bu k2 2
R
j^ 2 2 h(!)j j^u(!)j d!
= sup
^u2L2
k ^u k 2 2
max!2R j
sup
^u2L2
^ h(!)j2 k ^u k2 2
k ^u k2 2
= max
!2R jb h(!)j 2
max!2R j ^ h(!)j. Similarly, using Parseval's identity for
50 CHAPTER 5. SIGNAL SPACES AND NORMS
Gain dB
Phase deg
40
20
0
−20
10 −3
0
−90
−180
10 −3
periodic signals
10 −2
10 −2
k H k 2 pow = sup
P
10 −1
Frequency (rad/sec)
10 −1
Frequency (rad/sec)
Figure 5.2: A Bode diagram
= sup
P
sup
P
sup
Py
u is P -periodic Pu
sup
u is P -periodic
= max
!2R jb h(!)j 2
max
k2Z jb 2
h(2 k=P)j
10 0
10 0
P 1k=;1 jb h(2 k=P)ukj 2
P 1k=;1 jukj 2
showing that k H kpow max!2R jbh(!)j. Theorem 5.16 provides equality for the latter
inequalities. For periodic signals (statement 3) this can be seen as follows. Suppose that
!0 is such that
jbh(!0)j =max
!2R jbh(!)j Take a harmonic input u(t) =e j!0t and note that this signal has power Pu =1and line
spectrum u1 =1,uk =0for k 6= 1. From Example 5.14 it follows that the output y has
line spectrum y1 = b h(!0), and yk =0fork 6= 1, and using Parseval's identity, the output
has power Py = jb h(!0)j 2 . We therefore obtain that
k H kpow = b h(!0) = max
!2R j^ h(!)j
10 1
10 1

52 CHAPTER 5. SIGNAL SPACES AND NORMS
5.3. SYSTEMS AND SYSTEM NORMS 51
A stochastic interpretation of the H1 norm
as claimed. The proof of statement 2 is more involved and will be skipped here.
We conclude this subsection with a discussion on a stochastic interpretation of the H1
norm of a transfer function. Consider the set T of all stochastic (continuous time)
processes s(t) on the nite time interval [0T] for which the expectation
Eksk2 Z T
2T := E s
0
T (t)s(t) dt (5.18)
The transfer function associated with (5.12) is the Laplace transform of the impulse
response h. This object will be denoted by H(s) (which the careful reader perceives as
poor and ambiguous notation at this stage2 ). Formally,
Z 1
H(s) := h(t)e
;1
;st dt
is well de ned and bounded. Consider the convolution system (5.12) and assume that
h 2L1 (i.e. the system is stable) and the input u 2 T . Then the output y is a stochastic
process and we can introduce the \induced norm"
where the complex variable s is assumed to belong to an area of the complex plane where
the above integral is nite and well de ned. The Laplace transforms of signals are de ned
in a similar way andwehave that
Ekyk2 2T
Ekuk2 2T
kHk2 stochT := sup
u2 T
y = h u () ^y(!) = ^ h(!)^u(!) () Y (s) =H(s)U(s):
which depends on the length of the time horizon T . This is closely related to an induced
operator norm for the convolution system (5.12). We would like to extend this de nition
to the in nite horizon case. For this purpose it seems reasonable to de ne
If the Laplace transform exists in an area of the complex plane which includes the imaginary
axis, then the Fourier transform is simply bh(!) =H(j!).
(5.19)
1
E
T ksk2 2T
Eksk2 2 := lim
T !1
Remark 5.19 It is common engineering practice (the adjective `good' or `bad' is left
to your discretion) to denote Laplace transforms of signals u ambiguously by u. Thus
u(t) means something really di erent than u(s)! Whereas y(t) = H(u)(t) refers to the
convolution (5.12), the notation y(s) =Hu(s) istobeinterpreted as the product of H(s)
and the Laplace transform u(s) ofu(t). The notation y = Hu can therefore be interpreted
in two (equivalent) ways!
assuming that the limit exists. This expectation can be interpreted as the average power
of a stochastic signal. However, as motivated in this section, we would also like to work
with input and output spaces U and Y that are linear vector spaces. Unfortunately, the
class of stochastic processes for which the limit in (5.19) exists is not a linear space. For
this reason, the class of stochastic input signals U is set to
We return to our discussion of induced norms. The right-hand side of (5.14) and (5.15)
is de ned as the H1 norm of the system (5.12).
:=fs j ksk < 1g
De nition 5.20 Let H(s) be the transfer function of a stable single input single output
system with frequency response ^ h(!). The H1 norm of H, denotedkHk1 is the number
where
k H k1 := max
!2R j^ h(!)j: (5.17)
E 1
T ksk2 2T
ksk2 := lim sup
T !1
In this case, is a linear space of stochastic signals, but k k does not de ne a norm
on . This is easily seen as ksk =0for any s 2L2. However, it is a semi norm as it
satis es conditions 2 and 3 in Remark 5.6. With this class of input signals, we can extend
the \induced norm" kHkstochT to the in nite horizon case
The H1 norm of a SISO transfer function has therefore the interpretation of the maximal
peak in the Bode diagram of the frequency response ^ h of the system and can be directly
`read' from such a diagram. Theorem 5.16 therefore states that
k H(s) k1=k H k (22)=k H kpow :
kyk
kuk
In words, this states that
kHk2 stoch := sup
u2
the energy induced norm and the power induced normofH is equal to the H1
norm of the transfer function H(s).
which is bounded for stable systems H. The following result is the crux of this discussion
and states that kHkstoch is, in fact, equal to the H1 norm of the transfer function H.
Theorem 5.21 Let h 2 L1 and let H(s) be the transfer function of the system (5.12).
Then
kHk stoch = kHk1:
A proof of this result is beyond the scope of these lecture notes. The result can be
found in [18].
2 For we de nedH already as the mapping that associates with u 2Uthe element H(u). However,
from the context it will always be clear what we mean

54 CHAPTER 5. SIGNAL SPACES AND NORMS
5.3. SYSTEMS AND SYSTEM NORMS 53
(RMS) value of the output of the system when the input is a realization of a unit variance
white noise process. That is, let
(
a unit variance white noise process t 2 [0T]
u(t) =
0 otherwise
5.3.2 The H2 norm of a system
The notation H2 is commonly used for the class of functions of a complex variable that
do not have poles in the open right-half complex plane (they are analytic in the open
right-half complex plane) and for which the norm
n Z 1 1
o1=2 kskH2 := sup s ( + j!)s( + j!)d!
>0 2 ;1
and let y = h u be the corresponding output. Using the de nition of a nite horizon
2-norm from (5.18), we set
kHk2 Z 1
RMST := E y
;1
T (t)y(t)dt = Ekyk2 2T
is nite. The `H' stands for Hardy space. Thus,
< 1g
H2 = fs :C !C j sanalytic in 0 and kskH2
where E denotes expectation. Substitute (5.12) in the latter expression and use that
h( )h( ) d
E(u(t1)u(t2)) = (t1 ; t2) to obtain that
kHk2 Z T Z t
RMST = dt
0 t;T
Z T
. This \cold-hearted" de nition has, in fact, a very elegant system theoretic interpretation.
Before giving this, we rst remark that the H2 norm can be evaluated on the imaginary
axis. That is, for any s 2H2 one can construct a boundary function s(!) =lim #0 s( +
j!), which exists for almost all !. Moreover, this boundary function is square integrable,
Z T
i.e., s 2L2 and kskH2 = ksk2. Stated otherwise,
t(h( )h( )+h(; )h(;tau))d
h( )h( )d ;
= T
0
;T
If the transfer function is such that the limit
1
T kHk2 RMST
kHk2 RMS = lim
T !1
n 1 o1=2 kskH2 = s (!)s(!)
2
Thus, the supremum in the de nition of the H2 norm always occurs on the boundary
=0. It is for this reason that s is usually identi ed with the boundary function and the
bar in s is usually omitted.
remains bounded we obtain the in nite horizon RMS-value of the transfer function H. In
fact, it then follows that
kHk2 Z 1
RMS = h( )h( ) d
;1
= 1
Z 1
H(j!)H (j!) d!
2 ;1
= kHk2 H2
Deterministic interpretation
To interpret the H norm, consider again the convolution system (5.12) and suppose that
we areinterested only in the impulse response of this system. This means, that we take the
impulse (t) as the only candidate input for H. The resulting output y(t) =(Hu)(t) =h(t)
is an energy function so that Eh < 1. Using Parseval's identity we obtain
Thus, the H2 norm of the transfer function is equal to the in nite horizon RMS value of
the transfer function.
Another stochastic interpretation of the H2 norm can be given as follows. Let u(t) be
a stochastic process with mean 0 and covariance Ru( ). Taking such a process as input to
(5.12) results in the output y(t) which is a random variable for each time instant t 2 T . It
is easy to see that the output y has also zero mean. The condition that h 2L2 guarantees
that the output y has nite covariances y( ) = E[y(t)y(t ; )] and easy calculations4 show that the covariances Ry( ) are given by
Eh =k h k2 2=
1
p k b 2
h k2 2
n Z 1 1
= ^h (!)
2 ;1
^ o
h(!)d!
= kH(s)k2 H2
where H(s) is the transfer function associated with the input-output system. The square
of the H2 norm is therefore equal to the energy of the impulse response. To summarize:
Ry( )=E[y(t + )y(t)]
Z 1 Z 1
= h(s
;1 ;1
0 )Ru( + s 00 ; s 0 )h(s 00 )ds 0 ds 00
De nition 5.22 Let H(s) be the transfer function of a stable single input single output
system with frequency response ^ h(!). The H2 norm of H, denoted k H kH2 is the number
n Z 1 1
o1=2 k H kH2 :=
H(j!)H(;j!)d! : (5.20)
2 ;1
The latter expression is a double convolution which by taking Fourier transforms results
in the equivalent expression
Stochastic interpretation
y(!) = ^ h(j!) u(!) ^ h(;j!): (5.21)
The H2 norm of a transfer function has an elegant equivalent interpretation in terms of
stationary stochastic signals3 . The H2 norm is equal to the expected root-mean-square
4 Details are not important here.
3 The derivations in this subsection are not relevant for the course!

56 CHAPTER 5. SIGNAL SPACES AND NORMS
5.4. MULTIVARIABLE GENERALIZATIONS 55
y1
y2
y3
- --
H
- ----
u1
u2
u3
u4
u5
Figure 5.3: Amultivariable system.
in the frequency domain. We nowassume u to be a white noise process with u(!) = 1 for
all ! 2R. (This implies that Ru( )= ( ). Indeed the variance of this signal theoretically
equals (0) = (0) = 1. This is caused by the fact that all freqencies have equal power
(density) 1, which in turn is necessary to allow for in nitely fast changes of the signal
to make future values independent of momentary values irrespective of the small time
di erence. Of course in practice it is su cient if the \whiteness" is just in broadbanded
noise with respect to the frequency band of the plant under study.) Using (5.21), the
spectrum of the output is then given by
(5.22)
where the convolution kernel h(t) isnow, for every t 2R, a real matrix of dimension p m.
The transfer function associated with this system is the Laplace transform of h and is the
y(!) =j ^ h(!)j 2
Z 1
function
which relates the spectrum of the input and the spectrum of the output of the system
de ned by theconvloution (5.12). Integrating the latter expression over ! 2R and using
the de nition of the H2 norm yields that
h(t)e ;st dt:
H(s) =
;1
k ^ h k2 2
Z 1
^h(!)
;1
^ h(;!)d!
Z 1
;1 y(!)d!
1
=
2
k H(s) k 2 H2
Thus H(s) has dimension p m for every s 2C . We will again assume that the system
is stable in the sense that all entries [H(s)]ij of H(s) (i = 1::: p and j = 1::: m)
have their poles in the left half plane or, equivalently, that the ij-th element [h(t)]ij
of h, viewed as a function of t, belongs to L1. As in the previous section, under this
assumption H de nes an operator mapping bounded inputs to bounded outputs (but now
for multivariable signals!) and bounded energy inputs to bounded energy outputs. That
is,
(5.23)
= 1
2
= 1
2
= k Ry( ) k 2 2
H : Lm 1 ;! Lp 1
H : Lm 2 ;! L p
2
Thus the H2 norm of the transfer function H(s) has the interpretation of the L2 norm of
the covariance function Ry( ) of the output y of the system when the input u is taken to
be a white noise signal with variance equal to 1. From this it should now be evident that
when we de ne in this stochastic context the norm of a stochastic (stationary) signal s
where the superscripts p and m denote the dimensions of the signals. We will be mainly
interested in the L2-induced and power induced norm of such a system. These norms are
de ned as in the previous section
with mean 0 and covariance Rs( )tobe
o 1=2
nZ 1
E[s(t + )s(t)]d
k s k := k Rs( ) k2=
k y k2
k u k2
;1
k H k (22) := sup
u2Lm 2
P 1=2
y
P 1=2
u
k H kpow := sup
Pu6=0
then the H2 norm of the transfer function H(s) is equal to the norm k y k of the output y,
when taking white noise as input to the system. Note that above norm is rather a power
norm than an energy norm and that for a white noise input u we get
nZ 1 o1=2 k u k=k Ru( ) k2= ( )d =1:
where y = H(u) is the output signal.
Like in section 5.3, we wish to express the L2-induced and power-induced norm of
the operator H as an H1 norm of the (multivariable) transfer function H(s) and to
obtain (if possible) a multivariable analog for the maximum peak in the Bode diagram of
a transfer function. This requires some background on what is undoubtedly one of the
most frequently encountered decompositions of matrices: the singular value decomposition.
It occurs in numerous applications in control theory, system identi cation, modelling,
numerical linear algebra, time series analysis, to mention only a few areas. We will devote
a subsection to the singular value decomposition (SVD) as a refreshment.
;1
5.4 Multivariable generalizations
5.4.1 The singular value decomposition
In the previous section we introduced various norms to measure the relative size of a single
input single output system. In this section we generalize these measures for multivariable
systems. The mathematical background and the main ideas behind the de nitions and
characterizations of norms for multivariable systems is to a large extent identical to the
concepts derived in the previous section. Throughout this section we will consider an
input-output system with m inputs and p outputsasinFigure5.3.
Again, starting with a convolution representation of such a system, the output y is
In this section we will forget about dynamics and just consider real constant matrices of
dimension p m. Let H 2Rp m be a given matrix. Then H maps any vector u 2Rm to
avector y = Hu inRp according to the usual matrix multiplication.
Z 1
determined from the input u by
h(t ; )u( )d
y(t) =(Hu)(t) =h u =
;1

58 CHAPTER 5. SIGNAL SPACES AND NORMS
5.4. MULTIVARIABLE GENERALIZATIONS 57
acting on vectors u 2 Rm and producing vectors y = Hu 2 Rp according to the usual
matrix multiplication.
Let H = Y U T be a singular value decomposition of H and suppose that the m m
matrix U =(u1u2::: um) and the p p matrix Y =(y1y2 ::: yp) where ui and yj are
the columns of U and Y respectively, i.e.,
De nition 5.23 A singular value decomposition (SVD) of a matrix H 2Rp m is a decomposition
H = Y U T , where
Y 2Rp p is orthogonal, i.e. Y T Y = YY T = Ip,
U 2Rm m is orthogonal, i.e. U T U = UU T = Im,
ui 2Rm
0 0 where
2Rp m is diagonal, i.e. = ; 0 0
yj 2Rp
with i = 1 2::: m and j = 1::: p. Since U is an orthogonal matrix, the vectors
1
C
A
1 0 0 ::: 0
0 2 0 ::: 0
0
B
@
0 =diag( 1::: r) =
fuigi=1:::m constitute an orthonormal basis for Rm . Similarly, the vectors fyjgj=1:::p
T
constitute an orthonormal basis forRp . Moreover, since uj ui is zero except when i = j
(in which case uT . . . . .
0 0 0 0 r
i ui = 1), there holds
and
Hui = Y U T ui = Y ei = iyi:
In other words, the i-th basis vector ui is mapped in the direction of the i-th basis vector
yi and `ampli ed' by an amount of i. It thus follows that
1 2 ::: r > 0
Every matrix H has such a decomposition. The ordered positive numbers
k Hui k = i k yi k = i
1 2::: r
where we used that k yi k= 1. So, e ectively, if we have a general input vector u it
will rst be decomposed by U T along the various orthogonal directions ui. Next, these
decomposed components are multiplied by the corresponding singular values ( ) and then
(by Y ) mapped onto the corresponding directions yi. If the "energy" in u is restricted
to 1, i.e. k u k= 1, the "energetically" largest output y is certainly obtained if the u is
directed along u1 so that u = u1. As a consequence, it is easy to grasp that the induced
norm of H is related to the singular value decomposition as follows
are uniquely de ned and are called the singular values of H. The singular values of
H 2Rp m can be computed via the familiar eigenvalue decomposition because:
H T H = U Y T Y U T = U 2 U T = U U T
and:
HH T = U U T U Y T = Y 2 Y T = Y Y T
k Hu k
k u k = k Hu1 k
k u1 k = 1
k H k := sup
u2Rm Consequently, ifyou want to compute the singular values with pencil and paper, you can
use the following algorithm. (For numerically well conditioned methods, however, you
should avoid the eigenvalue decomposition.)
In other words, the largest singular value 1 of H equals the induced norm of H (viewed
as a function fromRm toRp ) whereas the input u1 2Rm de nes an `optimal direction' in
the sense that the norm of Hu1 is equal to the induced norm of H. The maximal singular
value 1, often denoted by , can thus be viewed as the maximal `gain' of the matrix
H, whereas the smallest singular value r, sometimes denoted as , can be viewed as the
minimal `gain' of the matrix under normalized `inputs' and provided that the matrix has
full rank. (If the matrix H has not full rank, it has a non-trivial kernel so that Hu =0
for some input u 6= 0).
Algorithm 5.24 (Singular value decomposition) Given a p m matrix H.
Construct the symmetric matrix H T H (or HH T if m is much larger than p)
Compute the non-zero eigenvalues 1::: r of H TH. Since for a symmetric matrix
the non-zero eigenvalues are positive numbers, we can assume the eigenvalues to be
ordered: 1 ::: r > 0
Remark 5.25 To verify the latter expression, note that for any u 2Rm , the norm
The k-th singular value of H is given by k = p k, k =1 2::: r.
T U T u
k Hu k 2 = u T H T Hu = u T U
= x T 0 0 x
where x = U T u. It follows that
mX
The number r is equal to the rank of H and we remark that the matrices U and Y need
not be unique. (The sign is not de ned and nonuniqueness can occur in case of multiple
singular values.)
The singular value decomposition and the singular values of a matrix have a simple
and straightforward interpretation in terms of the `gains' and the so called `principal
directions' of H. 5 For this, it is most convenient to view the matrix as a linear operator
2
i jxij
k 0 0 x k 2 =
k Hu k2 = max
kxk=1
max
kuk=1
i=1
which is easily seen to be maximal if x1 = 1 and xi = 0 for all i 6= 1.
5
In fact, one may argue why eigenvalues of a matrix have played such a dominant role in your linear
algebra course. In the context of linear mappings, singular values have amuch more direct and logical
operator theoretic interpretation.

5.4. MULTIVARIABLE GENERALIZATIONS 59
5.4.2 The H1 norm for multivariable systems
Consider the p m stable transfer function H(s) and let
H(j!) =Y (j!) (j!)U (j!)
be a singular value decomposition of H(j!) for a xed value of ! 2R. Since H(j!) is
in general complex valued, we have that H(j!) 2C p m and the singular vectors stored
in Y (j!) and U(j!) are complex valued. For each such !, the singular values, still being
real valued (i.e. i 2R), are ordered according to
1(!) 2(!) ::: r(!) > 0
where r is equal to the rank of H(s) and in general equal to the minimum of p and m.
Thus the singular values become frequency dependent! From the previous section we infer
that for each ! 2R
0
k H(j!)^u(!) k
k ^u(!) k
1(!)
or, stated otherwise,
k H(j!)^u(!) k 1(!) k ^u(!) k
so that (!) := 1(!) viewed as a function of ! has the interpretation of a maximal gain
of the system at frequency !. It is for this reason that a plot of (!) with ! 2R can be
viewed as a multivariable generalization of the Bode diagram!
De nition 5.26 Let H(s) be a stable multivariable transfer function. The H1 norm of
H(s) is de ned as
k H(s) k1:= sup
!2R (H(j!)):
With this de nition we obtain the natural generalization of the results of section 5.3 for
multivariable systems. Indeed, we have the following multivariable analog of theorem 5.16:
Theorem 5.27 Let T =R+ or T =R be the time set. For a stable multivariable transfer
function H(s) the L2 induced norm and the power induced normisequal to the H1 norm
of H(s). That is,
k H k (22)=k H k pow=k H(s) k1
The derivation of this result is to a large extent similar to the one given in (5.23). An
example of a \multivariable Bode diagram"' is depicted in Figure 5.4.
The bottom line of this subsection is therefore that the L2-induced operator norm and
the power-induced norm of a system is equal to the H1 norm of its tranfer function.
5.4.3 The H2 norm for multivariable systems
The H2 norm of a p m transfer function H(s) is de ned as follows.
De nition 5.28 Let H(s) be a stable multivariable transfer function of dimension p m.
The H2 norm of H(s) is de ned as
n Z 1 1
o1=2 k H(s) kH2= trace [H (;j!)H(j!)]d! :
2 ;1
60 CHAPTER 5. SIGNAL SPACES AND NORMS
Singular Values dB
40
30
20
10
0
−10
−20
−30
10 −3
10 −2
10 −1
Frequency (rad/sec)
10 0
Figure 5.4: The singular values of a transfer function
Here, the `trace' of a square matrix is the sum of the entries at its diagonal. The rationale
behind this de nition is a very simple one and very similar, in spirit, to the idea behind
the H2 norm of a scalar valued transfer function. For single-input single-output systems
the square of the H2 norm of a transfer function H(s) is equal to the energy in the impulse
response of the system. For a system with m inputs we can consider m impulse responses
by putting an impulsive input at the i-th input channel (i = 1::: m) and `watching'
the corresponding output, say y (i) , which is a p dimensional energy signal for each such
input. We will de ne the squared H2 norm of a multi-variable system as the sum of the
energies of the outputs y (i) as a re ection of the total \energy". Precisely, let us de ne m
impulsive inputs,the i-th being
u (i) (t) =
where the impulse (t) appears at the i-th spot. The corresponding output is a p dimensional
signal which we will denote by y (i) (t) and which has bounded energy if the system
0
B
@
0.
(t)
0
.
0
1
C
A
10 1

62 CHAPTER 5. SIGNAL SPACES AND NORMS
5.4. MULTIVARIABLE GENERALIZATIONS 61
5.5 Exercises
is assumed to be stable. The square of its two norm is given by
Z 1 pX
1. Consider the following continuous time signals and determine their amplitude (k
k1), their energy (k k2) and their L1 norm (k k1).
jy (i)
j (t)j2dt k y (i) k 2 2 :=
j=1
;1
0 for t 0.
where y (i)
j denotes the j-th component of the output due to an impulsive input at the i-th
input channel.
The H2 norm of the transfer function H(s) is nothing else than the square root of the
(c) x(t) = exp( jtj) for xed . Distinguish the cases where > 0 < 0 and
=0.
sum of the two-norms of these outputs. That is:
mX
2. This exercise is mainly meant to familiarize you with various routines and procedures
in MATLAB. This exercise involves a single input single output control scheme and
should be viewed as a `prelude' for the multivariable control scheme of an exercise
below.
k y (i) k 2 2 :
k H(s) k 2 H2 =
i=1
Consider a single input single output system described by the transfer function
;s
(s +1)(s +2) :
P (s) =
The system P is controlled by the constant controller C(s) = 1. We consider the
usual feedback interconnection of P and C as described earlier.
In a stochastic setting, an m dimensional (stationary) stochastic process admits an
m-dimensional mean = E[u(t)] which is is independent of t, whereas its second order
moments E[u(t1)u(t2) T ] now de ne m m matrices which only depend on the time difference
t1 ; t2. As in the previous section, we derive that the in nite horizon RMS value
equals the H2 norm of the system, i.e.,
kHk2 Z 1
RMS = trace[h(t)h
;1
T (t)] dt = kHk2 H2 :
(a) Determine the H1 norm of the system P .
Hint: You can represent the plant P in MATLAB by introducing the numerator
(`teller') and the denominator (`noemer') polynomial coe cients separately. Since
(s +1)(s +2) = s2 +3s +2 the denominator polynomial is represented by a variable
den=[1 3 2] (coe cients always in descending order). Similarly, thenumerator
polynomial of P is represented by num=[0 -1 0]. The H1 norm of P can now be
read from the Bode plot of P by invoking the procedure bode(num,den).
(b) Determine the H1 norm of the sensitivity S, the complementary sensitivity T
and the control sensitivity R of the closed loop system.
Hint: The feedback interconnection of P and C can be obtained by the MATLAB procedures
feedbk or feedback. After reading the help information about this procedure
(help feedbk) welearn that the procedure requires state space representations of P
and C and produces a state space representation of the closed loop system. Make sure
that you use the right `type' option to obtain S T and R, respectively. A state space
representation of P can be obtained, e.g., by invoking the routine tf2ss (`transfer-tostate-space').
Thus, [a,b,c,d]= tf2ss(num,den) gives a state space description of P .
If you prefer a transfer function description of the closed loop to determine the H1
norms, then try the conversion routine ss2tf. See the corresponding help information.

64 CHAPTER 6. WEIGHTING FILTERS
d
- y
- g ?
P (s)
-
u
C(s)
-
r - g e
6;
?g
Chapter 6
Figure 6.1: Multivariable feedback con guration
which maps the reference signal r to the (real) tracking error r ; y (6= e in Fig.6.1
because of !) and the disturbance d to y.
Weighting lters
The complementary sensitivity
T = PC(I + PC) ;1 = I ; S
6.1 The use of weighting lters
which maps the reference signal r to the output y and the sensor noise to y.
6.1.1 Introduction
The H1 norm of an input-output system has been shown to be equal to
The control sensitivity
R = C(I + PC) ;1
which maps the reference signal r, the disturbance d and the measurement noise
to the control input u.
The maximal singular values of each of these transfer functions S T and R play an
important role in robust control design for multivariable systems. As is seen from the
de nitions of these transfers, the singular values of the sensitivity S (viewed as function
of frequency ! 2R) determine both the tracking performance as well as the disturbance
attenuation quality of the closed-loop system. Similarly, the singular values of the complementary
sensitivity model the ampli cation (or attenuation) of the sensor noise to
the closed-loop output y for each frequency, whereas the singular values of the control sensitivity
give insight for which frequencies the reference signal has maximal (or minimal)
e ect on the control input u.
All our H1 control designs will be formulated in such away that
an optimal controller will be designed so as to minimize the H1 norm of a
multivariable closed-loop transfer function.
k H(u) k2
k H(s) k1 = sup (H(j!)) = sup
!2R u2L2 k u k2
The H1 norm therefore indicates the maximal gain of the system if the inputs are allowed
to vary over the class of signals with bounded two-norm. The frequency dependent
maximal singular value (!), viewed as a function of !, provides obviously more detailed
information about the gain characteristics of the system than the H1 norm only.
For example, if a system is known to be all pass, meaning that the two-norm of the
output is equal to the two-norm of the input for all possible inputs u, then at every
frequency ! the maximal gain (H(j!)) of the system is constant and equal to the H1
norm k H(s) k1. The system is then said to have a at spectrum. This in contrast to
low-pass or high-pass systems in which the function (!) vanishes(or is attenuated) at
high frequencies and low frequencies, respectively.
It is this function, (j!), that is extensively manipulated in H1 control system design
to meet desired performance objectives. These manipulations are carried out by choosing
appropriate weights on the signals entering and leaving a control con guration like for
example the one of Figure 6.1. The speci cation of these weights is of crucial importance
for the overall control design and is one of the few aspects in H1 robust control design that
can not be automated. The choice of appropriate weighting lters is a typical `engineering
skill' which is based on a few simple mathematical observations, and a good insight in the
performance speci cations one wishes to achieve.
Once a control problem has been speci ed as an optimization problem in which the H1
norm of a (multivariable) transfer function needs to be minimized, the actual computation
of an H1 optimal controller which achieves this minimum is surprisingly easy, fast and
reliable. The algorithms for this computation of H1 optimal controllers will be the subject
of Chapter 8. The most time consuming part for a well performing control system using
H1 optimal control methods is the concise formulation of an H1 optimization problem.
This formulation is required to include all our a-priori knowledge concerning signals of
interest, all the (sometimes con icting) performance speci cations, stability requirements
and, de nitely not least, robustness considerations with respect to parameter variations
and model uncertainty.
Let us consider a simpli ed version of an H1 design problem. Suppose that a plant P
is given and suppose that we are interested in minimizing the H1 norm of the sensitivity
6.1.2 Singular value loop shaping
Consider the multivariable feedback control system of Figure 6.1 As mentioned before,
the multivariable stability margins and performance speci cations can be quanti ed by
considering the frequency dependent singular values of the various closed-loop systems
which we can distinguish in Figure 6.1
In this con guration we distinguish various `closed-loop' transfer functions:
The sensitivity
S =(I + PC) ;1
63

66 CHAPTER 6. WEIGHTING FILTERS
6.1. THE USE OF WEIGHTING FILTERS 65
jV (j!)j
1
S =(I + PC) ;1 over all controllers C that stabilize the plant P . The H1 optimal control
problem then amounts to determine a stabilizing controller C opt such that
min
C stab k S(s) k1 = k (I + PC opt ) ;1 k1
;!r 0 !r
Figure 6.2: Ideal low pass lter
d
Such acontroller then deserves to be called H1 optimal. However,itisby no means clear
that there exists a controller which achieves this minimum. The `minimum' is therefore
usually replaced by an `in mum' and we need in general to be satis ed with a stabilizing
controller Copt such that
- y
- g ?
P (s)
-
u
C(s)
-
e
-r g 6
V (s)
-
r 0
opt := inf
C stab k S(s) k1 (6.1)
g
k(I + PCopt) ;1 k1
(6.2)
: (6.3)
where opt is a prespeci ed number which we like to (and are able to) choose as close
Figure 6.3: Application of an input weighting lter
as possible to the optimal value opt. For obvious reasons, C opt is called a suboptimal H1
controller, and this controller may clearly depend on the speci ed value of .
Suppose that a controller achieves that k S(s) k1 . It then follows that for all
frequencies ! 2R
S(s)V (s). In Figure 6.3 we see that this amounts to including the transfer function V (s)
in the diagram of Figure 6.1 and considering the `new' reference signal r0 as input instead
of r. Thus, instead of the criterion (6.4), we nowlook for a controller which achieves that
(S(j!)) kS(s) k1 (6.4)
k S(s)V (s) k1
Thus is an upperbound of the maximum singular value of the sensitivityateachfrequency ! 2R. Conclude from (6.4) and the general properties of singular values, that the tracking
error r ; y (interpreted as a frequency signal) then satis es
where 0. Observe that for the ideal low-pass lter V this implies that
(S(j!)V (j!))
k ^r(!) ; ^y(!) k (S(j!)) k ^r(!) k (6.5)
k ^r(!) k
(S(j!)) (6.6)
k S(s)V (s) k1 = max
!
= max
j!j !r
In this design, no frequency dependent a-priori information concerning the reference
signal r or frequency dependent performance speci cations concerning the tracking error
r ; y has been incorporated. The inequalities (6.5) hold for all frequencies.
Thus, is now an upperbound of the maximum singular value of the sensitivity for frequencies
! belonging to the restricted interval [;!r!r]! Conclude that with this ideal
lter V
The e ect of input weightings
the minimization of the H1 norm of the weighted sensitivity corresponds to
minimization of the maximal singular value (!) of the sensitivity function for
frequencies ! 2 [;!r!r].
The tracking error r ; y now satis es for all ! 2R the inequalities
Suppose that the reference signal r is known to have a bandwith [0!r]. Then inequality
(6.5) is only interesting for frequencies ! 2 [0!r], as frequencies larger than !r are
not likely to occur. However, the controller was designed to achieve (6.4) for all ! 2R
and did not take bandwith speci cations of the reference signal into account. If we de ne
a stable transfer function V (s) with ideal frequency response
(
1 if ! 2 [;!r!r]
V (j!) =
0 otherwise
(6.7)
k r(!) ; y(!) k = k S(j!)V (j!)r 0 (!) k
kS(j!) k kV (j!)r 0 (!) k
(S(j!)) k r(!) k
jV ;1 (j!)jkr(!) k
then the outputs of such a lter are band limited signals with bandwith [0!r], i.e. for
any r0 2L2 the signal
where r is now a bandlimited reference signal, and
r(s) =V (s)r 0 (s)
1
jV (j!)j
jV ;1 (j!)j =
has bandwidth [0!r]. (See Figure 6.2). Instead of minimizing the H1 norm of the
sensitivity S(s) we now consider minimizing the H1 norm of the weighted sensitivity

68 CHAPTER 6. WEIGHTING FILTERS
6.1. THE USE OF WEIGHTING FILTERS 67
where W is a (stable) transfer function whose frequency response is ideally de ned by the
band pass lter
(
1 if ! j!j !
W (j!) =
0 otherwise
which isto be interpreted as 1 whenever V (j!) =0. For those frequencies (j!j >!r in
this example) the designed controller does not put a limit to the tracking error for these
frequencies did not appear in the reference signal r.
The last inequality in (6.7) is the most useful one and it follows from the more general
observation that, whenever k S(s)V (s) k1 with V a square stable transfer function
whose inverse V ;1 is again stable, then for all ! 2R there holds
and depicted in Figure 6.4. Instead of minimizing the H1 norm of the sensitivity S(s)
jW (j!)j
[S(j!)] = [S(j!)V (j!)V ;1 (j!)]
[S(j!)V (j!)] [V ;1 (j!)]
6
1
[V ;1 (j!)]
We thus come to the important conclusion that
!
-
0 ! !
Figure 6.4: Ideal band pass lter
a controller C which achieves that the weighted sensitivity
k S(s)V (s) k1
we consider minimizing the H1 norm of the weighted sensitivity W (s)S(s). In Figure 6.5
it is shown that this amounts to including the transfer function W (s) in the diagram of
Figure 6.1 (where we put = 0) and considering the `new' output signal e0 . A controller
which achieves an upperbound on the weighted sensitivity
results in a closed loop system in which
(S(j!)) [V ;1 (j!)] (6.8)
k W (s)S(s) k1
accomplishes, as in (6.6), that
(W (j!)S(j!))
(S(j!)) (6.9)
k W (s)S(s) k1 = max
!
= max
Remark 6.1 This conclusion holds for any stable weighting lter V (s) whose inverse
V ;1 (s) is again a stable transfer function. This is questionable for the ideal lter V we
used here to illustrate the e ect, because for ! > !r the inverse lter V (j!) ;1 can be
quali ed as unstable. In practice we will therefore choose lters which have a rational
transfer function being stable, minimum phase and biproper. An alternative rst order
lter for this example could thus have been e.g. V (s) = (s+100!r)
100(s+!r) .
! ! !
Remark 6.2 It is a standard property of the singular value decomposition that,whenever
V ;1 (j!) exists,
which provides an upperbound of the maximum singular value of the sensitivity for frequencies
! belonging to the restricted interval ! ! !. The tracking error e satis es
again the inequalities (6.7), with V replaced by W and it should not be surprising that the
same conclusions concerning the upperbound of the spectrum of the sensitivity S hold. In
particular, we nd similar to (6.8) that for all ! 2R there holds
1
[V (j!)]
[V ;1 (j!)] =
where denotes the smallest singular value.
(S(j!)) [W ;1 (j!)] (6.10)
The e ect of output weightings
provided the stable weighting lter W (s) has an inverse W ;1 (s) which is again stable.
6 e0
d
W
- y
- g ?
P (s)
-
u
C(s)
r - g e -
6
6
In the previous subsection we considered the e ect of applying a weighting lter for an
input signal. Likewise, we can also de ne weighting lters on the output signals which
occur in a closed-loop con guration as in Figure 6.1.
We consider again (as an example) the sensitivity S(s) viewed as a mapping from
the reference input r to the tracking error r ; y = e, when we fully disregard for the
moment the measurement noise . A straightforward H1 design would minimize the H1
norm of the sensitivity S(s) and result in the upperbound (6.5) for the tracking error.
We could, however, be interested in minimizing the spectrum of the tracking error at
speci c frequencies only. Let us suppose that we are interested in the tracking error e at
frequencies ! ! ! only, where ! > 0 and !>0 de ne a lower and upperbound. As
in the previous subsection, we introduce a new signal
Figure 6.5: Application of an output weighting lter
e 0 (s) =W (s)e(s)

70 CHAPTER 6. WEIGHTING FILTERS
6.1. THE USE OF WEIGHTING FILTERS 69
for some > 0 which is as close as possible to filt (which depends on the plant P and
the choice of the weigthing lters V and W ). To nd such a larger than or equal to the
unknown and optimal lt is the subject of Chapter 8, but what is important here is that
the resulting sensitivity satis es (6.11).
By incorporating weighting lters to each input and output signal which isofinterest
in the closed-loop control con guration, we arrive at extended con guration diagrams such
as the one shown in Figure 6.6.
6.1.3 Implications for control design
In this section we willcommentonhowthe foregoing can be used for design purposes. To
this end, there are a few important observations to make.
d 0
?
6
u 0
For one thing, we showed in subsection 6.1.2 that by choosing the frequency response
of an input weighting lter V (s) so as to `model' the frequency characteristic of the
input signal r, the a-priori information of this reference signal has been incorporated
in the controller design. By doing so, the minimization of the maximum singular
value of the sensitivity S(s) has been re ned (like in (6.6)) to the frequency interval
of interest. Clearly, we can do this for any input signal.
Vd
Wu
-y Wy -y0
- f ?d
P (s)
v- u
C(s) -
6
f
6
-
r 0
Vr -r
0
?f V
- e0
- f- e ?
We
Secondly, we obtained in (6.8) and in (6.10) frequency dependent upperbounds for
the maximum gain of the sensitivity. Choosing V (j!) (or W (j!)) appropriately,
enables one to specify the frequency attenuation of the closed-loop transfer function
(the sensitivity in this case). Indeed, choosing, for example, V (j!)alowpass transfer
function implies that V ;1 (j!) is a high pass upper-bound on the frequency spectrum
of the closed-loop transfer function. Using (6.8) this implies that low frequencies of
the sensitivity are attenuated and that the frequency characteristic of V has `shaped'
the frequency characteristic of S. The same kind of `loop-shaping' can be achieved
by either choosing input or output weightings.
Figure 6.6: Extended con guration diagram
Thirdly, by applying weighting factors to both input signals and output signals
we can minimize (for example) the H1 norm of the two-sided weighted sensitivity
W (s)S(s)V (s), i.e., a controller could be designed so as to achieve that
k W (s)S(s)V (s) k1
General guidelines on how to determine input and output weightings can not be given,
for each application requires its own performance speci cations and a-priori information
on signals. Although the choice of weighting lters in uences the overall controller design,
the choice of an appropriate lter is to a large extent subjective. As a general warning,
however, one should try to keep the lters of as low a degree as possible. This, because the
order of a controller C that achieves inequality (6.12) is, in general, equal to the sum of the
order of the plant P , and the orders of all input weightings V and output weightings W .
The complexity of the resulting controller is therefore directly related to the complexity
of the plant and the complexity of the chosen lters. High order lters lead to high order
controllers, which maybe undesirable.
More about appropriate weighting lters and their interactive e ects on the nal solution
in a complicated scheme as Fig. 6.6 follows in the next chapters.
for some >0. Provided the transfer functions V (s) and W (s) havestable inverses,
this leads to a frequency dependent upperbound for the original sensitivity. Precisely,
in this case
(S(j!)) [V ;1 (j!)] [W ;1 (j!)] (6.11)
6.2 Robust stabilization of uncertain systems
from which we see that the frequency characteristic of the sensitivity is shaped
by both V as well as W . It is precisely this formula that provides you with a
wealth of design possibilities! Once a performance requirement for a closed-loop
transfer function (let's say the sensitivity S(s)) is speci ed in terms of its frequency
characteristic, this characteristic needs to be `modeled' by the frequency response
of the product V ;1 (j!)W ;1 (j!) by choosing the input and output lters V and
W appropriately. A controller C(s) that bounds the H1 norm of the weighted
sensitivity W (s)S(s)V (s) then achieves the desired characteristic by equation (6.11).
6.2.1 Introduction
The theory of H1 control design is model based. By this we mean that the design of a
controller for a system is based on a model of that system. In this course we will not
address the question how such a model can be obtained, but any modeling procedure
will, in practice, be inaccurate. Depending on our modeling e orts, we can in general
expect a large or small discrepancy between the behavior of the (physical) system which
we wish to control and the mathematical model we obtained. This discrepancy between
the behavior of the physical plant and the mathematical model is responsible for the fact
that a controller, we designed optimally on the basis of the mathematical model, need not
ful ll our optimality expectations once the controller is connected to the physical system.
It is easy to give examples of systems in which arbitrary small parameter variations of
The weighting lters V and W on input and output signals of a closed-loop transfer
function give therefore the possibility to shape the spectrum of that speci c closed-loop
transfer. Once these lters are speci ed, a controller is computed to minimize the H1
norm of the weighted transfer and results in a closed-loop transfer whose spectrum has
been shaped according to (6.11).
In the example of a weighted sensitivity, the controller C is thus computed to establish
that
lt := inf
C stab k W (s)S(s)V (s) k1 (6.12)
kW (s)(I + PC) ;1 V (s) k1
(6.13)
: (6.14)

72 CHAPTER 6. WEIGHTING FILTERS
6.2. ROBUST STABILIZATION OF UNCERTAIN SYSTEMS 71
Additive uncertainty
The simplest way to represent the discrepancy between the model and the true system is
by taking the di erence of their respective transfer functions. That is,
Pt = P + P (6.15)
plant parameters, in a stable closed loop system con guration, fully destroy the stability
properties of the system.
Robust stability refers to the ability of a closed loop stable system to remain stable in
the presence of modeling errors. For this, one needs to have some insight in the accuracy
of a mathematical model which represents the physical system we wishtocontrol. There
are many ways to do this:
where P is the the nominal model, Pt is the true or perturbed model and P is the additive
perturbation. In order to comply with notations in earlier chapters and to stress the relation
with the next input multiplicative perturbation description, we use the notation P as
one mathematical object to display the perturbation of the nominal plant P . Additive
perturbations are pictorially represented as in Figure 6.7.
One can take a stochastic approach and attach a certain likelihood or probability
to the elements of a class of models which are assumed to represent the unknown,
(often called `true') system.
One can de ne a class of models each of which is equally acceptable to model the
unknown physical system
P
-
One can select one nominal model together with a description of its uncertainty in
terms of its parameters, in terms of its frequency response, in terms of its impulse
response, etc. In this case, the uncertain part of a process is modeled separately
from the known (nominal) part.
-
+
?
- P - j
+
For each of these possibilities a quanti cation of model uncertainty is necessary and essential
for the design of controllers which are robust against those uncertainties.
In practice, the design of controllers is often based on various iterations of the loop
Figure 6.7: Additive perturbations
data collection ;! modeling ;! controller design ;! validation
Multiplicative uncertainty
in which improvement of performance of the previous iteration is the main aim.
In this chapter we analyze robust stability ofacontrol system. We introduce various
ways to represent model uncertainty and we will study to what extent these uncertainty
descriptions can be taken into account to design robustly stabilizing controllers.
Model errors may also be represented in the relative ormultiplicative form. We distinguish
the two cases
6.2.2 Modeling model errors
Pt = (I + )P = P + P = P + P (6.16)
Pt = P (I + )=P + P =P + P (6.17)
It may sound somewhat paradoxial to model dynamics of a system which one deliberately
decided not to take into account in the modeling phase. Our purpose here will be to only
provide upperbounds on modeling errors. Various approaches are possible
where P is the nominal model, Pt is the true or perturbed model and is the relative
perturbation. Equation (6.16) is used to represent output multiplicative uncertainty, equation(6.17)
represents input multiplicative uncertainty. Input multiplicative uncertainty is
well suited to represent inaccuracies of the actuator being incorporated in the transfer P .
Analogously, the output multiplicative uncertainty is a proper means to represent noise
e ects of the sensor. (However, the real output y should be still distinguishable from
the measured output y + ). The situations are depicted in Figure 6.8 and Figure 6.9,
respectively. Note that for single input single output systems these two multiplicative
uncertainty descriptions coincide. Note also that the products P and P in 6.16 and
6.17 can be interpreted as additive perturbations of P .
model errors can be quanti ed in the time domain. Typical examples include descriptions
of variations in the physical parameters in a state space model.
alternatively, model errors can be quanti ed in the frequency domain by analyzing
perturbations of transfer functions or frequency responses.
We will basically concentrate on the latter in this chapter. For frequency domain model
uncertainty descriptions one usually distinguishes two approaches, which lead to di erent
research directions:
Unstructured uncertainty: model uncertainty is expressed only in terms of upperbounds
on errors of frequency responses. No more information on the origin of the
modeling errors is used.
Remark 6.3 We also emphasize that, at least for single input single output systems, the
multiplicative uncertainty description leaves the zeros of the perturbed system invariant.
The popularityofmultiplicative model uncertainty description is for this reason di cult to
understand for it is well known that an accurate identi cation of the zeros of a dynamical
system is a non-trivial and very hard problem in system identi cation.
Structured uncertainty: apart from an upperbound on the modeling errors, also the
speci c structure in uncertainty of parameters is taken into account.
For the analysis of unstructured model uncertainty in the frequency domain there are
four main uncertainty models which we brie y review.

74 CHAPTER 6. WEIGHTING FILTERS
6.2. ROBUST STABILIZATION OF UNCERTAIN SYSTEMS 73
Coprime factor uncertainty
-
Coprime factor perturbations have beenintroduced to cope with perturbations of unstable
plants. Any (multivariable rational) transfer function P can be factorized as P = ND ;1
in such away that
both N and D are stable transfer functions.
-
+ ?
- P - j
+
D is square and N has the same dimensions as P .
there exist stable transfer functions X and Y such that
Figure 6.8: Output multiplicative uncertainty
XN + YD= I
which isknown as the Bezout equation, Diophantine equation or even Aryabhatta's
identity.
-
6
Such a factorization is called a (right) coprime factorization of P .
Remark 6.4 The terminology comes from number theory where two integers n and d are
called coprime if 1 is their common greatest divisor. It follows that n and d are coprime
if and only if there exist integers x and y such thatxn + yd =1.
? - P -
j
+
-
A left coprime factorization has the following interpretation. Suppose that a nominal
plant P is factorized as P = ND ;1 . Then the input output relation de ned by P satis es
Figure 6.9: Input multiplicative uncertainty
y = Pu = ND ;1 u = Nv
where we de ned v = D ;1u, or, equivalently, u = Dv. Now note that, since N and D are
stable, the transfer function
Feedback multiplicative uncertainty
In few applications one encounters feedback versions of the multiplicative model uncertainties.
They are de ned by
(6.20)
y
: v 7!
u
N
D
is stable as well. We haveseen that such a transfer matrix maps L2 signals to L2 signals.
We can thus interpret (6.20) as a way togenerate all bounded energy input-output signals
u and y which are compatible with the plant P . Indeed, any element v in L2 generates via
(6.20) an input output pair (u y) for which y = Pu,andconversely, any pair (u y) 2L2
satisfying y = Pu is generated by plugging in v = D ;1u in (6.20).
Pt = (I + ) ;1 P (6.18)
Pt = P (I + ) ;1 (6.19)
Example 6.5 The scalar transfer function P (s) = (s;1)(s+2)
(s;3)(s+4) has a coprime factorization
P (s) =N(s)D ;1 (s) with
s ; 1
; 3
N(s) = D(s) =s
s +4 s +2
Let P = ND ;1 be a right coprime factorization of a nominal plant P . Coprime factor
uncertainty refers to perturbations in the coprime factors N and D of P . We de ne a
perturbed model
and referred to as the output feedback multiplicative model error and the input feedback
multiplicative model error. We will hardly use these uncertainty representations in this
course, and mention them only for completeness. The situation of an output feedback
multiplicative model error is depicted in Figure 6.10. Note that the sign of the feedback
addition from is irrelevant, because the phase of will not be taken into account when
considering norms of .
Pt = (N + N)(D + D) ;1 (6.21)
-
- P - j ?
where
:= ; N D
Figure 6.10: Output feedback multiplicative uncertainty
re ects the perturbation of the coprime factors N and D of P . The next Fig. 6.11 illustrates
this right coprime uncertainty inablockscheme.

76 CHAPTER 6. WEIGHTING FILTERS
6.2. ROBUST STABILIZATION OF UNCERTAIN SYSTEMS 75
- -
D N
6
- y
Pt(s)
- C(s) - u
r - h
6
D ;1 ?
?
y
- - -
N
- -
v
u
Figure 6.12: Feedback loop with uncertain system
Figure 6.11: Right coprime uncertainty.
Find a controller C for the feedback con guration of Figure 6.12 such that C
1
stabilizes the perturbed plant Pt for all k k1 with > 0 as small as
possible (i.e., C makes the stability margin as large as possible).
Remark 6.6 It should be emphasized that the coprime factors N and D of P are by
no means unique! A plant P admits many coprime factorizations P = ND ;1 and it is
therefore useful to introduce some kind of normalization of the coprime factors N and D.
It is often required that the coprime factors should satisfy the normalization
Such a controller is called robustly stabilizing or optimally robustly stabilizing for the
perturbed systems Pt. Since this problem can be formulated for each of the uncertainty
descriptions introduced in the previous section, we can de ne four types of stability margins
D D + N N = I
The additive stability margin is the H1 norm of the smallest stable P for which
the con guration of Figure 6.12 with Pt de ned by (6.15) becomes unstable.
This de nes the normalized right coprime factorization of P and it has the interpretation
that the transfer de ned in (6.20) is all pass.
The output multiplicative stability margin is the H1 norm of the smallest stable
which destabilizes the system in Figure 6.12 with Pt de ned by (6.16).
6.2.3 The robust stabilization problem
For each of the above types of model uncertainty, the perturbation is a transfer function
which we assume to belong to a class of transfer functions with an upperbound on their
The input multiplicative stability margin is similarly de ned with respect to equation
(6.17) and
H1 norm. Thus, we assume that
1
The coprime factor stability margin is analogously de ned with respect to (6.21) and
the particular coprime factorization of the plant P .
k k1
The main results with respect to robust stabilization of dynamical systems follow in
a straightforward way from the celebrated small gain theorem. If we consider in the
con guration of Figure 6.12 output multiplicative perturbed plants Pt =(I + )P then
we can replace the block indicated by Pt by the con guration of Figure 6.8 to obtain the
system depicted in Figure 6.13.
where 0. 1 Large values of therefore allow for small upper bounds on the norm of
, whereas small values of allow for large deviations of P . Note that if !1then the
H1 norm of is required to be zero, in which case perturbed models Pt coincide with
the nominal model P .
For a given nominal plant P this class of perturbations de nes a class of perturbed
plants
1
Pt k k1
-
v w
? -y - h
P (s)
- C(s) - u
+
r - h
6;
Figure 6.13: Robust stabilization for multiplicative perturbations
To study the stability properties of this system we can equivalently consider the system
of Figure 6.14 in which M is the system obtained from Figure 6.13 by setting r =0and
`pulling' out the uncertainty block .
which depends on the particular model uncertainty structure.
Consider the feedback con guration of Figure 6.12 where the plant P has been replaced
by the uncertain plant Pt. We will assume that the controller C stabilizes this system if
= 0, that is, we assume that the closed-loop system is asymptotically stable for the
nominal plant P . An obvious question is how large k k1 can become before the closed
loop system becomes unstable. The H1 norm of the smallest (stable) perturbation
which destabilizes the closed-loop system of Figure 6.12 is called the stability margin of
the system.
We can also turn this question into a control problem. The robust stabilization problem
amounts to nding a controller C so that the stability margin of the closed loop system is
maximized. The robust stabilization problem is therefore formalized as follows:
1 The reason for taking the inverse ;1 as an upperbound rather than will turn useful later.

78 CHAPTER 6. WEIGHTING FILTERS
6.2. ROBUST STABILIZATION OF UNCERTAIN SYSTEMS 77
for all ! 2R. Stated otherwise
-
: (6.22)
1
[M(j!)]
[ (j!)] <
for all ! 2R.
M
6.2.4 Robust stabilization: main results
Robust stabilization under additive perturbations
Figure 6.14: Small gain con guration
Carrying out the above analysis for the case of additive perturbations leads to the following
main result on robust stabilization in the presence of additive uncertainty.
For the case of output multiplicative perturbations M maps the signal w to v and the
corresponding transfer function is easily seen to be
Theorem 6.8 (Robust stabilization with additive uncertainty) Acontroller C stabilizes
Pt = P + P for all k P k1< 1 if and only if
M = T = PC(I + PC) ;1
C stabilizes the nominal plant P
.
k C(I + PC) ;1 k1
i.e., M is precisely the complementary sensitivity transfer function2 . Since we assumed
that the controller C stabilizes the nominal plant P it follows that M is a stable transfer
function, independent of the perturbation but dependent on the choice of the controller
C.
The stability properties of the con guration of Figure 6.13 are determined by the small
gain theorem (Zames, 1966):
Remark 6.9 Note that the transfer function R = C(I + PC) ;1 is the control sensitivity
of the closed-loop system. The control sensitivity of a closed-loop system therefore re ects
the robustness properties of that system under additive perturbations of the plant!
The interpretation of this result is as follows
Theorem 6.7 (Small gain theorem) Suppose that the systems M and are both stable.
Then the autonomous system determined by the feedback interconnection of Figure
6.14 is asymptotically stable if
The smaller the norm of the control sensitivity, the greater will be the norm of the
smallest destabilizing additive perturbation. The additive stability margin of the
closed loop system is therefore precisely the inverse of the H1 norm of the control
sensitivity
k M k1< 1
For a given controller C the small gain theorem therefore guarantees the stability of
the closed loop system of Figure 6.14 (and thus also the system of Figure 6.13) provided
is stable and satis es for all ! 2R
(M(j!) (j!)) < 1:
1
k C(I + PC) ;1 k1
If we liketo maximize the additive stability margin for the closed loop system, then
we needtominimize the H1 norm of the control sensitivity R(s)!
For SISO systems this translates in a condition on the absolute values of the frequency
responses of M and . Precisely, for all ! 2R we shouldhavethat Theorem 6.8 can be re ned by considering for each frequency ! 2 R the maximal
allowable perturbation P which makes the system of Figure 6.12 unstable. If we assume
that C stabilizes the nominal plant P then the small gain theorem and (6.22) yields that
for all additive stable perturbations P for which
(M ) = jM j = jMjj j = (M) ( )
(where we omitted the argument j! in each transfer) to guarantee the stability of the closed
loop system. For MIMO systems we obtain, by using the singular value decomposition,
that for all ! 2R
1
[R(j!)]
[ P (j!)] <
(M ) = (YM MU M Y U ) (M) ( )
the closed-loop system is stable. Furthermore, there exists a perturbation P right on
the boundary (and certainly beyond) with
(where again every transfer is supposed to be evaluated at j!) and the maximum is
reached for Y = UM that can always be accomplished without a ecting the constraint
k k1< 1 . Hence, to obtain robust stability we have to guarantee for both SISO and
MIMO systems that
1
[R(j!)]
[ P (j!)] =
[M(j!)] [ (j!)] < 1
which destabilizes the system of Figure 6.12.
2 Like inchapter 3 actually M = ;T , but the sign is irrelevant as it can be incorporated in .

80 CHAPTER 6. WEIGHTING FILTERS
6.2. ROBUST STABILIZATION OF UNCERTAIN SYSTEMS 79
C stabilizes the nominal plant P
Robust stabilization under multiplicative perturbations
.
k (I + PC) ;1 k1
Remark 6.13 We recognize the transfer function S = (I + PC) ;1 = I ; T to be the
sensitivity of the closed-loop system.
For multiplicative perturbations, the main result on robust stabilization also follows as a
direct consequence of the small gain theorem, and reads as follows for the class of output
multiplicative perturbations.
The interpretation of this result is similar to the foregoing robustness theorem and not
included here.
Theorem 6.10 (Robust stabilization with multiplicative uncertainty) Acontroller
C stabilizes Pt =(I + )P for all k k1< 1 if and only if
C stabilizes the nominal plant P
6.2.5 Robust stabilization in practice
.
k PC(I + PC) ;1 k1
The robust stabilization theorems of the previous section can be used in various ways.
If there is no a-priori information on model uncertainty then the frequency responses
of the control sensitivity ( [R(j!)]), the complementary sensitivity ( [T (j!)]) and
the sensitivity ( [S(j!)]) provide precise information about the maximal allowable
perturbations [ (j!)] for which the controlled system remains asymptotically stable
under (respectively) additive, multiplicative and feedback multiplicative perturbations
of the plant P . Graphically, we can get insight in the magnitude of these
admissable perturbations by plotting the curves
1
add(!) =
[R(j!)]
Remark 6.11 We recognize the transfer function T = PC(I + PC) ;1 = I ; S to be
the complementary sensitivity of the closed-loop system. The complementary sensitivity
of a closed-loop system therefore re ects the robustness properties of that system under
multiplicative perturbations of the plant!
The interpretation of this result is similar to the foregoing robustness theorem:
The smaller the norm of the complementary sensitivity T (s), the greater will be the
norm of the smallest destabilizing output multiplicative perturbation. The output
multiplicative stability margin of the closed loop system is therefore the inverse of
the H1 norm of the complementary sensitivity
1
[T (j!)]
1
[S(j!)]
mult(!) =
1
k PC(I + PC) ;1 k1
feed(!) =
for all frequency ! 2R. (which corresponds to `mirroring' the frequency responses of
[R(j!)], [T (j!)], and [S(j!)] around the 0dB axis). The curves add(!), mult(!)
and feed(!) then provide an upperbound on the allowable additive, multplicative
and feedback multiplicative perturbations per frequency ! 2R.
By minimizing the H1 norm of the complementary sensitivity T (s) we achieve a
closed loop system which ismaximally robust against output multiplicative perturbations.
If, on the other hand, the information about the maximal allowable uncertainty of
the plant P has been speci ed in terms of one or more of the curves add(!), mult(!)
or feed(!) then we can use these speci cations to shape the frequency response of
either R(j!), T (j!) orS(j!) using the ltering techniques described in the previous
chapter. Speci cally, let us suppose that a nominal plant P is available together
with information of the maximal multiplicative model error mult(!) for ! 2 R.
We can then interpret mult as the frequency response of a weighting lter with
transfer function V (s), i.e. V (j!) = mult(!). The set of all allowable multiplicative
perturbations of the nominal plant P is then given by
Theorem 6.10 can also be re ned by considering for each frequency ! 2R the maximal
allowable perturbation which makes the system of Figure 6.12 unstable. If we assume
that C stabilizes the nominal plant P then all stable output multiplicative perturbations
for which
1
[T (j!)]
[ (j!)] <
leave the closed-loop system stable. Moreover, there exists a perturbation right onthe
boundary, so:
1
[T (j!)]
[ (j!)]
V
where V is the chosen weighting lter with frequency response mult and where
is any stable transfer function with k k1 < 1. Pulling out the transfer matrix
from the closed-loop con guration (as in the previous section) now yields a slight
modi cation of the formulas in Theorem 6.10. A controller C now achieves robust
stability against this class of perturbations if and only if it stabilizes P (of course)
and
which destabilizes the system of Figure 6.12.
Robust stabilization under feedback multiplicative perturbations
For feedback multiplicative perturbations, the main results are as follows
kPC(I + PC) ;1 V k1 = kTVk1 1:
Theorem 6.12 (Robust stabilization with feedback multiplicative uncertainty)
Acontroller C stabilizes Pt =(I + ) ;1P for all k k1< 1 if and only if

82 CHAPTER 6. WEIGHTING FILTERS
6.2. ROBUST STABILIZATION OF UNCERTAIN SYSTEMS 81
(a) Determine the H1 norm of P . At which frequency ! is the norm k P k1
attained?
- W - - V -
6
Hint: First compute a state space representation of P by means of the conversion algorithm
tfm2ss (`transfer-matrix-to-state-space'). Read the help information carefully!
(The denominator polynomial is the same as in Exercise 2, the numerator polynomials
are represented in one matrix: the rst row being [0 -47 2], the second [0 -42 0], etc.
Once you have a state space representation of P you can read its H1 norm from a plot
of the singular values of P . Use the routine sigma.
?
- P
- n -
Figure 6.15: Filtered additive perturbation.
(b) Use (6.24) as a controller for the plant P and plot the singular values of the
closed loop control-sensitivity C(I + PC) ;1 to investigate robust stability of
this system. Determine the robust stability margin of the closed loop system
under additive perturbations of the plant.
Hint: Use the MATLAB routine feedbk with the right `type' option as in exercise 6.1
to construct a state space representation of the control-sensitivity and use sigma to
read H1 norms of multivariable systems.
(c) Consider the perturbed controller
The latter expression is a constraint ontheH1 norm of the weighted complementary
sensitivity! We therefore need to consider the H1 optimal control design problem
so as to bound the H1 norm of the weighted complementary sensitivity TV by one.
So the next goal is to synthesize controllers which accomplish this upperbound. This
problem will be discussed in forthcoming chapters. In a more general and maybe
more familiar setting we can quantify our knowledge concerning the additive model
error by means of pre- and post lters V and W as schematized in Fig 6.15. Clearly,
in this case the additive model error P = V W . If satis es the norm constraint
1:13 0
0 :88
C(s) =
k k1< 1
and compute the closed-loop poles of this system. Conclusion?
Hint: Use again the procedure feedbk to obtain a state space representation of the
closed loop system. Recall that the closed loop poles are the eigenvalues of the `A'
matrix in any minimal representation of the closed loop system. See also the routine
minreal.
then for every frequency ! 2R we have that
( P (j!)) (W (j!)) (V (j!)):
3. Consider the linearized system of an unstable batch reactor described by the state
space model
Consequently, if we `pull out' the transfer from the closed loop yields that M =
WRV. To ful l the small gain constraint the control sensitivity R then needs to
satisfy
0 0
5:679 0 C
1:136 ;3:146A
1:136 0
u
1
C
0
B
@
0
B
@
k WRV k1 1:
x +
1
C
A
1:38 ;0:2077 6:715 ;5:676
;0:5814 ;4:29 0 0:675
1:067 4:273 ;6:654 5:893
0:048 4:273 1:343 ;2:104
_x =
6.2.6 Exercises
1. Derive a robust stabilization theorem in the spirit of Theorem 6.10 for
1 0 1 ;1
0 1 0 0
y =
(a) the class of input multiplicative perturbations.
(b) the class of input feedback multiplicative perturbations
(a) Verify (using MATLAB!) that the input output system de ned by this model
is unstable.
2. Consider a 2 2 system described by the transfer matrix
(b) Consider the controller with transfer function
0 ;2
8 0
3
7
56s
(s+1)(s+2)
;47s+2
(s+1)(s+2)
1
+
s
0 ;2
5 0
C(s) =
5 : (6.23)
2
6
4
P (s) =
50s+2
(s+1)(s+2)
;42s
(s+1)(s+2)
Using Matlab, interconnect the controller with the given plant and show that
the corresponding closed loop system is stable.
(c) Make a plot of the singular values (as function of frequency) of the complementary
sensitivity PC(I + PC) ;1 of the closed loop system.
(d) What are your conclusions concerning robust stability of the closed loop system?
The controller for this system is a diagonal constant gain matrix given by
(6.24)
1 0
0 1
C(s) =
We consider the usual feedback con guration of plant and controller.

84 CHAPTER 7. GENERAL PROBLEM.
while:
u = Ky (7.2)
denotes the controller. Eliminating u and y yields:
Chapter 7
z =[G11 + G12K(I ; G22K) ;1 G21]w def
= M(K)w (7.3)
An expression like (7.3) in K will be met very often and has got the name linear
fractional transformation abbreviated as LFT. Our combined control aim requires:
General problem.
k M(K) k1
(7.4)
= min
Kstabilising
k z k2
k w k2
min
Kstabilising sup
w2L2
Now that we have prepared all necessary ingredients in past chapters we are ready to
compose the general problem in such a structure, the so-called augmented plant, that the
problem is well de ned and therefore the solution straight forward to obtain. We will start
with a formal exposure and de nitions and next illustrate it by examples.
as the H1-norm is the induced operator norm for functions mapping L2;signals to
L2;signals, as explained in chapter 5. Of course we have tocheck whether stabilising controllers
indeed exist. This can best be analysed when we consider a state space description
of G in the following stylised form:
7.1 Augmented plant.
(7.5)
1
A (n+[w]+[u])
2R(n+[z]+[y])
0
@ A B1 B2
C1 D11 D12
G :
C2 D21 D22
The augmented plant contains, beyond the process model, all the lters for characterising
the inputs and weighting the penalised outputs as well as the model error lines. In Fig.
7.1 the augmented plant isschematised.
exogenous input w z output to be controlled
- -
- G(s)
-
6
control input u y measured output
where n is the dimension of the state space of G while [.] indicates the dimension of
the enclosed vector. It is evident that the unstable modes (i.e. canonical states) have to
be reachable from u so as to guarantee the existence of stabilising controllers. This means
that the pair fA B2g needs to be stabilisable. The controller is only able to stabilise, if
it can conceive all information concerning the unstable modes so that it is necessary to
require that fA C2g must be detectable. So, summarising:
There exist stabilising controllers K(s), i the unstable modes of G are both controllable
by u and observable from y which is equivalent to requiring that fA B2g is stabilisable and
fA C2g is detectable.
An illustrative example: sensitivity. Consider the structure of Fig. 7.2.
?
K(s)
Figure 7.1: Augmented plant.
~x ~n
- -
0
6
?
Vn
Wx
6
n
r =0 + e
? y ~y
- n - C - P - n - Wy
-
; 6
x
?
In order not to confuse the inputs and outputs of the augmented plant with those of
the internal blocks we will indicate the former ones in bold face. All exogenous inputs are
collected in w and are the L2;bounded signals entering the shaping lters that yield the
actual input signals such as reference, disturbance, system perturbation signals, sensor
noise and the kind. The output signals, that have to be minimised in L2;norm and
that result from the weighting lters, are collected in z and refer to (weighted) tracking
errors, actuator inputs, model error block inputs etc. The output y contains the actually
measured signals that can be used as inputs for the controller block K. Its output u
functions as the controller input, applied to the augmented system with transfer function
G(s). Consequently in s;domain we may write the augmented plant in the following,
properly partitioned form:
Figure 7.2: Mixed sensitivity structure.
(7.1)
w
u
z
y = G11 G12
G21 G22
83

86 CHAPTER 7. GENERAL PROBLEM.
7.2. COMBINING CONTROL AIMS. 85
m11 m12
(7.9)
1
C
A
.
. ..
. ..
m21
0
B
@
M =
Output disturbance n is characterised by lter Vn from exogenous signal ~n belonging
to L2. For the moment wetakethe reference signal r equal to zero and forget about the
model error, lter Wx, the measurement noise etc. , because we want to focus rst on
exclusively one performance measure. We would like to minimise ~y, i.e. the disturbance
in output y with weighting lter Wy so that equation (7.4) turns into:
.
It can be proved that:
= min
Cstabilising k Wy(I + PC) ;1 Vn k1= (7.6)
k ~y k2
k ~n k2
min
Cstabilising sup
~n2L2
k mij k1 k M k1 (7.10)
(7.7)
= min
Cstabilising k WySVn k1
Consequently the condition:
In the general setting of the augmented plant the structure would be as displayed in
Fig. 7.3.
k M k1< 1 (7.11)
G(s)
is su cient to guarantee that:
~y
+
- Vn
- - Wy
-
6+
;
?
- P
-
n
~n
8i j :k mij k1< 1 (7.12)
So the k : k1 of the full matrix M bounds the k : k1 of the various entries. Certainly,
it is not a necessary condition, as can be seen from the example:
;e
6
x
(7.13)
M =(m1 m2)
if k mi k1 1fori p =12 then k M k1 2
?
;C
so that it is advisory to keep the composed matrix M as small as possible. The most
trivial example is the so-called mixed sensitivity problem as represented in Fig. 7.2. The
reference r is kept zero again so that we have only one exogenous input viz. ~n and two
outputs ~y and ~x that yield a two blockaugmented system transfer to minimise:
Figure 7.3: Augmented plant for sensitivity alone.
The corresponding signals and transfers can be represented as:
(7.14)
k1
k M k1= k WySVn
WxRVn
G
z }| {
(7.8)
~n
x
WyVn WyP
Vn P
~y
;e =
The corresponding augmented problem setting is given in Fig. 7.4 and described by
the generalised transfer function G as follows:
;C = K
(7.15)
1
A ~n
x
0
@ WyVn WyP
0 Wx
P
=
1
A = G ~n
x
0
@ ~y
~x
;e
It is a trivial exercise to subsitute the entries Gij in equation (7.3), yielding the same
M as in equation (7.6).
Vn
By proper choice of Vn the disturbance can be characterised and the lter Wx should
guard the saturation range of the actuator in P . Consequently the lower term in M viz.
k WxRVn k1 1 represents a constraint. From Fig. 7.2 we also learn that we can think
the additive weighted model error 0 between ~x and ~n. Consequently if we end up with:
7.2 Combining control aims.
k WxRVn k1 k M k1< (7.16)
Along similar lines we could go through all kinds of separate and isolated criteria (as is
done in the exercises!). However, we are not so much interested in single criteria but much
more in con icting and combined criteria. This is usually realised as follows:
If we have several transfers properly weighted, they can be taken as entries mij in a
composed matrix M like:

88 CHAPTER 7. GENERAL PROBLEM.
7.3. MIXED SENSITIVITY PROBLEM. 87
3. Compute a stabilising controller C (see chapter 13) such that:
< (7.21)
W1SV1
W2NV2 1
z
where is as small as possible.
4. If >1, decrease W1 and/or V1 in gain and/or frequency band in order to relax the
performance aim and thereby giving more room to satisfy the robustness constraint.
Go back tostep3.
G(s)
- Wy
-
6
+
~n - Vn
- n -
6
Wx
+
;
?
x -
6
P
- ;e
? - 6
~y
- - ~x
6
5. If

90 CHAPTER 7. GENERAL PROBLEM.
7.4. A SIMPLE EXAMPLE. 89
a>0 (7.28)
V (s) = a
s + a
j (7.24)
V
1+PC
j
jM(s)j = sup
s2C +
jM(j!)j = sup
s2C +
sup
!
Since S = MV ;1 , the corresponding sensitivities can be displayed in a Bode diagram
as in Fig. 7.7.
The peaks inC + will occur for the extrema in S = (1 + PC) ;1 when P (bi) is zero.
These zeros put the bounds and it can be proved that a controller can be found such that:
jV (bi)j (7.25)
k M k1= max
i
If there exists only one right half plane zero b, we can optimise M by a stabilising
controller C1 in the 1-norm leading to optimal transfer M1. For comparison we can
also optimise the 2-norm by a controller C2 analogously yielding M2. Do not try to solve
this yourself. The solutions can be found in [11]). The ideal controllers are computed which
will turn out to be nonproper. In practice we can therefore only apply these controllers
in a su ciently broad band. For higher frequencies we have to attenuate the controller
transfer by adding a su cient number of poles to accomplish the so-called roll-o . For the
ideal controllers the corresponding optimal, closed loop transfers are given by:
Figure 7.7: Bode plot of tracking solution S.
M1 = jV (b)j (7.26)
Unfortunately, theS1 approaches in nity for increasing !,contrary to the S2. Remember
that we still study the solution for the ideal, nonproper controllers. Is this increasing
sensitivity disastrous? Not in the ideal situation, where we did not expect any reference
signal components for these high frequencies. However, in the face of stability robustness
and actuator saturation, this is a bad behaviour as we necessarily require that T is small
and because S + T = 1, inevitably:
(7.27)
M2 = V (b) 2b
s + b
as displayed in the approximate Bode-diagram Fig. 7.6.
lim
!!1 jS1j = 1 ) lim
!!1 jT1j = lim
!!1 j1 ; S1j = 1 (7.29)
Consequently robustness and saturation requirements will certainly be violated. But
it is no use complaining, as these requirements were not included in the criterion after all.
Inclusion can indeed improve the solution in these respects, but, like in the H2 solution,
we have topaythen by aworse sensitivity at the low pass band. This is another waterbed
e ect.
Figure 7.6: Bode plot of tracking solution M(K).
7.5 The typical compromise
Atypical weighting situation for the mixed sensitivity problem is displayed in Fig. 7.8.
Suppose the constraint isonN = T . Usually, W1V1 is low pass and W2V2 is high pass.
Suppose also that, by readjusting weights W1V1, wehaveindeed obtained:
inf
Kstabilising k M(K) k1= 1 (7.30)
Then certainly :
k W1SV1 k1< 1 )8! : jS(j!)j < jW1(j!) ;1 V1(j!) ;1 j (7.31)
k W2TV2 k1< 1 )8! : jT (j!)j < jW2(j!) ;1 V2(j!) ;1 j (7.32)
Notice that M1 is an all pass function. (From this alone we may conclude that the
ideal controller must be nonproper.) It turns out that, if somewhere on the frequency axis
there were a little hill for M, whose top determines the 1-norm, optimisation could still
be continued to lower this peak but at the cost of an increase of the bottom line until
the total transfer were at again. This e ect is known as the waterbed e ect. We also
note that this could never be the solution for the 2-norm problem as the integration of
this constant level jM1j from ! =0till ! = 1 would result in an in nitely large value.
Therefore, H2 accepts the extra costs at the low pass band for obtaining large advantage
after the corner frequency ! = b.
Nevertheless, the H2 solution has another advantage here, if we study the real goal:
the sensitivity. Therefore we haveto de ne the shaping lter V that characterises the type
of reference signals that we may expect for this particular tracking system. Suppose e.g.
that the reference signals live in a low pass band till ! = a so that we couldchoose lter
V as:

92 CHAPTER 7. GENERAL PROBLEM.
7.6. AN AGGREGATED EXAMPLE 91
nv
- o -
6~u
?
Vv
Wu
6 P
v
r
u
+
+
?
- Vr
- Cff - - P0
- -
6
+
+
nr
y
?
Cfb
Figure 7.8: Typical mixed sensitivity weights.
n ?
+ ;
? -
?
e
We
? ~e
as exempli ed in Fig. 7.8. Now it is crucial that the point of intersection of the
curves ! !jW1(j!)V1(j!)j and ! !jW2(j!)V2(j!)j is below the 0 dB-level. Otherwise,
there would be a con ict with S + T = 1 and there would be no solution! Consequently,
heavily weighted bands (> 0dB) for S and T should always exclude each other. This is
the basic e ect, that dictates how model uncertainty and actuator saturation, that puts
a constraint on T , ultimately bounds the obtainable tracking and disturbance reduction
band represented in the performance measure S.
Figure 7.9: Atwo degree of freedom controller.
7.6 An aggregated example
However, the bound for a particular subcriterion will mainly be e ected if all other
entries are zero. Inversely, if we would know beforehand that say k mij k1< 1 for
i 2 1 2::: nij 2 1 2::: nj, then the norm for the complete matrix k M k1 could still
become p max (ninj). Ergo, it is advantageous to combine most control aims.
In Fig. 7.10 the augmented plant/controller con guration is shown for the two degree
of freedom controlled system.
An augmented planted is generally governed by the following equations:
(7.34)
w
u
z
y = G11 G12
G21 G22
(7.35)
Till so far only very simple situations have been analysed. If we deal with more complicated
schemes where also more control blocks can be distinguished, the main lines remain valid,
but a higher appeal is done for one's creativity in combining control aims and constraints.
Also the familiar transfers take more complicated forms. As a straightforward example
we just take the standard control scheme with only a feedforward block extra as sketched
in Fig. 7.9.
This so-called two degree of freedom controller o ers more possibilities: tracking and
disturbance reduction are represented now by di erent transfers, while before, these were
combined in the sensitivity. Note also that the additive uncertainty P is combined with
the disturbance characterisation lter Vv and the actuator weighting lter Wu such that
P = Vv oWu under the assumption:
u = Ky (7.36)
8! 2R : j oj 1 ) j Pj jVvWuj (7.33)
that take for the particular system the form:
1
A (7.37)
0
@ nv
nr
u
1
C
A
;WeVv WeVr ;WePo
0 0 Wu
0
B
@
1
C
A =
~e
~u
y
r
0
B
@
Vv 0 Po
0 Vr 0
(7.38)
(7.39)
y
r
By properly choosing Vv and Wu we can obtain robustness against the model uncertainty
and at the same time prevent actuator saturation and minimise disturbance.
Certainly we then have to design the two lters Vv and Wu for the worst case bounds of
the three control aims and thus we likely have to exaggerate somewhere for each separate
aim. Nevertheless, this is preferable above the choice of not combining them and instead
adding more exogenous inputs and outputs. These extra inputs and outputs would increase
the dimensions of the closed loop transfer M and, the more entries M has, the more
conservative the bounding of the subcriteria de ned by these entries will be, because we
only have:
u = ; Cfb Cff
The closed loop system is then optimised by minimising:
if k M k1< then 8i j : k mij k1

94 CHAPTER 7. GENERAL PROBLEM.
7.6. AN AGGREGATED EXAMPLE 93
The respective, above transfer functions at the left and the right side of the inequality
signs can then be plotted in Bode diagrams for comparison so that we can observe which
constraints are the bottlenecks at which frequencies.
- We
-
~e
n
6
+
-
;
AugmentedP lant
6
z
- v
-
~u
- -
Vv
Wu
nv
6
w
- r
Vr - - Po
- -
6
+
y
-
r
? -
?
+
n
nr
y
6
u
u
Controller
?
Cff
? n
+
?
6
Cfb
+
Figure 7.10: Augmented plant/controller for two degree of freedom controller.
(7.40)
k1
k M k1=k G11 + G12K(I ; G22K) ;1 G21 k1=k M11 M12
M21 M22
and in particular:
1
A (7.41)
0
@ ;We(I ; PoCfb) ;1Vv WefI ; (I ; PoCfb) ;1PoCffgVr M =
Wu(I ; PoCfb) ;1 CffVr
WuCfb(I ; PoCfb) ;1 Vv
which canbeschematised as:
performance
1
C
A
tracking : ~e
nr
sensitivity : ~e
nv
(7.42)
0
B
@
constraints
stability robustness : ~u input saturation : nv
~u nr
Suppose that we can manage to obtain:
k M k1< 1 (7.43)
then it can be guaranteed that 8! 2R:
jI ; (I ; PoCfb) ;1 PoCffj < jWeVrj
j(I ; PoCfb) ;1 j < jWeVvj
(7.44)
1
C
A
0
B
@
j(I ; PoCfb) ;1 Cffj < jWuVrj
jCfb(I ; PoCfb) ;1 j < jWuVvj

96 CHAPTER 7. GENERAL PROBLEM.
7.7. EXERCISE 95
7.7 Exercise
~e 6~x
6~z
6
?
We Wx Wz V
6
6
6 n
r e x z ? +
~y
- Vr - k - C - P - l - Wy -
+
6{
+
y
? -
~r
For the given blockscheme we consider rst SISO-transfers from a certain input to a
certain output. It is asked to compute the linear fractional transfer, to explain the use of
the particular transfer, to name it (if possible) and nally to give the augmented plant in
blockscheme and express the matrix transfer G. Train yourself for the following transfers:
a) from to ~y (see example `sensitivity' in lecture notes)
b) from ~r to ~e
c) from to ~z (two goals!)
d) from to ~x (two goals!)
The same for the following MIMO-transfers:
e) from to ~y and ~z (three goals!)
We nowsplit the previously combined inputs in into two inputs 1 and 2 with respective
shaping lters V1 and V2:
f) from 1 and 2 to ~y and ~z.
Also for the next scheme:
-
~x
Wx
-
6
~r r + x y { e
~e
- Vr
- C1
- k -
P
- k - We
-
6+
6
+
?
C2
? -
g) from ~r to ~x and ~e.

98CHAPTER 8. PERFORMANCE ROBUSTNESS AND -ANALYSIS/SYNTHESIS.
Stability robustness. Because proper scaling was taken, it follows that stability
robustness can be guaranteed according to:
fk k1 1g\fkM11(K) k1< 1g (8.3)
Chapter 8
So the 1-norm of M11 determines robust stability.
Nominal performance. Without model errors taken into account (i.e. =0 and
thus h=0) k z k2 can be kept less than 1 provided that:
k M22(K) k1< 1 (8.4)
Performance robustness and
-analysis/synthesis.
So the 1-norm of M22 determines nominal performance.
This condition can be unambiguously translated into a stability condition, like for
stability robustness, by introducing a fancy feedback over a fancy block p as:
8.1 Robust performance
w = pz : fk p k1 1g\fkM22(K) k1< 1g (8.5)
There is now a complete symmetry and similarity in the two separate loops over
and p.
It has been shown how tosolveamultiple criteria problem where also stability robustness
is involved. But it is not since chapter 3 that we have discussed performance robustness
and then only in rather abstract terms where a small S had to watch robustness for T
and vice versa. It is time now to reconsider this issue, to quantify its importance and to
combine it with the other goals. It will turn out that we have practically inadvertently
incorporated this aspect as can be illustrated very easily with Fig. 8.1.
Robust performance. For robust performance we haveto guarantee that z stays
below 1 irrespective of the model errors. That is, in the face of a signal h unequal
to zero and k h k2 1, we require k z k2< 1. If we nowrequire that:
k M(K) k1< 1 (8.6)
we havea su cient condition to guarantee that the performance is robust.
proof: From equation 8.6 we have:
Figure 8.1: Performance robustness translated into stability robustness
(8.7)
h
w 2
<
g
z 2
The left block scheme shows the augmented plant where the lines, linking the model
error block, have been made explicit. When we incorporate the controller K, asshown in
the right blockscheme, the closed loop system M(K) isalsocontainig these lines, named
by g and h. With the proper partitioning the total transfer can be written as:
From k k1 1wemaystate:
(8.1)
h
w
g
z = M11 M12
M21 M22
k h k2 k g k2 (8.8)
Combination with the rst inequality yields:
h = g (8.2)
(8.9)
g
w 2
<
g
z 2
We suppose that a proper scaling of the various signals has been taken place such that
each of the output signals has 2-norm less than or equal to one provided that each of the
input components has 2-norm less than one. We can then make three remarks about the
closed loop matrix M(K):
97

100CHAPTER 8. PERFORMANCE ROBUSTNESS AND -ANALYSIS/SYNTHESIS.
8.2. -ANALYSIS 99
so that indeed:
k z k2

102CHAPTER 8. PERFORMANCE ROBUSTNESS AND -ANALYSIS/SYNTHESIS.
8.2. -ANALYSIS 101
when:
Consequently, an equivalent condition for stability is:
(M(j!)) def
=k M k (8.25)
sup
!
sup (M ) < 1 (8.17)
! R
represents a yet unknown measure. For obvious reasons, the is also called the
structured singular value. Because in general we can no longer have VWT = 2 it
will also be clear that
As we willshow, this condition takes the already encountered form:
for is unstructured : fk k1 1g\fkM k1< 1g (8.18)
(M) (M) (8.26)
This -value is certainly less than or equal to the maximum singular value of M,
because it incorporates the knowledge about the diagonal structure and should thus display
less conservatism. The father of is John Doyle and the symbol has generally been
accepted in control community for this measure. Equation 8.24 suggests that we can nd
a norm \k k " on exclusively matrix M that can function in a condition for stability.
First of all, the condition, and thus this -norm, cannot be independent on because the
special structural parameters (i.e. ni and mi) should be used. Consequently this so-called
-norm is implicitely taken for the special structure of . Secondly, we can indeed connect
a certain number to k M k , but it is not a norm \pur sang". It has all properties to
be a "distance" in the mathematical sense, but it lacks one property necessary to be a
norm, namely: k M k can be zero without M being zero itself (see example later on).
Consequently, \k k " is called a seminorm.
Because all above conditions and de nitions may be somewhat confusing by now, some
simple examples will be treated, to illustrate the e ects. We rst consider some matrices
M and for a speci c frequency !, which is not explicitly de ned.
We depart from one -matrix given by:
for the case that the block has no special structure. Note, that this is a condition
solely on matrix M.
proof:
Condition (8.18) for the unstructured can be explained as follows. The (M) indicates
the \maximum ampli cation" by mapping M. If M = W V represents the singular
value decomposition of M, we can always choose =VW because:
( ) = (VW )= p max(VW WV )= p max(I) =1 (8.19)
which isallowed. Consequently:
M =W W = W W ;1 ) (M) = (M ) = sup (M ) (8.20)
because the singular value decomposition happens here to be the eigenvalue decompostion
as well. So from equations 8.17 and 8.20 robust stability is a fact if we have for
each frequency:
f8 ( ) 1g\f (M) < 1g (8.21)
(8.27)
( 1) 1(,j 1j 1)
( 2) 1(,j 2j 1)
0
1
0 2
=
Next we study three matrices M in relation to this :
(8.28)
0
1
2
1 0 2
M =
If we apply this for each !, we end up in condition (8.18).
end proof.
However, if 2 has the special diagonal structure, then we can not (generally)
choose = VW . In other words, in such a case the system would not be robustly stable
for unstructured but could still be robustly stable for structured . So, it no longer
holds that sup (M ) = (M). But in analogy we de ne:
see Fig. 8.3.
The loop transfer consists of two independent loops as Fig. 8.3 reveals and that
follows from:
(M ) (8.22)
(M) def
= sup
2
and the equivalent stability condition for each frequency is:
(8.29)
1
2 1 0
1 0 2 2
M =
f8 2 g\f (M) < 1g (8.23)
Obviously (M) = max(M ) = 1
2 , which is less than one, so that robust stability
is guaranteed. But in this case also (M) = 1
2 so that there is no di erence between
the structured and the unstructured case. Because all matrices are diagonal, we are
just dealing with two independent loops.
In analogy we then have a similar condition on M for robust stability in the case of
the structured , by:
for is structured : f 2 g\fkM k < 1g (8.24)

104CHAPTER 8. PERFORMANCE ROBUSTNESS AND -ANALYSIS/SYNTHESIS.
8.2. -ANALYSIS 103
(8.32)
0 10
0 0 ) M = 0 10 2
0 0
M =
Now we deal with an open connection as Fig. 8.5 shows .
Figure 8.3: Two separate robustly stable loops
Figure 8.5: Robustly stable open loop.
It is clear that (M) = max(M ) = 0, although M 6= 0! Indeed is not a norm.
Nevertheless = 0 indicates maximal robustness. Whatever ( ) < 1= (M) =1 ,
the closed loop is stable, because M is certainly stable and the stable transfers are
not in a closed loop at all. On the other hand, the \conservative" 1-norm warns for
non-robustness as (M) =10> 1. From its perspective , supposing a full matrix,
this is correct since:
The equivalence still holds if we change M into:
(8.30)
2 0
0 1
M =
Then one learns:
(8.31)
0 10
0 0
M = 2 1 0
0 2
(8.33)
= 10 21 10 2
0 0
1 12
21 2
M =
so that Fig. 8.6 represents the details in the closed loop.
so that (M) = max(M )=2> 1 and stability is not robust. But also (M) =2
would have told us this and Fig. 8.4.
1 12
6
6
6
21 2
6
? - l - 10
-
-6 M
?
Figure 8.6: Detailed closed loop M with unstructured .
Clearly there is a closed loop now with looptransfer 10 21 where in worst case we can
have j 21j = 1 so that the system is not robustly stable. Correctly the (M) = 10 tells
us that for robust stability we require ( ) < 1= (M) =1=10 and thus j 21j < 1=10.
Figure 8.4: Two not robustly stable loops
Summarising we obtained merely as a de nition that robust stability is realised if:
Things become completely di erent ifweleave the diagonal matrices and study:

106CHAPTER 8. PERFORMANCE ROBUSTNESS AND -ANALYSIS/SYNTHESIS.
8.3. COMPUTATION OF THE -NORM. 105
(M) < 1g (8.34)
f 2 g\fkM k = sup
!
0
1
0
U 1
U 2
So a Bode plot could look like displayed in Fig. 8.7.
6
2
U 2
p
0
U 3
0
0
U1
U2
Figure 8.7: Bode plot of structured singular value.
MU
-
M(K)
?- -
U3
0
The actual computation of the -norm is quite another thing and appears to be
complicated, indirect and at least cumbersome.
8.3 Computation of the -norm.
Figure 8.8: Detailed structure of U related to .
The crucial observation at the basis of the computation, which will become an approximation,
is:
Because (M) will stay larger than (MU)even if we change U we can push this lower
bound upwards until it even equals the (M):
(M) (M) (M) (8.35)
Without proving these two-sided bounds explicitly, we will exploit them in deriving
tighter bounds in the next two subsections.
sup (MU)= (M) (8.38)
U
So in principle this could be used to compute , but unfortunately the iteration process,
to arrive at the supremum is a hard one because the function (MU) is not convex in the
entries uij.
So our hope is xed to lowering the upper bound.
8.3.1 Maximizing the lower bound.
8.3.2 Minimising the upper bound.
Again we apply the trick of inserting identities, consisting of matrices, into the loop. This
time both at the left and the right side of the block which wewant tokeep unchanged
as exempli ed in Fig. 8.9
Careful inspection of this Fig. 8.9 teaches that if is postmultiplied by DR and
premultiplied by D ;1
L it remains completely unchanged because of the "corresponding
identities structure" of DR and DL. This can be formalised as:
Without a ecting the loop properties we can insert an identity into the loop e ected by
UU = U U = I where U is a unitary matrix. A matrix is unitary if its conjugate
transpose U , is orthonormal to U, soU U =1. It is just a generalisation of orthonormal
matrices for complex matrices.
The lower bound can be increased by inserting such compensating blocks U and U in
the loop such that the -block isunchanged while the M-part is maximised in . The
is invariant under premultiplication by a unitary matrix U of corresponding structure as
shown in Fig. 8.8.
Let the matrix U consist of diagonal blocks Ui corresponding to the blocks i :
U U = fdiag(U1U2::: Up)j dim(Ui) =dim( i T i )UiU i = Ig (8.36)
DL 2 DL = fdiag(d1I1d2I2::: dpIp)j dim(Ii) = dim( i T i )di 2Rg (8.39)
as exempli ed in Fig. 8.8 Then, neither the stability nor the loop transfer is changed
if we insert I = UU into the loop. As U is unitary, we can also rede ne the dashed block
U as the new model error which also lives in set :
DR 2 DR = fdiag(d1I1d2I2::: dpIp)j dim(Ii) = dim( T i i)di 2Rg (8.40)
0 def
= U 2 (8.37)

108CHAPTER 8. PERFORMANCE ROBUSTNESS AND -ANALYSIS/SYNTHESIS.
8.3. COMPUTATION OF THE -NORM. 107
In practice one minimises for a su cient number of frequencies !j the maximum singular
value (DLMD ;1
R ) for all di(!j). Next biproper, stable and minimum phase lters
^di(j!) are tted to the sequence di(!j) and the augmented plant in a closed loop with
the controller K is properly pre- and postmultiplied by the obtained lter structure. In
that way we are left with generalised rational transfers again. This operation leads to the
following formal, shorthand notation:
DR D ;1
L 2
0
I1
1
d ;1
2 I1
0
0
I1
6
2
d2I1
d ;1
3 I2
0
p
0
d3I1
0
;1
DLMD R k1 k DLM(K) ^ D ^ ;1
R k1 ;! inf k DM(K)D
D ;1 k1
(8.44)
k inf
di(!)i=23:::p
where the distinction between DL and DR is left out of the notation as they are
linked in di anyhow. Also their rational lter structure is not explicitly indicated. As a
consequence we can write:
0
I1
0
I1
d2I1
- -
M(K)
d ;1
2 I1
?- -
A(M(!)) (8.45)
k M k inf k DMD
D ;1 k1 sup
!
d ;1
3 I1
0
d3I2
Consequently, if A remains below 1 for all frequencies, robust stability is guaranteed
and the smaller it is, the more robustly stable the closed loop system is. This nishes the
-analysis part: given a particular controller K the -analysis tells you about robustness
in stability and performance.
0
DLMD ;1
R
8.4 -analysis/synthesis
Figure 8.9: Detailed structure of D related to .
By equation (8.43) we have a tool to verify robustness of the total augmented plant in
a closed loop with controller K. The augmented plant includes both the model errorblock
and the arti cial, fancy performance block. Consequently robust stability should
be understood here as concerning the generalised stability which implies that also the
performance is robust against the plant perturbations. But this is only the analysis, given
a particular controlled block M which is still a function (LFT) of the controller K. For
the synthesis of the controller we were used to minimise the H1-norm:
If all i are square, the left matrix DL and a right matrix DR coincide. All coe cients
di can be multiplied by a free constant without a ecting anything in the complete loop.
Therefore the coe cient d1 is generally chosen to be one as a "reference".
Again the loop transfer and the stability condition are not in uenced by DL and DR
and we can rede ne the model error :
= 2 (8.41)
0 def
= DR D ;1
L
k M(K) k1
(8.46)
inf
Kstabilising
Again the is not in uenced so that we can vary all di and thereby pushing the upper
bound downwards:
but we have just found that this is conservative and that we should minimise:
;1 def
(DLMD R ) = A(M) (8.42)
(M) inf
dii=23:::p
(8.47)
inf
Kstabilising k DM(K)D;1 k1
However, for each newKthe subsequently altered M(K) involves a new minimisation for
D so that we haveto solve:
inf
Kstabilising inf
D k DM(K)D;1 k1 (8.48)
It turns out that this upper bound A(M) isvery close in practice to (M) and it even
equals (M) if the dimension of is less or equal to 3. And fortunately, the optimisation
with respect to di is a well conditioned one, because the function k DLMD ;1
R k1 appears
to be convex in di. So A is generally used as the practical estimation of . However, it
should be done for all frequencies ! which boils down to a nite, representative number
of frequencies ! and we nally have:
In practice this is tried to be solved by the following iteration procedure under the name
D-K-iteration process:
1. Put D = I
A(M(!)) (8.43)
;1
DLMD R k1= sup
!
k M k k inf
di(!)i=23:::p

110CHAPTER 8. PERFORMANCE ROBUSTNESS AND -ANALYSIS/SYNTHESIS.
8.5. A SIMPLE EXAMPLE 109
2. K-iteration. Compute optimal K for the last D.
3. D-iteration. Compute optimal D for the last K.
4. Has the criterion k DM(K)D ;1 k1 changed signi cantly during the last two steps?
If yes: goto K-iteration, ifno: stop.
In practice this iteration process appears to converge usually in not too many steps. But
there can be exceptions and in principle there is a possibility that it does not converge at
all.
This formally completes the very brief introduction into -analysis/synthesis. A few
extra remarks will be added before a simple example will illustrate the theory.
As a formal de nition of the structured singular value one often \stumbles" across
the following \mind boggling" expression in literature:
Figure 8.10: First order plant with parameter uncertainties.
(8.49)
(M) = [inff ( )j det(I ; M )=0g] ;1
where one has to keep in mind that the in mum is over which has indeed the same
structure as de ned in the set but not restricted to ( i) < 1. Nevertheless, the
de nition is equivalent with the one discussed in this section. In the exercises one
can verify that the three methods (if dim( ) 3) yield the same results.
ni mi It is tacitly supposed that all i live in the unity balls in C while we often
know thatonlyrealnumbers are possible. This happens e.g. when it concerns inaccuracies
in \physical" real parameters (see next section). Consequently not taking
into account this con nement to real numbers (R) will again give rise to conservatism.
Implicit incorporation of this knowledge asks more complicated numerical
tools though.
8.5 A simple example
Figure 8.11: Augmented plant for parameter uncertainties.
Consider the following rst order process:
while the outer loops are de ned by:
1 0 a1
(8.52)
0 2 a2
u = Ky = Cy (8.53)
=
a1
a2
=
b1
b2
Incorporation of a stabilising controller K, which is taken as a static feedback here,
we obtain for the transfer M(K):
P = K0
(8.50)
s +
where we havesome doubts about the correct values of the two parameters K0 and .
So let 1 be the uncertainty in the gain K0 and 2 be the model error of the pole value .
Furthermore, we assume a disturbance w at the input of the process. We want to minimise
its e ect at the output by feedback across controller C. For simplicity there are no shaping
nor weighting lters and measurement noise and actuator saturation are neglected. The
whole set up can then easily be presented by Fig. 8.10 and the corresponding augmented
plant byFig. 8.11.
The complete input-output transfer of the augmented plant Ge can be represented as:
1
A (8.54)
0
@ b1
b2
w
1
A
;K ;1 1
;K ;1 1
s + ;K0 K0
0
@
1
s + + K0K
1
A =
0
@ a1
a2
z
| {z }
1
C
A
b1
b2
w
u
0
B
@
1
C
A
;1
s+
;1
s+
;K0
s+
;K0
s+
1
s+
1
s+
K0
s+
K0
s+
;1 0 s+
;1 0 s+
M(K)
(8.51)
1 ;K0
s+
0
B
@
1
C
A =
a1
a2
z
y
0
B
@
The analysis for robustness of the complete matrix M(K) is rather complicated for
1 ;K0
s+

112CHAPTER 8. PERFORMANCE ROBUSTNESS AND -ANALYSIS/SYNTHESIS.
8.5. A SIMPLE EXAMPLE 111
information is lost in the H1-approach and this takes over to the -approach. Explicit
implementation of the phase information can only be done in such a simple example and
will appear to be the great winner. Because we know that 1 and 2 are real, the pole of
the system with proportional feedback K is given by:
analytical expressions so that we like to con ne to the robust stability in the strict sense
for changes in 1 and 2 that is:
(8.55)
;K ;1
;K ;1
1
s + + K0K
M11 =
;( + K0K + 2 + K 1) (8.66)
Since we did not scale, we may de ne the -analysis as:
Because K is such that nominal (for i = 0) stability is true, total stability is guaranteed
for:
k M11 k = (8.56)
2 = (diag( 1 2)j ( i) < 1 ) (8.57)
K 1 + 2 > ;( + K0K) (8.67)
This half space in 1 2-space is drawn in Fig. 8.12 for numerical values: =1K0 =
1K =2.
For (!) we get (the computation is an exercise):
(8.58)
jKj +1
p
! 2 +( + K0K) 2
(!) =
The supremum over the frequency axis is then obtained for ! = 0 so that:
= (8.59)
jKj +1
+ K0K
k M11 k =
because K stabilises the nominal plant so that:
+ K0K >0 (8.60)
Ergo, -analysis guarantees robust stability as long as :
(8.61)
= 1
+ K0K
jKj +1
for i =1:2 : j ij <
It is easy to verify (also an exercise) that the unstructured H1condition is:
Figure 8.12: Various bounds in parameter space.
The two square bounds are the -bound and the H1-bound. The improve of on
H1 is rather poor in this example but can become substantial for other realistic plants.
There is also drawn a circular bound in Fig. 8.12. This one is obtained by recognising that
signals a1 and a2 are the same in Fig. 8.11. This is the reason that M11 had so evidently
rank 1. By proper combination the robust stability can thus be established by a reduced
M11 that consists of only one row and then is no longer di erent from H1 both yielding
the circular bound with less computations. (This is an exercise.)
Another appealing result is obtained by letting K approach 1, then:
s
2(K2 +1)
(M(K !)) =
! 2 ! (8.62)
+( + K0K) 2
p
2(K2 +1)
k M11 k1=
+ K0K = 1 ! (8.63)
+ K0K
j ij < p =
2(K2 +1) 1
(8.64)
1
Indeed, the -analysis is less conservative than the H1-analysis as it is easy to verify
that:
(8.68)
; bound : j 1j (8.65)
(8.70)
true ; bound : 1 > ;K0
Finally we would like to compare these results with an even less conservativeapproach
where we make use of the phase information as well. As mentioned before, all phase

114CHAPTER 8. PERFORMANCE ROBUSTNESS AND -ANALYSIS/SYNTHESIS.
8.6. EXERCISES 113
8.6 Exercises
9.1: Show that, in case M12 =0orM21 = 0, the robust performance condition is ful lled
if both the robust stability and the performance for the nominal model are guaranteed.
Does this case, o -diagonal terms of M zero, make sense ?
9.2: Given the three examples in this chapter:
(8.71)
0 10
0 0
2 0
0 1
1=2 0
0 1=2
M =
according to the second de nition :
0
1
0 2
Compute the -norm if =
= [inff ( )j det (I ; M ) = 0g] ;1 (8.72)
9.3: Given:
(8.73)
0
1
0 2
=
;1=2 1=2
;1=2 1=2
M =
a) Compute and of M. Are these good bounds for ?
b) Compute in three ways.
9.4: Compute explicitly k M11 k1 and k M11 k for the example in this chapter where:
(8.74)
K ;1
K ;1
1
s + + K0K
M11 =
What happens if we use the fact that the the error block output signals a1 and a2 are
the same , so that can be de ned as =[ 1 2 ] T ? Show that the circular bound
of the last Fig. 8.12 results.

116 CHAPTER 9. FILTER SELECTION AND LIMITATIONS.
~d
-
dmax
-
Chapter 9
n
- rmax - n- 1
C - - ~P
- zmax -
umax
max - ?
d
P
r +
u ~u ~z z ?
6;
y
~
-
?
~r
Filter Selection and Limitations.
Figure 9.1: Range scaled controlled system.
An H1-analogon for such a zero frequency setup would be as follows. In H1 we
measure the inputs and outputs as k w k2 and k z k2 so that the induced norm is
k M k1. In zero frequency q setup it would be the Euclidean norm for inputs and outputs,
i.e. k w kE=k w k2= iw2 i and likewise for z. The induced norm is trivially the usual
matrix norm, so k M k1= maxi( i(M)) = (M) . Note that because of the scaling we
immediately have for all signals, inputs or outputs:
k s k2= j~sj 1 (9.2)
For instance a straightforward augmented plant could lead to:
In this chapter we will discuss several aspects of lter selection in practice. First we
will show how signal characteristics and model errors can be measured and how these
measurements together with performance aims can lead to e ective lters. E ective in
the sense, that solutions with k M k1< 1 are feasible without contradicting e.g.
"S+T=I" and other fundamental bounds.
Apart from the chosen lters there are also characteristics of the process itself, which
ultimately bound the performance, for instance RHP (=Right Half Plane) zeros and/or
poles, actuator and output ranges, less inputs than outputs etc. We will shortly indicate
their e ects such that one is able to detect the reason, why 1 could not be obtained
and what the best remedy or compromise can be.
1
A = Mw (9.3)
0
@ ~r
~d
~
9.1 A zero frequency set-up.
z = ~u
~e = WuRVr WuRVd WuRV
WeSVr WeSVd WeTV
9.1.1 Scaling
where as usual S =1=(1 + PC), T = PC=(1 + PC), R = C=(1 + PC) and e = r ; y.
In the one frequency set-up the majority of lters can be directly obtained from the
scaling:
1
A = Mw (9.4)
0
@ ~r
~d
~
1
umax Rrmax
1
umax Rdmax
1
umax R max
WeSrmax WeSdmax WeT max
~u
~e =
z =
9.1.2 Actuator saturation, parsimony and model error.
Suppose that the problem is well de ned and we would be able to nd a controller C R
such that
k M k1= (M) < 1 (9.5)
The numerical values of the various signals in a controlled system are usually expressed
in their physical dimensions like m, N, V , A, o , ::: . Next, depending on the size of the
signals, we also have a rough scaling possibility in the choice of the units. For instance
a distance will basically be expressed in meters, but in order to avoid very large or very
small numbers we can choose among km, mm, m, A or lightyears. Still this is too
rough a scaling to compare signals of di erent physical dimensions. As a matter of fact
the complete concept of mapping normed input signals onto normed output signals, as
discussed in chapter 5, incorporates the basic idea of appropriate comparison of physically
di erent signals by means of the input characterising lters V and output weighting
lters W . The lter choice is actually a scaling problem for each frequency. So let us
start in a simpli ed context and analyse the scaling rst for one particular frequency,
say ! = 0. Scaling on physical, numerically comparable units as indicated above is not
accurate enough and a trivial solution is simply the familiar technique of eliminating
physical dimensions by dividing by the maximum amplitude. So each signal s can then
be expressed in dimensionless units as ~s according to:
then this tells us e.g. that k z k2< 1, so certainly ~u

118 CHAPTER 9. FILTER SELECTION AND LIMITATIONS.
9.1. AZERO FREQUENCY SET-UP. 117
or, since weights are naturally chosen as positive numbers, we take:
(9.12)
pmax = umax
Consequently, an extra addition to the output of the plant representing the model
perturbation is realised by jpj pmax. In combining the output additions we get n =
;p ; d + r and:
By the applied scaling we can only guarantee that k w k2< p 3 so that disappointingly
follows u< p 3umax, which is not su cient toavoid actuator saturation. This e ect can
be weakened by choosing Wu = p 3=umax or we can try to eliminate it by diminishing
the number of inputs. This can be accomplished because both tracking and disturbance
reduction require a small sensitivity S. In the next Fig. 9.2 we showhowby rearrangement
reference signals, disturbances and model perturbations can be combined in one augmented
plant input signal.
~n
(9.13)
nmax = pmax + dmax + rmax
?
6
~u
Note that the sign of p and d, actually being a phase angle, does not in uence the
weighting. Also convince yourself of the substantial di erence of diminishing the number
of inputs compared with increasing the Wu with a factor p 3. We have j~nj 1 contrary
to the original three inputs j~pj 1, j ~ dj 1 and j~rj 1, implying a reduction of a factor
3 in stead of p 3. The 2-norm applied to w in the two blockschemes of Fig. 9.2 would
indeed yield the factor p q
3 as k ~p k2 + k d~ k2 + k ~r k2 p 3 contrary to p k ~n k2 1.
By reducing the number of inputs we have done so taking care that the maximum value
was retained. If several 2-normed signals are placed in a vector, the total 2-norm takes
the average of the energy or power. Consequently we are confronted again with the fact
that not H1 is suited for protection against actuator saturation, but l1-control is.
Note, that for the proper quantisation of the actuator input signal we had to actually
add the reference signal, the disturbance and the model error output. For robust stability
alone it is now su cient that:
nmax
~p d ~ ~r
-
? ? ?
6 -
1
umax
~u
n
?
pmax dmax rmax
; -
6? -
1
umax
P
u
e = r ; y
P
C
p
?
d r
? y
- - ; ?
-
? e = r ; y
- max -
~
)
?
6 -
?
u
- 6
max
-
C
~
Figure 9.2: Combining sensitivity inputs.
(9.14)
nmax = umax
The measuring of the model perturbation will be discussed later. Here we assume that
the general, frequency dependent, additive model error can be expressed as :
whatever the derivation of nmax might be. In the next section we will see that in the
frequency dependent case, a real prevention of actuator saturation can never be guaranteed
in H1-control. Actual practice will then be to combine Vd and Vr into Vn, heuristically
de ne a Wu and verify whether for robust stability the condition:
k P k1< (9.7)
The transfer from ~p to ~u in Fig. 9.2 is given by:
1
8! : jVnWuj (!) (9.15)
Rpmax k1< 1 (9.8)
k
umax
is ful lled. If not, either Vn or Wu should be corrected.
so that stability is robust for:
9.1.3 Bounds for tracking and disturbance reduction.
k k1< 1 (9.9)
Till sofar we have discussed all weights except for the error weight We. Certainly we
would like tochoose We as big and broad as possible in order to keep the error e as small
as possible. If we forget about the measurement noise for the moment and apply the
simpli ed right scheme of Fig. 9.2, we obtain a simple mixed sensitivity problem:
Combination yields that:
(9.10)
k1

120 CHAPTER 9. FILTER SELECTION AND LIMITATIONS.
9.1. AZERO FREQUENCY SET-UP. 119
Consequently, if we have obviously rmax zmax for a tracking system, the tracking
error e = r ; y can never become small.
For SISO plants this e ect is quite obvious, but for MIMO systems the same internal
scaling of plant P can be very revealing in detecting these kind of internal insu ciencies
as we will show later.
On the other hand, if the gain of the scaled plant P~ is larger than 1, one should not
think that the way is free to zero sensitivity S. For real systems, where the full frequency
dependence plays a role, we will see plenty oflimitinge ects.Only for ! =0we are used
to claim zero sensitivity in case of integrator(s) in the loop. In that case we haveindeed in nite gain (1=(j!)) similar to the previous example by taking C = 1. Nevertheless in
practice we always have to deal with the sensor and inevitable sensor noise . If we indeed
have S = 0, inevitably T = 1 and e = T = . So in its full extent the measurement
noise is present in the error, which simply re ects the trivial fact that you can never track
better than the accuracy of the sensor. So sensor noise bounds both traking error and
disturbance rejection and should be brought in properly by the weight max in our example
in order to minimise its e ect in balance with the other bounds and claims.
(9.17)
C = W 2 e Pu2 max
In order to keep (M) 1 to prevent actuator saturation we can put (M) =1for
the computed controller C yielding:
(9.18)
We =1= p n 2 max ; P 2 u 2 max
A special case occurs for jPumaxj = jnmaxj which simply states that the range of
the actuator is exactly su cient to cause the output z of plant P to compensate for the
"disturbance" n. So if actuator range and plant gain is su ciently large we can choose
We = 1 and thus C = 1 so that M becomes:
(9.19)
nmax
Pumax
0
M =
9.2 Frequency dependent weights.
9.2.1 Weight selection by scaling per frequency.
In the previous section the single frequency case servedasavery simple concept to illustrate
some fundamental limitations, that certainly exist in the full, frequency dependent
situation. All e ects take over, where we have to consider a similar kind of scaling but
actually for each frequency. Usually the H1-norm is presented as the induced norm of
the mapping from the L2 space to the L2 space. In engineering terms we then talk about
the (square-root of) the energy of inputs k w k2 towards the (square-root of) the energy of
outputs k z k2. Mathematically, this is ne, but in practice we seldomly deal with nite energy
signals. Fortunately, theH1-norm is also the induced norm for mapping powers onto
powers or even expected powers onto expected powers as explained in chapter 5. If one
considers a signal to be deterministic, where certain characteristics may vary, the power
can simply be obtained by describing that signal by a Fourier series, where the Fourier
coe cients directly represent the maximal amplitude per frequency. This maximum can
thus be used as a scaling for each frequency analogous to the one frequency example of
the previous section. On the other hand if one considers the signal to be stochastic (stationary,
one sample from an ergodic ensemble), one can determine the power density s
and use the square-root of it as the scaling. One can even combine the two approaches,
for instance stochastic disturbances and deterministic reference signals. In that case one
should bear in mind that the dimensions are fundamentally di erent and a proper constant
should be brought in for appropriate weighting. Only if one sticks to one kind of
approach, any scaling constant c is irrelevant as it disappears by the fundamental division
in the de nition:
and no error results while (M) =jm11j =1.
If jPumaxj > jnmaxj, their is plenty ofchoice for the controller and the H1 criterion is
minimised by decreasing jm11j more at the cost of a small increase of jm21j. Note that this
control design is di erent from minimising jm21j under the constraint of jm11j 1. For
this simple example the last problem can be solved, but the reader is invited to do this
and by doing so to obtain an impression of the tremendous task for a realistically sized
problem.
If jPumaxj < jnmaxj, it is principally impossible to compensate all "possible disturbance"
n. This is re ected in the maximal weight We we can choose that allows for a
(M) 1. Some algebra shows that:
(9.20)
s
1 ; P 2u2 max
n2 max
=
1
Wenmax
If e.g. only half the n can be compensated, i.e. jPumaxj = 1
2jnmaxj, we have jSj
1
Wenmax =
q
3
4 which is very poor. This represents the impossibility totrack better than
50% or reduce the disturbance more than 50%. If one increases the weight We one is
confronted with a similar increase of and no solution k M k1 1 can be obtained. One
can test this beforehand by analysing the scaled plant as indicated in Fig. 9.1. The plant
P has been normalised internally according to:
(9.21)
P = zmax ~ P 1
umax
(9.23)
k cMw kpower
k cw kpower
=sup
w
k Mw kpower
k w kpower
k M k1= sup
w
so that P~ is the transfer from ~u, maximally excited actuator normalised on 1, to
maximal, undisturbed, scaled output ~z. Suppose now that j Pj ~ < 1. It tells you that not
all outputs in the intended output range can be obtained due to the actual actuator. The
maximal input umax can only yield:
Furthermore, as we have learned from the ! = 0 scaling, the maxima (=range) of
the inputs scale and thus de ne the input characterising lters directly, while the output
lters are determined by theinverse so that we obtain e.g. for input v to output x:
(9.22)
umaxj = jzmax ~ Pj

122 CHAPTER 9. FILTER SELECTION AND LIMITATIONS.
9.2. FREQUENCY DEPENDENT WEIGHTS. 121
3. The lters should be preferably be biproper. Any pole zero excess would in fact cause
zeros at in nity, that make thelter uninvertable and inversion happens implicitly
in the controller design.
(9.24)
Mxvcvmax
Mxvvmax = 1
cxmax
WxMxvVv = 1
xmax
4. The dynamics of the generalised plant should not exceed about 5 decades on the
frequency scale for numerical reasons dependent on the length of the mantissa in
your computer. So double precision can increase the number of decades. In single
precision it thus means that the smallest radius (=distance to the origin) divided by
the largest radius of all poles and zeros of plant and lters should not be less than
10 ;5 .
5. The lters are preferably of low order. Not only the controller will be simpler as
it will have the total order of the augmented plant. Also lters very steep at the
border of the aimed tracking band will cause problems for the robustness as small
deviations will easily let the fast loops in Nyquist plot tresspass the hazardous point
-1.
So again the constant is irrelevant, unless input- and output lters are de ned with
di erent constants. In chapter 5 it has been illustrated how the constant relating the
deterministic power contents to a power density value can be obtained. It has been done
by explicitely computing the norms in both concepts for an example signal set that can
serve for both interpretations. From here on we suppose that one has chosen the one
or other convention and that we can continue with a scaling per frequency similar to
the scaling in the previous section. So smax(!) represents the square-root of any powerde
nition for signal s(j!), e.g. smax(!) = p ss(j!). Remember that the phase of lters
and thus of smax(!) is irrelevant. Straightforward implementation of scaling would then
lead to:
9.2.2 Actuator saturation: Wu
1
smax(!) s(!) ! Ws(j!)s(!) s(!) =smax(!)~s(!) ! Vs(j!)~s(!) (9.25)
~s(!) =
The characterisation or weighting lters of most signals can su ciently well be obtained
as described in the previous subsection. A characterisation per frequency is well in line
with practice. The famous exception is the lter Wu where we would like to bound the
actuator signal (and sometimes its derivative) in time. However, time domain bounds,
in fact L1-norms, are incompatible with frequency domain norms. This is in contrast
with the energy and power norms (k : k2) that relate exacty according to the theorem of
Parceval. Let us illustrate this, starting with the zero frequency set-up of the rst section.
As we were only dealing with frequency zero a bounded power would uniquely limit the
maximum value in time as the signal is simply a constant value:
Arrows have been used in above equations because immediate choice of e.g.Vs(j!) =
smax(!) = p ss(j!) would unfortunately rarely yield a rational transferfunction Vs(j!)
and all available techniques and algorithms in H1 design are only applicable for rational
weights. Therefore one has to come up with not too complicated rational weights Vs or
Ws satisfying:
e:g:
j = j p ss(j!)j (9.26)
1
smax(!)
jVs(j!)j jsmax(!)j e:g:
= j p ss(j!)j jWs(j!)j j
(9.27)
k s kL1= jsj = p s 2 =k s kpower
If the power can be distributed over more frequencies, a maximum peak in time can be
created by proper phase alignment of the various components as represented in Fig. 9.3.
The routine "magshape" in Matlab-toolbox LMI can help you with this task. There
you can de ne a number of points in the Bode amplitude plot where the routine provides
you with a low order rational weight function passing through these points. When you
have a series of measured or computed weights in frequency domain, you can easily come
up with a rational weight su ciently close (from above) to them.
Whether you use these routines or you do it by hand, you have watch the following
side conditions:
3
2.5
2
1.5
1
0.5
0
−0.5
−1
−1.5
−4 −3 −2 −1 0 1 2 3 4
1. The weighting lter should be stable and minimum phase. Be sure that there are no
RHP (=Right Half Plane) poles or zeros. Unstable poles would disrupt the condition
of stability for the total design, also for the augmented plant. Nonminimum phase
zeros would prohibit implicit inversion of the lters in the controller design.
Figure 9.3: Maximum sum of 3 properly phase aligned sine waves.
Suppose we have n sine waves:
s(t) =a1 sin(!1t + 1)+a2 sin(!2t + 2)+a3 sin(!3t + 3)+:::+ an sin(!nt + n)
(9.28)
2. Poles or zeros on the imaginary axis cause numerical problems for virtually the same
reason and should thus be avoided. If one wants an integral weighting, i.e. a pole
in the origin, in order to obtain an in nite weight at frequency zero and to force
the design to place an integrator in the controller, one should approximate this in
the lter. In practice it means that one positions a pole in the weight very close
to the origin in the LHP (Left Half Plane). The distance to the origin should be
very small compared to the distances of other poles and zeros in plant and lters.
Alternatively, one could properly include an integrator to the plant and separate it
out to the controller lateron, when the design is nished. In that case be thoughtful
about how theintegrator is included in the plant (not just concatenation!).

124 CHAPTER 9. FILTER SELECTION AND LIMITATIONS.
9.2. FREQUENCY DEPENDENT WEIGHTS. 123
zt
- -
Pt
with total power equal to one. If we distribute the power equally over all sine waves
we get:
6
+
? p
- -
;
u
(9.29)
q
1
n
n
i=1a2 i =1 8i : ai = a ) ai = a =
z
P
-
and consequently, with proper choice of phases i the peak in time domain equals:
(9.30)
r
n 1
i=1ai = n
n
Figure 9.4: Additive model error from p=u.
Certainly, for the continuous case, we have in nitely many frequencies so that n !1
and:
known inputs, the deviating transfers Pt for the respective frequencies can be computed.
Alternatively, one could use broadbanded input noise and compute the various tranfer
samples by crosscorrelation techniques.
Quite often these cumbersome measurements, that are contaminated by inevitable
disturbances and measurement noise and are very hard to obtain in case of unstable plants,
can be circumvented by proper computations. If the structure of the plant-transfer is very
well known but various parameter values are unclear, one can simply evaluate the transfers
for sets of expected parameters and treat these as possible model-deviating transfers.
Next, the various deviating transfers for a typical set of frequencies, obtained either by
measurements or by computations, should be evaluated in a polar (Nyquist) plot contrary
to what is often shown by means of a Bode plot. This is illustrated in Fig. 9.5.
= 1 (9.31)
lim
n!1 n
r
1
n
10
10 0
Frequency (rad/sec)
10 −1
−30
So the bare fact that we have in nitely many frequencies available (continuous spectrum)
will create the possibility of in nitely large peaks in time domain. Fortunately, this
very worst case will usually not happen in practice and we can put bounds in frequency
domain that will generally be su cient for the practical kind of signals that will virtually
exclude the very exceptional occurrence of above phase aligned sine waves. Nevertheless
fundamentally we cannot have any mathematical basis to choose the proper weight Wu
and we have to rely on heuristics. Usually an actuator will be able to follow sine waves
over a certain band. Beyond this band, the steep increases and decreases of the signals
cannot be tracked any more and in particular the higher frequencies cause the high peaks.
Therefore in most cases Wu has to have the character of a high pass lter with a level
equal to several times the maximum amplitude of a sine wave the actuator can track.
The design, based upon such a lter , has to be tested next in a simulation with realistic
reference signals and disturbances. If the actuator happens to saturate, it will be clear
that Wu should be increased in amplitude and/or bandwidth. If the actuator is excited
far from saturation the weight Wu can be softened. This Wu certainly forms the weakest
aspect in lter design.
0.6
0
−10
0.4
Gain dB
−20
0.2
10 1
0
Imag Axis
0
−0.2
X
−0.4
X M X
−0.6
X
0 0.5 1 1.5
Real Axis
−30
−60
Phase deg
10 1
10 0
Frequency (rad/sec)
10 −1
−90
9.2.3 Model errors and parsimony.
Figure 9.5: Additive model errors in Bode and Nyquist plots.
The model P is given by:
Like actuator saturation, also model errors put strict bounds, but they can fortunately be
de ned and measured directly in frequency domain. As an example we treat the additive
model error according to Fig. 9.4.
We can measure p = zt ; z =(Pt ; P )u. For each frequency we would like to obtain
the di erence jPt(j!) ; P (j!)j. In particular we are interested in the maximum deviation
(!) R such that:
(9.33)
P = 1
s +1
while deviating transfers Pt are taken as:
8! : jPt(j!) ; P (j!)j = j P (j!)j < (!) (9.32)
(9.34)
1:2
s+:8
:8
s+1:2 or
1:2
s+1:2 or
Pt = :8
s+:8 or
Given the Bode plot one is tended to take the width of the band in the gain plot as a
measure for the additive model error for each frequency. This would lead to:
Since P is a rational transfer, we would like to have the transfer Pt in terms of gain
and phase as function of the frequency !. This can be measured by o ering respective
sinewaves of increasing frequency to the real plant and measure amplitude and phase of
the output for long periods to monitor all changes that will usually occur. Given the

126 CHAPTER 9. FILTER SELECTION AND LIMITATIONS.
9.2. FREQUENCY DEPENDENT WEIGHTS. 125
If this condition is full lled, we don't have tointroduce an extra lter Vp for stability.
The extra exogenous input p can be prefered for proper quantisation of the control signal
u, but this can also be done by increasing Wu properly.
The best is to combine the exogeneous inputs d, r and p into a signal n, likewe did in
section 9.1.2, but now with appropriate combination for each frequency. This boils down
to nding a rational lter transfer Vn(j!) such that:
jjPtj;jPjj (9.35)
max
Pt
which is certainly wrong. In the Nyquist plot we haveindicated for ! =1the model
transfer by 'M' and the several deviating transfers by 'X'. The maximum model error is
clearly given by the radius of the smallest circle around 'M' that encompasses all plants
'X'. Then we really obtain the vectorial di erences for each !:
8! : jVn(j!)j jVd(j!)j + jVr(j!)j + jVp(j!)j (9.44)
Again the routine "magshape" in the LMI-toolbox can help here.
Pragmatically, one usually combines only Vd and Vr into Vn and cheques whether:
jPt(j!) ; P (j!)j (9.36)
(!) =max
Pt
8! : (!) < jWu(j!)Vn(j!)j (9.45)
The reader is invited to analyse how the wrong measure of equation 9.35 can be
distinguished in the Nyquistplot.
Finally we havethe following bounds for each frequency:
is satis ed. If not, the weighting lter Wu is adapted until the condition is satis ed.
9.2.4 We bounded by fundamental constraint: S + T = I
j P (j!)j < (!) (9.37)
Foratypical low-sized H1 problem like:
~n
~ = Mw (9.46)
z = ~u
~e = WuRVn WuRV
WeSVn WeTV
The signal p in Fig. 9.4 is that component in the disturbance free output of the
true proces Pt due to input u, that is not accounted for by the model output Pu. This
component can be represented by an extra disturbance at the output in the generalised
plant likein Fig. 9.2, but now withaweighting lter p = Vp(j!)~p. If the goal k M k1<
1wehave:
all weights have been discussed except for the performance weight We. The characterising
lters of the exogenous inputs ~n and ~ left little choice as these were determined
by the actual signals to be expected for the closed loop system. The control weighting
lter Wu was de ned by rigorous bounds derived from actuator limitations and model
perturbations. Now it is to be seen how good a nal performance can be obtained by
optimum choice of the error lter We. We would like to see that the nal closed loop system
shows good tracking behaviour and disturbance rejection for a broad frequency band.
Unfortunately, the We will appear to be restricted by many bounds, induced by limitations
in actuators, sensors, model accuracy and the dynamic properties of the plant tobe
controlled. The in uence of the plant dynamics will be discussed in the next section. Here
we will show how the in uences of actuator, sensor and model accuracy put restrictions
on the performance via respectively Wu, V and the combination of Wu, Vn and V .
Mentioned lters all bound the complementarity sensitivity T as a contraint:
k WuRVp k1< 1 ()8! : jWuRVpj < 1 (9.38)
For robust stability, based on the small gain theorem, we have as condition:
k R P k1< 1 ()8! : jR Pj < 1 () (9.39)
j Pj
8! : jWuRVpj < 1 (9.40)
jWuVpj
Given the bounded transfer of equation 9.38, a su cient condition is:
8! : j Pj < jWuVpj (9.41)
fk WuRVn k1< 1g,f8! : jWuRVnj = jWuP ;1 TVnj < 1g (9.47)
fk WuRV k1< 1g,f8! : jWuRV j = jWuP ;1 TV j < 1g (9.48)
fk WeTV k1< 1g,f8! : jWeTV j < 1g (9.49)
and this can be guaranteed if the weights are su ciently large such that the bounded
model perturbations of equation 9.37 can be brought in as:
8! : j Pj < (!) < jWuVpj (9.42)
In above inequalities the plant transfer P , that is not optional contrary to the controller
C, functions as part of the weights on T . Because the T is bounded accordingly the
freedom in the performance represented by the sensitivity S is bounded on the basis of
the fundamental constraint S + T = I.
The constraints on T can be represented as k W2TV2 k1< 1whereW2 and V2 represent
the various weight combinations of inequalities 9.47-9.49. Renaming the performance aim
as:
Of course, for stability also the other input weight lters Vd, Vr or even V in stead of
Vp could have been used, because they all combined with Wu limit the control sensitivity
R. Consequently, for robust stability it is su cient tohave:
8! : (!) < supfjWuVdj jWuVrj jWuV jg (9.43)

128 CHAPTER 9. FILTER SELECTION AND LIMITATIONS.
9.2. FREQUENCY DEPENDENT WEIGHTS. 127
k WeSVn k1 def
= k W1SV1 k1< 1 (9.50)
*
T
S
Now we can repeat the comments made in section 7.5.
The H1 design problem requires:
: ^
1
T
0
T
z
S
S
k W1SV1 k1< 1 ,8! : jS(j!)j < jW1(j!) ;1 V1(j!) ;1 j (9.51)
k W2TV2 k1< 1 ,8! : jT (j!)j < jW2(j!) ;1 V2(j!) ;1 j (9.52)
^
Atypical weighting situation for the mixed sensitivity problem is displayed in Fig. 9.6.
Figure 9.7: Possibilities for jSj < 1, jTj < 1 and S + T =1.
g (9.56)
1
jW2V2j
g\fjTj <
1
jW1V1j
fjSj <
then necessarily:
(9.57)
1
jW2V2j
< jTj <
1 ; S = T ) 1 ; 1
jW1V1j
Figure 9.6: Typical mixed sensitivity weights.
which essentially tells us that for aimed small S, enforced by jW1V1j, theweight jW2V2j
should be chosen less than 1 and vice versa.
Generally, this can be accomplished but an extra complication occurs when W1 = W2
and V1 and V2 have xed values as they characterise real signals. This happens in the
example under study where we have WeSVn and WeTV . This leads to an upper bound
for the lter We according to:
It is clear that not both jSj < 1=2 andjTj < 1=2 can be obtained, because S + T =1
for the SISO-case. Consequently the intersection point of the inverse weights should be
greater than 1:
> 1=2 (9.53)
1
jW2V2j
1
jW1V1j =
9! :
(9.58)
1
jWeVnj )j jV j)jWej < 1
jV j
> jV (1 ;
1
jWej
This is still too restrictive, because it is not to be expected that equal phase 0 can be
accomplished by any controller at the intersection point. To allow for su cient freedom
in phase it is usually required to take at least:
The better the sensor, the smaller the measurement lter jV j can be, the larger the
lter jWej can be chosen and the better the ultimate performance will be. Again this
re ects the fact that we can never control better than the accuracy of the sensor allows
us. We encountered this very same e ect before in the one frequency example. Indeed,
this e ect particularly poses a signi cant limiting e ect on the aim to accomplish zero
tracking error at ! =1. A nal zero error in the step-response for a control loop including
an integrator should therefore be understood within this measurement noise e ect.
1
jW1V1j =
1
> 1 , (9.54)
jW2V2j
9! : jW1V1j = jW2V2j < 1 (9.55)
9! :
9.3 Limitations due to plant characteristics.
In the previous subsections the weights V have been based on the exogenous input characteristics.
The weight Wu was determined by the actuator limits and the model perturbations.
Finally, limits on the weight We were derived based on the relation S + T = I.
Never, the characteristics of the plant itself were considered. It appears that these very
It can easily be understood that the S and T vectors for frequencies in the neighbourhood
of the intersection point can then only be taken in the intersection area of the two
circles in Fig. 9.7.
Consequently, it is crucial that the point ofintersection of the curves jW1(j!)V1(j!)j
and jW2(j!)V2(j!)j is below the 0 dB-level, otherwise there would be a con ict with
S + T =1and there would be no solution 1! Consequently, heavily weighted bands
(> 0dB) forS and T should always exclude each other.
Further away from the intersection point the condition S + T requires that for small
S the T should e ectively be greater than 1 and vice versa. If we want:

130 CHAPTER 9. FILTER SELECTION AND LIMITATIONS.
9.3. LIMITATIONS DUE TO PLANT CHARACTERISTICS. 129
g (9.67)
1
jWuVnj
8! : fjWuQVnj < 1g,fjQj <
dynamical properties put bounds on the nal performance. This is clear if one accepts
that some e ort is to be made to stabilise the plant, which inevitably will be at the cost of
the performance. We will see that not so much instability, but in particular nonminimum
phase zeros and limited gain can have detrimental e ects on the nal performance.
Above bound on jQj prohibits to take Q = P ;1 as jPj is too small for certain frequencies
! , so that we willalways have:
9.3.1 Plant gain.
Let us forget about the low measurement noise for the moment and concentrate on the
remaining mixed sensitivity problem:
> 0g (9.68)
8! :fjPQj < 1g)fj1 ; PQj > 1 ;jPQj > 1 ; jPj
jWuVnj
Consequently, we learn from the condition jWe(1 ; PQ)Vnj < 1:
(~n) =Mw (9.59)
WuRVn
=
WeSVn
z = ~u
~e
(9.69)
1
1
8! :jWej <
jP j
jWuj )
jVnj;
) =
jP j
jWuVnj
jVnj(1 ;
1
jVn(1 ; PQ)j <
From chapter 4 we know that for stable plants P we may use the internal model
implementation of the controller where Q = R and S =1; PQ. Very high weights jWej
for good tracking necessarily require:
and the best sensivity we can expect for such a weight We is necessarily close to its
upper bound given by:
(9.70)
=1; jPj
jWuVnj
jP j
jVnj; jWuj
=
jVnj
1
jWeVnj
8! :jSj <
8! : fjWeSVnj = jWe(1 ; PQ)Vnj < 1g, (9.60)
fj(1 ; PQ)Vnj < 1
0g) (9.61)
jWej
fQ = P ;1g (9.62)
9.3.2 RHP-zeros.
Even in the case that P is invertable, it needs to have su cient gain, since the rst
term in the mixed sensitivity problem yields:
For perfect tracking and disturbance rejection one should be able to choose Q = P ;1 .
In the previous section this was thwarted by the range of the actuator or by the model
uncertainty via mainly Wu. Another condition on Q is stability and here the nonminimum
phase or RHP (Right Half Plane) zeros are the spoil-sport. The crux is that no controller
C may compensate these zeros by RHP-poles as the closed loop system would become
internally unstable. So necessarily from the maximum modulus principle, introduced in
chapter 4, we get:
8! : fjWuRVnj = jWuP ;1 Vnj < 1g, (9.63)
fjP (j!)j > jWu(j!)Vn(j!)jg , (9.64)
1
fjP (j!)j
jWu(j!)j > jVn(j!)jg (9.65)
sup jWe(j!)S(j!)Vn(j!)j jWe(z)(1 ; P (z)Q(z))Vn(z)j = jWe(z)Vn(z)j (9.71)
!
where z is any RHP-zero where necessarily P (z) = 0 and jQ(z)j < 1. Unfortunately,
this puts an underbound on the weighted sensitivity. Because we want the weighted
sensitivity to be less than one, we should at least require that the weights satisfy:
The last constraint simply states that, given the bound on the actuator input by
j1=Wuj, the maximum e ect of an input u at the output, viz. jP=Wuj should potentially
compensate the maximum disturbance jVnj. That is, the gain of the plant P for each
frequency in the tracking band should be large enough to compensate for the disturbance
n as a reaction of the input u. In frequency domain, this is the same constraint as we
found in subsection 9.1.3
Typically, if we compare the lower bound on the plant with the robustness constraint
on the additive model perturbation, we get:
jWe(z)Vn(z)j < 1 (9.72)
This puts a strong constraint on the choice of the weight We because heavy weights
at the imaginary axis band, where we like to have a small S, will have to be arranged
by poles and zeros of We and Vn in the LHP and the "mountain peaks" caused by the
poles will certainly have their " mountain ridges" passed on to the RHP where at the
position of the zero z their height is limited according to above formula. This is quite an
abstract explanation. Let us therefore turn to the background of the RHP-zeros and a
simple example.
8! : jPj > jWuVnj > j Pj (9.66)
which says that modelling error larger than 100% will certainly prevent tracking and
disturbance rejection, as can easily be grasped.
All is well, but what should be done if the gain of P is insu cient, at least at certain
frequencies? Simply adapt your performance aim by decreasing the weight We as follows.
Starting from the constraint wehave:

132 CHAPTER 9. FILTER SELECTION AND LIMITATIONS.
9.3. LIMITATIONS DUE TO PLANT CHARACTERISTICS. 131
Step Response
From: U(1)
10 1
1.5
Nonminimum phase zeros, as the engineering name indicates, originate from some
strange internal phase characteristics, usually by contradictory signs of behaviour in certain
frequency bands. As an example may function:
1
PC/(PC+1)
10 0
P1C/s(P1C+1)
P2C/(P2C+1)
0.5
(9.73)
s ; 8
(s + 1)(s + 10)
= ;
PC/s(PC+1)
To: Y(1)
Amplitude
2
;
s +10
P (s) =P1(s)+P2(s) = 1
s +1
P1C/(P1C+1)
0
10 −1
−0.5
P2C/s(P2C+1)
0 0.5 1 1.5
−1
10 2
10 1
10 0
10 −1
10 −2
Time (sec.)
Figure 9.9: Closed loop of nonminimum phase plant and its components.
The two transfer components show competing e ects because of the sign. The the
sign of the transfer with the slowest pole at -1 is positive. The sign of the other transfer
with the faster dynamics pole at -10 is negative. Brought into one rational transfer this
e ect causes the RHP-zero at z =8. Note that the zero is right between the two poles in
absolute value. The zero could also have been occured in the LHP, e.g. by di erent gains
of the two rst order transfers (try for yourself). In that case a controller could easily
cope with the phase characteristic by putting a pole on this LHP-zero. In the RHP this
is not allowed because of internal stability requirement. So, let us take a straightforward
PI-controller that compensates the slowest pole:
As a consequence for the choice of We for such a system we cannot aim at a broader
frequency band than, as a rule of the thumb, ! (0 jzj=2) and also the gain of We is limited.
This limit is re ected in the above found limitation:
(9.74)
s +1
K
s
s ; 8
(s +1)(s + 10)
P (s)C(s) =;
jWe(z)Vn(z)j < 1 (9.75)
and take controller gain K such that we obtain equal real and imaginary parts for the
closed loop poles as shown in Fig. 9.8.
If, on the other hand, wewould like to obtain a good tracking for a band ! (2jzj 100jzj)
the controller can indeed wellbechosen to control the component ;2=(s + 10), while now
the other component 1=(s +1) is the nasty one. In a band ! (jzj=2 2jzj) we can never
track well, because the opposite e ects of both components of the plant are apparent in
their full extent.
If we havemore RHP-zeros zi, wehave asmanyforbidden tracking bands ! (jzij=2 2
jzij). Even zeros at in nity playarole as explained in the next subsection.
20
15
10
5
9.3.3 Bode integral.
0
Imag Axis
For strictly proper plants combined with strictly proper controllers we will have zeros
at in nity. It is irrelevant whether in nity is in the RHP. Zeros at in nity should be
treated like all RHP-zeros, simply because they cannot be compensated by poles. Because
in practice each system is strictly proper, we have that the combination of plant and
controller L(s) = P (s)C(s) has at least a pole zero excess (#poles ; #zeros) of two.
Consequently it is required:
−5
−10
−15
−20
−20 −15 −10 −5 0 5 10 15 20
Real Axis
Figure 9.8: Rootlocus for PI-controlled nonminimumphase plant.
jWe(1)Vn(1)j < 1 (9.76)
and we necessarily have:
j =1 (9.77)
1
1+L(s)
jSj = lim
s!1 j
lim
s!1
Any tracking band will necessarily be bounded. However, how can we see the in uence
of zeros at in nity at a nite band? Here the Bode Sensitivity Integral gives us an
impression (the proof can be found in e.g. Doyle [2]). If the pole zero excess is at least 2
and we haveno RHP poles, the following holds:
which leads to K =3. In Fig. 9.9 the step response and the bode plot for the closed
loop system is showed.
Also the results for the same controller applied to the one component P1(s) =1=(s+1)
or the other component P2(s) =;2=(s +10) is shown. The bodeplot shows a total gain
enclosed by the two separate components and the component ;2=(s +10) is even more
broadbanded. Alas, if we would have only this component, the chosen controller would
make the plant unstable as seen in the step response. For the higher frequencies the phase
of the controller is incorrect. For the lower frequencies ! (0 3:5) the phase of the controller
is appropriate and the plant iswell controlled. The e ect of the higher frequencies is still
seen at the initial time of the response where the direction (sign) is wrong.

134 CHAPTER 9. FILTER SELECTION AND LIMITATIONS.
9.3. LIMITATIONS DUE TO PLANT CHARACTERISTICS. 133
again because of internal stability, but in closed loop they have been displaced into the LHP
by means of the feedback. So in closed loop, they are no longer existent, and consequently
their e ect is not as severe as of the RHP-zeros. Nevertheless, their shift towards the LHP
has to be paid for, as we will see.
The e ect of RHP-poles cannot be analysed by means of the internal model, because
this concept can only be applied to stable plants P . The straightforward generalisation
of the internal model for unstable plants has been explained in chapter 11. Essentially,
the plant is rst fed back for stabilisation and next an extra external loop with a stable
controller Q is applied for optimisation. So the idea is rst stabilisation and on top of that
optimisation of the stable closed loop. It will be clear that the extra e ort of stabilisation
has to be paid for. The currency is the use of the actuator range. Part of the actuator range
will be occupied for the stabilisation task so that less is left for the optimisation compared
with a stable plant, where we can use the whole range of the actuator for optimisation.
This can be illustrated by a simple example represented in Fig. 9.11.
Z 1
ln jS(j!)jd! =0 (9.78)
0
The explanation can best be done with an example:
(9.79)
K
s(s +100)
L(s) =P (s)C(s) =
so that the sensitivity in closed loop will be:
(9.80)
s(s +100)
s 2 + 100s + K
S =
r +
u
- - 1
K
- -
s a
6;
n
For increasing controller gain K = f2100 21000 210000g the tracking band will be
broader but we have topaywith higher overshoot in both frequency and time domain as
Fig. 9.10 shows.
10 1
?
10 0
Figure 9.11: Example for stabilisation e ort.
The plant has either a pole in RHP at a > 0 or a pole in LHP at ;a < 0. The
proportional controller K is bounded by the range of juj < umax, while the closed loop
should be able to track a unit step. The control sensitivity is given by:
K=2100
10 −1
K=21000
10 −2
K=210000
(9.81)
s a + K
= K(s a)
K
1+ K
s a
R = r
u =
10 −3
For stability we certainly need K > a. The maximum juj for a unit step occurs at
t = 0 so:
10 −4
10 4
10 3
10 2
10 1
10 0
10 −1
10 −5
(9.82)
max(u)
=u(0) = lim R(s) =K = umax
t
s!1
Figure 9.10: Sensitivity for looptransfer with pole zero excess 2 and no RHP-poles.
So it is immediately clear that, limited by theactuator saturation, the pole in closed
loop can maximally be shifted umax to the left. Consequently, for the unstable plant, a
part a is used for stabilisation of the plant and only the remainder K ; a can be used for
a bandwidth K ; a as illustrated in Fig. 9.12.
Note that the actuator range should be large enough, i.e. umax > a. Otherwise
stabilisation is not possible. It de nes a lower bound on K > a. With the same e ort
we obtain a tracking band of K + a for the stable plant. Also the nal error for the step
response is smaller:
The Bode rule states that the area of jS(j!)j under 0dB equals the area above it.
Note that we haveas usual a horizontal logarithmic scale for ! in Fig. 9.10 which visually
disrupts the concepts of equal areas. Nevertheless the message is clear: the less tracking
error and disturbance we want to obtain over a broader band, the more we have to pay
for this by a more than 100% tracking error and disturbance multiplication outside this
band.
9.3.4 RHP-poles.
(9.83)
a
K a
s a
s a + K =
e = Sr ) e(1) = lim
s!0
and certainly
The RHP-zeros play a fundamental role in the performance limitation because they cannot
be compensated by poles in the controller and will thus persist in existence also in the
closed loop. Also the RHP-poles cannot be compensated by RHP-zeros in the controller,

136 CHAPTER 9. FILTER SELECTION AND LIMITATIONS.
9.3. LIMITATIONS DUE TO PLANT CHARACTERISTICS. 135
three times a weighted complementary sensitivity, of which two are explicitly weighted
control sensitivities:
K = umax
X
a
<
X
;a
;K ; a ;K + a
<
WeTV ) WT VT = WeV (9.87)
(9.88)
K = umax
(9.89)
WuRVn ) WT VT = WuVn
P
WuRV ) WT VT = WuV
P
Only the rst entry yields bounds on the weights according to:
Figure 9.12: Rootloci for both plants 1=(s a).
jWe(p)V ((p)j < 1 (9.90)
because for the other two holds:
j =0< 1 (9.91)
j Wu(p)Vn[ (p)
P (p)
a
K + a <
a
(9.84)
K ; a
being the respective absolute nal errors. Let us show these e ects by assuming some
numerical values: K = umax =5,a = 1, which leads to poles of respectively -4 and -6 and
dito bandwidths. The nal errors are respectively 1=6 and 1=4. Fig. 9.13 shows the two
step responses. Also the two sensitivities are shown, where the di erences in bandwidth
and the nal error (at ! = 0) are evident.
as jP (p)j = 1.
The condition of inequality 9.90 is only a poor condition, because measurement noise
is usually very small. This is not the e ect we are looking for, but alas I have not been
able to nd it explicitly. You are invited to express the stabilisation e ort explicitly in the
weights.
In the Bode integral the e ect of RHP-poles is evident, because if we haveNp unstable
poles pi the Bode integral changes into:
Step Response
From: U(1)
10 0
1.4
1.2
P=1/(s−1) K=5
1
Z 1
0.8
P=1/(s+1) K=5
Np
i=1

138 CHAPTER 9. FILTER SELECTION AND LIMITATIONS.
9.3. LIMITATIONS DUE TO PLANT CHARACTERISTICS. 137
actuator range could not be made explicit in a bound on the allowable weighting lters
for the left over performance. If you have a good idea yourself, you will certainly get a
good mark for this course.
where we used that h(0) = 0 when the closed loop system is strictly proper and that
h(1) is nite.Because it is straightforward that
Z 1
Z 1
9.3.5 RHP-poles and RHP-zeros
(9.97)
de ;pt = ; e;pt
p j1 0 = 1
p
0
e ;pt dt = ; 1
p
0
It goes without saying that, when a plant has both RHP-zeros and RHP-poles, the limitations
of both e ects will at least add up. It will be more because the stabilisation e ort
will be larger. RHP-zeros will attract rootloci to the RHP, while we want to pull the
rootloci over the imaginary axis into the LHP. The stabilisation is in particular a heavy
task when we haveto deal with alternating poles and zeros on the positive real axis. These
plants are infamous, because they can only be stabilised by unstable and nonminimum
phase controllers that add to the limitations again. These plants are called "not strongly
stabilisable". Take for instance a plant with a zero z>0, p>0 and an integrator pole at
0. If z0 K0 K O < X >
0
z p
Equation 9.98 restricts the attainable step responses: the integral of the step response
error, weighted by e ;pt must vanish. As h(t) is below 1 for small values of t, this area must
be compensated by values above 1 for larger t, and this compensation is discounted for
t !1by theweight e ;pt and even more so if the steady state error happens to be zero
by integral control action. So the step response cannot show an in nitesimally small error
for a long time to satisfy 9.98. The larger p is, the shorter the available compensation
time will be, during which the response is larger than 1. If an unstable pole and actuator
limitations are both present, the initial error integral of the step response is bounded from
below, and hence there must be a positive control error area which is at least as large
as the initial error integral due to the weight e ;pt . Consequently either large overshoot
and rapid convergence to the steady state value or small overshoot and slow convergence
must occur. For our example P =1=(s ; 1) we can choose C = 5, as we did before, or
C =5(s+1)=s to accomplish zero steady state error and still avoiding actuator saturation.
The respective step responses are displayed in Fig. 9.14 together with the weight e ;t .
1.4
Figure 9.15: Rootloci for a plant, which is not strongly stabilisable.
C=5
1.2
Only if we add RHP-zeros and RHP-poles in the controller such thatwe alternatingly
have pairs of zeros and poles on the real positive axiswecan accomplish that the rootloci
leave the real positive axis and can be drawn to the LHP as illustrated in Fig. 9.16.
1
C=5(s+1)/s
0.8
K>0
K>0
6
K>0 K

140 CHAPTER 9. FILTER SELECTION AND LIMITATIONS.
9.3. LIMITATIONS DUE TO PLANT CHARACTERISTICS. 139
9.3.6 MIMO.
The previous subsections were based on the silent assumption of a SISO plant P . For
MIMO plants fundamentally the same restrictions hold but the interpretation is more
complicated. For example the plant gain is multivariable and consequent limitations need
further study. For a m input m output plant the situation is sketched in Fig 9.17, where
e.g. m =3.
passing through the LHP rst. In Skogestadt & Postlethwaite [15] this is formalised in
the following bounding theorem:
Theorem: Combined RHP-poles and RHP-zeros. Suppose that P (s) has Nz
RHP-zeros zj and has Np RHP-poles pi. Then for closed-loop stability the weighted sensitivity
function must satisfy for each RHP-zero zj:
~r1 ~r2 ~r3
6~u1 6~u2 6~u3
1 (9.99)
jzj +pij
jzj ; pij
k WSSVS k1 c1jjWS(zj)VS(zj)j c1j = Np
i=1
? ?
?
and the weighted complementary sensitivity function must satisfy for each RHP-pole
Vr3
Vr2
Vr1
Wu3
Wu2
Wu1
pi:
+ r1 r2 r3
?
- - n
- We1
-
;
e1 ~e1
+
?
- P
- n - We2
-
;
e2 ~e2
+ ?
- - n - We3
-
; e3 ~e3
6
6 6
1 (9.100)
jzj + pij
jzj ; pij
k WT TVT k1 c2jjWT (pi)VT (pi)j c2j = Nz
j=1
u1
u2
where WS and VS are sensitivity weighting lters like the pair fWeVng. Similarly,
WT and VT are complementary sensitivity weighting lters like the pair fWeV g. If we
want theinnitynorms to be less than 1, above inequalities put upper bounds on the
weight lters. On the other hand if we apply the theorem without weights we get:
u3
(9.101)
c2i
c1j k T k1 max
i
k S k1 max
j
Figure 9.17: Scaling of a 3x3-plant.
This shows that large peaks for S and T are unavoidable if we have a RHP-pole and
RHP-zero located close to each other.
The scaled tracking error ~e as a function of the scaled reference ~r and the scaled control
signal ~u is given by:
1
A =
0
@ ~e1
~e2
~e =
1
A
0
@ ~r1
~r2
1
A
0
@ Vr1 0 0
1
A
~e3
0
0 Vr2 0
0 0 Vr3
1
A =
0
@ ~u1
~u2
~u3
1
A
~r3
;1
W
0
@
1
A
0
u1 0 0
0
@ P11 P12 P13
P21 P22 P23
P31 P32 P33
1
A
0 W ;1
u2
0 0 W ;1
u3
= @ We1 0 0
0 We2 0
0 0 We3
0
; @ We1 0 0
0 We2 0
0 0 We3 ;
= We Vr~r ; PW ;1
u ~u
(9.102)
Note that we haveas usual diagonal weights where jWui(j!)j stands for the maximum
range of the corresponding actuator for the particular frequency !. Also the aimed range
of the reference ri is characterised by jVri(j!)j and should at least correspond to the
permitted range for the particular output zi for frequency !. For heavy weights We in
order to make ~e 0we need:
Vr~r = PW ;1
u ~u = r = Pu (9.103)

142 CHAPTER 9. FILTER SELECTION AND LIMITATIONS.
9.3. LIMITATIONS DUE TO PLANT CHARACTERISTICS. 141
Let the input u = F being the horizontal force exerted to the carriage and let the
outputs be the angle of the pendulum and x the position of the carriage. So we have 1
input and 2 outputs and we would like to track a certain reference for the carriage and at
the same time keep the infuence of disturbance on the pendulum angle small according to
The ranges of the actuators should be su ciently large in order to excite each output
up to the wanted amplitude expressed by:
Fig. 9.19.
~u = WuP ;1 Vr~r (9.104)
-
e
6
so that
6
-
+ ? -
u +
- n - - P1 -
-
6;
P2
; C1 C2
(9.105)
k ~u k2 k WuP ;1Vr k1k ~r k2 1
,k WuP ;1Vr k1 1
6+
r
Because (A ;1 )= (A) wemay write:
?
-
d
(9.106)
8! : (WuP ;1Vr) 1 ,
8! : (V ;1
r PW;1 u ) 1
Figure 9.19: Keeping e and small in the face of r and d.
That is, we like tomake the total sensitivity small:
(9.107)
d
r
which simply states that the gains of the scaled plant in the form of the singular values
should all be larger than 1. The plant is scaled with respect to each allowd input ui and
each aimed output zj for each frequency !. A singular value less than one implies that a
certain aimed combination of outputs indicated by the corresponding left singular vector
cannot be achieved by anyallowed input vector ~u 1.
We presented the analysis for the tracking problem. Exactly the same holds of course
for the disturbance rejection for which Vd should be substituted for Vr. Note, that the
di erence in sign for r and d does not matter. Also the additive model perturbation, i.e.
Vp and Wu, can be treated in the same way and certainly the combination of tracking,
disturbance reduction and model error robustness by means of Vn.
In above derivation we assumed that all matrices were square and invertible. If we
have m inputs against p outputs where p > m (tall transfer matrix) we are in trouble.
We actually have p ; m singular values equal to 0 which iscertainly less than 1. It says
that certain output combinations cannot be controlled independent from other output
combinations as we haveinsu cient inputs. We can only aim at controlling p ; m output
combinations. Let us show this with a well known example: the pendulum on a carriage
of Fig. 9.18.
;1
C1
C2
= P1u + d
x = P2u
u = C1 + C2e
e = r ; x
)
e
=
1 0
0 1 + ; P1 P2
If we want both the tracking of x and the disturbance reduction of better than
without control we need:
< 1 (9.108)
1
(I + PC)
(S) = ((I + PC) ;1 )=
The following can be proved:
y
(I + PC) 1+ (PC) (9.109)
6
Since the rank of PC is 1 (1 input u) wehave (PC) = 0 so that:
j
2l
h
(I + PC) 1 ) (S) 1 (9.110)
?
This result implies that we can never control both outputs e and appropriately in the
same frequency band! It does not matter how the real transfer functions P1 and P2 look
like. Also instability is not relevant here. The same result holds for a rocket, when the
pendulum is upright, or for a gantry crane, when the pendulum is hanging. The crucial
limitation is the fact that we haveonly one input u. The remedy is therefore either to add
more independent inputs (e.g. a torque on the pendulum) or require less by weighting the
tracking performance heavily and leaving only determined by stabilisation conditions.
F
M
x
Figure 9.18: The inverted pendulum on a carriage.

144 CHAPTER 9. FILTER SELECTION AND LIMITATIONS.
9.4. SUMMARY 143
1. (A B2) is not stabilisable. Unstable modes cannot be controlled. Usually not well
de ned plant. Be sure that all your weights are stable and minimum phase.
2. (A C2) is not detectable. Unstable modes cannot be observed. Usually not well
de ned plant. Again, all your weights should be stable and minimum phase.
3. D12 does not have full rank equal to the number of inputs ui. This means that not
all inputs ui are penalised in the outputs z by means of the weights Wui. They
should be penalised for all frequencies so that biproper weights Wui are required. If
not all ui are weighted for all frequencies, the e ect is the same as when we have in
LQG-control a weight matrix R which is singular and needs to be inverted in the
solution algorithm. In chapter 13 we sawthat for the LQG-problem D12 = R 1
2 .
In above example of the pendulum we treated the extreme case that (P ) = 0 but
certainly simular e ects occur approximately if (P ) :
0
y = C2x + D21w + D22u

146 CHAPTER 9. FILTER SELECTION AND LIMITATIONS.
9.4. SUMMARY 145
6. In case of both RHP-poles and RHP-zeros test on basis of theorem equations 9.99
and 9.100.
Still no solution? Find an expert.

148 CHAPTER 10. DESIGN EXAMPLE
(10.3)
(s + :06 + 6j)(s + :06 ; 6j)
(s + :05 + 5j)(s + :05 ; 5j)
(s + :125)
(s + 1)(s ; 1)
Pa(s) =Ka
Finally, we haveM(s) =P (s) as basic model and Ps(s) ; M(s) and Pa(s) ; M(s) as
possible additive model perturbations. The Bode plots are shown in Fig. 10.1. As the
errors exceed the nominal plant at! 5:5 the control band will certainly be less wide.
Chapter 10
|M|,|M−Ps|,|M−Pa|
10 2
10 1
Design example
10 0
10 −1
10 −2
10 −3
10 −4
The aim of this chapter is to synthesize a controller for a rocket model with perturbations.
First, a classic control design will be made so as to compare the results with H1-control
and -control. The use of various control toolboxes will be illustrated. The program
les which will be used can be obtained from the ftp-site nt01.er.ele.tue.nl or via the
internet home page of this course. We refer to the \readme" le for details.
10 −5
10 −6
10.1 Plant de nition
10 2
10 1
10 −1
10 0
The plant and its perturbations
10 −2
10 −3
10 −7
Figure 10.1: Nominal plant and additive perturbations.
In Matlab, the plant de nition can be implemented as follows
% This is the script file PLANTDEF.M
%
% It first defines the model M(s)=-8(s+.125)/(s+1)(s-1)
% from its zero and pole locations. Subsequently, it introduces
% the perturbed models Pa(s)=M(s)*D(s) and Ps(s) = M(s)/D(s) where
% D(s) has poles and zeros nearby the imaginary axis
z0=-.125 p0=[-11]
zs=[-.125-.05+j*5-.05-j*5] ps=[-11-.06+j*6-.06-j*6]
za=[-.125-.06+j*6-.06-j*6] pa=[-11-.05+j*5-.05-j*5]
[numm,denm]=zp2tf(z0,p0,1)
[nums,dens]=zp2tf(zs,ps,1)
[numa,dena]=zp2tf(za,pa,1)
% adjust the gains:
km=polyval(denm,0)/polyval(numm,0)
ks=polyval(dens,0)/polyval(nums,0)
ka=polyval(dena,0)/polyval(numa,0)
numm=numm*kmnums=nums*ksnuma=numa*ka
The model has been inspired by a paper on rocket control from Enns [17]. Booster rockets
y through the atmosphere on their way to orbit. Along the way, they encounter aerodynamic
forces which tend to make the rocket tumble. This unstable phenomenon can be
controlled with a feedback of the pitch rate to thrust control. The elasticity of the rocket
complicates the feedback control. Instability can result if the control law confuses elastic
motion with rigid body motion. The input is a thrust vector control and the measured
output is the pitch rate. The rocket engines are mounted in gimbals attached to the bottom
of the vehicle to accomplish the thrust vector control. The pitch rate is measured
with a gyroscope located just below the center of the rocket. Thus the sensor and actuator
are not co-located. In this example we have an extra so called \ ight path zero" in the
transfer function on top of the well known, so called \short period pole pair" which are
mirrored with respect to the imaginary axis. The rigid body motion model is described
by the transfer function
(10.1)
(s + :125)
(s + 1)(s ; 1)
M(s) =;8
Note that M(0) =1. We we will use the model M as the basic model P in the control
design.
The elastic modes are described by complex, lightly damped poles associated with
zeros. In this simpli ed model we only take thelowest frequency mode yielding:
% Define error models M-Pa and M-Ps
[dnuma,ddena]=parallel(numa,dena,-numm,denm)
[dnums,ddens]=parallel(nums,dens,-numm,denm)
% Plot the bode diagram of model and its (additive) errors
w=logspace(-3,2,3000)
magm=bode(numm,denm,w)
mags=bode(dnums,ddens,w)
dmaga=bode(dnuma,ddena,w)
(10.2)
(s + :05 + 5j)(s + :05 ; 5j)
(s + :06 + 6j)(s + :06 ; 6j)
(s + :125)
(s +1)(s ; 1)
Ps(s) =Ks
The gain Ks is determined so that Ps(0) = 1. Fuel consumption will decrease distributed
mass and sti ness of the fuel tanks. Also changes in temperature play a role. As
a consequence, the elastic modes will change. We have taken the worst scenario in which
poles and zeros change place. This yields:
147

150 CHAPTER 10. DESIGN EXAMPLE
10.2. CLASSIC CONTROL 149
Nyquist MC
5
4
3
2
1
0
−1
−2
−3
−4
−5
−5 −4 −3 −2 −1 0 1 2 3 4 5
Real Axis
rootlocus MC
4
3
2
dmags=bode(dnums,ddens,w)
loglog(w,magm,w,dmags,w,dmaga)
title('|M|,|M-Ps|,|M-Pa|')
xlabel('The plant and its perturbations')
1
Imag Axis
0
Imag Axis
−1
10.2 Classic control
−2
−3
−4
−4 −3 −2 −1 0 1 2 3 4
Real Axis
Figure 10.3: Root locus and Nyquist plot for low order controller.
rootloci PtsC and PtaC
The plant is a simple SISO-system, so we should be able to design a controller with classic
tools. In general, this is a good start as it gives insight into the problem and is therefore
of considerable help in choosing the weighting lters for an H1-design.
For the controlled system we wish to obtain a zero steady state, i.e., integral action,
while the bandwidth is bounded by the elastic mode at approximately 5.5 rad/s, as we
require robust stability and robust performance for the elastic mode models. Some trial
and error with a simple low order controller, leads soon to a controller of the form
10
(10.4)
(s +1)
s(s +2)
C(s) = 1
2
5
In the bode plot of this controller in Fig. 10.2, we observe that the control band is
bounded by ! 0:25rad/s.
0
Imag Axis
50
−5
0
Gain dB
−10
10 2
10 1
10 −1
10 0
Frequency (rad/sec)
bodeplots controller
10 −2
10 −3
−50
−10 −5 0 5 10
Real Axis
−250
Figure 10.4: Root loci for the elastic mode models.
−255
−260
Phase deg
Nyquist PtaC
5
4
3
2
1
0
−1
−2
−3
−4
−5
−5 −4 −3 −2 −1 0 1 2 3 4 5
Real Axis
Nyquist PtsC
5
4
3
2
1
0
−1
−2
−3
−4
−5
−5 −4 −3 −2 −1 0 1 2 3 4 5
Real Axis
−265
10 2
10 1
10 −1
10 0
Frequency (rad/sec)
10 −2
10 −3
−270
Imag Axis
Imag Axis
Figure 10.2: Classic low order controller.
Figure 10.5: Nyquist plots for elastic mode models.
and the elastic mode models in Fig. 10.6. Still, we notice some high frequent oscillations,
that occur for the model Pa, as the poles have been shifted closer to the imaginary axis
by the feedback and consequently the elastic modes are less damped.
We can do better by taking care that the feedback loop shows no or very little action
just in the neighborhood of the elastic modes. Therefore we include a notch lter, which
The root locus and the Nyquist plot look familiar for the nominal plant in Fig. 10.3,
but we could have done much better by shifting the pole at -2 to the left and increasing
the gain.
If we study the root loci for the two elastic mode models of Fig. 10.4 and the Nyquist
plots in Fig. 10.5, it is clear why sucharestricted low pass controller is obtained. Increase
of the controller gain or bandwidth would soon cause the root loci to pass the imaginary
axis to the RHP for the elastic mode model Pa. This model shows the most nasty dynamics.
It has the pole pair closest to the origin. The root loci, which emerge from those poles,
loop in the RHP. Also in the corresponding right Nyquist plot we see that an increase of
the gain would soon lead to an extra and forbidden encircling of the point -1by the loops
originating from the elastic mode.
By keeping the control action strictly low pass, the elastic mode dynamics will hardly
be in uenced, as we mayobserve from the closed loop step responses of the nominal model

152 CHAPTER 10. DESIGN EXAMPLE
10.2. CLASSIC CONTROL 151
Nyquist MC
5
4
3
2
1
0
−1
−2
−3
−4
−5
−5 −4 −3 −2 −1 0 1 2 3 4 5
Real Axis
step disturbance for M, Pts or Pta in loop
1
0.5
Imag Axis
rootlocus MC
10
8
6
4
2
0
−2
−4
−6
−8
−10
−10 −8 −6 −4 −2 0 2 4 6 8 10
Real Axis
Imag Axis
0
−0.5
Amplitude
−1
Figure 10.8: Root locus and Nyquist plot controller with notch lter.
−1.5
−2
0 10 20 30 40 50 60 70 80 90 100
Time (secs)
the e ect because apparently the gain should be very large in order to derive the exact
track of the root locus.
Figure 10.6: Step responses for low order controller.
rootloci PtsC and PtaC
10
8
6
4
2
0
−2
−4
−6
−8
−10
−10 −8 −6 −4 −2 0 2 4 6 8 10
Real Axis
has a narrow dip in the transfer just at the proper place:
(10.5)
+ :055 + 5:5j)(s + :055 ; 5:5j)
150(s
(s +50+50j)(s +50; 50j)
(s +1)
s(s +2)
C(s) = 1
2
Imag Axis
We have positioned zeros just in the middle of the elastic modes pole-zero couples.
Roll o poles have been placed far away, where they cannot in uence control, because at
! =50theplanttransfer itself is very small. We clearly discern this dip in the bode plot
of this controller in Fig. 10.7.
50
0
Figure 10.9: Root loci for the elastic mode models with notch lter.
Gain dB
−50
This is also re ected in the Nyquist plots in Fig.10.10. Because of the notch lters,
the loops due to the elastic modes have been substantially decreased in the loop transfer
and consequently there is little chance left that the point -1 is encircled.
10 2
10 1
10 −1
10 0
Frequency (rad/sec)
bodeplots controller
10 −2
10 −3
−100
0
Nyquist PtaC
5
4
3
2
1
0
−1
−2
−3
−4
−5
−5 −4 −3 −2 −1 0 1 2 3 4 5
Real Axis
−90
−180
Phase deg
−270
Imag Axis
Nyquist PtsC
5
4
3
2
1
0
−1
−2
−3
−4
−5
−5 −4 −3 −2 −1 0 1 2 3 4 5
Real Axis
Imag Axis
10 2
10 1
10 −1
10 0
Frequency (rad/sec)
10 −2
10 −3
Figure 10.7: Classic controller with notch lter.
Figure 10.10: Nyquist plots for elastic mode models with notch lter.
Finally, as a consequence, the step responses of the two elastic models show no longer
elastic mode oscillations and they di er hardly from the rigid mass model as shown in
The root locus and the Nyquist plot for the nominal plant in Fig. 10.8 are hardly
changed close to the origin. Further away, where the roll-o poles lay, the root locus is not
interesting and has not been shown. The poles remain su ciently far from the imaginary
axis, as expected, given the small plant transfer at those high frequencies.
Studying the root loci for the two elastic mode models of Fig. 10.9 it can be seen that
there is hardly any shift of the elastic mode poles. Even Matlab had problems in showing

154 CHAPTER 10. DESIGN EXAMPLE
10.2. CLASSIC CONTROL 153
[numcls,dencls]=feedback(1,1,numls,denls,-1)
[numcla,dencla]=feedback(1,1,numla,denla,-1)
step(numcl,dencl) hold
step(numcls,dencls) step(numcla,dencla)
title('step disturbance for M, Pts or Pta in loop')
pause hold off
Fig. 10.11.
step disturbance for M, Pts or Pta in loop
1
0.5
% Improved classic controller C(s)=[.5(s+1)/s(s+2)]*
% 150(s+.055+j*5.5)(s+.055-j*5.5)/(s+50-j*50)(s+50-j*50)
0
−0.5
Amplitude
numc=conv(-[.5 .5],[1 .1 30.2525]*150)
denc=conv([1 2 0],[1 100 5000])
bode(numc,denc,w) title('bodeplots controller')
pause
numl=conv(numc,numm) denl=conv(denc,denm)
rlocus(numl,denl)
axis([-10,10,-10,10]) title('rootlocus MC')
pause
nyquist(numl,denl,w)
set(figure(1),'currentaxes',get(gcr,'plotaxes'))
axis([-5,5,-5,5]) title('Nyquist MC')
pause
[numls,denls]=series(numc,denc,nums,dens)
[numla,denla]=series(numc,denc,numa,dena)
rlocus(numls,denls) hold rlocus(numla,denla)
axis([-10,10,-10,10]) title('rootloci PtsC and PtaC')
pause hold off
nyquist(numls,denls,w)
set(figure(1),'currentaxes',get(gcr,'plotaxes'))
axis([-5,5,-5,5]) title('Nyquist PtsC')
pause
nyquist(numla,denla,w)
set(figure(1),'currentaxes',get(gcr,'plotaxes'))
axis([-5,5,-5,5]) title('Nyquist PtaC')
pause
[numcl,dencl]=feedback(1,1,numl,denl,-1)
[numcls,dencls]=feedback(1,1,numls,denls,-1)
[numcla,dencla]=feedback(1,1,numla,denla,-1)
step(numcl,dencl)
hold
step(numcls,dencls)step(numcla,dencla)
title('step disturbance for M, Pts or Pta in loop')
pause hold off
−1
−1.5
−2
0 2 4 6 8 10 12 14 16 18 20
Time (secs)
Figure 10.11: Step responses for controller with notch lter.
You can replay all computations, possibly with modi cations, by running raketcla.m
as listed below:
% This is the script file RAKETCLA.M
%
% In this script file we synthesize controllers
% for the plant (defined in plantdef) using classical
% design techniques. It is assumed that you ran
% *plantdef* before invoking this script.
%
% First try the classic control law: C(s)=.5(s+1)/s(s+2)
10.3 Augmented plant and weight lter selection
Being an example we want tokeep the control design simple so that we propose a simple
mixed sensitivity set-up as depicted in Fig. 10.12.
The exogenous input w = ~ d stands in principle for the aerodynamic forces acting on
the rocket in ight for a nominal speed. It will also represent the model perturbations
together with the weight on the actuator input u = u. The disturbed output of the rocket,
the pitch rate, should be kept to zero as close as possible. Because we can see it as an
error, we incorporate it, in a weighted form ~e, asa component oftheoutputz =(~u ~e) T .
At the same time, the error e is used as the measurement y = e for the controller. Note
numc=-[.5 .5] denc=[1 2 0]
bode(numc,denc,w) title('bodeplots controller')
pause
numl=conv(numc,numm) denl=conv(denc,denm)
rlocus(numl,denl) title('rootlocus MC')
pause
nyquist(numl,denl,w)
set(figure(1),'currentaxes',get(gcr,'plotaxes'))
axis([-5,5,-5,5]) title('Nyquist MC')
pause
[numls,denls]=series(numc,denc,nums,dens)
[numla,denla]=series(numc,denc,numa,dena)
rlocus(numls,denls) hold
rlocus(numla,denla) title('rootloci PtsC and PtaC')
pause hold off
nyquist(numls,denls,w)
set(figure(1),'currentaxes',get(gcr,'plotaxes'))
axis([-5,5,-5,5]) title('Nyquist PtsC')
pause
nyquist(numla,denla,w)
set(figure(1),'currentaxes',get(gcr,'plotaxes'))
axis([-5,5,-5,5]) title('Nyquist PtaC')
pause
[numcl,dencl]=feedback(1,1,numl,denl,-1)

156 CHAPTER 10. DESIGN EXAMPLE
10.3. AUGMENTED PLANT AND WEIGHT FILTER SELECTION 155
solution because Vd is equally involved in both terms of the mixed sensitivity problem.
The bode plot of lter is displayed in Fig. 10.13.
z = ~u
~e
-
Wu
-
-
6
- Vd
-
6
? -
w = ~ d
Weighting parameters in control configuration
10 4
10 2
?
- - -
-
y = e
-
P n We
u = u
-
10 0
?
6
G
10 −2
?
K
10 −4
10 2
10 1
10 −1
10 0
|Vd|, |We|, |Wu|
10 −2
10 −3
10 −6
Figure 10.12: Augmented plant for rocket.
Figure 10.13: Weighting lters for rocket.
that we did not pay attention to measurement errors. The mixed sensitivity isthus de ned
by:
Based on the exercise of classic control design, we cannot expect a disturbance rejection
over a broader band than 2rad/s. Choosing again a biproper lter for We, we cannot go
much further with the zero than the zero at 100 for Vd. Keeping We on the 0 dB line for
low frequencies we thus obtain:
!
~d (10.6)
w = WuRVd
WeSVd
WuKVd
1;PK
WeVd
1;PK
z = ~u
~e =
(10.8)
+ 100
= :02s
s +2
:02s +2
s +2
We =
as displayed in Fig. 10.13.
Concerning Wu, we again know very little about the actuator consisting of a servosystem
driving the angle of the gimbals to direct the thrust vector. Certainly,theallowed band
will be low pass. So all we can do is to choose a high pass penalty Wu such that the expected
model perturbations will be covered and hope that this is su cient to prevent from actuator
saturation. The additive model perturbations jPs(j!);M(j!)j and jPa(j!);M(j!)j
are shown in Fig. 10.14 and should be less than WR(j!) = Wu(j!)Vd(j!)j, which are
displayed as well.
We have chosen two poles in between the poles and zeros of the exible mode of the
rocket just at the place where we have chosen zeros in the classic controller. We will see
that, by doing so, indeed the mixed sensitivity controller will also contain zeros at these
positions showing the same notch lter. In order to make the Wu biproper again, we
now haveto choose zeros at the lower end of the frequency range, i.e., at .001rad/s. The
gain of the lter has been chosen such that the additive model errors are just covered by
WR(j!) =Wu(j!)Vd(j!)j. Finally we have forWu:
The disturbance lter Vd represents the aerodynamic forces. Since these are forces
which act on the rocket, like the actuator does by directing the gimballs,itwould be more
straightforward to model d as an input disturbance. To keep track with the presentation of
disturbances at the output throughout the lecture notes and to cope more easily with the
additive perturbations by means of VdWu,wehavechosen to leave it an output disturbance.
As we knowvery little about the aerodynamic forces, a at spectrum seems appropriate as
we see no reason that some frequencies should be favoured. Passing through the process,
of which the predominant behaviour is dictated by two poles and one zero, there will be
a decay for frequencies higher than 1rad/s with -20dB/decade. We could then choose a
rst order lter Vd with a pole at -1. We like to shift the pole to the origin. In that
way we will penalise the tracking error via WeSVd in nitely heavily at ! = 0, so that
the controller will necessarily contain integral action. For numerical reasons we have to
take theintegration pole somewhat in the LHP at a distance which is small compared to
the poles and zeros that determine the transfer P . Furthermore, if we choose Vd to be
biproper, we avoid problems with inversions, where we will see that in the controller a lot
of pole-zero cancellations with the augmented plant will occur, in particular for the mixed
sensitivity problems. So, Vd has been chosen as:
Wu = 1 100s
3
2 + :2s + :0001
s2 100 (s + :001)
=
+ :1s +30:2525 3
2
(10.9)
(s + :05 + j5:5)(s + :05 ; j5:5)
Having de ned all lters, we can now test, whether the conditions with repect to
S + T =1are satis ed. Therefore we display WS = WeVd as the weighting lter for the
sensitivity S in Fig. 10.15.
(10.7)
s + 100
= :01
s + :0001
:01s +1
s + :0001
Vd =
Note that the pole and zero lay 6 decades apart, which will be on the edge of numerical
power. The gain has been chosen as .01, which appeared to give least numerical problems.
As there are no other exogenous inputs, there is no problem of scaling. If we increase
the gain of Vd we will just have a larger in nity norm bound , but no di erent optimal

158 CHAPTER 10. DESIGN EXAMPLE
10.3. AUGMENTED PLANT AND WEIGHT FILTER SELECTION 157
magWu=bode(numWu,denWu,w)
loglog(w,magVd,w,magWe,w,magWu)
xlabel('|Vd|, |We|, |Wu|')
title('Weighting parameters in control configuration')
pause
magWS=magVd.*magWe
magWR=magVd.*magWu
magWT=magWR./magm
loglog(w,magWS,w,magWR,w,magWT)
xlabel('|WS|, |WR| and |WT|')
title('Sensitivity, control and complementary sensitivity weightings')
pause
loglog(w,magWR,w,dmags,w,dmaga)
title('Compare additive modelerror weight and "real" additive errors')
pause
echo off
Compare additive modelerror weight and "real" additive errors
10 4
10 2
10 0
10 −2
10 −4
10 −6
10 2
10 1
10 0
10 −1
10 −2
10 −3
10 −8
Figure 10.14: WR encompasses additive model error.
10.4 Robust control toolbox
Sensitivity, control and complementary sensitivity weightings
10 3
The mixed sensitivity problem is well de ned now. With the Matlab Robust Control
toolbox we can compute a controller together with the associated . This toolbox can
only be used for a simple mixed sensitivity problem. The con guration structure is xed,
only the weighting lters corresponding to S, T and/or R have to be speci ed. The
example which we study in this chapter ts in such a framework, but we emphasize that
the toolbox lacks the exibility for larger, or di erent structures. For the example, it nds
=1:338 which issomewhat too large, so that we should adapt the weights once again.
The frequency response of the controller is displayed in Fig. 10.16 and looks similar to the
controller found by classical means.
10 2
10 1
10 0
10 −1
10 −2
10 −3
100
10 2
10 1
10 −1
10 0
|WS|, |WR| and |WT|
10 −2
10 −3
10 −4
50
Gain dB
0
Figure 10.15: Weighting lters for sensitivities S R and T .
10 4
10 2
10 −2
10 0
Frequency (rad/sec)
10 −4
10 −6
−50
270
180
90
Phase deg
10 4
10 2
10 −2
10 0
Frequency (rad/sec)
10 −4
10 −6
0
Figure 10.16: H1 controller found by Robust Control Toolbox.
Similarly the weight for the control sensitivity R is WR = WuVd and from that we derive
that for the complementary sensitivity T the weight equals WT = WuVd=P represented in
Fig. 10.15. We observe that WS is lowpass and WT is high pass and, more importantly,
they intersect below the 0dB-line.
In this example, the above reasoning seems to suggest that one can derive and synthesize
weighting lters. In reality this is an iterative process, where one starts with certain
lters and adapts them in subsequent iterations such that they lead to a controller which
gives acceptable behaviour of the closed loop system. In particular, the gains of the various
lters need several iterations to arrive at proper values.
The proposed lter selection is implemented in the following Matlab script.
Nevertheless, the impulse responses displayed in Fig. 10.17 still show the oscillatory
e ects of the elastic modes. More trial and error for improving the weights is therefore
necessary. In particular, has to be decreased.
Finally in Fig. 10.18 the sensitivity and the control sensitivity are shown together with
their bounds which satisfy:
% This is the script WEIGHTS.M
numVd=[.01 1] denVd=[1 .001]
numWe=[.02 2] denWe=[1 2]
numWu=[100 .2 .0001]/3 denWu=[1 .1 30.2525]
magVd=bode(numVd,denVd,w)
magWe=bode(numWe,denWe,w)

160 CHAPTER 10. DESIGN EXAMPLE
10.4. ROBUST CONTROL TOOLBOX 159
1
% Define the weigthing parameters
weights
0.8
0.6
% Next we need to construct the augmented plant. To do so,
% the robust control toolbox allows to define *three weigths* only.
% (This may be viewed as a severe handicap!) These weights will be
% called W1, W2, and W3 and represent the transfer function weightings
% on the controlled system sensitivity (S), control sensitivity (R)
% and complementary sensitivity (T), respectively. From our configuration
% we find that W1 = Vd*We, W2=Vd*Wu and W3 is not in use. We specify
% this in state space form as follows.
[aw1,bw1,cw1,dw1]=tf2ss(conv(numVd,numWe),conv(denVd,denWe))
ssw1=mksys(aw1,bw1,cw1,dw1)
[aw2,bw2,cw2,dw2]=tf2ss(conv(numVd,numWu),conv(denVd,denWu))
ssw2=mksys(aw2,bw2,cw2,dw2)
ssw3=mksys([],[],[],[])
0.4
0.2
0
Amplitude
−0.2
−0.4
−0.6
−0.8
−1
0 10 20 30 40 50 60 70 80 90 100
Time (secs)
Figure 10.17: Step responses for closed loop system with P = M, Ps or Pa and H1
controller.
% the augmented system is now generated with the command *augss*
% (sorry, it is the only command for this purpose in this toolbox...)
[tss]=augss(syg,ssw1,ssw2,ssw3)
% Controller synthesis in this toolbox is done with the routine
% *hinfopt*. Check out the help information on this routine and
% find out that we actually compute 1/gamma where gamma is
% the `usual' gamma that we use throughout the lecture notes.
[gamma,ssf,sscl]=hinfopt(tss,[1:2],[.001,1,0])
gamma=1/gamma
disp('Optimal H-infinity norm is approximately ')
disp(num2str(gamma))
(10.10)
8! : jS(j!)j < jW ;1
S (j!)j = jWe(j!)Vd(j!)j
8! : jR(j!)j < jW ;1
R (j!)j = jWu(j!)Vd(j!)j
|S|, |R| and their bounds
10 4
10 3
10 2
% Next we evaluate the robust performance of this controller
[af,bf,cf,df]=branch(ssf) % returns the controller in state space form
bode(af,bf,cf,df) pause
[as,bs,cs,ds]=tf2ss(nums,dens) % returns Ps in state space form
[aa,ba,ca,da]=tf2ss(numa,dena) % returns Pa in state space form
[alm,blm,clm,dlm]=series(af,bf,cf,df,ag,bg,cg,dg)
[als,bls,cls,dls]=series(af,bf,cf,df,as,bs,cs,ds)
[ala,bla,cla,dla]=series(af,bf,cf,df,aa,ba,ca,da)
[acle,bcle,ccle,dcle]=feedback([],[],[],1,alm,blm,clm,dlm,-1)
[aclu,bclu,cclu,dclu]=feedback(af,bf,cf,df,ag,bg,cg,dg,-1)
[acls,bcls,ccls,dcls]=feedback([],[],[],1,als,bls,cls,dls,-1)
[acla,bcla,ccla,dcla]=feedback([],[],[],1,ala,bla,cla,dla,-1)
step(acle,bcle,ccle,dcle)
hold
step(acls,bcls,ccls,dcls)
step(acla,bcla,ccla,dcla)
pause
hold off
boundR=gamma./magWR boundS=gamma./magWS
magcle=bode(acle,bcle,ccle,dcle,1,w)
magclu=bode(aclu,bclu,cclu,dclu,1,w)
loglog(w,magcle,w,magclu,w,boundR,w,boundS)
title('|S|, |R| and their bounds')
10 1
10 0
10 −1
10 −2
10 2
10 1
10 0
10 −1
10 −2
10 −3
10 −3
Figure 10.18: jSj and jRj and their bounds =jWSj resp. =jWRj.
Note that for low frequencies the sensitivity S is the limiting factor, while for high
frequencies the control sensitivity R puts the contraints. At about 1

162 CHAPTER 10. DESIGN EXAMPLE
10.5. H1 DESIGN IN MUTOOLS. 161
Planta=nd2sys(numa,dena)
systemnames='Contr Planta'
inputvar='[d]'
outputvar='[-Planta-dContr]'
input_to_Planta='[Contr]'
input_to_Contr='[-Planta-d]'
sysoutname='realclpa'
cleanupsysic='yes'
sysic
10.5 H1 design in mutools.
In the \ -analysis and synthesis toolbox", simply indicated by \Mutools", we haveplenty of freedom to de ne the structure of the augmented plant ourselves. The listing for the
example under study raketmut.m is given as:
%
% SCRIPT FILE FOR THE CALCULATION AND EVALUATION
% OF CONTROLLERS USING THE MU-TOOLBOX
%
% This script assumes that you ran the files plantdef and weights
%
% CONTROLLER AND CLOSED LOOP EVALUATION
[ac,bc,cc,dc]=unpck(Contr)
bode(ac,bc,cc,dc)
pause
[acl,bcl,ccl,dcl]=unpck(realclp)
[acls,bcls,ccls,dcls]=unpck(realclps)
[acla,bcla,ccla,dcla]=unpck(realclpa)
step(acl,bcl,ccl,dcl)
hold
pause
step(acls,bcls,ccls,dcls)
pause
step(acla,bcla,ccla,dcla)
pause
hold off
boundR=gamma./magWR
boundS=gamma./magWS
[magcl,phasecl,w]=bode(acl,bcl,ccl,dcl,1,w)
loglog(w,magcl,w,boundR,w,boundS)
title('|S| , |R| and their bounds')
% REPRESENT SYSTEM BLOCKS IN INTERNAL FORMAT
Plant=nd2sys(numm,denm)
Vd=nd2sys(numVd,denVd)
We=nd2sys(numWe,denWe)
Wu=nd2sys(numWu,denWu)
% MAKE GENERALIZED PLANT USING *sysic*
systemnames='Plant Vd We Wu'
inputvar='[dwu]'
outputvar='[WeWu-Plant-Vd]'
input_to_Plant='[u]'
input_to_Vd='[dw]'
input_to_We='[-Plant-Vd]'
input_to_Wu='[u]'
sysoutname='G'
cleanupsysic='yes'
sysic
Running this script in Matlab yields =1:337 and a controller that deviates somewhat
from the robust control toolbox controller for high frequencies ! > 103rad/s. The step
responses and sensitivities are virtually the same. It shows that the controller is not
unique as it is just a controller in the set of controllers that obey k G k1< with G
stable. As long as is not exactly minimal, the set of controllers contains more than
one controller. For MIMO-plants even for minimal the solution for the controller is not
unique. Furthermore there are aberrations due to numerical anomalies.
% CALCULATE CONTROLLER
[Contr,fclp,gamma]=hinfsyn(G,1,1,0,10,1e-4)
10.6 LMI toolbox.
% MAKE CLOSED LOOP INTERCONNECTION FOR MODEL
systemnames='Contr Plant'
inputvar='[d]'
outputvar='[-Plant-dContr]'
input_to_Plant='[Contr]'
input_to_Contr='[-Plant-d]'
sysoutname='realclp'
cleanupsysic='yes'
sysic
The \LMI toolbox" provides a very exible way for synthesizing H1 controllers. The
toolbox has its own format for the internal representation of dynamical systems which,
in general, is not compatible with the formats of other toolboxes (as usual). The toolbox
can handle parameter varying systems and has a user friendly graphical interface for the
design of weighting lers. As for the latter, we refer to the routine
magshape
The calculation of H1 optimal controllers proceeds as follows.
% Script file for the calculation of H-infinity controllers
% in the LMI toolbox. This script assumes that you ran the files
% *plantdef* and *weights* before.
% MAKE CLOSED LOOP INTERCONNECTION FOR Ps
Plants=nd2sys(nums,dens)
systemnames='Contr Plants'
inputvar='[d]'
outputvar='[-Plants-dContr]'
input_to_Plants='[Contr]'
input_to_Contr='[-Plants-d]'
sysoutname='realclps'
cleanupsysic='yes'
sysic
% MAKE CLOSED LOOP INTERCONNECTION FOR Pa

164 CHAPTER 10. DESIGN EXAMPLE
10.7. DESIGN IN MUTOOLS 163
6j
K
% FIRST REPRESENT SYSTEM BLOCKS IN INTERNAL FORMAT
Ptsys=ltisys('tf',numm,denm)
Vdsys=ltisys('tf',numVd,denVd)
Wesys=ltisys('tf',numWe,denWe)
Wusys=ltisys('tf',numWu,denWu)
5j
U
;:05
;:06
Figure 10.19: Variability of elastic mode.
% MAKE GENERALIZED PLANT
inputs = 'dwu'
outputs = 'WeWu-Pt-Vd'
Ptin='Pt : u'
Vdin='Vd : dw'
Wein='We : -Pt-Vd'
Wuin='Wu : u'
G=sconnect(inputs,outputs,[],Ptin,Ptsys,Vdin,Vdsys,...
Wein,Wesys,Wuin,Wusys)
% CALCULATE H-INFTY CONTROLLER USING LMI SOLUTION
[gamma,Ksys]=hinflmi(G,[1 1],0,1e-4)
(10.12)
(10.13)
;8(s + :125)
Pt(s) =
(s ; 1)(s +1)
(s + :055 ; :005 ; j(5:5 ; :5 ))(s + :055 ; :005 + j(5:5 ; :5 ))
(s + :055 + :005 ; j(5:5+:5 ))(s + :055 + :005 + j(5:5+:5 ))
K0
% MAKE CLOSED-LOOP INTERCONNECTION FOR MODEL
Ssys = sinv(sadd(1,smult(Ptsys,Ksys)))
Rsys = smult(Ksys,Ssys)
where the extra constant K0 is determined by Pt(0) = 1: the DC-gain is kept on 1.
If we de ne the nominal position of the poles and zeros by a0 = :055 and b0 = 5:5
rearrangement yields:
f1+ multg (10.14)
s + :125
Pt(s) =;8
s2 ; 1
; (:02s + :02a0 +2b0)
(10.15)
s 2 +(2a0 + :01 )s + a 2 0 + b2 0 + (:01a0 + b0)+:250025 2
mult = k0
(10.16)
k0 = a2 0 + b2 0 + :250025 2 + (:01a0 + b0)
a2 0 + b2 0 + :250025 2 ; (:01a0 + b0)
The factor F = 1+ mult can easily be brought into a state space description with
fA B C Dg:
% EVALUATE CONTROLLED SYSTEM
splot(Ksys,'bo',w) title('Bodeplot of controller')
pause
splot(Ssys,'sv') title('Maximal singular value of Sensitivity')
pause
splot(Ssys,'ny') title('Nyquist plot of Sensitivity')
pause
splot(Ssys,'st') title('Step response of Sensitivity')
pause
splot(Rsys,'sv') title('Maximal sv of Control Sensitivity')
pause
splot(Rsys,'ny') title('Nyquist plot of Control Sensitivity')
pause
splot(Rsys,'st') title('Step response of Control Sensitivity')
pause
10.7 design in mutools
(10.17)
0 1
A = A1 + dA =
;a2 0 ; b2 0 0
+
0 ;2a0 ; ; 2 :01
C = C1 + dC = ; 0 0 + a2 0 +b2 0 + + 2
a2 0 +b2 ;
0 ; + 2 ; ;:02
B = B1 + dB = 0 0
+
1 0
D = D1 + dD =1+0 =11:0011 =5:50055 = :250025
In -design we pretend to model the variability of the exible mode very tightly by means
of speci c parameters in stead of the rough modelling by an additive perturbation bound
by WuVd. In that way we hopetoobtain a less conservative controller. We suppose that
the poles and zeros of the exible mode shift along a straight line in complex plane between
the extreme positions of Ps and Pa as illustrated in Fig. 10.19.
Algebraically this variation can then be represented by one parameter according to:
Note that for =0we simply have F =1+ mult =1.
If we let = (s) with j (j!)j 1 we have given the parameter delta much more
freedom, but the whole description then ts with the -analysis. We havefor the dynamic
transfer F (s):
(10.11)
8 R ;1 1
poles : ;:055 ; :005 j(5:5+:5 )
zeros : ;:055 + :005 j(5:5 ; :5 )
(10.18)
sx = A1x + B1u1 + dA(s)x + dB(s)u1 dB(s) =0
y1 = C1x + D1u1 + dC(s)x + dD(s)u1 dD(s) =0
So that the total transfer of the plant including the perturbation is given by:

166 CHAPTER 10. DESIGN EXAMPLE
10.7. DESIGN IN MUTOOLS 165
=30:2502 we rewrite:
With = a 2 0 + b2 0
dA = B2(I ; D22) ;1 C2 (10.22)
dB = B2(I ; D22) ;1 D12 (10.23)
dC = D12(I ; D22) ;1 C2 (10.24)
dD = D12(I ; D22) ;1 D21 (10.25)
dC = ; 1+ ; + 2 0 (10.19)
0 0
; ; 2 ;:02
dA =
and with some patience one can derive that:
Next we can de ne 5 extra input lines in a vector u2 and correspondingly 5 extra
output lines in a vector y2 that are linked in a closed loop via u2 = y2 with:
0
@ A B1 B2
C1 D11 D12
(10.20)
1
C
A
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0
B
@
=
(10.26)
1
C
A
0 1 0 0 0 0 0 0
; ;:11 1 ; 0 0 ; ;:01
;2
0 0 1 ; 0 0 ;:02
1 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 1 ; 0 0
0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 0
0
B
@
1
A =
C2 D21 D22
and let F be represented by:
This multiplicative error structure can be embedded in the augmented plantassketched
in Fig. 10.21.
1
A (10.21)
0
@ x
u1
u2
1
A
0
@ A B1 B2
C1 D11 D12
1
A =
C2 D21 D22
0
@ _x
y1
y2
6
I5
so that we have obtained the structure according to Fig. 10.20.
6 -
~e
~d
-
Vd
-
u2
6
y2
u2
y2
? -
-
F ?
- P - - l - m - We
-
6
u1
y1
6
? -
- -
6
:0001
- ? -
G
K
?
j
F
- -
D22
-
D12
6
-
-6 -
D21
?
6 -
C2
B2
1
s I
6
y1
- B1
- h?
- - C1
- ? i -
6
6
u1
Figure 10.21: Augmented plant for -set-up.
?
A1
Note that we have skipped the weighted controller output ~u. We had no real bounds
on the actuator ranges and we actually determined Wu in the previous H1-designs such
that the additive model perturbations are covered. In this -design under study the model
perturbations are represented by the -block so that in principle we can skip Wu. If we
do so, the direct feedthrough of the augmented plant D12 has insu cient rank. We have
to penalise the input u and this is accomplished by the extra gain block with value .0001.
This weights u very lightly via the output error e. It is just su cient toavoid numerical
anomalies without in uencing substantially the intended weights.
? - -
D1
Figure 10.20: Dynamic structure of multiplicative error.
The two representations correspond according to linear fractional transformation LFT:

168 CHAPTER 10. DESIGN EXAMPLE
10.7. DESIGN IN MUTOOLS 167
blkp=[1 11 11 11 11 11 1]
[bnds1,dvec1,sens1,pvec1]=mu(clp1_g,blkp)
vplot('liv,m',vnorm(clp1_g),bnds1)
pause
Unfortunately, the -toolbox was not yet ready to process uncertainty blocks in the
form of I so that we haveto proceed with 5 independent uncertainty parameters i and
thus:
% FIRST mu-CONTROLLER
(10.27)
1
C
A
1 0 0 0 0
0 2 0 0 0
0 0 3 0 0
0 0 0 4 0
0 0 0 0 5
0
B
@
[dsysL1,dsysR1]=musynfit('first',dvec1,sens1,blkp,1,1)
mu_inc1=mmult(dsysL1,GMU,minv(dsysR1))
[k2,clp2]=hinfsyn(mu_inc1,1,1,0,100,1e-4)
clp2_g=frsp(clp2,omega)
[bnds2,dvec2,sens2,pvec2]=mu(clp2_g,blkp)
vplot('liv,m',vnorm(clp2_g),bnds2)
pause
[ac,bc,cc,dc]=unpck(k2)
bode(ac,bc,cc,dc)
pause
=
As a consequence the design will be more conservative, but the controller will become
more robust. The commands for solving this design in the -toolbox are given in the next
script:
% MAKE CLOSED LOOP INTERCONNECTION FOR MODEL
systemnames='k2 Plant'
inputvar='[d]'
outputvar='[-Plant-dk2]'
input_to_Plant='[k2]'
input_to_k2='[-Plant-d]'
sysoutname='realclp'
cleanupsysic='yes'
sysic
% Let's make the system DMULT first
alpha=11.0011 beta=30.25302
gamma=5.50055 epsilon=.250025
ADMULT=[0,1-beta,-.11]
BDMULT=[0,0,0,0,0,01,-gamma,0,0,-epsilon,-.01]
CDMULT=[0,0,1,00,00,00,00,1]
DDMULT=[1,-alpha,0,-2*alpha*gamma/beta,0,-.02 ...
0,0,0,0,0,0 ...
0,0,0,1,0,0 ...
0,1,-epsilon/beta,gamma/beta,0,0, ...
0,1,0,0,0,0 ...
0,0,0,0,0,0]
mat=[ADMULT BDMULTCDMULT DDMULT]
DMULT=pss2sys(mat,2)
% MAKE CLOSED LOOP INTERCONNECTION FOR Ps
Plants=nd2sys(nums,dens)
systemnames='k2 Plants'
inputvar='[d]'
outputvar='[-Plants-dk2]'
input_to_Plants='[k2]'
input_to_k2='[-Plants-d]'
sysoutname='realclps'
cleanupsysic='yes'
sysic
% MAKE GENERALIZED MUPLANT
systemnames='Plant Vd We DMULT'
inputvar='[u2(5)dwx]'
outputvar='[DMULT(2:6)We+.0001*x-DMULT(1)-Vd]'
input_to_Plant='[x]'
input_to_Vd='[dw]'
input_to_We='[-DMULT(1)-Vd]'
input_to_DMULT='[Plantu2(1:5)]'
sysoutname='GMU'
cleanupsysic='yes'
sysic
% MAKE CLOSED LOOP INTERCONNECTION FOR Pa
Planta=nd2sys(numa,dena)
systemnames='k2 Planta'
inputvar='[d]'
outputvar='[-Planta-dk2]'
input_to_Planta='[k2]'
input_to_k2='[-Planta-d]'
sysoutname='realclpa'
cleanupsysic='yes'
sysic
% CALCULATE HINF CONTROLLER
[k1,clp1]=hinfsyn(GMU,1,1,0,100,1e-4)
% PROPERTIES OF CONTROLLER
% Controller and closed loop evaluation
[acl,bcl,ccl,dcl]=unpck(realclp)
[acls,bcls,ccls,dcls]=unpck(realclps)
[acla,bcla,ccla,dcla]=unpck(realclpa)
step(acl,bcl,ccl,dcl)
omega=logspace(-2,3,100)
spoles(k1)
k1_g=frsp(k1,omega)
vplot('bode',k1_g)
pause
clp1_g=frsp(clp1,omega)
blk=[1 11 1 1 11 11 1]

170 CHAPTER 10. DESIGN EXAMPLE
10.7. DESIGN IN MUTOOLS 169
% Controller and closed loop evaluation
[acl,bcl,ccl,dcl]=unpck(realclp)
[acls,bcls,ccls,dcls]=unpck(realclps)
[acla,bcla,ccla,dcla]=unpck(realclpa)
step(acl,bcl,ccl,dcl)
hold
pause
step(acls,bcls,ccls,dcls)
pause
step(acla,bcla,ccla,dcla)
pause
hold off
hold
pause
step(acls,bcls,ccls,dcls)
pause
step(acla,bcla,ccla,dcla)
pause
hold off
% SECOND mu-CONTROLLER
spoles(k2)
k2_g=frsp(k2,omega)
vplot('bode',k2_g)
pause
First the H1-controller for the augmented plant is computed. The = 33:0406,
much too high. Next one is invited to choose the respective orders of the lters that
approximate the D-scalings for a number of frequencies. If one chooses a zero order, the
rst approximate = = 19:9840 and yields un Pa unstable at closed loop. A second
iteration with second order approximate lters even increases = = 29:4670 and Pa
remains unstable.
A second try with second order lters in the rst iteration brings = down to 5.4538
but still leads to an unstable Pa. In second iteration with second order lters the program
fails altogether.
Stimulated nevertheless by the last attempt we increase the rst iteration order to 3
which produces a = = 4:9184 and a Pa that just oscillates in feedback. A second
iteration with rst order lters increases the = to 21.2902, but the resulting closed
loops are all stable.
Going still higher we takeboth iterations with 4-th order lters and the = take the
respective values 4.4876 and 10.8217. In the rst iteration the Pa still shows a ill damped
oscillation, but the second iteration results in very stable closed loops for all P , Ps and
Pa. The cost is a very complicated controller of the order 4+10*4+10*4=44!
[dsysL2,dsysR2]=musynfit(dsysL1,dvec2,sens2,blkp,1,1)
mu_inc2=mmult(dsysL2,mu_inc1,minv(dsysR2))
[k3,clp3]=hinfsyn(mu_inc2,1,1,0,100,1e-4)
clp3_g=frsp(clp3,omega)
[bnds3,dvec3,sens3,pvec3]=mu(clp3_g,blkp)
vplot('liv,m',vnorm(clp3_g),bnds3)
pause
[ac,bc,cc,dc]=unpck(k3)
bode(ac,bc,cc,dc)
pause
% MAKE CLOSED LOOP INTERCONNECTION FOR MODEL
systemnames='k3 Plant'
inputvar='[d]'
outputvar='[-Plant-dk3]'
input_to_Plant='[k3]'
input_to_k3='[-Plant-d]'
sysoutname='realclp'
cleanupsysic='yes'
sysic
% MAKE CLOSED LOOP INTERCONNECTION FOR Ps
Plants=nd2sys(nums,dens)
systemnames='k3 Plants'
inputvar='[d]'
outputvar='[-Plants-dk3]'
input_to_Plants='[k3]'
input_to_k3='[-Plants-d]'
sysoutname='realclps'
cleanupsysic='yes'
sysic
% MAKE CLOSED LOOP INTERCONNECTION FOR Pa
Planta=nd2sys(numa,dena)
systemnames='k3 Planta'
inputvar='[d]'
outputvar='[-Planta-dk3]'
input_to_Planta='[k3]'
input_to_k3='[-Planta-d]'
sysoutname='realclpa'
cleanupsysic='yes'
sysic

172 CHAPTER 11. BASIC SOLUTION OF THE GENERAL PROBLEM
jsI ; A ; B2Fj =0 jsI ; A ; HC2j =0 (11.1)
The really new component istheblocktransfer Q(s) as an extra feedback operating on
the output error e. If Q =0,wejust have the stabilising LQG-controller that we will call
here the nominal controller Knom. For analysing the e ect of the extra feedback by Q, we
can combine the augmented plant and the nominal controller in a block T as illustrated
in Fig. 11.2.
- G - - G - - -
-
6
T
Knom
?
K
Q Q
-
-
6
?
-
- -
-
6 6
?
-
w z w z w z
u y u y
=
=
v e v e
?
Chapter 11
Basic solution of the general
problem
Figure 11.2: Combining Knom and G into T .
Originally, we had as optimisation criterion:
In this chapter we will present the principle of the solution of the general problem. It
o ers all the insight into the problem that we need. The computational solution follows
a somewhat di erent direction (nowadays) and will be presented in the next chapter 13.
The fundamental solution discussed here is a generalisation of the previously discussed
\internal model control" for stable systems.
The set of all stabilising controllers, also for unstable systems, can be derived from the
blockscheme in Fig. 11.1.
(11.2)
min
Kstabilising k G11 + G12K(I ; G22K) ;1 G21 k1
\Around" the stabilising controller Knom, incorporated in block T , we get a similar
criterion in terms of Tij that highly simpli es into the next a ne expression:
(11.3)
min
Qstabilising k T11 + T12QT21 k1
because T22 appears to be zero! As illustrated in Fig. 11.3, T22 is actually the transfer
between output error e and input v of Fig. 11.1. To understand that this transfer is
zero, we have to realise that the augmented plant is completely and exactly known. It
incorporates the nominal plant model P and known lters. Although the real process
may deviate and cause a model error, for all these e ects one should have taken care by
appropriately chosen lters that guard the robustness. It leaves the augmented plant fully
and exactly known. This means that the model thereafter, that is used in the nominal
controller, ts exactly. Consequently, ifw=0, the output error e, only excited by v, must
be zero! And the corresponding transfer is precisely T22. From the viewpoint ofQ: it sees
no transfer between v and e.
If T22 = 0, the consequent a ne expression in controller Q can be interpreted then
very easily as a simple forward tracking problem as illustrated in Fig. 11.3.
Because Knom stabilised the augmented plant for Q = 0, we can be sure that all
transfers Tij will be stable. But then the simple forward tracking problem of Fig. 11.3
can only remain stable, if Q itself is a stable transfer. As a consequence we nowhavethe set of all stabilising controllers by just choosing Q stable. This set is then clustered on
the nominal controller Knom, de ned by F and H, and certainly the ultimate controller
Figure 11.1: Solution principle
The upper major block represents the augmented plant. For reasons of clarity, wehave
skipped the direct feedthrough block D. The lower, major block can easily be recognised
as a familiar LQG-control where F is the state feedback control block and H functions
as a Kalman gain block. Neither F nor H have to be optimal yet, as long as they cause
stable poles from:
171

174 CHAPTER 11. BASIC SOLUTION OF THE GENERAL PROBLEM
173
(The remainder of this chapter might poseyou to some problems, if you are not well introduced into \functional analysis".
Then just try to make the best out of it as it is only one page.)
It appears that we can now use the freedom, left in the choices for F andH, and it can
be proved that F and H can be chosen (for square transfers) such that :
T12 T12 = I (11.5)
T21 T21 = I (11.6)
In mathematical terminology these matrices are therefore called inner, while engineers
prefer to denote them as all pass transfers. These transfers all possess poles in the left
half plane and corresponding zeros in the right half plane exactly symmetric with respect
to the imaginary axis. If the norm is restricted to the imaginary axis, which is the case
for the 1-norm and the 2-norm, we maythus freely multiply by the conjugated transpose
of these inners and obtain:
Figure 11.3: Resulting forward tracking problem.
min
Qstable k T11 + T12QT21 k= min
Qstable k T12 T11T21 + T12 T12QT21T21 k= (11.7)
k L + Q k (11.8)
def
= min
Qstable
K can be expressed in the \parameter" Q. This expression, which we will not explicitly
give here for reasons of compactness, is called the Youla parametrisation after its inventor.
This is the momenttostepbackfor a moment and memorise the internal model control
where we were also dealing with a comparable transfer Q. Once more Fig. 11.4 pictures
that structure with comparable signals v and e.
By the conjugation of the inners into Tij ,wehavee ectively turned zeros into poles
and vice versa, thereby causing that all poles of L are in the right halfplane. For the
norm along the imaginary axis there is no objection but more correctly we have to say
now that we deal with the L1 and the L2 spaces and norms. As outlined in chapter 5 the
(Lebesque) space L1 combines the familiar (Hardy) space H1 of stable transfers and the
complementary H ; 1 space, containing the antistable or anticausal transfers that have all
? l
d +
+
-
+
Pt
? +
m
-
;
P
-
e v 6
- l - Q -
+
6;
? -
r
their poles in the right half plane. Transfer L is such a transfer. Similarly the space L2
consists of both the H2 and the complementary space H ? 2 of anticausal transfers. The
question then arises, how to approximate an anticausal transfer L by a stable, causal Q in
the complementary space where the approximation is measured on the imaginary axis by
the proper norm. The easiest solution is o ered in the L2 space, because this is a Hilbert
space and thus an inner product space which implies that H2 and H ? 2 are perpendicular
(that induced the symboling). Consequently Q is perpendicular to L and can thus never
\represent" a componentofL in the used norm and will thus only contribute to an increase
of the norm unless it is taken zero. So in the 2-norm the solution is obviously: Q =0.
Unfortunately, for the space L1, where we are actually interested in, the solution is
not so trivial, because L1 is a Banach space and not an inner product space. This famous
problem :
?
Figure 11.4: Internal model control structure.
Indeed, for P = Pt and the other external input d (to be compared with w) being zero,
thetransferseenby Q between v and e is zero. Furthermore, we also obtained, as a result
of this T22 = 0, a ne expressions for the other transfers Tij, being the bare sensitivity
and complementary sensitivity by then. So the internal model control can be seen as a
particular application of a much more general scheme that we study now. In fact Fig. 11.1
turns into the internal model of Fig. 11.4 by choosing F =0andH =0,whichisallowed,
because P and thus G is stable.
The remaining problem is:
(11.9)
k L + Q k1
L 2H ; 1 : min
Q2H1
has been given the name Nehari problem to the rst scientist, studying this problem. It
took considerable time and energy to nd solutions one of which is o ered to you in chapter
8, as being an elegant one.But maybe you already got some taste here of the reasons why
it took so long to formalise classical control along these lines. As nal remarks we can
add:
(11.4)
min
Qstable k T11 + T12QT21 k1
Generically minQ2H1 (L + Q) is all pass i.e. constant for all ! 2R. T12 and T21
were already taken all pass, but also the total transfer from w to z viz. T11+T12QT21
will be all pass for the SISO case, due to the waterbed e ect.
Note that the phrase Qstabilising is now equivalent with Qstable! Furthermore we
may aswell take other norms provided that the respective transfers live in the particular
normed space. We could e.g. translate the LQG-problem in an augmented plant and then
require to minimise the 2-norm in stead of the 1-norm. As Tij and Q are necessarily stable
they live inH2 as well so that we can also minimise the 2-norm for reasons of comparison.(
If there is a direct feed through block D involved, the 2-norm is not applicable, because a
constant transfer is not allowed in L2.)

176 CHAPTER 11. BASIC SOLUTION OF THE GENERAL PROBLEM
175
11.1 Exercises
7.1:Consider the following feedback system:
For MIMO systems the solution is not unique, as we just consider the maximum
singular value. The freedom in the remaining singular values can be used to optimise
extra desiderata.
Plant: y = P (u + d)
Controller: u = K(r ; y)
Errors: e1 = W1u and e2 = W2(r ; y)
It is known that k r k2 1 and k d k2 1, and it is desired to design K so as to minimise:
k2
k e1
e2
a) Show that this can be formulated as a standard H1problem and compute G.
b) If P is stable, rede ne the problem a ne in Q.
7.2: Take the rst blockscheme of the exercise of chapter 6. To facilitate the computations
we just consider a SISO-plant and DC-signals (i.e. only for ! = 0!) so that we avoid
complex computations due to frequency dependence. If there is given that k k2< 1 and
P =1=2 then it is asked to minimise k ~y k2 under the condition k x k2< 1 while is the
only input.
a) Solve this problem by means of a mixed sensitivity problem iteratively adapting Wy
renamed as . Hint: First de ne V and Wx. Sketch the solution in terms of controller
C and compute the solution directly as a function of .
b) Solve the problem exactly: minimise k ~y k2 while k x k2< 1. Why is there a di erence
with the solution sub a) ? Hint: For this question it is easier to de ne the problem
a ne in Q.

178 CHAPTER 12. SOLUTION TO THE GENERAL H1 CONTROL PROBLEM
control algorithms, which we brie y describe in a separate section and which you are
probably familiar with.
12.2 The computation of system norms
We start this chapter by considering the problem of characterizing the H2 and H1 norms
of a given (multivariable) transfer function H(s) in terms of a state space description of
the system. We will consider the continuous time case only for the discrete time versions
of the results below are less insightfull and more involved.
Let H(s) beastable transfer function of dimension p m and suppose that
Chapter 12
H(s) =C(Is; A) ;1 B + D
Solution to the general H1 control
problem
where A B C and D are real matrices de ning the state space equations
(12.1)
_x(t) =Ax(t)+Bw(t)
z(t) =Cx(t)+Dw(t):
12.1 Introduction
Since H(s) is stable, all eigenvalues of A are assumed to be in the left half complex plane.
We suppose that the state space has dimension n and, to avoid redundancy, we moreover
assume that (12.1) de nes a minimal state space representation of H(s) in the sense that
n is as small as possible among all state space representations of H(s).
Let us recall the de nitions of the H2 and H1 norms of H(s):
k H(s) k2 Z 1
2 := 1=2 trace(H(j!)H (j!))d!
;1
k H(s) k1 := sup
!2R (H(j!))
where denotes the maximal singular value.
12.2.1 The computation of the H2 norm
We haveseen in Chapter 5 that the (squared) H2 norm of a system has the simple interpretation
as the sum of the (squared) L2 norms of the impulse responses which we can
extract from (12.1). If we assume that D = 0 in (12.1) (otherwise the H2 norm is in nite
so H =2H2) and if bi denotes the i-th column of B, then the i-th impulse response is given
by
In previous chapters we havebeen mainly concerned with properties of control con gurations
in which a controller is designed so as to minimize the H1 norm of a closed loop
transfer function. So far, we did not address the question how such a controller is actually
computed. This has been a problem of main concern in the early 80-s. Various mathematical
techniques have been developed to compute `H1-optimal controllers', i.e., feedback
controllers which stabilize a closed loop system and at the same time minimize the H1
norm of a closed loop transfer function. In this chapter we treat a solution to a most
general version of the H1 optimal control problem which is now generally accepted to
be the fastest, simplest, and computationally most reliable and e cient way to synthesize
H1 optimal controllers.
The solution which we present here is the result of almost a decenium of impressive
research e ort in the area of H1 optimal control and has received widespread attention
in the control community. An amazing number of scienti c papers have appeared (and
still appear!) in this area of research. In this chapter we will treat a solution of the
general H1 control problem which popularly is referred to as the `DGKF-solution', the
acronym standing for Doyle, Glover, Khargonekar and Francis, four authors of a famous
and prize winning paper in the IEEE Transactions on Automatic Control1 . From a mathematical
and system theoretic point of view, this so called `state space solution' to the
H1 control problem is extremely elegant and worth a thorough treatment. However, for
practical applications it is su cient to know the precise conditions under which the state
space solution `works' so as to have a computationally reliable way to obtain and to design
H1 optimal controllers. The solution presented in this chapter admits a relatively
straightforward implementation in a software environment likeMatlab. The Robust Control
Toolbox hasvarious routines for the synthesis of H1 optimal controllers and we will
devote a section in this chapter on how to use these routines.
This chapter is organized as follows. In the next section we rst treat the problem
of how to compute the H2 norm and the H1 norm of a transfer function. These results
will be used in subsequent sections, where we present the main results concerning H1
controller synthesis in Theorem 12.7. We will make a comparison to the H2 optimal
zi(t) =Ce At bi:
Since H(s) has m inputs, we have m of such responses, and for i = 1::: m, their L2
norms satisfy
k zi k2 Z 1
2 = b
0
T
i eAT t T At
C Ce bidt
= b T
Z 1
i e
0
ATt T At
C Ce dtbi
= b T
i Mbi:
Z 1
Here, we de ned
e AT t C T Ce At dt
M :=
1 \State Space Solutions to the Standard H2 and H1 Control Problems", by J.Doyle, K. Glover, P.
Khargonekar and B. Francis, IEEE Transactions on Automatic Control, August 1989.
0
177

180 CHAPTER 12. SOLUTION TO THE GENERAL H1 CONTROL PROBLEM
12.2. THE COMPUTATION OF SYSTEM NORMS 179
12.2.2 The computation of the H1 norm
which is a square symmetric matrix of dimension n n which is called the observability
gramian of the system (12.1). Since xTMx 0 for all x 2Rn we have that M is nonnegative
de nite 2 . In fact, the observability gramian M satis es the equation
The computation of the H1 norm of a transfer function H(s) is slightly more involved.
We will again present an algebraic algorithm, but instead of nding an exact expression
for k H(s) k1, we will nd an algebraic condition whether or not
MA+ A T M + C T C =0 (12.2)
k H(s) k1 < (12.4)
for some real number 0. Thus, we will set up a test so as to determine whether (12.4)
holds for certain value of 0. By performing this test for various values of we may
get arbitrarily close to the norm k H(s) k1.
We will brie y outline the main ideas behind this test. Recall from Chapter 5, that
the H1 norm is equal to the L2 induced norm of the transfer function, i.e.,
k Hw k2
k H(s) k1 = sup :
w2L2 k w k2
This means that k H(s) k1 if and only if
k Hw k2 2 ; 2 k w k2 2=k z k2 2 ; 2 k w k2 0: (12.5)
which is called a Lyapunov equation in the unknown M. Since we assumed that the state
space parameters (A B C D) de ne a minimal representation of the transfer function
H(s), the pair (A C) isobservable3 , and the matrix M is the only symmetric non-negative
de nite solution of (12.2). Thus, M can be computed from an algebraic equation, the
Lyapunov equation (12.2), which isamuch simpler task than solving the in nite integral
expression for M.
The observability gramian M completely determines the H2 norm of the system H(s)
as is seen from the following characterization.
Theorem 12.1 Let H(s) be a stable transfer function of the system described by the
state space equations (12.1). Suppose that (A B C D) is a minimal representation of
H(s). Then
for all w 2 L2. (Indeed, dividing (12.5) by k w k2 2 gives you the equivalence). Here,
z = Hw is the output of the system (12.1) when the input w is applied and when the
initial state x(0) is set to 0.
Now, suppose that 0 and the system (12.1) is given. Motivated by the middle
expression of (12.5) weintroduce for arbitrary initial conditions x(0) = x0 and any w 2L2,
the criterion
1. k H(s) k2 < 1 if and only if D =0.
2. If M is the observability gramian of (12.1) then
mX
b T i Mbi:
k H(s) k 2 2= trace(B T MB)=
(12.6)
J(x0w):=kz k2 2 ; 2 k w k2 2
Z 1
= jz(t)j
0
2 ; 2jw(t)j2 dt
i=1
where z is the output of the system (12.1) when the input w is applied and the initial
state x(0) is taken to be x0.
For xed initial condition x0 we willbeinterested in maximizing this criterion over all
possible inputs w. Precisely, for xed x0, we look for an optimal input w 2L2 such that
Thus the H2 norm of H(s) isgiven by a trace formula involving the state space matrices
A B C, from which the observability gramian M is computed. The main issue here is
that Theorem 12.1 provides an algebraic characterization of the H2 norm which proves
extremely useful for computational purposes.
There is a `dual' version of theorem 12.1 which is obtained from the fact that k
H(s) k2=k H (s) k2. We state it for completeness
J(x0w) J(x0w ) (12.7)
for all w 2L2. We will moreover require that the state trajectory x(t) generated by this
so called worst case input is stable in the sense that the solution x(t) of the state equation
_x = Ax + Bw with x(0) = x0 satis es limt!0 x(t) =0.
The solution to this problem is simpler than it looks. Let us take > 0 such that
2 T I ; D D is positive de nite (and thus invertible) and introduce the following Riccati
Theorem 12.2 Under the same conditions as in theorem 12.1,
k H(s) k 2 2=trace(CWC T )
equation
where W is the unique symmetric non-negative de nite solution of the Lyapunov equation
A T K + KA +(B T K ; D T C) T [ 2 I ; D T D] ;1 (B T K ; D T C)+C T C = 0: (12.8)
AW + WA T + BB T =0: (12.3)
It is then a straightforward exercise in linear algebra4 to verify that for any real symmetric
solution K of (12.8) there holds
(12.9)
J(x0w)=x T
0 Kx0;kw +[ 2 I ; D T D] ;1 (B T K ; D T C)x k2 ( 2I;DTD) The square symmetric matrix W is called the controllability gramian of the system (12.1).
Theorem 12.2 therefore states that the H2 norm of H(s) can also be obtained by computing
the controllability gramian associated with the system (12.1).
4 A `completion of the squares' argument. If you are interested, work out the derivative d
dt xT (t)Kx(t)
using (12.1), substitute (12.8) and integrate over [0 1) to obtain the desired expression (12.9).
2which isnot thesameassaying that all elements of M are non-negative!!!
3 At that is, Ce x0 =0forallt 0 only if the initial condition x0 =0.

182 CHAPTER 12. SOLUTION TO THE GENERAL H1 CONTROL PROBLEM
12.2. THE COMPUTATION OF SYSTEM NORMS 181
OUTPUT: the value approximating the H1 norm of H(s) within .
The second step of this algorithm involves the investigation of the existence of stabilizing
solutions of (12.8), which is a standard routine in Matlab. We will not go into the details of
an e cient algebraic implementation of the latter problem. What is of crucial importance
here, though, is the fact that the computation of the H1 norm of a transfer function (just
like the computation of the H2 norm of a transfer function) has been transformed to an
algebraic problem. This implies a fast and extremely reliable way to compute these system
norms.
for all w 2 L2 which drive the state trajectory to zero for t ! 1. Here, k f k2 Q with
Q = QT > 0 denotes the `weighted' L2 norm
k f k2 Z 1
Q := f T (t)Qf(t)dt: (12.10)
0
Now, have a look at the expression (12.9). It shows that for all w 2 L2, (for which
limt!1 x(t) = 0) the criterion J(x0w) is at most equal to xT 0 Kx0, and equality is obtained
by substituting for w the state feedback
12.3 The computation of H2 optimal controllers
w (t) =;[ 2 I ; D T D] ;1 (B T K ; D T C)x(t) (12.11)
The computation of H2 optimal controllers is not a subject of this course. In fact, H2
optimal controllers coincide with the well known LQG controllers which some of you may
be familiar with from earlier courses. However, for the sake of completenes we treat the
controller structure of H2 optimal controllers once more in this section.
We consider the general control con guration as depicted in Figure 13.1. Here,
which then maximizes J(x0w) over all w 2 L2. This worst case input achieves the
inequality (12.7) (again, provided the feedback (12.11) stabilizes the system (12.1)). The
only extra requirement for the solution K to (12.8) is therefore that the eigenvalues
fA +[ 2 I ; D T D] ;1 (B T K ; D T C)g C ;
-
z
-
w
G
-
all lie in the left half complex plane. The latter is precisely the case when the solution K to
(12.8) is non-negative de nite and for obvious reasons we call such a solution a stabilizing
solution of (12.8). One can show that whenever a stabilizing solution K of (12.8) exists,
it is unique. So there exists at most one stabilizing solution to (12.8).
For a stabilizing solution K, wethus obtain that
u y
J(x0w) J(x0w ) = x T
0 Kx0
K
for all w 2L2. Now, taking x0 = 0 yields that
0
J(0w)=k z k 2 2 ; 2 k w k 2 2
Figure 12.1: General control con guration
for all w 2 L2. This is precisely (12.5) and it follows that k H(s) k1 . These
observations provide the main idea behind the proof of the following result.
w are the exogenous inputs (disturbances, noise signals, reference inputs), u denote the
control inputs, z is the to be controlled output signal and y denote the measurements.
The generalized plant G is supposed to be given, whereas the controller K needs to be
designed. Admissable controllers are all linear time-invariant systems K that internally
stabilize the con guration of Figure 13.1. Every such admissible controller K gives rise
to a closed loop system which maps disturbance inputs w to the to-be-controlled output
variables z. Precisely, if M denotes the closed-loop transfer function M : w 7! z, then
with the obvious partitioning of G,
Theorem 12.3 Let H(s) be represented by the (minimal) state space model (12.1). Then
1. k H(s) k1 < 1 if and only if eigenvalues (A) C ;
2. k H(s) k1 < if and only if there exists a stabilizing solution K of the Riccati
equation (12.8).
How does this result convert into an algorithm to compute the H1 norm of a transfer
function? The following bisection type of algorithm works in general extremely fast:
M = G11 + G12K(I ; G22K) ;1 G21:
The H2 optimal control problem is formalized as follows
Algorithm 12.4 INPUT: stopping criterion " > 0 and two numbers l h satisfying
l < k H(s) k1 < h.
Synthesize a stabilizing controller K for the generalized plant G such that
k M k2 is minimal.
Step 1. Set =( l + h)=2.
Step 2. Verify whether (12.8) admits a stabilizing solution.
The solution of this important problem is split into two independent problems and makes
use of a separation structure:
Step 3. If so, set h = . If not, set l = .
Step 4. Put " = h ; l
First, obtain an \optimal estimate" ^x of the state variable x, based on the measurements
y.
Step 5. If " " then STOP, elsogo to Step 1.

184 CHAPTER 12. SOLUTION TO THE GENERAL H1 CONTROL PROBLEM
12.3. THE COMPUTATION OF H2 OPTIMAL CONTROLLERS 183
The resulting state space description then is:
1
C
A
Second, use this estimate ^x as if the controller would have perfect knowledge of the
full state x of the system.
v 0 B
Q 1
2 0 0 0
A R 1
2
1
A =
G =
0 0 0 R 1
2
0
B
@
0
@ A B1 0 B2
C1 0 0 D
C2 0 I 0
0
C 0 R 1
2
w
As is well known, the Kalman lter is the optimal solution to the rst problem and the
state feedback linear quadratic regulator is the solution to the second problem. We will
devote a short discussion on these two sub-problems.
Let the transfer function G be described in state space form by the equations
_x = Ax + B1w1 + B2u
8
><
The celebrated Kalman lter is a causal, linear mapping taking the control input u
and the measurements y as its inputs, and producing an estimate ^x of the state x in such
away thattheH2 norm of the transfer function from the noise w to the estimation error
e = x ; ^x is minimal. Thus, using our deterministic interpretation of the H2 norm of a
transfer function, the Kalman lter is the optimal lter in the con guration of Figure 12.3
for which theL2 norm of the impulse response of the estimator Me : w 7! e is minimized.
It is implemented as follows.
(12.12)
z = C1x + Du
>:
y = C2x + w2
x
-
+ h -
; 6^x
e
z (not - used)
-
w
-
Plant
-
u
Filter
-
y
u
where the disturbance input w = ; w1
w2 is assumed to be partitioned in a component w1
acting on the state (the process noise) and an independent component w2 representing
measurement noise. In (12.12) we assume that the system G has no direct feedthrough
in the transfers w ! z (otherwise M =2 H2) and u ! y (mainly to simplify the formulas
below). We further assume that the pair (A C2) is detectable and that the pair (A B2)
is stabilizable. The latter two conditions are necessary to guarantee the existence of
stabilizing controllers. All these conditions are easy to grasp if we compare the set of
equations (12.12) with the LQG-problem de nition as proposed e.g. in the course \Modern
Control Theory":
Consider Fig.12.2
w2
6~x
w1
6~u
Figure 12.3: The Kalman lter con guration
?
?
R 1
2
w
Q 1
2
R 1
2
v
R 1
2
Theorem 12.5 (The Kalman lter.) Let the system (12.12) be given and assume that
the pair (A C2) is detectable. Then
B [sI ; A] ;1
v
6
w
?
- - n - x
? y
- C - n -
6
u
1. the optimal lter which minimizes the H2 norm of the mapping Me : w ! e in the
con guration of Figure 12.3is given by the state space equations
d^x
dt (t) =A^x(t)+B2u(t)+H(y(t) ; C2^x(t)) (12.13)
=(A ; HC2)^x(t)+B2u(t)+Hy(t) (12.14)
Figure 12.2: The LQG problem.
(12.15)
where H = YCT 2 and Y is the unique square symmetric solution of
0=AY + YA T ; YC T 2 C2Y + B1B T 1
which has the property that (A ; HC2) C ;
.
where v and w are independent, white, Gaussian noises of variance respectively Rv and
Rw. They represent the direct state disturbance and the measurement noise. In order to
cope with the requirement of the equal variances of the inputs they are inversely scaled
1
1
; ; 2
2
by blocks R v and R w to obtain inputs w1 and w2 that have unit variances. The output
of this augmented plant is de ned by:
!
2. The minimal H2 norm of the transfer Me : w 7! e is given by k Me k2 2 = trace Y .
~x
~u
=
1
Q 2 x
R 1
2 u
z =
The solution Y to the Riccati equation (12.15) or the gain matrix H = YCT 2 are sometimes
referred to as the Kalman gain of the lter (12.13). Note that Theorem 12.5 is put
completely in a deterministic setting: no stochastics are necessary here.
For our second sub-problem we assume perfect knowledge of the state variable. That
is, we assume that the controller has access to the state x of (12.12) and our aim is to nd
a state feedback control law of the form
in order to accomplish that
k z k2 Z 1
2= fx
0
T Qx + u T Rugdt
(compare the forthcoming equation (12.16)). The other inputs and outputs are given by:
u(t) =Fx(t)
u = u y = y:
w = w1
w2

186 CHAPTER 12. SOLUTION TO THE GENERAL H1 CONTROL PROBLEM
12.3. THE COMPUTATION OF H2 OPTIMAL CONTROLLERS 185
K(s)
y
^x
Regulator Filter
u
u
such thattheH2 norm of the state controlled, closed-loop, transfer function Mx : w 7! z
is minimized. In this sub-problem the measurements y and the measurement noise w2
evidently do not play arole. Since k Mx k2 is equal to the L2 norm of the corresponding
impulse response, our aim is therefore to nd a control input u which minimizes the
i dt (12.16)
h
T T
x (t)C1 C1x(t)+2u T (t)D T C1x(t)+u T (t)D T Du(t)
Z 1
criterion
k z k 2 2 =
0
Figure 12.4: Separation structure for H2 optimal controllers
12.4 The computation of H1 optimal controllers
subject to the system equations (12.12). Minimization of equation 12.16 yields the so
called quadratic regulator and supposes only an initial value x(0) and no inputs w. The
solution is independent oftheinitialvalue x(0) and thus such an initial value can also be
accomplished by dirac pulses on w1. This is similar to the equivalence of the quadratic
regulator problem and the stochastic regulator problem as discussed in the course \Modern
Control Theory". The nal solution is as follows:
In this section we will rst present the main algorithm behind the computation of H1
optimal controllers. From Section 12.2 we learned that the characterization of the H1
norm of a transfer function is expressed in terms of the existence of a particular solution
to an algebraic Riccati equation. It should therefore not be a surprise6 to see that the
computation of H1 optimal controllers hinges on the computation of speci c solutions of
Riccati equations. In this section we present the main algorithm and we will resist the
temptation to go into the details of its derivation. The background and the main ideas
behind the algorithms are very similar to the ideas behind the derivation of Theorem 12.3
and the cost criterion (12.6). We defer this background material to the next section.
We consider again the general control con guration as depicted in Figure 13.1 with
the same interpretation of the signals as given in the previous section. All variables may
be multivariable. The block G denotes the \generalized system" and typically includes
a model of the plant P together with all weighting functions which are speci ed by the
`user'. The block K denotes the \generalized controller" and includes typically a feedback
controller and/or a feedforward controller. The block G contains all the `known' features
(plant model, input weightings, output weightings and interconnection structures), the
block K needs to be designed. Admissable controllers are all linear, time-invariant systems
K that internally stabilize the con guration of Figure 13.1. Every such admissible
controller K gives rise to a closed loop system which maps disturbance inputs w to the tobe-controlled
output variables z. Precisely, ifM denotes the closed-loop transfer function
M : w 7! z, then with the obvious partitioning of G,
Theorem 12.6 (The state feedback regulator.) Let the system (12.12) be given and
assume that (A B2) is stabilizable. Then
1. the optimal state feedback regulator which minimizes the H2 norm of the transfer
Mx : w ! z is given by
u(t) =Fx(t) =;[D T D] ;1 (B T 2 X + DT C1)x(t) (12.17)
where X is the unique square symmetric solution of
0=A T X + XA ; (B T 2 X + DTC1) T [D T D] ;1 (B T 2 X + DTC1)+C T 1 C1 (12.18)
which has the property that (A ; B2F ) C ;
.
2. The minimal H2 norm of the transfer Mx : w 7! z is given by k R k2 2 = trace BT 1 XB1.
The result of Theorem 12.6 is easily derived by using a completion of the squares
argument applied for the criterion (12.16). If X satis es the Riccati equation (12.18) then
a straightforward exercise in rst-years-linear-algebra gives you that
k z k2 2 = x T
0 Xx0+ k u ; [D T D] ;1 (B T
2 X + D T C1)x kDTD M = G11 + G12K(I ; G22K) ;1 G21
and the H1 control problem is formalized as follows
where X is the unique solution of the Riccati equation (12.18) and where we used the
notation of (12.10). From the latter expression it is immediate that k z k2 is minimized if
u is chosen as in (12.17).
The optimal solution of the H2 optimal control problem is now obtained by combining
the Kalman lter with the optimal state feedback regulator. The so called certainty
equivalence principle or separation principle 5 implies that an optimal controller K which
minimizes k M(s) k2 is obtained by
Synthesize a stabilizing controller K such that
k M k1 <
replacing the state x in the state feedback regulator (12.17) by the Kalman lter
estimate ^x generated in (12.13).
for some value of >0. 7
Note that already at this stage of formalizing the H1 control problem, we can see
that the solution of the problem is necessary going to be of a `testing type'. The synthesis
algorithm will require to
The separation structure of the optimal H2 controller is depicted in Figure 12.4. In
equations, the optimal H2 controller K is represented in state space form by
( d^x
(12.19)
dt (t) =(A + B2F ; HC2)^x(t)+Hy(t)
u(t) = F ^x(t)
6
Although it took about ten years of research!
7Strictly speaking, this is a suboptimal H1 control problem. The optimal H1 control problem amounts
to minimizing k M(s) k1 over all stabilizing controllers K. Precisely, if 0 := inf K k M(s) k1then
stabilizing
the optimal control problem is to determine 0 and an optimalK that achieves this minimal norm. However,
this problem isvery hard to solve in this general setting.
where the gains H and F are given as in Theorem 12.5 and Theorem 12.6.
5 The word `principle' is an incredible misnamer at this place for a result which requires rigorous mathematical
deduction.

188 CHAPTER 12. SOLUTION TO THE GENERAL H1 CONTROL PROBLEM
12.4. THE COMPUTATION OF H1 OPTIMAL CONTROLLERS 187
The synthesis of H1 suboptimal controllers is based on the following two Riccati
equations
Choose a value of >0.
See whether there exist a controller K such thatk M(s) k1 < .
(12.22)
1 C1
0=A T X + XA ; X[B2B T
2 ; ;2 B1B T
1 ]X + C T
If yes, then decrease . If no, then increase .
0=AY + YA T ; Y [C T
2 C2 ; ;2 C T
1 C1]Y + B1B T
1 : (12.23)
Observe that these de ne quadratic equations in the unknowns X and Y . The unknown
matrices X and Y are symmetric and have both dimension n n where n is the dimension
of the state space of (12.21). The quadratic terms are inde nite in both equations (both
quadratic terms consist of the di erence of two non-negative de nite matrices), and we
moreover observe that both equations (and hence their solutions) depend on the value of .
We will be particularly interested in the so called stabilizing solutions of these equations.
We call a symmetric matrix X a stabilizing solution of (12.22) if the eigenvalues
(A ; B2B T
2 X + ;2 B1B T ;
1 X) C :
Similarly, a symmetric matrix Y is called a stabilizing solution of (12.23) if
(A ; YC2C T
2 + ;2 YC T
1 C1) ; C :
To solve this problem, consider the generalized system G and let
_x = Ax + B1w + B2u
(12.20)
z = C1x + D11w + D12u
8
><
>:
y = C2x + D21w + D22u
be a state space description of G. Thus,
G11(s) =C1(Is; A) ;1 B1 + D11 G12(s) =C1(Is; A) ;1 B2 + D12
G21(s) =C2(Is; A) ;1 B1 + D21 G22(s) =C2(Is; A) ;1 B2 + D22:
With some sacri ce of generality we make the following assumptions.
A-1 D11 = 0 and D22 =0.
It can be shown that whenever stabilizing solutions X or Y of (12.22) or (12.23) exist,
then they are unique. In other words, there exists at most one stabilizing solution X of
(12.22) and at most one stabilizing solution Y of (12.23). However, because these Riccati
equations are inde nite in their quadratic terms, it is not at all clear that stabilizing
solutions in fact exist. The following result is the main result of this section, and has been
considered as one of the main contributions in optimal control theory during the last 10
years. 8
A-2 The triple (A B2C2) is stabilizable and detectable.
A-3 The triple (A B1C1) is stabilizable and detectable.
A-4 DT 12 (C1 D12) =(0 I).
DT 21 )=(0 I).
A-5 D21(BT 1
Theorem 12.7 Under the conditions A-1{A-5, there exists an internally stabilizing controller
K that achieves
k M(s) k1 <
if and only if
1. Equation (12.22) has a stabilizing solution X = X T 0.
2. Equation (12.23) has a stabilizing solution Y = Y T 0.
Assumption A-1 states that there is no direct feedthrough in the transfers w 7! z and
u 7! y. The second assumption A-2 implies that we assume that there are no unobservable
and uncontrollable unstable modes in G22. This assumption is precisely equivalent to saying
that internally stabilizing controllers exist. Assumption A-3 is a technical assumption
made on the transfer function G11. Assumptions A-4 and A-5 are just scaling assumptions
that can be easily removed, but will make all formulas and equations in the remainder of
this chapter acceptably complicated. Assumption A-4, simply requires that
k z k2 Z 1
2= jC1x + D12uj2 Z 1
dt = (x T C T
1 C1x + u T u)dt:
3. (XY ) < 2 .
0
0
Moreover, in that case one such controller is given by
(
_ ;2 =(A + B1BT 1 X) + B2u + ZH(C2 ; y)
(12.24)
u = F
In the to-be controlled output z, we thus have a unit weight on the control input signal
u, a weight C T
1 C1 on the state x and a zero weight on the cross terms involving u and
x. Similarly, assumption A-5 claims that state noise (or process noise)is independent of
measurement noise. With assumption A-5 we can partition the exogenous noise input w
as w = ; w1
w2 where w1 only a ects the state x and w2 only a ects the measurements y.
The foregoing assumptions therefore require our state space model to take the form
where
F := ;B T 2 X
H := YC T 2
Z := (I ; ;2 YX) ;1
_x = Ax + B1w1 + B2u
(12.21)
z = C1x + D12u
8
><
>:
y = C2x + w2
8 We hope you like it:::
where w = ; w1
w2 .

190 CHAPTER 12. SOLUTION TO THE GENERAL H1 CONTROL PROBLEM
12.4. THE COMPUTATION OF H1 OPTIMAL CONTROLLERS 189
Step 2. Let := ( l + h)=2 and verify whether there exists matrices X = X T and
Y = Y T satisfying the conditions 1{3 of Theorem 12.7.
A few crucial observations need to be made.
Step 3. If so, then set h = . If not, then set l = .
Step 4. Put " = h ; l.
Theorem 12.7 claims that three algebraic conditions need to be checked before we
can conclude that there exists a stabilizing controller K which achieves that the
closed loop transfer function M has H1 norm less than . Once these conditions
are satis ed, one possible controller is given explicitly by the equations (12.24) which
we putinobserver form.
Step 5. If ">"then go to Step 2.
Step 6. Put = h and let
(
_ ;2 =(A + B1B T
1 X + B2F + ZHC2) ; ZHy
u = F
de ne the state space equations of a controller K(s).
Note that the dynamic order of this controller is equal to the dimension n of the
state space of the generalized system G. Incorporating high order weighting lters
in the internal structure of G therefore results in high order controllers, which may
be undesirable. The controller (12.24) has the block structure as depicted in Figure
12.5. This diagram shows that the controller consists of a dynamic observer
which computes a state vector on the basis of the measurements y and the control
input u and a memoryless feedback F which maps to the control input u.
OUTPUT: K(s) de nes a stabilizing controller which achieves k M(s) k1 < .
12.5 The state feedback H1 control problem
It is interesting to compare the Riccati equations of Theorem 12.7 with those which
determine the H2 optimal controller. In particular, we emphasize that the presence
of the inde nite quadratic terms in (12.22) and (12.23) are a major complication to
guarantee existence of solutions to these equations. If we let ! 1 we see that
the inde nite quadratic terms in (12.22) and (12.23) become de nite in the limit
and that in the limit the equations (12.22) and (12.23) coincide with the Riccati
equations of the previous section.
The results of the previous section can not fully be appreciated if no further system
theoretic insight is given in the main results. In this section we will treat the state
feedback H1 optimal control problem, which is a special case of Theorem 12.7 and which
provides quite some insight in the structure of optimal H1 control laws.
In this section we will therefore assume that the controller K(s) has access to the full
state x, i.e., we assume that the measurements y = x and we wish to design a controller
K(s) for which the closed loop transfer function, alternatively indicated here by Mx :
w 7! z satis esk Mx k1 < . The procedure to obtain such a controller is basically an
interesting extension of thearguments we put forward in section 12.2.
The criterion (12.6) de ned in section 12.2 only depends on the initial condition x0
of the state and the input w of the system (12.1). Since we are now dealing with the
system (12.21) with state measurements (y = x) and two inputs u and w, we should treat
the criterion
K(s)
y
u
u
F H1 lter
Figure 12.5: Separation structure for H1 controllers
(12.26)
J(x0uw) := k z k2 2 ; 2 k w k2
Z 1
= jz(t)j
0
2 ; 2jw(t)j2 dt
A transfer function K(s) of the controller is easily derived from (12.24) and takes the
explicit state space form
as a function of the initial state x0 and both the control inputs u as well as the disturbance
inputs w. Here z is of course the output of the system (12.21) when the inputs u and w
are applied and the initial state x(0) is taken to be x0.
We will view the criterion (12.26) as a game between two players. One player, u, aims
to minimize the criterion J, while the other player, w, aims to maximize it. 9 We call a
pair of strategies (u w ) optimal with respect to the criterion J(x0uw) if for all u 2L2
and w 2L2 the inequalities
(12.25)
_ ;2
=(A + B1B T
1 X + B2F + ZHC2) ; ZHy
u = F
which de nes the desired map K : y 7! u.
Summarizing, the H1 control synthesis algorithm looks as follows:
Algorithm 12.8 INPUT: generalized plant G in state space form (13.16) or (12.21)
tolerance level ">0.
J(x0u w) J(x0u w ) J(x0uw ) (12.27)
ASSUMPTIONS: A-1tillA-5.
9
Just like a soccer match where instead of administrating the number of goals of each team, the
di erence between the number of goals is taken as the relevant performance criterion. After all, this is the
only relevant criterion which counts at the end of a soccer game ::: .
Step 1. Find l h such thatM : w 7! z satis es
l < k M(s) k1 < h

192 CHAPTER 12. SOLUTION TO THE GENERAL H1 CONTROL PROBLEM
12.6. THE H1 FILTERING PROBLEM 191
h -
6;
^z
e
-
z +
-
w
-
y
Plant
-
u
Filter
are satis ed. Such apair(uw ) de nes a saddle point for the criterion J. We maythink of u as a best control strategy, while w is the worst exogenous input. The existence
of such a saddle point is guaranteed by the solutions X of the Riccati equation (12.22).
Speci cally, under the assumptions made in the previous section, for any solution X
of (12.22) a completion of the squares argument will give you that for all pairs (u w) of
(square integrable) inputs of the system (12.21) for which limt!1 x(t) = 0 there holds
-
u
: (12.28)
J(x0uw)=x T
0 Xx0+ k w ; ;2 B T
1 Xx k2 2 ;ku + B T
2 Xx k2 2
Figure 12.6: The H1 lter con guration
a lter mapping (u y) 7! ^z such that the for overall con guration with transfer function
Me : w 7! e the H1 norm
Thus, if both `players' u and w have access to the state x of (12.21) then (12.28) gives us
immediately a saddle point
(
T
u (t) := ;B2 Xx(t)
w (t) := ;2B1Xx(t) which satis es the inequalities (12.27). We see that in that case the saddle point
(12.30)
k e k 2 2
k w1 k 2 2 + k w2 k 2 2
k Me(s) k2 1 = sup
w1w22L2
J(x0u w )=x T
0 Xx0
2 is less than or equal to some pre-speci ed value .
The solution to this problem is entirely dual to the solution of the state feedback H1
problem and given in the following theorem.
which gives a nice interpretation of the solution X of the Riccati equation (12.22). Now,
taking the initial state x0 =0gives that the saddle point J(0u w ) = 0 which, by (12.27)
gives that for all w 2L2
Theorem 12.9 (The H1 lter.) Let the system (12.29) be given and assume that the
assumptions A-1 till A-5 hold. Then
J(0u w) J(0u w )=0
1. there exists a lter which achieves that the mapping Me : w ! e in the con guration
of Figure 12.6 satis es
As in section 12.2 it thus follows that the closed loop system Mx : w 7! z obtained by
applying the static state feedback controller
u (t) = ;B T
2 Xx(t)
k Me k1 <
if and only if the Riccati equation (12.23) has a stabilizing solution Y = Y T 0.
results in k Mx(s) k1 . We moreover see from this analysis that the worst case
disturbance is generated by w .
2. In that case one such lter is given by the equations
(
_ ;2 =(A + B1BT 1 X) + B2u + H(C2 ; y)
12.6 The H1 ltering problem
(12.31)
^z = C1 + D21u
where H = YC T 2 .
Just like we splitted the optimal H2 control problem into a state feedback problem and a
ltering problem, the H1 control problem admits a similar separation. The H1 ltering
problem is the subject of this section and can be formalized as follows.
We reconsider the state space equations (12.21):
Let us make a few important observations
_x = Ax + B1w1 + B2u
8
><
We emphasize again that this lter design is carried out completely in a deterministic
setting. The matrix H is generally referred to as the H1 lter gain and clearly
depends on the value of (since Y depends on ).
(12.29)
z = C1x + D12u
>:
y = C2x + w2
It is important to observe that in contrast to the Kalman lter, the H1 lter depends
on the to-be-estimated signal. This, because the matrix C1, which de nes the to-beestimated
signal z explicitly, appears in the Riccati equation. The resulting lter
therefore depends on the to-be-estimated signal.
under the same conditions as in the previous section.
Just like the Kalman lter, the H1 lter is a causal, linear mapping taking the control
input u and the measurements y as its inputs, and producing an estimate ^z of the signal z
in suchaway that the H1 norm of the transfer function from the noise w to the estimation
error e = z ; ^z is minimal. Thus, in the con guration of Figure 12.6, we wish to design

194 CHAPTER 12. SOLUTION TO THE GENERAL H1 CONTROL PROBLEM
12.7. COMPUTATIONAL ASPECTS 193
Exercise 3. Suppose that a stable transfer function H(s) admits the state space representation
12.7 Computational aspects
_x = Ax + Bw
z = Cx + Dw:
2 T Show that k H(s) k1 < implies that I ; D D is positive de nite. Give an
example of a system for which the converse is not true, i.e., give an example for
which 2I ; DTD is positive de nite and k H(s) k1 > .
Exercise 4. This exercise is a more extensive simulation exercise. Using MATLAB and
the installed package MHC (Multivariable H1 Controller design) you should be able
to design a robust controller for the following problem. You may also like to use the
MHC package that has been demonstrated and for which amanual is available upon
request.
The system considered in this design is a satellite with two highly exible solar arrays
attached. The model for control analysis represents the transfer function from the
torque applied to the roll axis of the satellite to the corresponding satellite roll angle.
In order to keep the model simple, only a rigid body mode and a single exible mode
are included, resulting in a four state model. The state space system is described by
_x = Ax + Bu + Bw
y = Cx
where u is the control torque (in units Nm), w is a constant disturbance torque
(Nm), and y is the roll angle measurement (in rad). The state space matrices are
The Robust Control Toolbox in Matlab includes various routines for the computation
of H2 optimal and H1 optimal controllers. These routines are implemented with the
algorithms described in this chapter.
The relevant routine in this toolbox for H2 optimal control synthesis is h2lqg. This
routine takes the parameters of the state space model (12.12) or the more general state
space model (13.16) (which itconverts to (12.12)) as its input arguments and produces the
state space matrices (Ac Bc Cc Dc) of the optimal H2 controller as de ned in (12.19)
as its outputs. If desired, this routine also produces the state space description of the
corresponding closed-loop transfer function M as its output. (See the corresponding help
le).
For H1 optimal control synthesis, the Robust Control Toolbox includes an e cient implementation
of the result mentioned in Theorem 12.7. The Matlab routine hinf takes the
state space parameters of the model (13.16) as its input arguments and produces the state
space parameters of the so called central controller as speci ed by the formulae (12.24) in
Theorem 12.7. The routine makes use of the two Riccati solution as presented above. Also
the state space parameters of the corresponding closed loop system can be obtained as
an optional output argument. The Robust Control Toolbox provides features to quickly
generate augmented plants which incorporate suitable weighting lters. An e cient use of
these routines, however, requires quite some programming e ort in Matlab. Although we
consider this an excellent exercise it is not really the purpose of this course. The package
MHC (Multivariable H1 Control Design) has been written as part of a PhD study by
one of the students of the Measurement and Control Group at TUE, and has been customized
to easily experiment with lter design. During this course we will give asoftware
demonstration of this package.
0
B
@
0
B
@
given by
0
1:7319 10 ;5
0
3:7859 10 ;4
C
A
1
0 1 0 0
0 0 0 0
A =
0 0 0 1
0 0 ;! 2 B =
;2 !
C = ; 1 0 1 0 D =0:
1
C
A
12.8 Exercises
where ! =1:539rad=sec is the frequency of the exible mode and =0:003 is the
exural damping ratio. The nominal open loop poles are at
Exercise 0. Take the rst blockscheme of the exercise of chapter 6. De ne a mixed
sensitivity problem where the performance is represented by good tracking. Filter
We is low pass and has to be chosen. The robustness term is de ned by abounded
additive model error: k Wx ;1 P k1< 1.
Furthermore,k r k2< 1, P =(s ; 1)=(s + 1) and Wx = s=(s + 3). What bandwidth
can you obtain for the sensitivity being less than .01 ? Use the tool MHC!
;0:0046 + 1:5390j ;0:0046 ; 1:5390j 0
and the nite zeros at
Exercise 1. Write a routine h2comp in MATLAB which computes the H2 norm of a
transfer function H(s). Let the state space parameters (A B C D) be the input to
this routine, and the H2 norm
;0:0002 + 0:3219j ;0:0002 ; 0:3219j:
Because of the highly exible nature of this system, the use of control torque for
attitude control can lead to excitation of the lightly damped exural modes and
hence loss of control. It is therefore desired to design a feedback controller which
increases the system damping and maintains a speci ed pointing accuracy. That
is, variations in the roll angle are to be limited in the face of torque disturbances.
In addition the sti ness of the structure is uncertain, and the natural frequency, !,
can only be approximately estimated. Hence, it is desirable that the closed loop be
robustly stable to variations in this parameter.
k C(Is; A) ;1 B + D k2
its output. Build in su cient checks on the matrices (A B C D) to guarantee a
`fool-proof' behavior of the routine.
Hint: Use the Theorem 12.1. See the help les of the routines lyap in the control system
toolbox tosolvetheLyapunov equation (12.2). The procedures abcdchk, minreal, eig, or
obsv may provehelpful. The design objectives are as follows.
Exercise 2. Write a block diagram for the optimal H2 controller and for the optimal H1
controller.

196 CHAPTER 12. SOLUTION TO THE GENERAL H1 CONTROL PROBLEM
12.8. EXERCISES 195
1. Performance: required pointing accuracy due to 0:3Nm step torque disturbance
should be y(t) < 0:0007 rad for all t > 0. Additionally, the response
time is required to be less than 1 minute (60sec).
2. Robust stability: stable response for about 10% variations in the natural
frequency !.
3. Control level: control e ort due to 0:3Nm step torque disturbances u(t) <
0:5 Nm.
We will start the design by making a few simple observations
Verify that with a feedback control law u = ;Cy the resulting closed-loop
transfer U := (I + PC) ;1P maps the torque disturbance w to the roll angle y.
Note that, to achieve a pointing accuracy of 0:0007rad in the face of 0:3Nm
torque input disturbances, we require that U satis es the condition
= 0:0021 rad=Nm (12.32)
(U) = (I + PC) ;1 P < 0:0007
0:3
at least at low frequencies.
Recall that, for a suitable weighting function W we can achieve thatjU(j!)j
jW (j!)j for all !, where is the usual parameter in the ` {iteration' of the H1
optimization procedure.
Consider the weighting lter
(12.33)
s +0:4
s +0:001
Wk(s) =k
where k is a positive constant.
1. Determine a value of k so as to achieve the required level of pointing accuracy
in the H1 design. Try to obtain a value of which is more or less equal to 1.
Hint: Set up a scheme for H1 controller design in which the output y +10 ;5w is used
as a measurement variable and in which the to be controlled variables are
Wky
10 ;5u z =
(the extra output is necessary to regularize the design). Use the MHC package to
compute a suboptimal H1 controller C which minimizes the H1 norm of the closed
loop transfer w 7! z for various values of k > 0. Construct a 0:3Nm step torque
input disturbance w to verify whether your closed-loop system meets the pointing
speci cation. See the MHC help facility to get more details.
2. Let Wk be given by the lter (12.33) with k as determined in 2. Let V (s) be
a second weighting lter and consider the weighted control sensitivity M :=
WkUV = Wk(I + PC) ;1PV. Choose V in such away that an H1 suboptimal
controller C which minimizes k M k1 meets the design speci cations.
Hint: Use the same con guration as in part 2 and compute controllers C by using the
package MHC and by varying the weighting lter V .
3. After you complete the design phase, make Bode plots of the closed-loop response
of the system and verify whether the speci cations are met by perturbing
the parameter ! and by plotting the closed{loop system responses of the signals
u and y under step torque disturbances of 0:3Nm.

198 CHAPTER 13. SOLUTION TO THE GENERAL H1 CONTROL PROBLEM
example, by observing that the system is an interconnection of dissipative components, or
by considering systems in which a loss of energy is inherent to the behavior of the system
(due to friction, optical dispersion, evaporation losses, etc.).
In this chapter we will formalize the notion of a dissipative dynamical system for the
class of linear time-invariant systems. It will be shown that linear matrix inequalities
(LMI's) occur in a very natural way in the study of linear dissipative systems. Solutions
of these inequalities have a natural interpretation as storage functions associated with a
dissipative dyamical system. This interpretation will play a key role in understanding
the relation between LMI's and questions related to stability, robustness, and H1 controller
design. In recent years, linear matrix inequalities have emerged as apowerful tool
to approach control problems that appear hard if not impossible to solve in an analytic
fashion. Although the history of LMI's goes back to the fourties with a major emphasis of
their role in control in the sixties (Kalman, Yakubovich, Popov, Willems), only recently
powerful numerical interior point techniques have beendeveloped to solve LMI's in a practically
e cient manner (Nesterov, Nemirovskii 1994). Several Matlab software packages
are available that allow a simple coding of general LMI problems and of those that arise
in typical control problems (LMI Control Toolbox, LMI-tool).
Chapter 13
Solution to the general H1 control
problem
13.1.2 Dissipativity
Consider a continuous time, time-invariant dynamical system described by the equations1
(
_x = Ax + Bu
:
(13.1)
y = Cx + Du
As usual, x is the state which takes its values in a state space X = Rn , u is the input
taking its values in an input space U =Rm and y denotes the output of the system which
assumes its values in the output space Y =Rp . Let
s : U Y !R
be a mapping and assume that for all time instances t0t1 2R and for all input-output
pairs u y satisfying (13.1) the function
In previous chapters we have been mainly concerned with properties of control con gurations
in which a controller is designed so as to minimize the H1 norm of a closed
loop transfer function. So far, we did not address the question how such a controller is
actually computed. This has been a problem of main concern in the early 80-s. Various
mathematical techniques have been developed to compute H1-optimal controllers, i.e.,
feedback controllers which stabilize a closed loop system and at the same time minimize
the H1 norm of a closed loop transfer function. In this chapter we treat a solution to a
most general version of the H1 optimal control problem. We willmakeuseofatechnique
which isbasedonLinear Matrix Inequalities (LMI's). This technique is fast, simple, and
at the same time a most reliable and e cient way to synthesize H1 optimal controllers.
This chapter is organized as follows. In the next section we rst treat the concept of
a dissipative dynamical system. We will see that linear dissipative systems are closepy
related to Linear Matrix Inequalities (LMI's) and we will subsequently show howtheH1
norm of a transfer function can be computed by means of LMI's. Finally, we consider the
synthesis question of how to obtain a controller which stabilizes a given dynamical system
so as to minimize the H1 norm of the closed loop system. Proofs of theorems are included
for completeness only. They are not part of the material of the course and can be skipped
upon rst reading of the chapter.
s(t) :=s(u(t)y(t))
13.1 Dissipative dynamical systems
js(t)jdt < 1. The mapping s will be referred to as the supply
is locally integrable, i.e., R t1
t0
13.1.1 Introduction
function.
De nition 13.1 (Dissipativity) The system with supply rate s is said to be dissipative
if there exists a non-negative function V : X !R such that
Z t1
s(t)dt V (x(t1)) (13.2)
V (x(t0)) +
t0
for all t0 t1 and all trajectories (u x y) which satisfy (13.1).
1 Much of what is said in this chapter can be applied for (much) more general systems of the form
_x = f(x u), y = g(x u).
The notion of dissipativity (or passivity) is motivated by the idea of energy dissipation in
many physical dynamical systems. It is a most important concept in system theory and
dissipativity plays a crucial role in many modeling questions. Especially in the physical
sciences, dissipativity is closely related to the notion of energy. Roughly speaking, a
dissipative system is characterized by the property that at any time the amount of energy
which the system can conceivably supply to its environment can not exceed the amount
of energy that has been supplied to it. Stated otherwise, when time evolves a dissipative
system absorbs a fraction of its supplied energy and transforms it for example into heat,
an increase of entropy, mass, electromagnetic radiation, or other kinds of energy `losses'.
In many applications, the question whether a system is dissipative or not can be answered
from physical considerations on the way the system interacts with its environment. For
197

200 CHAPTER 13. SOLUTION TO THE GENERAL H1 CONTROL PROBLEM
13.1. DISSIPATIVE DYNAMICAL SYSTEMS 199
the two basic laws of thermodynamics state that for all system trajectories (T Q W ) and
all time instants t0 t1
Z t1
Q(t)+W (t) dt = E(x(t1))
E(x(t0)) +
t0
(which is conservation of thermodynamical energy) and the second law of thermodynamics
states that the system trajectories satisfy
Z t1
dt S(x(t1))
; Q(t)
T (t)
S(x(t0)) +
t0
Interpretation 13.2 The supply function (or supply rate) s should be interpreted as the
supply delivered to the system. This means that in a time interval [0t] work has been
done on the system whenever R t
0 s( )d is positive, while work is done by the system
if this integral is negative. The non-negative function V is called a storage function
and generalizes the notion of an energy function for a dissipative system. With this
interpretation, inequality (13.2) formalizes the intuitive idea that a dissipative system is
characterized by the property that the change of internal storage (V (x(t1)) ; V (x(t0)))
in any time interval [t0t1] will never exceed the amount of supply that ows into the
system (or the `work done on the system). This means that part of what is supplied to
the system is stored, while the remaining part is dissipated. Inequality (13.2) is known as
the dissipation inequality.
for a storage function S. Here, E is called the internal energy and S the entropy. The
rst law promises that the change of internal energy is equal to the heat absorbed by the
system and the mechanical work which is done on the system. The second law states that
the entropy decreases at a higher rate than the quotient of absorbed heat and temperature.
Note that thermodynamical systems are dissipative with respect to more than one supply
function!
Remark 13.3 If the function V (x( )) with V a storage function and x :R ! X a state
trajectory of (13.1) is di erentiable as a function of time, then (13.2) can be equivalently
written as
_V (t) s(u(t)y(t)): (13.3)
Example 13.7 As another example, the product of forces and velocities is a candidate
supply function in mechanical systems. For those familiar with the theory of bond-graphs
we remark that every bond-graph can be viewed as a representation of a dissipative dynamical
system where input and output variables are taken to be e ort and ow variables
and the supply function s is invariably taken to be the product of these two variables.
A bond-graph is therefore a special case of a dissipative system (and not the other way
around!).
Remark 13.4 (this remark may be skipped) There is a re nement of De nition 13.1
which isworth mentioning. The system is said to be conservative (or lossless) if there
exists a non-negative functionV : X !R such that equality holds in (13.2) for all t0 t1
and all (u x y) whichsatisfy (13.1). .
Example 13.8 Typical examples of supply functions s : U Y !R are
s(u y) =u T y (13.4)
s(u y) =kyk2 ;kuk2 (13.5)
s(u y) =kyk2 + kuk2 (13.6)
s(u y) =kyk2 (13.7)
Example 13.5 Consider an electrical network with n external ports. Denote the external
voltages and currents of the i-th port by(ViIi) and let V and I denote the vectors of length
n whose i-th component is Vi and Ii, respectively. Assume that the network contains (a
nite number of) resistors, capacitors, inductors and lossless elements such as transformers
and gyrators. Let nC and nL denote the number of capacitors and inductors in the network
and denote by VC and IL the vectors of voltage drops accrioss the capacitors and currents
through the inductors of the network. An impedance description of the system then takes
the form (13.1), where u = I, y = V and x = ; V T
C I T T
L . For such a circuit, a natural
supply function is
which arise in network theory, bondgraph theory, scattering theory, H1 theory, game
theory, LQ-optimal control and H2-optimal control theory.
s(V (t)I(t)) = V T (t)I(t):
This system is dissipative and
If is dissipative with storage function V , then we will assume that there exists a
reference point x 2 X of minimal storage, i.e. there exists x 2 X such that V (x ) =
minx2X V (x). You can think of x as the state in which the system is `at rest', an
`equilibrium state' for which noenergy is stored in the system. Given a storage function
V ,itsnormalization (with respect to x )isdenedasV (x) :=V (x) ; V (x ). Obviously,
V (x ) = 0 and V is a storage function of whenever V is. For linear systems of the form
(13.1) we usually take x =0.
nLX
nCX
LiI 2 Li
CiV 2 Ci +
V (x) :=
i=1
i=1
is a storage function of the system that represents the total electrical energy in the capacitors
and inductors.
13.1.3 A rst characterization of dissipativity
Instead of considering the set of all possible storage functions associated with a dynamical
system , we will restrict attention to the set of normalized storage functions. Formally,
the set of normalized storage functions (associated with ( s)) is de ned by
V(x ):=fV : X !R+ j V (x ) = 0 and (13.2) holdsg:
Example 13.6 Consider a thermodynamic system at uniform temperature T on which
mechanical work is being done at rate W and which is being heated at rate Q. Let
(TQW) be the external variables of such a system and assume that {either by physical
or chemical principles or through experimentation{ the mathematical model of the thermodynamic
system has been decided upon and is given by the time invariant system (13.1).
The rst and second law of thermodynamics may then be formulated in the sense of De -
nition 13.1 by saying that the system is conservative with respect to the supply function
s1 := (W + Q) and dissipative with respect to the supply function s2 := ;Q=T . Indeed,

202 CHAPTER 13. SOLUTION TO THE GENERAL H1 CONTROL PROBLEM
13.1. DISSIPATIVE DYNAMICAL SYSTEMS 201
Taking the supremum over all t1 0 and all such trajectories (u x y) (with x(0) = x0) yields that
Vav(x0) V (x0) < 1. To provethe converse implication it su ces to show thatVav is a storage
function. To see this, rst note that Vav(x) 0 for all x 2 X (take t1 = 0 in (13.8a)). To prove
that Vav satis es (13.2), lett0 t1 t2 and (u x y) satisfy (13.1). Then
The existence of a reference point x of minimal storage implies that for a dissipative
system
Z t1
Z t2
Z t1
s(u(t)y(t)) dt 0
0
s(u(t)y(t))dt:
s(u(t)y(t))dt ;
Vav(x(t0)) ;
t 1
t 0
Since the second term in the right hand side of this inequality holds for arbitrary t2 t1 and
arbitrary (u x y)j [t1t2] (with x(t1) xed), we can take the supremum over all such trajectories to
conclude that
Z t1
s(u(t)y(t))dt ; Vav(x(t1)):
Vav(x(t0)) ;
for any t1 0 and any (u x y) satisfying (13.1) with x(0) = x . Stated otherwise, any
trajectory of the system which emanates from x has the property that the net ow of
supply is into the system. In many treatments of dissipativity this property is often taken
as de nition of passivity.
We introduce two mappings Vav : X !R+ [1and Vreq : X !R [ f;1g which will
play a crucial role in the sequel. They are de ned by
t 0
Z t1
which shows that Vav satis es (13.2).
2a. Suppose that is dissipative and let V be a storage function. Then V (x) :=V (x);V (x ) 2
V(x ) so that V(x ) 6= . Observe that Vav(x ) 0 and Vreq(x ) 0 (take t1 = t;1 = 0 in
(13.8)). Suppose that the latter inequalities are strict. Then, using controllability of the system,
s(t) dt j t1 0 (u x y) satisfy (13.1) with x(0) = x0
Vav(x0) :=sup ;
0
(13.8a)
there exists t;1 0 t1 and a state trajectory x with x(t;1) = x(0) = x(t1) = x such
that ; R t1 0 s(t)dt > 0 and R 0
s(t)dt < 0. But this yields a contradiction with (13.2) as both
(Z 0
s(t) dt j t;1 0 (u x y) satisfy (13.1) with (13.8b)
Vreq(x0) := inf
t;1
R t1
0 s(t)dt 0andR 0
s(t)dt 0. Thus, Vav(x )=Vreq(x )=0. We already proved that Vav is a
t;1
storage function so that Vav 2V(x ). Along the same lines one shows that also Vreq 2V(x ).
t;1
x(0) = x0 and x(t;1) =x g
Z 0
s(u(t)y(t))dt
s(u(t)y(t))dt V (x0)
2b. If V 2V(x ) then
Z t1
;
t;1
0
Interpretation 13.9 Vav(x) denotes the maximal amount of internal storage that may
be recovered from the system over all state trajectories starting from x. Similarly, Vreq(x)
re ects the minimal supply the environment has to deliver to the system in order to excite
the state x via any trajectory in the state space originating in x .
for all t;1 0 t1 and (u x y) satisfying (13.1) with x(t;1) =x and x(0) = x0. Now takethe supremum and in mum over all such trajectories to obtain that Vav V Vreq.
13.2 Dissipative systems with quadratic supply functions
13.2.1 Quadratic supply functions
We refer to Vav and Vreq as the available storage and the required supply, respectively. Note
that in (13.8b) it is assumed that the point x0 2 X is reachable from the reference pont
x , i.e. it is assumed that there exist a control input u which brings the state trajectory
x from x at time t = t;1 to x0 at time t = 0. This is possible when the system is
controllable.
In this section we will apply the above theory by considering systems of the form (13.1)
with quadratic supply functions s : U Y !R, de ned by
Theorem 13.10 Let the system be described by (13.1) and let s be a supply function.
Then
T
(13.9)
y
u
Qyy Qyu
Quy Quu
s(u y) = y
u
1. is dissipative if and only if V av(x) is nite for all x 2 X.
2. If is dissipative and controllable then
Here,
(a) V avV req 2V(x ).
Q := Qyy Qyu
Quy Quu
(b) fV 2V(x )g ) fFor all x 2 X there holds 0 V av(x) V (x) V req(x)g.
is a real symmetric matrix (i.e. Q = QT ) which is partitioned conformally with u and
y. Note that the supply functions given in Example 13.8 can all be written in the form
(13.9).
Interpretation 13.11 Theorem 13.10 gives a necessary and su cient condition for a
system to be dissipative. It shows that both the available storage and the required supply
are possible storage functions. Moreover, statement (b) shows that the available storage
and the required supply are the extremal storage functions in V(x ). In particular, for any
state of a dissipative system, the available storage is at most equal to the required supply.
Remark 13.12 Substituting the output equation y = Cx + Du in the supply function
(13.9) shows that (13.9) can equivalently be viewed as a quadratic function in the variables
u and x. Indeed,
Proof. 1. Let be dissipative, V a storage function and x0 2 X. From (13.2) it then follows
that for all t1 0 and all (u x y) satis ng (13.1) with x(0) = x0,
x
u
Qxx Qxu
Qux Quu
T
s(u y) =s(u Cx + Du) = x
u
Z t1
s(u(t)y(t))dt V (x0) < 1:
;
0

204 CHAPTER 13. SOLUTION TO THE GENERAL H1 CONTROL PROBLEM
13.2. DISSIPATIVE SYSTEMS WITH QUADRATIC SUPPLY FUNCTIONS 203
Since (13.12) holds for all t0 t1 and all inputs u this reduces to the requirement that K 0
satis es the LMI F (K) 0.
(3)2). Conversely, if there exist K 0 such that F (K) 0 then (13.12) holds and it follows
that V (x) =xTKx is a storage function which satis es the dissipation inequality.
(1,5). If ( s) is dissipative thenbyTheorem (13.10), Vreq is a storage function. Since Vreq
is de ned as an optimal cost corresponding to a linear quadratic optimization problem, Vreq is
quadratic. Hence, if the reference point x =0,Vreq(x) is of the form xTK+x for some K+ 0.
Conversely, if Vreq = xTK+x, K+ 0, then it is easily seen that Vreq satis es the dissipation
inequality (13.2) which implies that ( s)isdissipative.
(1,6). Let ! 2 R be such that det(j!I ; A) 6= 0and consider the harmonic input u(t) =
exp(j!t)u0 with u0 2 Rm . De ne x(t) :=exp(j!t)(j!I ; A) ;1Bu0 and y(t) := Cx(t) +Du(t).
Then y(t) = exp(j!t)G(j!)u0 and the triple (u x y) satis es (13.1). Moreover,
where
C D
0 I
Qyy Qyu
Quy Quu
T
:
0 I
= C D
Qxx Qxu
Qux Quu
13.2.2 Complete characterizations of dissipativity
The following theorem is the main result of this section. It provides necessary and su cient
conditions for dissipativeness.
Theorem 13.13 Suppose that the system described by (13.1) is controllable and let
G(s) =C(Is; A) ;1B + D be thecorresponding transfer function. Let the supply function
s be de ned by (13.9). Then the following statements are equivalent.
u0
G(j!)
I
Qyy Qyu
Quy Quu
G(j!)
I
s(u(t)y(t)) = u 0
1. ( s) is dissipative.
which is a constant for all time t 2 R. Now suppose that ( s) is dissipative. For non-zero
frequencies ! the triple (u x y) is periodic with period P =2 =!. In particular, there must exist
a time instant t0 such that x(t0) = x(t0 + kP) = 0, k 2 Z. Since V (0) = 0, the dissipation
inequality (13.2) reads
2. ( s) admits a quadratic storage function V (x) :=x T Kx with K = K T 0.
3. There exists K = K T 0 such that
C D
0 I
T Qyy Qyu
Z t1
Z t1
0:
0 I
+ C D
Quy Quu
F (K) :=; ATK + KA KB
BTK 0
u0
G(j!)
I
Qyy Qyu
Quy Quu
G(j!)
I
u 0
s(u(t)y(t)) dt =
t 0
t 0
(13.10)
0
u0
G(j!)
I
Qyy Qyu
Quy Quu
G(j!)
I
=(t1 ; t0)u 0
0 such that V av(x) =x T K;x.
4. There exists K; = K T ;
for all t1 >t0. Since u0 and t1 >t0 are arbitrary this yields that statement 6 holds.
The implication 6 ,1 ismuch more involved and will be omitted here.
0 such that V req(x) =x T K+x.
5. There exists K+ = K T +
6. For all ! 2R with det(j!I ; A) 6= 0, there holds
Interpretation 13.14 The matrix F (K) is usually called the dissipation matrix. The
inequality F (K) 0 is an example of a Linear Matrix Inequality (LMI) in the (unknown)
matrix K. The crux of the above theorem is that the set of quadratic storage functions
in V(0) is completely characterized by the inequalities K 0 and F (K) 0. In other
words, the set of normalized quadratic storage functions associated with ( s) coincides
with those matrices K for which K = K T 0 and F (K) 0. In particular, the available
storage and the required supply are quadratic storage functions and hence K; and K+
also satisfy F (K;) 0 and F (K+) 0. Using Theorem 13.10, it moreover follows that
any solution K = K T 0ofF (K) 0 has the property that
0 (13.11)
G(j!)
I
Qyy Qyu
Quy Quu
G(j!)
I
Moreover, if one of the above equivalent statements holds, then V (x) := xTKx is a
quadratic storage function in V(0) if and only if K 0 and F (K) 0.
Proof. (1)2,4). If ( s) is dissipative then we infer from Theorem 13.10 that the available
storage Vav(x) is nite for any x 2 Rn . We claim that Vav(x) is a quadratic function of x. This is
a standard result from LQ optimization. Indeed, s is quadratic and
Z Z t1
t1
Vav(x) = sup ; s(t)dt = ; inf s(t)dt
0 K; K K+:
0
0
In other words, among the set of positive semi-de nite solutions K of the LMI F (K) 0
there exists a smallest and a largest element. Statement 6 provides a frequency domain
characterization of dissipativity. For physical systems, this means that whenever the
system is dissipative with respect to a quadratic supply function (and quite some physical
systems are), then there is at least one energy function which is a quadratic function of
the state variable, this function is in general non-unique and squeezed in between the
available storage and the required supply. Any physically relevant energy function which
happens to be of the form V (x) =xTKx will satisfy the linear matrix inequalities K > 0
and F (K) 0.
denotes the optimal cost of a linear quadratic optimization problem. It is well known that this
in mum is a quadratic form in x.
(4)1). Obvious from Theorem (13.10).
(2)3). If V (x) =xTKx with K 0 is a storage function then the dissipation inequality can
be rewritten as
Z t1
; d
dt x(t)TKx(t)+s(u(t)y(t)) dt 0:
t 0
Substituting the system equations (13.1), thisisequivalent to
o
x(t)
u(t)
T n T ;A K ; KA ;KB
;BT +
K 0
C D
T
Qyy Qyu C D
0 I Quy Quu 0 I
| {z }
x(t)
u(t)
Z t1
dt 0:
For conservative systems with quadratic supply functions a similar characterization
can be given. The precize formulation is evident from Theorem 13.13 and is left to the
reader.
t 0
F (K)
(13.12)

206 CHAPTER 13. SOLUTION TO THE GENERAL H1 CONTROL PROBLEM
13.2. DISSIPATIVE SYSTEMS WITH QUADRATIC SUPPLY FUNCTIONS 205
13.3 Dissipativity and H1 performance
13.2.3 The positive real lemma
Let us analyze the importance of the last result, Corollary 13.17, for H1 optimal control. If
is dissipative with respect to the supply function (13.13), then we infer from Remark 13.3
that for any quadratic storage function V (x) =xTKx, 2 u T u ; y T y: (13.14)
_V
We apply the above results to two quadratic supply functions which play an important
role in a wide variety of applications. First, consider the system (13.1) together with the
quadratic supply function s(u y) = yTu. This function satis es (13.9) with Quu = 0,
Qyy =0and Quy = QT yu =1=2I. With these parameters, the following is an immediate
consequence of Theorem 13.13.
Suppose that x(0) = 0, A has all its eigenvalues in the open left-half complex plane (i.e. the
system is stable) and the input u is taken from the set L2 of square integrable functions.
Then both the state x and the output y of (13.1) are square integrable functions and
limt!1 x(t) =0. We can therefore integrate (13.14) from t = 0 till 1 to obtain that for
all u 2L2
Corollary 13.15 Suppose that the system described by (13.1) is controllable and has
transfer function G. Let s(u y) =yTu be a supply function. Then equivalent statements
are
1. ( s) is dissipative.
2. the LMI's
0
2 kuk 2 2 ;kyk 2 2
where the norms are the usual L2 norms. Equivalently,
K = K T
0
;ATK ; KA ;KB + CT ;BTK + C D + DT 0
have a solution.
: (13.15)
kyk2
kuk2
sup
u2L2
3. For all ! 2R with det(j!I ; A) 6= 0G(j!) + G(j!) 0.
Now recall from Chapter 5, that the left-hand side of (13.15) is the L2-induced norm or
L2-gain of the system (13.1). In particular, from Chapter 5 we infer that the H1 norm
of the transfer function G is equal to the L2-induced norm. We thus derived the following
result.
Moreover, V (x) =xTKx de nes a quadratic storage function if and only if K satis es the
above LMI's.
Theorem 13.18 Suppose that the system described by(13.1) is controllable, stable and
has transfer function G. Let s(u y) = 2uTu ; yTy be asupply function. Then equivalent
statements are
Remark 13.16 Corollary 13.15 is known as the Kalman-Yacubovich-Popov or the positive
real lemma and has played a crucial role in questions related to the stability of control
systems and synthesis of passive electrical networks. Transfer functions which satisfy the
third statement are generally called positive real.
13.2.4 The bounded real lemma
1. ( s) is dissipative.
Second, consider the quadratic supply function
.
2. kGkH1
s(u y) = 2 u T u ; y T y (13.13)
3. The LMI's
where 0. In a similar fashion we obtain the following result as an immediate consequence
of Theorem 13.13.
0
K = K T 0
ATK + KA + CTC KB + CTD BTK + DTC DTD ; 2I Corollary 13.17 Suppose that the system described by (13.1) is controllable and has
transfer function G. Let s(u y) = 2uTu ; yTy be a supply function. Then equivalent
statements are
have a solution.
1. ( s) is dissipative.
2. The LMI's
Moreover, V (x) =xTKx de nes a quadratic storage function if and only if K satis es the
above LMI's.
0
K = K T 0
ATK + KA + CTC KB + CTD BTK + DTC DTD ; 2I Interpretation 13.19 Statement 3 of Theorem 13.18 therefore provides a test whether
or not the H1-norm of the transfer function G is smaller than a prede ned number >0.
We can compute the L2-induced gain of the system (which istheH1 norm of the transfer
function) by minimizing > 0 over all variables and K > 0 that satisfy the LMI's
of statement 3. The issue here is that such a test and minimization can be e ciently
performed in the LMI-toolbox asimplemented in MATLAB.
have a solution.
2 I.
3. For all ! 2R with det(j!I ; A) 6= 0G(j!) G(j!)
Moreover, V (x) =xTKx de nes a quadratic storage function if and only if K satis es the
above LMI's.

208 CHAPTER 13. SOLUTION TO THE GENERAL H1 CONTROL PROBLEM
13.4. SYNTHESIS OF H1 CONTROLLERS 207
Controllers are therefore simply parameterized by the matrices Ac, Bc, Cc, Dc. The
controlled or closed-loop system then admits the description
(
_ = A + Bw
(13.18)
z = C + Dw
-
z
-
w
G
-
u y
where
1
A : (13.19)
0
@ A + BDcC BCc B1 + BDcF
BcC Ac BcF
C1 + EDcC ECc D + EDcF
K
A B
C D =
Figure 13.1: General control con guration
The closed-loop transfer matrix M can therefore be represented as M(s) =C(Is;A) ;1B+ D.
The optimal value of the H1 controller synthesis problem is de ned as
13.4 Synthesis of H1 controllers
; kMk1:
= inf
(AcBcCcDc) suchthat (A) C
Clearly, the number is larger than if and only if there exists a controller such that
;
(A) C and kMk1 < :
The optimal H1 value is then given by theminimal for which acontroller can still
be found.
By Theorem 13.182 , the controller (AcBcCcDc) achieves that (A) C ; and the
H1 norm kMkH1 < if and only if there exists a symmetric matrix X satisfying
< 0 (13.20)
ATX + XA+ CTC XB+ CTD BTX + DTC DTD; 2I X = X T > 0
In this section we present the main algorithm for the synthesis of H1 optimal controllers.
Consider the general control con guration as depicted in Figure 13.1. Here, w are the exogenous
inputs (disturbances, noise signals, reference inputs), u denote the control inputs,
z is the to be controlled output signal and y denote the measurements. All variables may
be multivariable. The block G denotes the \generalized system" and typically includes a
model of the plant together with all weighting functions which arespeci ed by the user.
The block K denotes the \generalized controller" and includes typically a feedback controller
and/or a feedforward controller. The block G contains all the known features (plant
model, input weightings, output weightings and interconnection structures), the block K
needs to be designed. Admissable controllers are all linear time-invariant systems K that
internally stabilize the con guration of Figure 13.1. Every such admissible controller K
gives rise to a closed loop system which maps disturbance inputs w to the to-be-controlled
output variables z. Precisely, if M denotes the closed-loop transfer function M : w 7! z,
then with the obvious partitioning of G,
The corresponding synthesis problem therefore reads as follows: Search controller parameters
(AcBcCcDc) and an X > 0 such that (13.20) holds.
Recall that A depends on the controller parameters since X is also a variable, we
observe that XA depends non-linearly on the variables to be found. There exist a clever
transformation so that the blocks in (13.20) which depend non-linearly on the decision
variables X and (AcBcC ; c Dc), is transformed to an a ne dependence of a new set
of decision variables
M = G11 + G12K(I ; G22K) ;1 G21:
The H1 control problem is formalized as follows
Synthesize a stabilizing controller K such that
kMkH1 <
for some value of >0.
K L
M N
v := X Y
Since our ultimate aim is to minimize the H1 norm of the closed-loop transfer function
M, we wish to synthesize an admissible K for as small as possible.
To solve this problem, consider the generalized system G and let
For this purpose, de ne
_x = Ax + B1w + Bu
8
><
Y I
I X
(13.16)
z = C1x + Dw + Eu
X(v) :=
>:
y = Cx + Fw
1
A
0
@ AY + BM A + BNC B1 + BNF
K AX + LC XB1 + LF
C1Y + EM C1 + ENC D + ENF
:=
A(v) B(v)
C(v) D(v)
2 With a slight variation.
be a state space description of G. An admissible controller is a nite dimensional linear
time invariant system described as
(
_xc = Acxc + Bcy
(13.17)
u = Ccxc + Dcy

210 CHAPTER 13. SOLUTION TO THE GENERAL H1 CONTROL PROBLEM
13.4. SYNTHESIS OF H1 CONTROLLERS 209
the calculations to reconstruct the controller out of the decision variable v. In particular,
one should avoid that the parameters v get too large, and that I ; XY is close to singular
what might render the controller computation ill-conditioned.
With these de nitions, the inequalities (13.20) can be replaced by the inequalities
1
0
A < 0: (13.21)
B(v) T ; I D(v) T
@ A(v)T + A(v) B(v) C(v) T
X(v) > 0
13.5 H1 controller synthesis in Matlab
C(v) D(v) ; I
The result of Theorem 13.20 has been implemented in the LMI Control Toolbox ofMatlab.
The LMI Control Toolbox supports continuous- and discrete time H1 synthesis
using either Riccati- or LMI based approaches. (The Riccati based approach had not been
discussed in this chapter). While the LMI approach is computationally more involved
for large problems, it has the decisive merit of eliminating the so called regularity conditions
attached to the Riccati-based solutions. Both approaches are based on state space
calculations. The following are the main synthesis routines in the LMI toolbox.
The one-one relation between the decision variables in (13.20), the decision variables in
(13.21) and solutions of the H1 control problem are now given in the following main
result.
Theorem 13.20 (H1 Synthesis Theorem) The following statements are equivalent.
1. There exists a controller (AcBcCcDc) and an X satisfying (13.20)
such that the inequalities (13.21) hold.
K L
M N
2. There exists v := X Y
Riccati-based LMI-based
continuous time systems hinfric hinflmi
discrete time systems dhinfric dhinflmi
Moreover, for any such v, the matrix I ; XY is invertible and there exist nonsingular U,
V such that I ; XY = UV T . The unique solutions X and (AcBcCcDc) are then given
by
Riccati-based synthesis routines require that
I 0
X U
;1
Y V
I 0
X =
1. the matrices E and F have full rank,
;1
2. the transfer functions G12(s) :=C(Is;A) ;1B1+F and G21(s) :=C1(Is;A) ;1B+E have no zeros on the j! axis.
:
V T 0
CY I
;1
K ; XAY L
M N
0 I
= U XB
Ac Bc
Cc Dc
LMI synthesis routines have no assumptions on the matrices which de ne the system
(13.16). Examples of the usage of these routines will be given in Chapter 10. We refer to
the corresponding help- les for more information.
In the LMI toolbox the command
We have obtained ageneralprocedure for deriving from analysis inequalities the corresponding
synthesis inequalities and for construction of the corresponding controllers.
The power of Theorem 13.20 lies in its simplicity and its generality. Virtually all analysis
results that are based on a dissipativity constraint with respect to a quadratic supply
function can be converted with ease into the corresponding synthesis result.
G = ltisys(A, [B1 B], [C1 C], [D E F zeros(dy,du)])
de nes the state space model (13.16) in the internal LMI format. Here dy and du are the
dimensions of the measurementvector y and the control input u, respectively. Information
about G is obtained by typing sinfo(G), plots of responses of G are obtained through
splot(G, 'bo') for a Bode diagram, splot(G, 'sv') for a singular value plot, splot(G,
'st') for a step response, etc. The command
[gopt, K] = hinflmi(G,r)
Remark on the controller order. In Theorem 13.20 we have not restricted the
order of the controller. In proving necessity of the solvability of the synthesis inequalities,
the size of Ac was arbitrary. The speci c construction of a controller in proving su ciency
leads to an Ac that has the same size as A. Hence Theorem 13.20 also include the side result
that controllers of order larger than that of the plant o er no advantage over controllers
that have the same order as the plant. The story is very di erent in reduced order control:
Then the intention is to include a constraint dim(Ac) k for some k that is smaller
than the dimension of A. It is not very di cult to derive the corresponding synthesis
inequalities however, they include rank constraints that are hard if not impossible to
treat by current optimization techniques.
then returns the optimal H1 performance in gopt and the optimal controller K in K.
The state space matrices (AcBcCcDc) which de ne the controller K are returned by
the command
[ac,bc,cc,dc] = ltiss(K).
Remark on strictly proper controllers. Note that the direct feed-through of
the controller Dc is actually not transformed we simply have Dc = N. If we intend to
design a strictly proper controller (i.e. Dc = 0), we can just set N =0to arrive at the
corresponding synthesis inequalities. The construction of the other controller parameters
remains the same. Clearly, the same holds if one wishes to impose an arbitrary more
re ned structural constraint on the direct feed-through term as long as it can be expressed
in terms of LMI's.
Remarks on numerical aspects. After having veri ed the solvability of the synthesis
inequalities, we recommend to take some precautions to improve the conditioning of

212 BIBLIOGRAPHY
[17] D.F. Enns, \Structured Singular Value Synthesis Design Example: Rocket Stabilisation",
American Control Conf., Vol.3, pp.2514-2520, 1990.
[18] M.A. Peters and A.A. Stoorvogel, \Mixed H2=H1 Control in a Stochastic Framework,"
Linear Algebra and its Applications, Vol. 205-206, pp. 971-996, 1984.
Bibliography
[1] J.M. Maciejowski, \Multivariable Feedback Design," Addison Wesley, 1989.
[2] J. Doyle, B. Francis, A. Tannenbaum, \Feedback Control Theory," McMillan Publishing
Co., 1990.
[3] M. Morari, E. Za riou, \Robust Process Control," Prentice Hall Inc., 1989.
[4] B.A. Francis, \A Course in H1 Control Theory," Lecture Notes in Control and Information
Sciences, Vol. 88, Springer, 1987.
[5] D.C. Mc.Farlane and K. Glover, \Robust Controller Design using Normalized Coprime
Plant Descriptions," Lecture Notes in Control and Information Sciences, Vol. 138,
Springer, 1990.
[6] A. Packard and J. Doyle, \The Complex Structured Singular Value," Automatica, Vol.
29, pp.71{109, January 1993.
[7] A. Weinmann, \Uncertain Models and Robust Control," Springer, 1991.
[8] I. Postlethwaite, \Robust Control of Multivariable Systems using H1 Optimization,"
Journal A, Vol 32, No. 4, pp 8{19, 1991.
[9] B.Ross Barmish, \New Tools fo Robustness of Linear Systems", Macmillan Publishing
Company,1994.
[10] Doyle and Stein,\Robustness with Observers", IEEE AC-24, no.4, August 1997.
[11] G.Zames and B.A.Francis,\Feedback ,Minimax Sensitivity and Optimal Robustness",IEEE
AC-28,no.5,May 1983.
[12] M. Green and D.J.N. Limebeer, \Linear Robust Control", Prentice Hall Information
and System Science Series, New Yersey, 1995.
[13] S.P. Bhattacharyya, H. Chapellat andL.H.Keel,\Robust Control: The Parametric
Approach ",Prentice Hall Information and Science Series, New Yersey, 1995.
[14] K. Zhou, with J.C. Doyle and K. Glover,\Robust and Optimal Control," Prentice
Hall Information and Science Series, New Yersey,1996.
[15] S. Skogestad and I. Postlethwaite,\Multivariable Feedback Control," John Whiley
and Sons, Chichester, 1996.
[16] S. Engell, \Design of Robust Control Systems with Time-Domain Speci cations",
Control Eng. Practice, Vol.3, No.3, pp.365-372, 1995.
211