Soft Computing Methods in Flight Control System Design

Soft Computing Methods in
Flight Control System Design
Marcel Oosterom

Soft Computing Methods in
Flight Control System Design
Proefschrift
ter verkrijging van de graad van doctor
aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof.dr.ir. J.T. Fokkema,
voorzitter van het College van Promoties,
in het openbaar te verdedigen op maandag 13 juni 2005,
om 15.30 uur door
Marcel Laurens Jean OOSTEROM
ingenieur luchtvaart en ruimtevaart
geboren te Deurne

Dit proefschrift is goedgekeurd door de promotor:
Prof. dr. R. Babuˇska, M.Sc.
Samenstelling promotiecommissie:
Rector Magnificus voorzitter
Prof. dr. R. Babuˇska, M.Sc. Technische Universiteit Delft, promotor
Prof. dr. ir. J.A. Mulder Technische Universiteit Delft
Prof. ir. H.B. Verbruggen Technische Universiteit Delft
Prof. dr. ir. M. Verhaegen Technische Universiteit Delft
Prof. R.J. Patton, Ph.D. The University of Hull
Dr. ir. G. Schram SKF Reliability Systems
K. Rosenberg, M.Sc. BAE SYSTEMS Avionics Limited
ISBN 90-8559-060-4
Copyright c○ 2005 by M. Oosterom.
No part of this publication may be reproduced or transmitted in any form or by any means,
electronic or mechanical, including photocopy, recording, or any information storage and retrieval
system, without permission in writing from the author.

to Dani

Contents
Summary xi
Samenvatting xvii
0 Notations and Abbreviations 1
0.1 List of Notations ............................ 1
0.2 List of Abbreviations .......................... 2
0.3CoordinateSystems .......................... 4
1 Introduction 7
1.1Background............................... 7
1.2 Problem statement ........................... 8
1.3Newdevelopments ........................... 9
1.4 Research aims and motivation of the methods used ......... 11
1.5Overviewofthethesis ......................... 14
2 Digital Fly-By-Wire Flight Control Systems 17
2.1HistoricalOverview........................... 17
2.2 Description of the Digital Fly-By-Wire Flight Control System . . . 19
2.3 Advantages and Disadvantages of Digital Fly-By-Wire ....... 21
3 Fuzzy Clustering for Partitioning of the Flight Envelope 25
3.1NonlinearControl ........................... 25
3.2 Fuzzy Clustering ............................ 27
3.3 Fuzzy Partitioning of the Flight Envelope .............. 28
3.4Conclusions............................... 40
4 Scheduled Classical Control 43
4.1 Stability and Control Augmentation System . . . .......... 43
4.2 Partition ................................. 46
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viii Contents
4.3 Automatic Tuning Procedure ..................... 46
4.4 The Scheduler .............................. 48
4.5 Evaluation ................................ 50
4.6Conclusions............................... 54
5 Scheduled Robust Multivariable Control 55
5.1 Introduction ............................... 55
5.2 Overview of Scheduled Robust MV Control ............. 56
5.3 General Description of the Robust Control Problem ........ 57
5.4 Robust Multivariable Flight Control Design ............. 59
5.5 Partition of the Flight Envelope .................... 66
5.6 Parameter Scheduled Robust Multivariable Control ......... 67
5.7Conclusions............................... 75
6 Virtual Angle-of-Attack Sensor 77
6.1 Introduction ............................... 77
6.2DesignRequirements.......................... 78
6.3 Structure of the Virtual Angle-of-Attack Sensor ........... 79
6.4 Design of the TS Fuzzy Model and the NN Model ......... 83
6.5 Validation of the Virtual AoA Sensor . . ............... 86
6.6Conclusions............................... 90
7 Soft Sensor Management and Virtual Sensors for FDIR 91
7.1 Introduction ............................... 91
7.2 Conventional Sensor Management and FCL Reconfiguration .... 92
7.3 Sensor Management and FCL Reconfiguration Based on Soft Computing..................................
97
7.4 Virtual Sensor for FDIR ........................ 104
7.5Conclusions............................... 109
8 Conclusions 111
8.1 Thesis Contributions .......................... 111
8.2 Efficiency Improvement of the Design of the System ........ 114
8.3 Enhancement of Flight Safety ..................... 115
8.4 Recommendations for Further Research ............... 115
A Synthetic Environment and Real-Time Code Generation 119
A.1SyntheticEnvironment......................... 119

A.2 Real-Time Code Generation ...................... 124
B Short-Period Approximation 125
B.1 Derivation of the Short-Period Approximation . . .......... 125
B.2 Derivation of Short-Period Related Equations . . .......... 126
C Performance Measures and Cross Validation 129
C.1PerformanceMeasures......................... 129
C.2 Cross Validation ............................ 130
D Soft Computing Techniques 131
D.1 Fuzzy Clustering ............................ 131
D.2 Identification via Fuzzy Clustering .................. 135
D.3NeuralNetworks ............................ 138
E Genetic Algorithms 143
E.1 Introduction ............................... 143
E.2HowDoTheyWork?.......................... 143
E.3 Theoretical Foundation ........................ 151
E.4 Application Areas ........................... 152
F Linear Matrix Inequalities for Control 153
F.1 Output-feedback H∞ ControlProblem................ 153
F.2LMIApproach ............................. 154
F.3PolePlacement............................. 156
Acknowledgements 159
Curriculum Vitae 161
List of Publications 163
Bibliography 165
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x Contents

Soft Computing Methods in
Flight Control System Design
Marcel Oosterom
Summary
The Digital Fly-By-Wire (DFBW) flight control system has many advantages over
the mechanical Flight Control System (FCS) in terms of weight, fuel consumption,
flexibility in the configuration of the bare airframe, etc. The most powerful feature
of the DFBW FCS is the wide range of capabilities that can be programmed into
the Flight Control Computer (FCC). The major drawback is the additional cost
associated with the design and initial acquisition of the system. Without a mechanical
back-up, the DFBW FCS is a safety-critical system. Stringent requirements
therefore apply with respect to the integrity and availability of the system. Fault
tolerance is achieved through hardware redundancy and dissimilarity among the
redundant hardware (and software) components is used to avoid common mode
failures.
For military (fighter) aircraft and large commercial aircraft, the advantages of
the DFBW FCS justify the additional cost of the system. However, this is not so
obvious for small commercial aircraft. The (relative) additional cost of the DFBW
FCS is much higher, while the advantages are not so evident as for large commercial
aircraft.
A radical change in current practices is required in order to bring DFBW
FCS technologies to the small commercial aircraft market at affordable cost whilst
maintaining the stringent safety requirements. The possibilities for a cost-effective
application of this technology to small commercial aircraft have been investigated
within the European project “Affordable Digital Fly-By-Wire Flight Control Systems
for Small Commercial Aircraft” (Phases I and II). Partners were involved
from industry, research institutes, and universities. The work described in this
thesis is for a large part performed within this project.
The aim of the research described in this thesis is to improve the efficiency of
flight control system design and/or to improve the flight safety. The improvement
of the efficiency of the FCS design results in reduced development cost, which is a
key factor in making DFBW technology affordable for small commercial aircraft.
xi

xii Summary
Besides the economical aspect, the improvement of flight safety is also an important
argument to justify bringing DFBW to the small commercial aircraft market.
In this thesis the focus has been put on two topics, namely:
1. Automated design techniques.
2. Sensor management, including virtual sensors.
The first item is specifically of interest with respect to nonlinear flight control
law design, while the second item focuses mainly on sensor management (voting/monitoring).
These two application areas are discussed in more detail in the
remainder of this section.
The applications described in this thesis are developed in the Synthetic Environment
(SE), which is defined as an ultra-high fidelity simulation tool that is
structurally representative of the practical implementation of a DFBW aircraft.
The SE has been developed within the ADFCS project.
Automated design techniques
Gain scheduling has been perhaps the most common systematic approach to control
of nonlinear systems in practice. In the aerospace industry, e.g. for the design
of the FCLs for commercial DFBW aircraft, and for a wide range of other application
areas. Even with the introduction of powerful control strategies such as model
predictive control and feedback linearization, gain scheduling remains an attractive
control strategy because of its simplicity and practical use. The design of the
scheduler is an iterative process, where each iteration consists of the identification
of the operating points, the tuning of the controller parameters, the design of the
scheduler and the evaluation of the global performance of the system. This is a
slow and costly procedure, however, despite this drawback almost no effort has
been spent on the development of a systematic approach to identify the operating
points and to design the corresponding scheduler.
In Chapter 3, an automated procedure is proposed for the identification of the
operating points for which the local flight control law parameters need to be tuned.
This procedure is based on the application of fuzzy clustering to a data set which
represents the variation of the dominant aircraft dynamics over the flight envelope.
The resulting cluster centers serve as the operating points. A singleton TS
fuzzy model is constructed to provide the interpolation mechanism for the Flight
Control Law (FCL) parameters in order to obtain a global nonlinear controller.
The main advantage of this approach is that the operating points are identified
simultaneously, in contrast to the iterative trail-and-error approach that is carried
out by the flight control engineer. Besides the reduction in the design effort with
respect to identifying the operating points, the fuzzy clustering approach results
in fewer operating points. This reduces the design effort for the tuning of the FCL
parameters. By using the same scheduling variables and operating points for all
FCL parameters that require scheduling, the scheduler becomes more transparent

and the (local) effects of parameter modifications are more predictable. This does
not necessarily mean that the best performance is achieved by scheduling all FCL
parameters in exactly the same way.
The fuzzy clustering approach should be considered as an interactive tool for
the flight control engineer. It supports the flight control engineer in deciding on
the number and the location of the operating points. The iterations in the design
procedure are automated, which simplifies the design procedure. It should be
noted that the identification of the operating points through fuzzy clustering does
not put any restrictions on the scheduling mechanism to be used.
The application of the automated design procedure to the classical FCLs that are
available in the SE is described in Chapter 4. The scheduler of the six most relevant
controller parameters is replaced by the scheduler designed by using the fuzzy
clustering approach. The selected scheduling variables are the Mach number and
the dynamic pressure. Eight operating points are identified for the flight envelope
in clean configuration and two for the flight envelope in landing configuration.
Pilot-in-the-loop simulations demonstrated that the performance of the FCLs designed
by using the automated design procedure is equivalent to the performance
of the default FCLs. Even though the fuzzy gain scheduler used fewer operating
points than in the conventional approach, it can be concluded that its performance
is comparable.
A significant contribution in reducing the DFBW development costs, without
compromising the safety requirements, can be obtained by replacing the classical
single-loop frequency response and root-locus design techniques by advanced
MultiVariable (MV) control design techniques for the development of the flight
control laws. The MV design approach has the advantage of reducing time and
cost for flight control design and refinement, and at the same time of a priori taking
into account robustness with respect to model uncertainties. In Chapter 5 the
local controllers are designed using robust MV control techniques. The challenge
is to combine the resulting MV controllers with gain scheduling, because of the
complexity and the opaque structure of such controllers. The combination of fuzzy
clustering and robust multivariable control is new and potentially very effective
in reducing the design effort since both the identification of the operating points
and the scheduler as well as the design of the local linear controllers are highly
automated.
The local H∞ controllers are designed using a model-matching approach. The
design is performed in continuous-time using Linear Matrix Inequalities (LMIs).
Order reduction is performed to reduce the number of parameters to be scheduled.
First a Hankel model reduction is applied and then the high-frequency modes are
removed from the controller. After Tustin discretization, the local controllers are
transformed to the δ-operator form to reduce their coefficient-pole sensitivity. The
latter is important with respect to gain scheduling.
The gain-scheduled robust MV controller has been evaluated off-line (linear
and nonlinear simulations, stability analysis) and through pilot-in-the-loop simulations.
The results of the off-line evaluation are satisfactory, although additional
xiii

xiv Summary
tuning is required to further improve the performance and stability characteristics.
The test pilots gave a Cooper-Harper (CH) rating of 1 in all flight conditions
concerning the aircraft dynamics. However, due to the high control force needed
to maneuver the aircraft, especially in the low dynamic pressure region, the overall
CH ratings were between 2 and 3. This can be corrected by adjusting the reference
model.
In conclusion, the contribution of the fuzzy clustering approach to improving the
efficiency of the design of flight control laws is significant. It is a global approach,
all the operating points are identified simultaneously, which results in a transparent
scheduling scheme with fewer operating points. Moreover, it is a model-based
approach that uses the nonlinear dynamics in the design phase and not only in
the evaluation phase. In combination with modern MV control techniques for the
design of the local controllers, the reduction in the design effort for the design of
the flight control laws is even more evident.
Sensor Management
Sensor management based on majority voting and point consolidation of like signals,
i.e. signals from different sensors measuring the same variable, is a proven
technology in modern fly-by-wire flight control systems. The assumption is that
the majority of like signals represents the truth and that any single dissimilar signal
is the result of a failure. Such a signal must be disconnected as soon as the
failure is detected. This principle fails in the event that there are two like signals
left (duplex operation), since there is no longer a majority. In the conventional approach,
the decision whether a sensor has failed or not is crisp. In order to reduce
the sensitivity of this decision to uncertainties like quantization and measurement
noise, a properly adjusted threshold is used.
Two opportunities have been identified to improve the flight safety (and comfort).
The introduction of a virtual sensor that makes use of non-like signals (analytical
redundancy) and can be used as an arbitrator during duplex operation.
Furthermore, an alternative sensor management procedure which improves the
consolidated signal and reduces failure-induced transients is introduced.
Many applications of analytical redundancy for FDI in flight control systems are reported.
Most frequently applied are observer-based techniques, parity-space methods,
and parameter-estimation schemes. The use of virtual sensors in aerospace
applications has not been widely investigated yet, although this technique has been
successfully applied in other fields like process control and engine control.
The approach investigated in Chapter 6 of this thesis is based on a combination
of white-box and black-box modelling and is demonstrated through the
design of a virtual angle-of-attack (AoA) sensor. The philosophy is to first achieve
the maximum performance out of a white-box modelling approach, using the well
known relations of the linearized aircraft dynamics. In the second step, the remaining
estimation error is further reduced by adding a black-box model that is
designed to fit the estimation error of the white-box model. The inputs of this

nonlinear, black-box model are determined using a nonlinear input selection approach.
The virtual AoA sensor makes use of another virtual sensor that estimates
the aircraft weight and the position of the center of gravity. This virtual sensor
does not make use of AoA sensor readings.
The performance of the virtual sensor is demonstrated by a large number of
nonlinear simulations for which the flight conditions and maneuvers are selected
randomly. The performance of the virtual sensor is good, with maximum estimation
errors for the angle-of-attack of less than 0.8 degrees.
The conventional sensor management system makes use of crisp thresholds. The
consolidated signal is computed by taking the (weighted) average of each sensor
reading. The mid-value signal is taken as a reference and the two extreme-value
signals are limited in their deviation from the mid-value (crisp threshold). The
monitor compares each sensor signal with the consolidated signal. If the absolute
difference exceeds a predefined (crisp) threshold, the corresponding monitor count
is increased. If the count value has reached the failure declaration value, a failure is
declared and the signal is latched. The locations of the thresholds is a compromise
between the minimization of false alarms and minimization of failure-induced transients.
While the first dictates large thresholds, the latter dictates small thresholds.
This compromise can be circumvented by introducing soft thresholds instead of
crisp thresholds, increasing flight safety and/or comfort.
Fuzzy logic is an excellent technique to implement this concept. Although these
techniques have been implemented in other application domains, such as the process
industry, their application in flight control systems has not been extensively
investigated yet. In Chapter 7 of this thesis a sensor management procedure
based on soft computing is proposed. In this soft sensor management system a
weight between (and including) zero and one (soft threshold) is assigned to each
sensor reading. This weight is computed based on the smallest absolute difference
between the other sensor readings. The consolidated signal is the weighted
average of the signals. When the weight of a sensor reading is equal to zero, the
corresponding signal is not contributing to the consolidated signal and the corresponding
monitor count is increased. The main difference from the conventional
monitoring scheme is that the monitor count rate is not a function of the difference
between the ith sensor reading and the consolidated signal, but a function of the
difference between the ith sensor reading and the other like sensor readings.
The soft sensor management system including a normal acceleration virtual
sensor has been demonstrated by means of closed-loop simulation examples in the
SE and on the flight simulator with pilot-in-the-loop simulations. In the conventional
approach, signal consolidation and signal monitoring are performed separately.
With the introduction of soft thresholds, signal consolidation and sensor
monitoring are integrated. This results in a more accurate consolidated signal and
reduces the failure induced transients, which contributes not only to flight safety,
but also to passenger comfort. Furthermore, it is demonstrated how virtual sensors
can be used to identify the faulty sensor in the case of a discrepancy between two
like sensor signals. When also the last sensor fails, the signal is no longer available
and the FCLs reconfigure to not using this signal. Furthermore, the FCL reconfiguration
is smoother as a direct result of the soft sensor management strategy.
xv

xvi Summary
In conclusion, the contribution of the soft computing to increase flight safety is
significant. With a virtual sensor it is possible to identify the faulty signal when a
discrepancy is detected between the signals from two physical like sensors. Moreover,
it is possible to detect a failure on the last remaining physical sensor. The
virtual sensor increases the integrity of the FCS and therefore contributes to flight
safety. Clearly virtual sensors can be designed using other techniques as well.
However, soft computing allows the designer to use well-known linear techniques
to create a global nonlinear system, which adds to the accuracy and reliability of
the virtual sensor. The contribution of soft sensor management to flight safety is
less significant than for the virtual sensor, but it does improve the system at no
extra cost.
Future research is suggested in the direction of optimization of the scheduling
mechanism for the FCL parameters making use of genetic algorithms, after the
operating points have been identified through fuzzy clustering. Furthermore, a
more integrated design of the scheduled robust MV controller is suggested, where
the tuning of the weighting functions and the scheduling mechanism are optimized
in an iterative manner taking into account global stability and performance
requirements. Still an open issue is the FCL reconfiguration when using robust MV
controllers. With respect to virtual sensors, a topic that needs further attention is
how to integrate the virtual sensor in the sensor management system and how to
deal with the limitations of a virtual sensor.

Soft Computing Methods in
Flight Control System Design
Marcel Oosterom
Samenvatting
Het Digitale Fly-By-Wire vliegtuigbesturingssysteem (DFBW) heeft vele voordelen
ten opzichte van mechanische besturing met betrekking tot gewicht, brandstofverbruik,
flexibiliteit in de vliegtuigconfiguratie, enzovoort. De sterkste eigenschap
van DFBW is de uitgebreide functionaliteit die in de besturingscomputer
geprogrammeerd kan worden. Het voornaamste nadeel van DFBW wordt gevormd
door de extra kosten die zijn gemoeid met het ontwerp en de initiële aankoop. Zonder
de aanwezigheid van een mechanisch back-up systeem is de beschikbaarheid
en goede werking van DFBW essentieel voor de veilige operatie van het vliegtuig.
Strenge eisen gelden derhalve voor de integriteit en beschikbaarheid van
het systeem. Tolerantie ten aanzien van defecten (fault tolerance) wordt bereikt
door middel van hardware redundantie. Om zogenaamde common mode storingen
te voorkomen, wordt ongelijkheid tussen redundante hardware (en software)
toegepast.
Voor militaire (jacht)vliegtuigen en grote passagiersvliegtuigen zijn de voordelen
van DFBW dermate groot, dat de extra kosten van het systeem gerechtvaardigd
zijn. Dit is echter niet zo duidelijk het geval voor kleine passagiersvliegtuigen.
De (relatieve) extra kosten van DFBW zijn vele malen groter voor kleine passagiersvliegtuigen,
terwijl aan de andere kant de voordelen niet zo overtuigend
zijn als voor grote passagiersvliegtuigen. Bij militaire (jacht)vliegtuigen spelen de
kosten een veel kleinere rol en gaat het voornamelijk om de extra functionaliteit.
Om DFBW besturingstechnologie tegen redelijke kosten op de markt voor
kleine passagiersvliegtuigen te kunnen brengen, waarbij de strenge veiligheidseisen
worden gehandhaafd, is een fundamentele verandering in het huidige ontwerpproces
noodzakelijk. In het Europese project genaamd Affordable Digital Fly-By-Wire
Flight Control Systems for Small Commercial Aircraft (ADFCS) zijn de mogelijkheden
voor een rendabele toepassing van DFBW in kleine passagiersvliegtuigen
onderzocht. Partners uit de industrie, onderzoekscentra en universiteiten hebben
aan dit project deelgenomen.
xvii

xviii Samenvatting
Het werk dat in dit proefschrift is beschreven is voor een groot deel uitgevoerd
binnen dit project. Het had als doel mogelijkheiden te onderzoeken om het ontwerpen
van vliegtuigbesturingssystemen efficiënter te maken en/of de vliegveiligheid
te verbeteren. Een grotere efficiëntie bij het ontwerpen van het vliegtuigbesturingssysteem
resulteert in lagere ontwikkelingskosten, hetgeen een belangrijk
aspect is met betrekking tot het economisch haalbaar maken van DFBW
voor kleine passagiersvliegtuigen. Naast het economische aspect is de verbetering
van de vliegveiligheid een belangrijk argument om de extra kosten te rechtvaardigen.
In dit proefschrift is de nadruk gelegd op twee onderwerpen, namelijk:
1. Geautomatiseerde ontwerptechnieken.
2. Sensor management, inclusief virtuele sensoren.
Het eerste onderwerp is specifiek van belang bij het ontwerpen van de niet-lineaire
vliegtuigregelaar, terwijl bij het tweede onderwerp de aandacht voornamelijk wordt
gericht op voting en monitoring van sensorsignalen. Deze twee toepassingsgebieden
worden in de rest van deze samenvatting nader besproken.
De toepassingen die in dit proefschrift zijn beschreven, zijn ontwikkeld in het
zogenaamde Synthetic Environment (SE), welke is gedefinieerd als een zeer natuurgetrouwe
simulatie tool die structureel representatief is voor de praktische
implementatie van een DFBW vliegtuig. Het SE is ontwikkeld binnen het ADFCS
project.
Geautomatiseerde ontwerptechnieken
Gain scheduling is wellicht de meest toegepaste systematische methode in de praktijk
voor het regelen van niet-lineaire systemen. In de luchtvaartindustrie wordt
het bijvoorbeeld gebruikt voor het ontwerpen van de vliegtuigregelaar, maar er
zijn vele andere toepassingsgebieden. Zelfs met de introduktie van krachtige regelontwerptechnieken
zoals model predictive control en feedback linearisatie, blijft gain
scheduling een aantrekkelijke regelstrategie omdat het simpel is en praktisch in het
gebruik. Het ontwerp van de scheduler, met andere woorden het interpolatiemechanisme,
is een iteratief proces, waarbij iedere iteratie bestaat uit het identificeren
van de ontwerppunten, het afstellen van de regelparameters, het ontwerpen van
de scheduler en het evalueren van de globale prestaties van het systeem. Dit een
tijdrovend en kostbaar proces, maar ondanks dit nadeel is er tot op heden nauwelijks
aandacht besteed aan het ontwikkelen van een systematische methode voor
het identificeren van de ontwerppunten en het ontwerpen van de bijbehorende
scheduler.
In Hoofdstuk 3 wordt een geautomatiseerde procedure voor de identificatie van
ontwerppunten geïntroduceerd, waarin de parameters van de vliegtuigregelaar
moeten worden afgesteld. Deze procedure is gebaseerd op het toepassen van fuzzy
clustering op een data set die de verandering van de dominante vliegtuigdynamica
in het operationele vlieggebied representeert. De centra van de daaruit voort-

komende clusters dienen als ontwerppunten. Een singleton TS fuzzy model wordt
geconstrueerd voor de interpolatie van de parameters van de vliegtuigregelaar om
zo een globaal niet-lineaire regelaar te verkrijgen.
Het voornaamste voordeel van deze methode is dat de ontwerppunten gelijktijdig
geïdentificeerd worden, hetgeen niet het geval is bij de iteratieve trial-anderror
methode die meestal wordt toegepast door vliegtuigregeltechnici. Naast de
vermindering van de ontwerpinspanning met betrekking tot het identificeren van
de ontwerppunten, resulteert de fuzzy clustering methode ook in een kleiner aantal
ontwerppunten. Dit resulteert wederom in minder inspanning voor het afstellen
van de parameters van de vliegtuigregelaar. Door dezelfde scheduling variabelen
en ontwerppunten te gebruiken voor alle parameters van de vliegtuigregelaar die
onderhevig zijn aan scheduling, wordt de scheduler transparanter en zijn de locale
effecten van parameter modificaties voorspelbaarder. Dit betekent overigens niet
dat de beste prestaties worden behaald door alle parameters op exact dezelfde
manier te interpoleren.
De fuzzy clustering methode moet gezien worden als een interactief gereedschap.
Het ondersteunt vliegtuigregeltechnici bij het bepalen van het aantal en de
locatie van de ontwerppunten. De iteraties in de ontwerpprocedure zijn geautomatiseerd,
hetgeen de ontwerpprocedure versimpelt. De identificatie van de ontwerppunten
met behulp van fuzzy clustering legt geen restricties op met betrekking tot
het te gebruiken scheduling mechanisme.
De toepassing van de geautomatiseerde ontwerpprocedure op de klassieke vliegtuigregelaar
die beschikbaar is in het SE wordt beschreven in Hoofdstuk 4. De scheduler
van de zes meest relevante regelparameters is vervangen door de scheduler
die is ontworpen door middel van de fuzzy clustering methode. De geselecteerde
scheduling variabelen zijn het Mach getal en de dynamische druk. Acht ontwerppunten
zijn geïdentificeerd voor het operationele vlieggebied in de kruisvluchtconfiguratie
en twee ontwerppunten voor het operationele vlieggebied in de landingsconfiguratie.
Simulaties met de piloot in de lus tonen aan dat de prestaties van
de vliegtuigregelaar ontworpen met de geautomatiseerde ontwerpmethode equivalent
zijn aan de prestaties van de vliegtuigregelaar die als standaard dient in het
ADFCS project. Dit ondanks het feit dat de vliegtuigregelaar die is ontworpen
met fuzzy clustering minder ontwerppunten gebruikt.
Een significante bijdrage aan de kostenreductie van een DFBW vliegtuigbesturingssysteem,
zonder daarbij de veiligheidseisen te schenden, kan worden bereikt
door de klassieke single-loop frequentie respons en root-locus ontwerptechnieken
te vervangen voor geavanceerde Multi-Variabele (MV) technieken voor het ontwerpen
van de vliegtuigregelaar. De MV regelontwerptechniek heeft als voordeel
het reduceren van tijd en kosten bij het ontwerpen en verfijnen van de vliegtuigregelaar,
terwijl tegelijkertijd rekening wordt gehouden met de robuustheid
met betrekking tot model onzekerheden. In Hoofdstuk 5 zijn de lokale regelaars
ontworpen met behulp van een robuuste MV regelontwerptechniek. De uitdaging
is om gain scheduling toe te passen op de resulterende MV regelaars, omdat dergelijke
regelaars geen fysisch interpreteerbare en/of duidelijke structuur hebben. De
xix

xx Samenvatting
combinatie van fuzzy clustering en robuuste MV regelontwerptechnieken is nieuw
en heeft de potentie om zeer effectief de ontwerpinspanning te reduceren doordat
zowel de identificatie van de ontwerppunten als het ontwerpen van de lokale lineaire
regelaars in grote mate geautomatiseerd zijn.
De lokale H∞ regelaars zijn ontworpen met behulp van de model-matching
methode. Het ontwerpen is uitgevoerd in continue tijd, gebruik makend van Linear
Matrix Inequalities (LMIs). De orde van de lokale regelaars wordt gereduceerd om
het aantal parameters voor het scheduling mechanisme te reduceren. Eerst wordt
er een Hankel model reductie uitgevoerd en vervolgens worden de hoogfrequente
polen en nulpunten verwijderd uit de regelaars. Na Tustin discretisatie wordt de
regelaar omgezet in de δ-operator vorm om de zogenaamde pool-coefficïent gevoeligheid
van de regelaar verder te reduceren. Dat laatste is erg belangrijk in verband
met gain scheduling.
De gain-scheduled robuuste MV regelaar is zowel off-line (lineaire en nietlineaire
simulaties en stabiliteitsanalyse) en on-line geëvalueerd door middel van
simulaties met de piloot in de lus. De resultaten van de off-line analyse zijn bevredigend,
alhoewel aanvullende afstelling nodig is om de prestaties en de stabiliteitseigenschappen
verder te verbeteren. De testpiloten gaven een Cooper-Harper (CH)
rating van 1 in alle vliegcondities voor de vliegtuigdynamica. Echter, door de
hoge stuurkrachten die nodig waren om het vliegtuig te manoeuvreren, vooral in
het vlieggebied met lage dynamische druk, variëerden de uiteindelijke CH ratings
tussen de 2 en 3. De benodigde stuurkracht kan eenvoudig gecorrigeerd worden
door het referentiemodel aan te passen.
De bijdrage van de fuzzy clustering methode om de efficïentie van het ontwerp
van vliegtuigregelaars te verbeteren is significant. Het is een globale techniek,
alle ontwerppunten worden gelijktijdig geïdentificeerd, hetgeen resulteert in een
transparant scheduling mechanisme met minder ontwerppunten. Bovendien is het
een modelgebaseerde techniek die de niet-lineaire dynamica gebruikt in de ontwerpfase
en niet alleen tijdens de evaluatie. In combinatie met moderne MV
regelontwerptechnieken voor het ontwerpen van de lokale regelaars is de reductie
in de ontwerpinspanning voor het ontwerpen van vliegtuigregelaars nog groter.
Sensor Management
Sensor management gebaseerd op majority voting en consolidatie van gelijksoortige
signalen, dat wil zeggen signalen van verschillende sensoren die dezelfde variabele
meten, is een bewezen technologie in moderne DFBW vliegtuigbesturingssystemen.
De aanname die daarbij wordt gemaakt, is dat de meerderheid van de gelijksoortige
signalen de waarheid representeren en dat een enkele afwijkend signaal het
gevolg is van een storing. Een dergelijk signaal moet ontkoppeld worden zodra de
storing gedetecteerd is. Dit principe faalt in het geval dat er nog maar twee gelijksoortige
signalen beschikbaar zijn (duplex operatie), omdat er in dat geval geen
sprake van een meerderheid kan zijn. In de conventionele methode is de beslissing
of er wel of niet een storing is opgetreden eenduidig, met andere woorden er is wèl
of geen storing. Om de gevoeligheid van deze beslissing te verminderen met het
oog op onzekerheden zoals quantisatie en meetruis, wordt een zorgvuldig afgestelde

drempelwaarde gebruikt.
Twee mogelijkheden zijn geïdentificeerd om de vliegveiligheid (en het vliegcomfort)
te verbeteren. Ten eerste is er de introductie van een virtuele sensor
die gebruik maakt van ongelijksoortige signalen (analytische redundantie) en die
gebruikt kan worden als een arbitrator tijdens duplex operatie. Daarnaast is een
alternatieve sensor management procedure geïntroduceerd die het geconsolideerde
signaal verbetert en storingsgeïnduceerde overgangsbewegingen verkleint.
Vele toepassingen van analytische redundantie voor Fout Detectie en Isolatie (FDI)
in vliegtuigbesturingssystemen zijn gerapporteerd. Meestal worden observergebaseerde
technieken, parity-space methoden en parameterschattingsschema’s gebruikt.
De toepassing van virtuele sensoren in de luchtvaartindustrie is nog niet
goed onderzocht, alhoewel deze techniek succesvol is gebleken in andere toepassingsgebieden
zoals procesbesturing en motorbesturing.
De methode die is onderzocht in Hoofdstuk 6 van dit proefschrift is gebaseerd
op de combinatie van white-box en black-box modellering en wordt gedemonstreerd
aan de hand van het ontwerp van een virtuele invalshoek sensor. De werkwijze die
hierbij is gebruikt is om eerst de maximale prestatie uit de white-box modelleringsmethode
te halen, gebruik makend van bekende, lineaire vliegtuigdynamica. In de
tweede stap wordt de resterende schattingsfout verkleind door een black-box model
toe te voegen dat is ontworpen om de schattingsfout van het white-box model te
modelleren. De ingangen van dit niet-lineaire, black-box model worden bepaald
met behulp van een niet-lineaire ingangsselectie procedure. De virtuele invalshoeksensor
maakt gebruik van een andere virtuele sensor die het vliegtuiggewicht en
de positie van het zwaartepunt van het vliegtuig schat. Deze virtuele sensor maakt
geen gebruik van het invalshoeksignaal zelf.
De prestatie van de virtuele sensor is gedemonstreerd door middel van een groot
aantal niet-lineaire simulaties waarvan de vliegcondities en manoeuvres voor iedere
simulatie opnieuw willekeurig werden geselecteerd. De prestatie van de virtuele sensor
is goed, met een maximale schattingsfout van de invalshoek van minder dan
0,8 graden.
Het conventionele sensor management systeem maakt gebruik van harde drempelwaarden.
Het geconsolideerde signaal wordt berekend door het (gewogen) gemiddelde
te nemen van alle gelijksoortige sensor signalen. De middelste waarde wordt
als referentie genomen en de afwijking van de twee uiterste signalen met betrekking
tot de referentie waarde wordt gelimiteerd (harde drempelwaarde). De
monitor vergelijkt elk signaal met het geconsolideerde signaal. Als het absolute
verschil een bepaalde van tevoren gedefinieerde drempelwaarde overschrijdt, wordt
de bijbehorende monitor teller bij elke computer cyclus verhoogd. Als de teller de
failure declaration value heeft bereikt (bijvoorbeeld na tien cycli), wordt er een
storingsmelding gegeven en wordt het betreffende signaal afgesloten. De posities
van de drempelwaarden is een compromis tussen het willen minimaliseren van het
aantal valse storingsmeldingen en het minimaliseren van de storingsgeïnduceerde
overgangsbeweging. Het eerste vereist een grote drempelwaarde, terwijl het laatste
juist een kleine drempelwaarde voorschrijft. Dit compromis kan worden omzeild
xxi

xxii Samenvatting
door vage drempelwaarden in plaats van harde drempelwaarden te gebruiken en
op die manier de vliegveiligheid en/of het comfort te verhogen.
Fuzzy Logic (FL) is een uitstekende techniek om dit concept te implementeren.
Alhoewel FL technieken in andere gebieden veelvuldig zijn toegepast, zoals in de
procesindustrie, is de toepassing ervan in de vliegtuigindustrie nog niet uitgebreid
bestudeerd. In Hoofdstuk 7 van dit proefschrift is een sensor management procedure
voorgesteld die is gebaseerd op soft computing. In dit zogenaamde soft sensor
management systeem krijgt elk sensor signaal een gewicht tussen (en inclusief) nul
en één (vage drempelwaarde) die wordt berekend met behulp van het kleinste absolute
verschil met betrekking tot de andere gelijksoortige sensor signalen. Wanneer
het gewicht van een sensor signaal gelijk aan nul is, wordt het betreffende signaal
niet meer meegenomen bij het berekenen van het geconsolideerde signaal en
wordt de betreffende monitor teller bij elke computer cyclus verhoogd. Het verschil
tussen het conventionele en het soft sensor management systeem is dat bij
het conventionele systeem de monitor teller wordt geactiveerd op basis van het
verschil tussen het ide sensor signaal en het geconsolideerde signaal, terwijl dit bij
het soft sensor management systeem gebeurt op basis van het verschil tussen het
ide sensor signaal en de andere gelijksoortige sensor signalen.
Het soft sensor management systeem, inclusief een virtuele normaalversnellingssensor,
is gedemonstreerd door middel van closed-loop simulatie voorbeelden in
het SE en door middel van vliegsimulaties met de piloot in de lus. Bij de conventionele
methode worden signaal consolidatie en signaal monitoring gescheiden
uitgevoerd. Met de introductie van vage drempelwaarden zijn deze twee functies
geïntegreerd. Dit resulteert in een nauwkeuriger geconsolideerd signaal en
reduceert de storingsgeïnduceerde overgangsbewegingen, hetgeen niet alleen bijdraagt
aan de vliegveiligheid, maar ook aan het passagierscomfort. Verder is
getoond hoe virtuele sensoren gebruikt kunnen worden om foutieve sensoren te
identificeren voor het geval dat er een discrepantie is tussen twee gelijksoortige
sensor signalen. Als er ook een fout optreedt in de laatste sensor, is het signaal
niet langer beschikbaar en wordt de vliegtuigregelaar zodanig gereconfigureerd dat
het betreffende signaal niet langer nodig is. Ook de reconfiguratie van de vliegtuigregelaar
verloopt soepeler ten gevolge van de soft sensor management strategie.
In conclusie, de bijdrage van soft computing voor het verbeteren van de vliegveiligheid
is significant. Met een virtuele sensor is het mogelijk om foutieve signalen
te identificeren wanneer er een discrepantie is tussen twee signalen van
twee fysieke gelijksoortige sensoren. Bovendien is het mogelijk om een fout te
detecteren bij de laatste overgebleven fysieke sensor. De virtuele sensor verhoogd
de betrouwbaarheid van het vliegtuigbesturingssysteem en derhalve de vliegveiligheid.
Virtuele sensoren kunnen ook met andere technieken ontworpen worden,
echter, met behulp van FL is het mogelijk om lineaire technieken te gebruiken
voor het ontwerpen van een globaal niet-lineair systeem, hetgeen bijdraagt aan
de accuratesse en betrouwbaarheid van de virtuele sensor. De bijdrage van soft
sensor management voor de vliegveiligheid is minder groot dan de bijdrage van de
virtuele sensor, maar het levert een verbetering van het systeem op meerprijs.

In Hoofdstuk 8 wordt onderzoek voorgesteld in de richting van het optimaliseren
van het scheduling mechanisme voor de parameters van de vliegtuigregelaar, nadat
de ontwerppunten zijn geïdentificeerd met behulp van fuzzy clusteren. Verder
wordt voorgesteld om de mogelijkheid van een meer geïntegreerd ontwerp van
de gain-scheduled robuuste multivariabele regelaar te onderzoeken, waarbij het
afstellen van de gewichten en het scheduling mechanisme worden geoptimaliseerd
met behulp van een (geautomatiseerd) iteratief proces en waarbij globale stabiliteit
en prestatievoorschriften worden meegenomen. De reconfiguratie van de vliegtuigregelaar
bij het gebruik van robuuste multivariabele regelsystemen vereist nog
veel onderzoek. Ten slotte dient meer aandacht besteed te worden aan de vraag
hoe de virtuele sensor het best geïntegreerd kan worden in het (soft) sensor management
systeem en hoe het beste omgegaan kan worden met de beperkingen van
de virtuele sensor.
xxiii

xxiv Samenvatting

0.1 List of Notations
0.1.1 General Parameters
Notations and Abbreviations
c number of clusters
g gravitational acceleration [m s−2 ]
m fuzziness exponent
n number of states
number of inputs
ni
no
number of outputs
Nr number of rules
s Laplace operator
w weight factor
Greek symbols
β firing degree
∆ threshold / difference indication
∆ model uncertainty block
µ membership degree
ρ air density [kg m−3 ]
ω natural frequency [rad s−1 ]
ζ natural damping
Superscripts
T transposed
0.1.2 Aircraft Related Parameters
b wing span [m]
c rate-of-climb [ft min −1 ]
c mean aerodynamic chord [m]
h altitude [ft]
1
0

2 Chapter 0. Notations and Abbreviations
m aircraft mass [lbs]
M Mach number
ny lateral acceleration [m s −2 ]
nz normal acceleration [m s −2 ]
p roll rate [deg s −1 ]
q pitch rate [deg s −1 ]
q dynamic pressure [mbar]
r roll rate [deg s −1 ]
S surface area [m 2 ]
u forward speed [knots]
V velocity [knots]
VC calibrated airspeed [knots]
VT true airspeed [knots]
w downward speed [m s −1 ]
W aircraft weight [lbs]
Greek symbols
α angle-of-attack [deg]
β sideslip angle [deg]
δ (control surface) deflection [deg]
γ flight-path angle [deg]
φ bank angle [deg]
θ pitch attitude [deg]
ψ heading [deg]
χ tracking angle [deg]
Subscripts
a aileron
b body-axes system
c column
e elevator
fl flap
ph phugoid motion
r rudder
s stabilator
sl slat
sp short-period motion
th throttle
v vehicle-carried axes system
0.2 List of Abbreviations
ACE Actuator Control Electronics
ADC Air Data Computer
ADFCS Affordable Digital fly-by-wire Flight Control System
AIM Aircraft Inertial Matrix

0.2. List of Abbreviations 3
ANN Artificial Neural Network
AoA Angle-of-Attack
CAP Control Anticipation Parameter
CC Clean Configuration
CG Center-of-Gravity
CGS Conventional Gain Scheduler
CH Cooper-Harper
DEL Direct Electrical Link
DFBW Digital Fly-By-Wire
DOF Degree-Of-Freedom
EFCS Electronic Flight Control System
EGPWS Enhanced Ground Proximity Warning System
EVM Error Validity Measure
FAR Federal Aviation Regulations
FC Flight Condition
FCC Flight Control Computer
FCL Flight Control Law
FCS Flight Control System
FDI Fault Detection and Isolation
FDIR Fault Detection, Isolation and Reconfiguration
FEPS Flight Envelope Protection System
FGS Fuzzy Gain Scheduling
FHV Fuzzy Hyper-Volume
FL Fuzzy Logic
GA Genetic Algorithm
GK Gustafson-Kessel
GM Gain Margin
GS Gain-Scheduled
HIRF High Intensity Radiated Field
HW HardWare
HQR Handling Quality Requirement
ISA International Standard Atmosphere
JAR Joint Aviation Requirements
LC Landing Configuration
LG Landing Gear
LMI Linear Matrix Inequality
LPV Linear Parameter-Varying
LS Least Squares
mac mean aerodynamic chord
MF Membership Function
MOSAIC Model-Oriented Software Automatic Interface Converter
MV MultiVariable
NLR National Aerospace Laboratory
NN Neural Network
PM Phase Margin
PROSIM Programme and Real-time Operations SIMulation
RFS Research Flight Simulator

4 Chapter 0. Notations and Abbreviations
RMSE Root Mean-Squared Error
ROC Rate Of Climb
SCA Small Commercial Aircraft
SCAS Stability and Control Augmentation System
SE Synthetic Environment
SFENA Société Française d’ Équipement pour la Navigation Aérienne
SP Short-Period
SW SoftWare
SXB Xie-Beni validity measure
TS Takagi-Sugeno
VAF Variance Accounted For
VS Virtual Sensor
VSTOL Very Short Take-Off and Landing
WCD Within-Cluster Distance
0.3 Coordinate Systems
Several reference systems are being used in aeronautics. One of the most important
is the Earth axis system (XE,YE,ZE), whose origin is fixed at the center of the
Earth. The XE is pointing north, the YE axis is pointing east and the orthogonal
triad is completed with the ZE pointing down. This reference system is primarily
used to express gravitational effects, altitude, horizontal distance and the orientation
of the aircraft. Furthermore, it serves and as a basic frame of reference, to
which any other axis frames are referred.
Also the aircraft itself must have a suitable axis system and the choice of this
axis system dictates the from taken by the equations of motion. Parallel to the
Earth reference system, but with the origin in the center-of-gravity of the aircraft
is the vehicle-carried reference system (XV ,YV ,ZV ), see also Figures 0.1 to 0.3.
Figure 0.1: Side view of the SCA model.
However, by using a system of axes fixed in the aircraft the inertia terms, which

0.3. Coordinate Systems 5
Figure 0.2: Top view of the SCA model.
appear in the equations of motion, can be considered constant. The body axis
system (XB,YB,ZB) is such an axis system and is therefore used to govern the
equations of motion. The origin of the body axis system coincides with the centerof-gravity
of the aircraft and has the XB axis pointing forward out of the nose of
the aircraft, the YB axis pointing out through starboard and the ZB axis pointing
down (see Figure 0.1 to 0.3). In this axis system the aerodynamic forces and
moments depend only upon the angles α and β, which orient the vector of the
true airspeed VT in relation to the body axis system.

6 Chapter 0. Notations and Abbreviations
Figure 0.3: Front view of the SCA model.

1
Introduction
In this chapter a brief description of digital fly-by-wire flight control systems
is given, together with its advantages and disadvantages with respect to the
mechanical flight control system. The problem statement is described and a
range of potential solutions are addressed. Some of those are investigated in
this thesis and are therefore discussed in more detail. The chapter concludes
with an outline of the remainder of the thesis.
This chapter is organized as follows: A brief background in the history of flight
control is given in Section 1.1 (see Chapter 2 for a more elaborate discussion on
digital fly-by-wire flight control systems). In Section 1.2 the problems involved with
the introduction of Digital Fly-By-Wire (DFBW) to small commercial aircraft are
discussed. Section 1.3 describes the new developments that are currently taking
place to reduce the cost of DFBW. The research aim and the motivation of the
methods used in this thesis are described in Section 1.4. In Section 1.5 an overview
of the thesis is given.
1.1 Background
The first powered flight took place on December 17, 1903 at Kitty Hawk, North
Carolina. In the first decades of aviation the pilot controlled the aircraft directly
through the application of manual force. As engine power and speeds increased,
more force was needed to move the control surfaces and hydraulically-boosted control
systems emerged. As the electronic era grew in the 1950’s, so did the idea of
aircraft with Electronic Flight Control Systems (EFCSs). Full authority electronic
flight control systems forced the issues of safety, availability and integrity to be
addressed. The EFCS now had the capability to hazard the aircraft under fault
or failure conditions and became subject to the airworthiness requirements. The
first generation systems were based on analog technology but, more importantly,
retained the hydraulic-mechanical system as a back-up. Later generations abandoned
the hydraulic-mechanical back-up to become full-time fly-by-wire.
The introduction of DFBW in the primary Flight Control System (FCS) was a
landmark development in the aerospace industry. The possibility to decouple the
7

8 Chapter 1. Introduction
Figure 1.1: Digital fly-by-wire flight control system (Source: (Collinson 1999)).
requirements for flight stability from the basic airframe configuration is especially
useful for military (fighter) aircraft. The first test of a DFBW system in an aircraft
took place in 1972 on a modified F-8 Crusader at the Dryden Flight Research
Center. Further advantages of DFBW technology are increased functionality, safety
and maintainability. These features make this technology also attractive for commercial
aircraft. The Airbus 320 is the first commercial aircraft with DFBW for
the primary FCS and is certified in 1988.
The DFBW FCS is illustrated in Figure 1.1 in simple diagrammatic form. The
core of the system is the Flight Control Computer (FCC). The output of the FCC
is the commanded control surface deflection, which is computed based on the pilot
command and the aircraft motion and air data sensor information. The actuator
drives the control surface to the commanded position through the actuator control
electronics. The corresponding aircraft response is again sensed by the motion and
air data sensors and is fed back to the FCC.
The DFBW FCS has many advantages over the mechanical FCS in terms of weight,
fuel consumption, flexibility in the configuration of the bare airframe, etc. The most
powerful feature of the DFBW FCS is the wide range of capabilities that can be
programmed into the FCC. Examples are stability and control augmentation, flight
envelope protection and flight control law reconfiguration. However, the design,
testing and certification of such systems is costly. It is up to the manufacturer
to determine which of these possible capabilities have enough added value to be
included in the FCS.
1.2 Problem statement
The major drawback of DFBW is the additional cost associated with design and
initial acquisition of the system compared to its mechanical counterpart. The additional
cost is for a large part related to (dissimilar) hardware replication, development
of the flight control system and the generation of flight safety critical
software.
The DFBW FCS without mechanical back-up is a safety critical system, there-

1.3. New developments 9
fore, stringent requirements are put on the integrity and availability of this system.
Fault tolerance is achieved by installing redundant sensors, flight control computers,
actuators and power supplies. Moreover, dissimilarity among the redundant
hardware and software is used to avoid common mode failures.
Flight safety critical software has to be developed and tested extensively while
following strict and elaborate procedures. In order to reduce the risk of generic
failures, typically the SW is developed separately for each FCC by different companies.
The development of the stability and control augmentation system, sensor and
actuator monitoring system, control reconfiguration system, flight envelope protection
system, etc., which are all incorporated in the flight control computer,
accounts for a large portion of the development cost of the aircraft. The reason for
this is the increased complexity of these systems, while the design and validation
methods that are used in the aerospace industry lag behind with the state-of-theart
technologies.
For military (fighter) aircraft and large commercial aircraft, the advantages of
the DFBW FCS justify the additional cost of the system. However, this is not so
obvious for small commercial aircraft. The (relative) additional cost of the DFBW
FCS is much higher for small commercial aircraft, while the advantages are not so
evident as for large commercial aircraft.
1.3 New developments
A radical challenge to current practices is required in order to bring DFBW FCS
technologies to the small commercial aircraft market at affordable cost whilst maintaining
the stringent safety requirements. The possibilities for a cost-effective application
of this technology to small commercial aircraft have been investigated within
the European project “Affordable Digital Fly-by-wire Flight Control Systems for
Small Commercial Aircraft” (ADFCS). Partners were involved from industry, research
institutes, and universities. The following topics have been identified as
having potential to reduce the cost of fly-by-wire flight control systems for small
commercial aircraft.
Novel design techniques
Classical design techniques are typically based on a single-loop iterative procedure
which is labor-intensive and therefore expensive. The application of novel, more
automated, design and validation techniques have great potential to reduce the
design effort of, for example, the flight control laws. In this respect a well defined
set of design requirements, such as the handling qualities requirements, is essential.
Handling qualities requirements
The military set of Handling Quality Requirements (HQRs) clearly distinguish
between the handling qualities for large transport aircraft and small fighter aircraft.
The civil equivalent is not in existence. There are HQRs that have been
developed for large civil aircraft but these have not been validated or re-derived

10 Chapter 1. Introduction
for small commercial aircraft applications. In order to ensure the efficiency of the
control law generation process it is important to have a clear and complete set of
requirements against which the process can be invoked.
Synthetic Environment
There is an increasing trend towards the systems engineering approach to testing
and validating top level conceptual designs through simulation. The purpose is
to identify potential problems at an early stage such that solutions can be found
prior to manufacture, avoiding time and cost penalties associated with rework
later in the life-cycle. Wherever possible reliance will be placed on the use of the
Synthetic Environments (SE) to design, develop, and assess options. This is considered
to be extremely cost-effective in the development and integration of large,
multi-component systems.
Pilot monitor
Although more related to safety than affordability, a relatively large number of
safety-related incidents recorded for small aircraft applications are attributed to
pilot error or omission. The feasibility of applying an advanced technology solution
to the task of monitoring the pilot actions and the aircraft state has great potential
in improving the flight safety record. Such a system could provide a warning in the
event that the pilot makes an unexpected action or fails to make an expected action
System integration
The current practice mitigates against high levels of integration with equipment
suppliers only considering that equipment which is under their direct control.
Specifically, sensors for flight control purposes tend to be dedicated and no use is
made of information that can be provided by other equipment such as navigation
systems. This leads to additional, costly replication of equipment and also limits
the scope for novel cross-equipment synergistic savings to be realized. In order to
realize the potential benefits and cost savings of integrated systems, a close cooperation
between equipment suppliers is essential.
More integrated systems, with the corresponding fault-tolerant strategy, will
allow for a leaner hardware structure without decreasing the safety of the system.
For example, in current DFBW FCS each computer is dedicated to perform
a specific task. This is especially true for flight safety critical applications. The
challenge is to guarantee that each application can never interfere with the other
applications that are running on the same computer. Also the sharing of sensor
information can lead to a smaller number of hardware components. This can be
achieved by introducing Fault Detection and Isolation (FDI) technologies that also
use information from non-like sensors to monitor the signal of interest.
Mixed actuation
The DFBW FCS puts stringent requirements on the availability of electrical and
hydraulic power to ensure continued control capability following loss of power
generation capabilities from both engines. The ultimate fall back is to retain the
mechanical control links as a reversion, but this negates one of the primary cost

1.4. Research aims and motivation of the methods used 11
saving benefits of installing a DFBW system. A more effective solution is to consider
alternative power sources, removing the reliance on hydraulic power supplies.
A further option, which is to be investigated, is to redistribute the primary control
surface function to other non-hydraulic dependent secondary control effectors such
as surface tabs.
1.4 Research aims and motivation of the methods used
The research aims are to investigate the possibilities to improve the efficiency of
flight control system design and/or to improve the flight safety. The improvement
of the efficiency of the FCS design results in reduced development cost, which is a
key factor in making DFBW technology affordable for small commercial aircraft.
On the other hand, besides the economical aspect, the improvement of flight safety
is also an important argument to justify bringing DFBW to the small commercial
aircraft market. In order to achieve the research aims, the focus has been put on
two topics:
1. Automated design techniques.
2. Replacement of hardware redundancy by analytical redundancy.
The first point is specifically of interest with respect to nonlinear flight control
law design, while the second point focusses mainly on sensor management (voting/monitoring).
These two application areas are discussed in more detail in the
remainder of this section.
1.4.1 Automated design techniques
Gain scheduling has been perhaps the most common systematic approach to control
of nonlinear systems in practice (Stengel et al. 1978, Shamma and Athans 1990,
Shamma and Athans 1992, Murray-Smith and Johansen 1997). Even with the introduction
of powerful control strategies such as model predictive control (Garcia
et al. 1989) and feedback linearization (Isidori 1989), gain scheduling remains an
attractive control strategy because of its simplicity and practical use. Applications
of fuzzy logic for gain scheduling in flight control demonstrate the feasibility
and flexibility of the approach to provide adequate control performance across the
flight envelope (Gonsalves and Zacharias 1994, Fujimori et al. 1997, Schram 1998).
The gain-scheduled controller consists of two elements, namely the scheduler
and the parameters obtained from tuning the local linear controllers in the operating
points. Automation of the design process for both elements and being able
to combine the results, are key factors in reducing the design effort.
Fuzzy clustering for gain scheduling
In spite of the fact that gain scheduling has been around for a long time, almost
no effort has been put in the development of a systematic approach to identify
the operating points and to design the corresponding scheduler. The design of the

12 Chapter 1. Introduction
scheduler is an iterative process, where each iteration consists of the identification
of the operating points, tuning of the controller parameters, design of the
scheduler and evaluation of the global performance of the system. This is a slow
and costly procedure due to the lack of automation. Moreover, different scheduling
variables and operating points are identified for different parameters, which makes
the overall system very complex and the design process difficult to manage. The
more complex the system to be controlled, the more time-consuming the design of
the scheduler. In this thesis it is proposed to apply fuzzy clustering for the identification
of the operating points.
Clustering is a technique to identify subsets in a data set. With crisp clustering
each data point belongs to a single subset, while with fuzzy clustering a data point
can belong to several subsets. In the latter case a membership degree is assigned to
each data point with respect to all subsets. The total membership degree is equal
to one. In this thesis fuzzy clustering is applied to partition the flight envelope
in a number of flight regimes. The objective is to automatically identify a (small)
number of operating points for which the flight control law parameters are tuned.
Fuzzy clustering is used to make sure that there is a smooth transition from one
operating regime to the other.
The application of fuzzy clustering for the partitioning of the flight envelope automates
the design procedure, which significantly reduces the number of iterations
that need to be performed. Moreover, the resulting controller is more transparent,
since the same scheduling variables and operating points are used for all controller
parameters.
Scheduled robust multivariable control
For most DFBW aircraft flying today the control laws have been developed by using
essentially classical single-loop frequency response and root-locus design techniques.
A significant contribution in reducing the DFBW development costs, without
compromising the efficiency and safety requirements, can be obtained by using
advanced multivariable control design techniques for the development of control
laws (Magni et al. 1997, Amato et al. 2001). The multivariable design approach
has the advantage of reducing the time and cost for flight control design and refinement,
and at the same time of a priori taking into account robustness with
respect to model uncertainties.
The challenge is to combine the resulting multivariable controllers with gain
scheduling. One of the difficulties with implementing gain-scheduled multivariable
control laws is the complexity of such control laws (Ly et al. 1985, Nichols
et al. 1993, Hyde and Glover 1993). Multivariable control design techniques have
been developed that either attempt to design a robust global controller which can
operate over a wide range of the plant operation without scheduling, see for example
(Perez and Nwokah 1991), or directly synthesize a gain-scheduled controller
using multiple models of the plant in the control design (Ostroff 1992, Reichert
1992, Apkarian and Gahinet 1995). Due to the large operating range and model
uncertainties in the case of an aircraft, a single multivariable controller will not
satisfy the design requirements.
Two categories of gain-scheduled robust multivariable controllers can be found

1.4. Research aims and motivation of the methods used 13
in the literature, namely scheduling of the controller output matrix and scheduling
of both the controller dynamics and the controller output matrix.
An example of the first category can be found in (Garg 1997). A nominal controller
is designed that gives a stable closed-loop system in the entire operating
range. The parameters of the output matrix are optimized such that the closedloop
system at the off-design points closely matches the closed-loop system in the
design point.
Many examples of the second category can be found in the literature, showing a
wide variety of design approaches (Nichols et al. 1993, Hyde and Glover 1993, Pellanda
et al. 2000, Lin and Khammash 2001). In these approaches an attempt is
made to reduce the order of the controller and/or to impose a certain structure on
the controller in order to simplify the scheduling problem. The modifications that
need to be made to the local linear controllers in order to improve their “schedulability”
impairs the performance of the local controllers and therefore also that
of the global controller.
The combination of fuzzy clustering and robust multivariable control is new and
potentially very effective in reducing the design effort. In this case both the identification
of the operating points and the scheduler as well as the design of the
local linear controllers are highly automated. Also here it is important to find a
suitable method to schedule the parameters of the local linear models.
1.4.2 Replacement of hardware redundancy by analytical redundancy
Sensor management based on majority voting and point consolidation of like signals
is a proven technology in modern fly-by-wire flight control systems (Rosenberg
1998). The assumption is that the majority of like signals represents the truth and
that any single dissimilar signal is the result of a failure. Such a signal must be
disconnected as soon as the failure is detected. This principle fails in the event
that there are only two like signals left (duplex operation), since there is no longer
a majority. In the conventional approach, the computation of the consolidated signal
(voting) and the monitoring of the sensor signals are performed separately. In
order to minimize the sensitivity of the sensor monitor to uncertainties like quantization
and measurement noise, a properly adjusted (crisp) threshold is used.
Two opportunities have been identified to improve the flight safety (and comfort).
First of all the introduction of a virtual sensor that makes use of non-like
signals (analytical redundancy) and can be used as an arbitrator during duplex
and simplex operation. Furthermore, the integration of the voting and monitoring
functions and the introduction of soft thresholds have the potential to improve
the consolidated signal and to increase the robustness of the sensor management
system with respect to uncertainties.
Virtual sensors
In the literature, many applications of analytical redundancy for FDI in flight control
systems are reported. Most frequently applied are observer-based techniques
(Menke and Maybeck 1995, Patton et al. 1989), parity-space methods (Chow

14 Chapter 1. Introduction
and Willsky 1984, Patton and Chen 1992, Gopisetty and Stengel 1998, Schram
et al. 1998), and parameter-estimation schemes (Isermann 1984, Frank 1990). The
use of virtual sensors in aerospace applications has not been widely investigated
yet, although this technique has been successfully applied in other fields like process
control and engine control (Leal et al. 1997, Hanzevack et al. 1997).
In this thesis the design of a virtual angle-of-attack is described making use of
fuzzy logic and neural networks. Fuzzy logic is used to compute the parameters of
a Linear Parameter-Varying (LPV) model. The neural network is used to account
for those dynamics that cannot be explained with the LPV model.
Voting/monitoring
Conventional sensor management is based on cross comparison of sensor signals.
The computation of the consolidated signal and the monitoring of the signals is
performed separately. In the first step the consolidated signal is computed using
all the sensor signals and in the second step the monitoring of the sensor signals is
performed based on the consolidated signal. This approach has the effect of a loworder
filter. In a more direct approach, where the monitoring of the sensor signals
is performed using the sensor signals themselves, sensor faults can be detected
faster.
The location of the (crisp) threshold is a compromise between minimization
of false alarms and minimization of failure-induced transients. The first dictates
large thresholds, while the latter dictates small thresholds. This compromise can be
circumvented by introducing soft thresholds instead of crisp thresholds, increasing
flight safety and/or comfort. Fuzzy logic is an excellent technique to implement
this concept. Although FL techniques have been implemented in other application
domains, such as the process industry (Schneider and Frank 1996, Frank and
Marcu 1999), their application in flight control systems has not been extensively
investigated yet.
1.5 Overview of the thesis
The outline of this thesis is as follows: In Chapter 2 several aspects of the digital
fly-by-wire flight control systems are described. An historic overview of flight
control systems is given after which the DFBW FCS is addressed from a functionality
point of view. The latter is important in order for the reader to be able to
understand the context of the applications described in this thesis with respect to
the entire flight control system. The advantages and disadvantages of the DFBW
FCS are described in more detail.
With Chapter 3 the description of the control part of the performed research
starts. In this chapter the partitioning of the flight envelope using fuzzy clustering
is introduced, which serves as the basis for gain scheduling. Two applications of
gain scheduling are described in this thesis for which this approach is used. A
classical aircraft control example is described in Chapter 4. Here the baseline
structure of the flight control laws in the SE is kept, where the gain scheduling
mechanism is replaced by the scheduling system introduced in Chapter 3. A robust
multivariable control problem with gain scheduling is described in Chapter 5.

1.5. Overview of the thesis 15
With Chapter 6 the sensor fault detection and isolation part of the thesis
starts. In this chapter the design of a virtual sensor for angle-of-attack is described.
The virtual sensor can serve either as an additional sensor or as an arbitrator in
case of sensor failures. The voting/monitoring system based on soft computing
is described in Chapter 7. Also the use of a virtual sensor as part of the voting/monitoring
system is introduced in this chapter.
In Chapter 8 the concluding remarks and recommendations for further research
are discussed.
Six appendices are added to provide the reader with background information
on the synthetic environment and the generation of the real-time code (Appendix
A), the short-period motion approximation (Appendix B), performance
measures (Appendix C), soft computing techniques (Appendix D), genetic algorithms
(Appendix E) and linear matrix inequalities for control (Appendix F).

16 Chapter 1. Introduction

Digital Fly-By-Wire Flight Control
Systems
In this chapter the digital fly-by-wire flight control system concept is described.
The introduction of DFBW brought about an immense shift in the design
philosophy for the bare airframe and the flight control system. Stability and
control requirements can now be met independently of the bare airframe and
the DFBW FCS enables a range of capabilities that were not possible before.
This chapter provides the background information that enables the reader to
understand the context in which the applications described in this thesis should
be placed.
This chapter is organized as follows: An historical overview of the development of
flight control systems is given in Section 2.1. In Section 2.2 the basic architecture
of the digital fly-by-wire flight control system is addressed and the functionality of
its modules is described. Section 2.3 discusses the advantages and disadvantages
of the DFBW FCS compared to its mechanical counterpart.
2.1 Historical Overview
The first powered flight took place on December 17, 1903 at Kitty Hawk, North
Carolina. The Flyer, built by Wilbur and Orville Wright, flew for 120 feet in 12
seconds, see Figure 2.1. The Wright brothers flew three more times that day, the
longest flight being over 852 feet in 59 seconds.
In the first decades of aviation the pilot controlled the aircraft directly through
the application of manual force, moving control sticks and rudder pedals linked by
cables and pushrods that pivoted control surfaces on the wings and tails.
As engine power and speeds increased, more force was needed to move the control
surfaces and hydraulically-boosted control systems emerged. Soon, all highperformance
and large aircraft had hydraulic-mechanical flight control systems.
This provided the means to include augmentation and automation systems that
17
2

18 Chapter 2. Digital Fly-By-Wire Flight Control Systems
Figure 2.1: Kitty Hawk, NC, December 17, 1903. Orville Wright’s famous first
airplane flight. (Source: http://www.wam.umd.edu/∼stwright/WrBr/wrights/
1903.html).
provided inputs additional to those of the pilot. In these early systems safety was
assured by severely restricting the control authority such that total failure could
not hazard the aircraft. These flight control systems restricted designers in the
configuration and design of aircraft because of the need for inherent flight stability.
As the electronic era grew in the 1950’s, so did the idea of aircraft with Electronic
Flight Control Systems (EFCSs). Full authority electronic flight control
systems forced the issues of safety, availability and integrity to be addressed.
The EFCS now had the capability to hazard the aircraft under fault or failure
conditions and became subject to the safety requirements of the airworthiness
authorities. The first generation systems were based on analog technology but,
importantly, retained the hydraulic-mechanical system as a back-up. Later generations
abandoned the hydraulic-mechanical back-up to become full-time fly-bywire.
This represented a watershed achievement that to a large extent decoupled
the requirements for flight stability from the basic airframe configuration (Schmitt
et al. 1998, Collinson 1999). In parallel, the possibilities of digital host computing
were being investigated under the expectation of increased flexibility and reduced
cost by decoupling the flight control system functionality (software) from the host
computer. Neither expectation was fully achieved, but current generation FCS implementations
are almost universally based on digital technology. The first test of
a digital fly-by-wire system in an aircraft was in 1972 on a modified F-8 Crusader
at the Dryden Flight Research Center. It was the forerunner of the DFBW flight
control systems now used on the space shuttles and modern military and civil
aircraft to make them safer, more manoeuvrable, and more efficient.
The first electrical flight control system for a civil aircraft was subcontracted
by Aérospatiale to Elliot Brothers (London) Ltd (now BAE SYSTEMS) in Britain
and SFENA (now Sextant Avionique) in France and was installed on Concorde.
This is an analog, full-authority system for all control surfaces. The first application
of electrical signalling of secondary flight controls with digital technology
appeared on several civil aircraft at the start of the 1980s with the Airbus A310
program (Briere et al. 1995). These systems control the slats, flaps and spoilers.

2.2. Description of the Digital Fly-By-Wire Flight Control System 19
The Airbus A320 (certified in early 1988) is the first example of a second generation
of civil fly-by-wire aircraft, rapidly followed by the A340 aircraft (certified
at the end of 1992). The first DFBW aircraft of Boeing, the B777, is certified in
April 1995.
2.2 Description of the Digital Fly-By-Wire Flight Control
System
In this section the digital fly-by-wire flight control system is discussed in more
detail. First the architecture of the flight control system is described and then the
functionality of the flight control computer is addressed.
Flight control system architecture
The digital fly-by-wire flight control system is illustrated in Figure 1.1 in simple
diagrammatic form. The core of the system is the flight control computer. The
output of the FCC is the commanded control surface deflection, which is computed
based on the pilot command and the aircraft motion and air data sensor
information. The actuator drives the control surface to the commanded position
through the actuator control electronics. The corresponding aircraft response is
again sensed by the motion and air data sensors and fed back to the FCC.
Safety is defined in terms of the probability of the occurrence of a catastrophic failure
per flight hour. According to the airworthiness requirements for civil transport
aircraft (JAR-25 and FAR-25), the probability of a catastrophic failure should be
less than 10 −9 per flight hour. In general, the reliability of single components (computers,
sensors, actuators, etc) is not sufficient to meet these requirements. In order
to be able to guarantee the integrity of the DFBW FCS, redundant components
are installed in parallel (hardware redundancy) together with a fault detection and
isolation system. The required level of redundancy is dependent of the reliability
of the single components and the safety requirement of the corresponding system.
Safety critical systems are subject to the failure probability of 10 −9 per flight hour,
since this corresponds to a catastrophic failure.
In a triplex redundant FCS each input signal of the FCC is measured with
three independent sensors, see Figure 2.2. These three signals are consolidated
into one signal, the consolidated or voted signal, such that all three flight control
computers receive the same input values. Through cross-comparison of these three
like-signals a possible sensor failure can be detected. The failure detection scheme
is based on the assumption that the probability of the failure of two like-sensors at
the same time is extremely remote. The sensor with an output that is dissimilar
from the output of its like-sensors, is therefore assumed to be the failed sensor.
When all three sensors are working properly, the consolidated signal is, for
example, the average of the three sensor signals. The three FCCs have exactly
the same functionality and their outputs are subject to cross-comparison and consolidation
in the same way as the sensor signals. The consolidated values of the
computer outputs, the consolidated control surface angle commands, are fed to

20 Chapter 2. Digital Fly-By-Wire Flight Control Systems
Figure 2.2: Lane processing scheme of a triplex flight control system (Adopted
from: (Collinson 1999)).
the actuator control electronics, see also Figure 1.1. For a triplex system, the system
continues to operate after the first like-sensor failure. However, in case of a
second like-sensor failure, it is not possible to identify the failed sensor by crosscomparison
since there is no longer a “majority”. In this case the whole system
must be isolated and, if the corresponding system is essential for the safe operation
of the aircraft, the second like-sensor failure could have catastrophic consequences.
This implies that the probability of a second like-sensor failure should be lower or
equal to the allowed probability of a catastrophic failure.
The failure survival philosophy goes even further. Besides the hardware redundancy,
also dissimilar redundancy is built in the flight control system in order to
avoid generic or common-mode failures. When all three flight control computers
are exactly the same, it is possible that they will exhibit undesirable behavior at
the same time due to for example a design or coding error. In order to minimize
the risk of generic faults, dissimilar software and hardware are implemented in
the flight control system. It should be noted that dissimilar hardware and software
will not prevent generic faults that are introduced in the design specifications
themselves.
Functionality of the flight control computer
The number of functionalities that can be implemented in the FCC of the DFBW
FCS is virtually unlimited. The three most important and/or common functionalities
are:
1. Sensor Management.
2. Stability and Control Augmentation.
3. Flight Envelope Protection.
The functioning of the corresponding systems will be discussed in more detail below.

2.3. Advantages and Disadvantages of Digital Fly-By-Wire 21
Sensor Management
The sensor management system serves as the portal of the FCC and is essential
for the safe operation of the aircraft. It has to make sure that the sensor information
that is used by the aircraft systems is correct. Therefore the outputs of the
(redundant) sensors are continuously checked through cross-comparison. In case
of a discrepancy amongst like-sensor signals, the outlier is identified and a failure
is declared on the corresponding sensor. The signals from the healthy sensors are
used to compute the consolidated signals. The consolidated signals are then used
to perform the other tasks of the FCC and are transmitted to all other aircraft
systems that require sensor information.
Stability and control augmentation
With a mechanical flight control system the stability is inherently provided by the
airframe or, if it is lacking or insufficient, by a dedicated Stability and Control
Augmentation System (SCAS). Through the SCAS, the flight control engineer is
able to modify the dynamics of the bare airframe (open-loop aircraft) such that
they comply with the design and/or handling quality requirements.
Flight envelope protection
The Flight Envelope Protection System (FEPS) is designed such that the operational
and structural limits, e.g. maximum angle-of-attack, maximum bank angle,
maximum speed or Mach number, maximum load factor, etc., will not be exceeded.
In case the pilot is about to exceed one of the limits, compromising the safe operation
of the aircraft, the control surface deflections commanded by the SCAS
are modified to prevent this from happening. This system increases the safety and
allows the pilot to react rapidly and/or strongly if necessary, without having to
worry about exceeding the operational/structural limits of the aircraft (care-free
handling). For example, when the Enhanced Ground Proximity Warning System
(EGPWS) issues a warning, the pilot should climb as fast as possible. With the
FEPS the pilot can pull back the column to the maximum position without exceeding
the maximum attitude and apply full throttle.
2.3 Advantages and Disadvantages of Digital Fly-By-Wire
The digital fly-by-wire flight control system has many advantages over the mechanical
flight control system, both in terms of implementation as well as functionality.
The disadvantages will be discussed at the end of this section.
Advantages of the DFBW FCS system
From the implementation point of view, the DFBW FCS has replaced the cables,
pushrods, springs, bob-weights, etc. by sensors, electrical wires, and computers
(Schmitt et al. 1998). This change had a revolutionary impact on the flight
control system design compared to its mechanical counterpart:

22 Chapter 2. Digital Fly-By-Wire Flight Control Systems
• Wires replacing cables and pushrods give designers greater flexibility in configuration
and size and placement of components such as tail surfaces and
wings.
• The DFBW system is smaller, more reliable, and in military aircraft the
system is less vulnerable to battle damage and jamming.
• DFBW permits the pilot’s command inputs (signals) to be transmitted electrically
over considerable distance without distortion.
• Temperature changes have little effect on electrical signaling designs, whereas
mechanical systems, especially those using cables and operating over a wide
temperature range, are likely to encounter problems.
• The system is easy to install and maintain; it requires less adjustments or
devices to ensure proper operation once installed.
• A DFBW system is more responsive to pilot control inputs. The result is more
efficient, safer aircraft with improved performance and design. Furthermore,
a DFBW system enables easier tuning of aircraft response to pilot inputs. It
therefore simplifies obtaining the same or similar handling characteristics in
a family of aircraft with the same DFBW system, like in the Airbus family
A320, A330 and A340, thus enabling reduced pilot training effort.
From the functionality point of view the DFBW FCS offers potential capabilities
that were not possible before with mechanical flight control systems:
• The SCAS provides additional design options to tailor the aircraft dynamics.
Besides the advantages mentioned earlier for military fighters, also commercial
aircraft benefit from the use of relaxed stability airframes. Less stable
means less negative lift and smaller fins and therefore less fuel consumption
(at the price of increased control activity). When it is still possible for the
pilot to fly the aircraft in the “conventional” way, a Direct Electrical Link
(DEL) could be used as a back-up.
• The FEPS increases the safety of the aircraft operations, both in the sense
of correcting undesired excursions outside the normal flight envelope as well
as carefree handling in emergency situations and thus obtaining maximum
performance more easily and consistently.
• Since the FCLs are embedded in the SW, the FCS is very flexible with respect
to changing its architecture. This property can be used for reconfiguration
purposes, both with respect to the loss of a signal as well as the loss of a
control surface. In both cases the architecture of the FCLs can be modified to
minimize the effect of the corresponding failure. In this way the effect of the
loss of a signal or control surface can be minimized. Also adaptive techniques
could be used to compensate for structural damage. This capability can
either be used to increase the safety or to reduce the number of hardware
components.

2.3. Advantages and Disadvantages of Digital Fly-By-Wire 23
The FCC allows for a number of features that are not possible in a mechanical flight
control system. A major design issue is deciding which features to include, which
offer the greatest return in terms of improved safety and cost efficiency, and what
the design and performance requirements are in order to bound the design activity.
Disadvantages of the DFBW FCS system
The most important disadvantage of the fly-by-wire flight control system is the
initial cost of procurement of the system and the increased complexity.
The digital fly-by-wire flight control system is completely dependent on electric
and hydraulic power, which makes these systems critical for the safe operation
of the aircraft. This is already the case for (large) commercial aircraft with
hydraulically-boosted controls without manual reversion. Small commercial aircraft
typically have manual controls or hydraulically-boosted controls with manual
reversion. The introduction of a DFBW system in a small commercial aircraft
would therefore mean that additional systems need to be installed in order to ensure
the availability of electric and hydraulic power.
The design of a mechanical flight control system is relatively simple, because
of the limited tools available. For the digital fly-by-wire flight control systems the
design freedom is unlimited so to speak. The flight control laws are now purely
embedded in software and the only hard boundary is the structural limit of the
airframe itself. It is now possible to add features to the FCLs such as pilot command
shaping, flight envelope protection, etc.
In order to ensure the availability of the system, redundant sensors, actuators,
flight control computers, etc. are implemented and their performance is crosschecked
continuously. The software of each flight control computer may have to
be written by a different team and compiled with different compilers. Dissimilarity
in software and hardware reduces the risk of generic failures, i.e. failures that
occur in all similar redundant components at the same time. Fault detection and
identification schemes need to be added to the system in order to manage all the
redundant hardware. This adds to the cost of the DFBW FCS compared to the
mechanical flight control system.
Another disadvantage of the DFBW system is the sensitivity for High Intensity
Radiated Fields (HIRF). The electrically transmitted signals could be distorted by
radiation fields. In order to prevent this, the electrical wires need to be protected.
Conclusion
For large commercial aircraft the advantages outweigh the disadvantages. For small
commercial aircraft this is not so obvious and the architecture of the FCS and
the design methodologies need to be optimized in order to reach an acceptable
compromise.

24 Chapter 2. Digital Fly-By-Wire Flight Control Systems

Fuzzy Clustering for Partitioning of
the Flight Envelope
The use of a linear design technique in combination with gain scheduling is
the most common systematic approach to the design of nonlinear flight control
laws. However, the selection of the operating points and the design of the interpolation
scheme remains a time-consuming procedure. In order to reduce the
design effort, an automated procedure has been developed and applied to the
design of a longitudinal flight control law in a fly-by-wire flight control system.
The number of operating points and their locations are determined automatically
by using fuzzy clustering on the basis of the changes in the aerodynamic
characteristics over the flight envelope. This approach also directly provides the
interpolation mechanism (membership functions) for the local flight control law
parameters.
This chapter is organized as follows: A brief introduction into nonlinear control
is given in Section 3.1. In Section 3.2 the basics of fuzzy clustering are discussed
(see Appendix D for a more detailed description on fuzzy clustering and related
subjects). The partitioning of the flight envelope using fuzzy clustering is described
in Section 3.3. Concluding remarks are given in Section 3.4.
3.1 Nonlinear Control
The general function of the flight control laws in the fly-by-wire flight control system
is to improve the handling qualities of the bare aircraft, in particular with
respect to stability, control and flight envelope protection (Tischler 1996). The design
of the flight control laws is a nonlinear control problem due to the nonlinear
aircraft dynamics, which also vary with the flight condition and aircraft configuration.
As a single linear controller cannot meet the design requirements over the
entire operating range, there are two principle solutions to this nonlinear control
problem:
1. Nonlinear design techniques, such as feedback linearization (Isidori 1989),
25
3

26 Chapter 3. Fuzzy Clustering for Partitioning of the Flight Envelope
provide a single nonlinear controller that meets the design requirements over
the entire operating range. The application of nonlinear design techniques is
often complicated due to the restrictive conditions that have to be met by the
system to be controlled. In many cases these conditions are too restrictive
and the corresponding nonlinear design technique cannot be applied at all.
Moreover, stability and robustness analysis still requires the use of linear
techniques.
2. The use of a linear design technique in combination with gain scheduling
is perhaps the most common systematic approach to the control of nonlinear
systems (Stengel et al. 1978, Shamma and Athans 1990, Shamma and
Athans 1992, Murray-Smith and Johansen 1997). Linear controllers are designed
for a number of design points and their parameters are subject to
interpolation for the flight conditions in between the design points. In this
way a global nonlinear controller is obtained. Despite recent advances in nonlinear
control, gain scheduling remains an attractive control strategy because
of its simplicity and practical usefulness.
In this thesis gain scheduling is considered. The scheduler design problem consists
of two tasks: selection of the scheduling variables and design of the parameter
scheduling algorithm (Stengel et al. 1978). Scheduling variables are usually selected
on the basis of the following two heuristics (Shamma and Athans 1990, Shamma
and Athans 1992):
• The scheduling variables should capture the nonlinearities of the system to
be controlled.
• The scheduling variables should be slowly time-varying compared to the
desired bandwidth of the closed-loop system.
Although many tools are available for the design of the local linear controllers,
the selection of the operating points and the design of the scheduling mechanism
are less straightforward. Typically, it results from an iterative procedure based on
past experience of the control engineer. During each iteration step, the following
tasks are performed:
1. Selection of a set of operating points.
2. Tuning of the flight control law parameters.
3. Design of the scheduler.
4. Validation by linear analysis and nonlinear simulation.
This process, which is completed when the closed-loop system dynamics are satisfactory
over the entire operating range, is time-consuming and often leads to more
operating points than necessary.
In order to reduce the design effort, an automated procedure for the design of
gain scheduled controllers has been developed using fuzzy logic techniques. The

3.2. Fuzzy Clustering 27
potential of fuzzy gain-scheduled flight control has been demonstrated in a number
of articles, see (Fujimori et al. 1997, Gonsalves and Zacharias 1994, Schram 1998)
among others. In these works however, the operating points and/or membership
functions are selected quite arbitrarily. The approach proposed in this chapter
goes one step further: the number of operating points and their locations are
determined automatically by using fuzzy clustering on the basis of changes in the
system dynamics over the operating range.
3.2 Fuzzy Clustering
An effective approach to the identification of complex nonlinear systems is to
partition the available data into subsets and approximate each subset by a linear
model. Fuzzy clustering can be used as a tool to obtain a partition of data where the
transitions between the subsets are gradual rather than abrupt. Fuzzy partitions
can be seen as a generalization of hard partitions, which is formulated in terms of
classical subsets.
3.2.1 Clustering algorithms
Clustering techniques are unsupervised methods that can be used to organize data
into groups based on similarities among the individual data items. Most clustering
algorithms do not rely on assumptions common to conventional statistical methods,
such as the underlying statistical distribution of data, and therefore they are
useful in situations where little prior knowledge exists.
A large family of fuzzy clustering algorithms is based on the minimization of
an objective function J formulated as:
J(Z; U, V, Ai) =
c
N
i=1 k=1
(µik) m D 2 ikA i
, (3.1)
where Z is the n×N data matrix, U is the c×N partition matrix, and V the n×c
matrix of cluster prototypes (centers). The number of variables in the data set is
denoted by n, N is the number of data samples, and c is the number of clusters.
DikA i is the distance measure, which is computed as follows:
D 2 ikA =(zk− vi)
i
T Ai(zk − vi), (3.2)
where zk is the kth data point and vi the ith cluster center. The n × n norminducing
matrix of the ith cluster is denoted by Ai, µik is the membership degree
of the kth data point with respect to the ith cluster and m is the ‘fuzziness’
exponent.
Iteratively solving the minimization problem results in c operating points. Each
operating point represents the center of an operating regime in which the data are
close to each other with respect to the objective function.

28 Chapter 3. Fuzzy Clustering for Partitioning of the Flight Envelope
3.2.2 Tunable parameters and validation measures
The clustering parameters influence the clustering results. The two basic parameters
that need to be tuned are the number of clusters c and the fuzziness exponent
m.
Number of clusters
In order to determine the optimal number of clusters, the fuzzy clustering results
are evaluated for different values of c. Clustering algorithms generally aim at locating
well-separated and compact clusters. When the number of clusters is chosen
equal to the number of groups that actually exist in the data, it can be expected
that the clustering algorithm will identify them correctly.
In (Gath and Geva 1989), it is suggested to assess the goodness of the obtained
partition by evaluating the separation between the clusters, the volume of
the clusters, and the number of data points concentrated in the vicinity of the cluster
prototype. There are many cluster validity measures and in this case we use the
Fuzzy Hyper-Volume (FHV) (Gath and Geva 1989), the Within-Cluster Distance
(WCD) (Krishnapuram 1994), and the Xie-Beni validity measure (SXB) (Xie and
Beni 1991). Good partitions are indicated by small values of all three validity measures.
Fuzziness exponent
The fuzziness exponent influences the fuzziness of the resulting partition. As m
approaches one from above, the partition becomes hard rather than fuzzy. The
higher the fuzziness exponent, the fuzzier the partition, and hence more overlap
between the clusters. Usually, m = 2 is initially chosen.
3.3 Fuzzy Partitioning of the Flight Envelope
Few examples of an automated approach to the identification of operating points
can be found in literature. In (Garduno-Ramirez and Lee 2000) the eigenvalue with
the largest magnitude variation is used to determine the operating points. Each
partition covers 20% of the range of the corresponding eigenvalue. Two articles
have been found that describe an automated procedure for the design of a gainscheduled
controller (Wada and Osuka 1997, McNichols and Fadali 2003). Here
it concerns an iterative procedure of scheduling, controller design and closed-loop
evaluation.
The fuzzy partitioning of the flight envelope for gain-scheduled flight control using
fuzzy clustering is described step by step in this section. An example of the design
of a gain-scheduled state feedback controller is given to illustrate the design
procedure.

3.3. Fuzzy Partitioning of the Flight Envelope 29
Altitude [ft]
45
40
35
30
25
20
15
10
5
V C = 150 Kts
M MAX = 0.85
V C = 375 Kts
0
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Mach number [−]
Figure 3.1: The operating range of the SCA in clean configuration, defined as a
function of Mach number, altitude and calibrated airspeed.
3.3.1 Generation of the data set
The data set used for fuzzy clustering must capture those aircraft dynamics that
are important from the control point of view. Figure 3.1 illustrates the flight envelope
of the Small Commercial Aircraft (SCA) model in clear configuration, which
is defined in terms of Mach number, altitude and calibrated airspeed. The SCA
model is modified from the model of an existing business jet aircraft and is implemented
in the SE that is used in the ADFCS project, see Appendix A.
When considering the longitudinal controller, the phugoid motion dynamics are of
minor importance. This has long been recognized, and trivial or no specifications
are placed on the phugoid mode in handling quality requirements. However, the
Short-Period (SP) motion dynamics are of major importance. It is their deficiencies
or those of their equivalents in DFBW aircraft, that have caused many difficulties
and even accidents (Gibson 1999). The linearized model for longitudinal motion is
described as follows (see also Appendix A):
⎡ ⎤ ⎡
⎤ ⎡ ⎤ ⎡ ⎤
⎢
⎣
˙u
˙w ⎥
˙q ⎦
˙θ
=
Xu Xw Xq Xθ u
⎢ Zu Zw Zq Zθ ⎥ ⎢
⎢
⎥ ⎢w
⎥
⎣ ˜Mu
˜Mw
˜Mq
˜Mθ ⎦ ⎣ q ⎦
0 0 1 0 θ
+
⎢
⎣
Xδe
Zδe
˜Mδe
0
⎥
⎦ δe, (3.3)
where u denotes the forward speed, w denotes the downward speed, q denotes the
pitch rate, θ denotes the pitch attitude and δe denotes the elevator deflection. For
the short-period approximation, only the downward velocity w and the pitch rate
q are considered, while the forward speed is assumed to be constant ( ˙u = 0):
˙w Zw Zq w Zδe
=
+ δe. (3.4)
˙q ˜Mw ˜Mq q ˜Mδe

30 Chapter 3. Fuzzy Clustering for Partitioning of the Flight Envelope
Altitude [kft]
27
26
25
24
23
22
21
0.44 0.46 0.48 0.5
Mach number [−]
(a) Grid points as a function of Mach
number and altitude.
−0.045
M δe
−0.05
−0.055
−0.5
−0.6
M q
−4
M α
(b) Data points zk =[ ˜ Mα ˜ Mq ˜ Mδe ] corresponding
to the grid points.
Figure 3.2: Construction of the data set. For the grid points in the 2-dimensional
space defined by Mach number and altitude (left), points in the 3-dimensional space
defined by ˜ Mα, ˜ Mq and ˜ Mδe are obtained via linearization of the nonlinear aircraft
model (right).
The aim of fuzzy clustering is to identify regions in the flight envelope where
the short-period motion dynamics can be approximated by a single linear model.
Therefore, the variables selected for the data set should contain information with
respect to the short-period motion dynamics.
Example
The most important aerodynamic derivatives with respect to the shortperiod
motion are ˜ Mα, ˜ Mq, and˜ Mδe (McLean 1990), where α denotes the
angle-of-attack. It should be noted that ˜ Mα = U0 ˜ Mw, where U0 is the
forward speed in the corresponding trimmed condition. The data set is
defined as:
Z = {zk|k =1, 2, ..., N},
where N is the number of data samples. To generate the data, a grid is
defined in the operating range with steps of 0.01 in Mach number and steps
of 1000 ft in altitude. At each grid point the nonlinear aircraft model is
trimmed and linearized and subsequently the aerodynamic derivatives ˜ Mα,
˜Mq, and˜ Mδe are obtained from the linear model (see also Figure 3.2):
zk =[ ˜ Mα(Mk,hk) ˜ Mq(Mk,hk) ˜ Mδe (Mk,hk)] T ,
where (Mk,hk) denotes the kth grid point in terms of Mach number and
altitude, respectively. This data set is generated automatically and the total
number of data samples is equal to N = 1805.
In the example only the aerodynamic derivatives are used for clustering. One can
also choose to add variables that describe the corresponding flight condition, for
−3
−2

3.3. Fuzzy Partitioning of the Flight Envelope 31
example Mach number (M) and angle-of-attack (α), to the data set, in which case
the data set would be defined as:
zk =[Mk hk ˜ Mα(Mk,hk) ˜ Mq(Mk,hk) ˜ Mδe (Mk,hk)] T . (3.5)
Explicitly including such variables increases the probability of obtaining convex
clusters with respect to the flight condition. If there are for example two operating
regimes with more or less the same aerodynamic derivatives but well separated
in terms of Mach number, two separate clusters will be identified. In case Mach
number is not part of the data set, it is likely that one cluster is identified that
results in a non-convex Membership Function (MF) as a function of Mach number.
However, the fuzzy partition is no longer solely based on the aircraft dynamics.
Convex clusters can always be obtained by sufficiently increasing the number of
clusters to be identified.
3.3.2 Fuzzy clustering
In this case, the objective of fuzzy clustering is to partition the available data
into subsets and to approximate each subset by a single linear model as given in
Equation 3.4. The rationale being that regimes for which the aircraft dynamics
can be approximated by a single linear model, are also suitable to control the
aircraft using a single linear controller. The most relevant elements of the linear
model, i.e. the most relevant aerodynamic derivatives, are used for fuzzy clustering.
3.3.3 The number of clusters and fuzziness exponent
Besides the standard validity measures FHV, WCD and SXB, a fourth validity
measure is introduced in order to find the suitable number of clusters. This is the
Error Validity Measure (EVM), which is a modelling performance measure. To
evaluate the EVM, the fuzzy partition is used to construct a singleton Takagi-
Sugeno fuzzy model (see Appendix D) of each of the aerodynamic derivatives in
the data set as a function of the scheduling variables:
Ri : If Z1 is Z1i and Z2 is Z2i then M = Mi i =1,...,Nr, (3.6)
where in this case Z1 and Z2 denote the scheduling variables, Z1i and Z2i denote
the corresponding membership functions, Nr denotes the number of rules and Mi
denotes the vector of aerodynamic derivatives. The degree of fulfillment of the ith
rule is computed by the product of the membership degrees of each statement in
that rule:
βi = µZ1i
µZ2i . (3.7)
The weight wi of each rule Ri is computed as follows:
wi =
βi
Nr j=1 βj
. (3.8)

32 Chapter 3. Fuzzy Clustering for Partitioning of the Flight Envelope
FHV
SXB
0.05
0.04
0.03
2.5
1.5
2 4 6 8 10 12
Number of clusters
2
1
2 4 6 8 10 12
Number of clusters
WCD
5000
4000
3000
2000
ERR
2 4 6 8 10 12
Number of clusters
x 10 4
3
2
1
2 4 6 8 10 12
Number of clusters
Figure 3.3: Performance of the validity measures as a function of the number of
clusters.
The output ˆ M of the singleton model is then:
Nr
ˆM = wiMi. (3.9)
i=1
The reason to use a singleton model is that eventually we are going to use the
TS fuzzy model for gain scheduling. The resulting scheduler is in fact a singleton
model and therefore it is decided to use the singleton model for the error validity
measure as well. The error validity measure is evaluated by summation of the
squared difference between the outputs of the TS fuzzy models at each grid point
( ˆ Mk) and their corresponding true values in the data set (Mk):
EVM = 1
N
(Mk −
N
ˆ Mk) 2 . (3.10)
k=1
The validity measures are used to determine the number of clusters in the partition.
When the number of clusters is fixed, the fuzzy partition is evaluated for different
values of the fuzziness exponent m. The choice of the fuzziness exponent is made
after visual inspection of the partition for different values of m. Of most importance
is that the clusters are convex in terms of the potential scheduling variables, e.g.
Mach number and altitude.
Example, continued
In this example the Gustafson-Kessel (GK) clustering algorithm is used.
The advantage of this algorithm is that the norm-inducing matrices Ai
are subject to the optimization. This clustering algorithm is able to identify
clusters of different shape and orientation, and is therefore more likely

3.3. Fuzzy Partitioning of the Flight Envelope 33
to discover the true partition of the data set. The fuzzy c-means algorithm
for example is limited to identifying circular clusters of equal volume
(Bezdek 1980), which makes the algorithm less flexible in correctly
identifying the clusters that are present in the data set.
In Figure 3.3 the validation measures are illustrated as a function of the
number of clusters. Good performance is indicated by small values of all
four validity measures. In this case both the Xie-Beni validity measure and
the modelling error show a local minimum for c = 8, while both the fuzzy
hyper-volume and within-cluster distance show a local minimum for c =9.
However, the rate of descent of the within-cluster distance has significantly
decreased for c > 8. The fuzzy hyper-volume shows a similar behavior.
Moreover, the local minimum of the Xie-Beni validity measure is most pronounced
compared to those of the other three validity measures and the
number of clusters is therefore set to c =8.Itshouldbenotedthatinall
these cases the fuzziness exponent was set to m =2.
While keeping the number of clusters fixed to c = 8, the fuzzy partition is
evaluated for different values of the fuzziness exponent m. Finally m =1.8
is selected to be used here.
3.3.4 Partition and membership functions
The resulting fuzzy partition is defined by the fuzzy partition matrix U, see also
Appendix D.1. In the fuzzy partition matrix each data point is assigned a membership
degree with respect to each cluster. The sum of the membership degrees
per data point is equal to one.
In order to be able to construct the scheduler as a TS fuzzy model, the multidimensional
fuzzy partition is approximated by the Cartesian product of onedimensional
membership functions. This is obtained through the following procedure:
Selection of the scheduling variables.
The fuzzy partition is approximated by the membership functions and the product
operator. Taking into account that the membership functions are obtained by
orthogonal projection of the clusters onto the axes of the scheduling variables, the
approximation of the fuzzy partition is most accurate when the orientation of the
fuzzy clusters are aligned with the scheduling variables.
Projection of the clusters onto the axes of the scheduling variables.
After selecting the scheduling variables, the orthogonal projections of the clusters
onto each of their axes are obtained.

34 Chapter 3. Fuzzy Clustering for Partitioning of the Flight Envelope
(a) Mach number vs altitude. (b)Machnumbervsdynamicpressure.
Figure 3.4: Fuzzy clustering result for c =8and m =1.8.
Approximation of the projections.
The membership functions are obtained by fitting the orthogonal projections with
a parametric membership function. Examples of shapes of membership functions
are triangular, trapezoidal and sigmoidal.
A posteriori modification of the membership functions.
It is necessary to modify the membership functions such that each operating point
fully and exclusively belongs to its corresponding cluster. In this way it is made
sure that in each operating point the scheduler reproduces those FCL parameters
as tuned for the corresponding operating point. The membership degree of each
operating point is thus equal to one for its corresponding cluster and zero for all
other clusters. Finally the kernel of each membership function is reduced to the
corresponding operating point only.
Example, continued
In Figure 3.4 the maximum membership degree for each data point is plotted.
Light areas indicate a maximum membership degree close to one, while
dark areas indicate lower maximum membership degree. The latter areas
indicate those regimes that belong to more than one cluster. Since each data
point is obtained by linearizing the nonlinear aircraft model, their corresponding
Mach number and altitude or dynamic pressure can be used to
construct Figures 3.4a and 3.4b, respectively.
In Figure 3.4 it can be seen that the orientation of the fuzzy clusters is
diagonal with respect to Mach number and altitude and (more or less)
orthogonal with respect to Mach number and dynamic pressure. The approximation
of the fuzzy partition is most accurate when the orientation
of the fuzzy clusters are aligned with the scheduling variables. This makes

3.3. Fuzzy Partitioning of the Flight Envelope 35
Membership degree
1
0.5
0
250
200
150
MN 5
Dynamic pressure [mbar]
100
50
0.2
0.4
0.6
0.8
1
DP 5
Mach number [−]
Figure 3.5: Projection of the fuzzy clusters onto the axes of the scheduling variables
(dashed-dotted) and their corresponding approximations by sigmoidal membership
functions (continuous).
Membership degree
Membership degree
1
0.5
0
1
0.5
0.3 0.4 0.5 0.6 0.7 0.8
Mach number [−]
0
50 100 150 200 250
Dynamic pressure [mbar]
(a) Membership functions.
Membership degree
Membership degree
1
0.5
0
1
0.5
Figure 3.6: Membership functions.
0.3 0.4 0.5 0.6 0.7 0.8
Mach number [−]
0
50 100 150 200 250
Dynamic pressure [mbar]
(b) Modified membership functions.
Mach number and dynamic pressure the most suitable scheduling variables.
An example of the orthogonal projection of the clusters onto the axes of the
scheduling variables is denoted in Figure 3.5 with the dash-dotted lines. For
example, the dash-dotted line as a function of Mach number is constructed
by evaluating the maximum membership degree of all dynamic pressures
for each Mach number.
In Figure 3.5 the sigmoidal membership functions are denoted with the
continuous line. There are eight membership functions for each scheduling
variable resulting from the projection of the eight fuzzy clusters, see Figure
3.5a.

36 Chapter 3. Fuzzy Clustering for Partitioning of the Flight Envelope
Figure 3.7: Decomposition of the flight envelope into eight operating regimes.
Table 3.1: The operating points obtained through fuzzy clustering.
FC Mach Altitude Dynamic
number [kft] pressure
[-] [mbar]
1 0.83 38.0 118
2 0.59 32.3 72
3 0.65 11.0 220
4 0.39 5.9 90
5 0.33 15.4 44
6 0.59 23.2 108
7 0.55 9.8 160
8 0.79 25.8 185
The modified membership functions are illustrated in Figure 3.5b. The partition
resulting from the membership functions is illustrated in Figure 3.7.
The corresponding operating points are given in Table 3.1.
3.3.5 The local controller design
When the singleton model of Equation 3.6 is established, the rule-base and membership
functions can be used for gain scheduling. Each membership function along
Mach number is connected to one membership function along dynamic pressure,
see also Figure 3.5, which results in a so-called diagonal rule-base. The local linear
controller parameters are tuned in the operating points and putting these param-

3.3. Fuzzy Partitioning of the Flight Envelope 37
Dynamic pressure [mbar]
260
240
220
200
180
160
140
120
100
80
60
7,12
10
6
1
11
40
0.3 0.4 0.5 0.6 0.7
Mach number [−]
0.8 0.9
Figure 3.8: Test flight regime. The gray dots denote the test flight conditions.
eters in the singleton TS fuzzy model results in the global nonlinear controller:
Ri : If Z1 is Z1i and Z2 is Z2i then K = Ki, i =1,...,Nr, (3.11)
where in this case Z1 denotes Mach number, Z2 denotes dynamic pressure and Ki
denotes the local controller.
The fuzzy gain scheduling approach does not put any restrictions on the structure
of the linear (flight) control laws. In Chapters 4 and 5 the classical and the robust
multivariable control techniques are applied, respectively.
Example, continued
The flight regime for which the gain-scheduled state feedback controller
will be designed is enclosed by the four operating points 1, 6, 7 and 8,
see Figure 3.8. The set of grid points that are within the flight regime of
interest are considered. From this set, only those grid points that have a
nonzero membership degree corresponding to one or more of the four operating
points 1, 6, 7 and 8 are selected. The grid points that have a nonzero
membership degree corresponding to one of the other operating points are
removed from the set. The resulting set of Flight Conditions (FCs) is denoted
by gray dots in Figure 3.8. These flight conditions will be referred to
as the test flight conditions. A fixed controller and a second gain-scheduled
controller are designed for comparison. The fixed controller is designed in
FC9 and the operating points for the second gain-scheduled controller are
FC10 to FC13. These operating points are chosen arbitrarily and the corresponding
membership functions are determined accordingly. The details
on the design flight conditions are given in Table 3.2.
In each of the nine design flight conditions a state feedback controller is de-
9
8
13

Imaginary axis
38 Chapter 3. Fuzzy Clustering for Partitioning of the Flight Envelope
4
3
2
1
0
−1
−2
−3
−4
−4 −3 −2 −1 0
Real axis
(a) Fixed controller.
Imaginary axis
4
3
2
1
0
−1
−2
−3
−4
−4 −3 −2 −1 0
Real axis
(b) FGS controller.
Figure 3.9: Closed-loop poles in the test flight conditions. The black x-marks denote
the closed-loop poles in the test flight conditions. The gray +-marks denote the desired
closed-loop poles.
Table 3.2: Design points and state feedback controller gains.
FC Mach Alt. Dyn. K
nr. [kft] press.
[-] [mbar]
1 0.83 38.0 118 [ 0.251 −0.124 −75.10 −53.26 ]
6 0.59 23.2 108 [ 0.084 −0.008 −57.86 −36.13 ]
7 0.55 9.8 160 [ 0.030 −0.194 −30.79 −15.69 ]
8 0.79 25.8 185 [ 0.017 −0.344 −35.74 −17.23 ]
9 0.70 25.9 140 [ 0.099 −0.149 −46.58 −26.25 ]
10 0.55 19.6 108 [ 0.066 −0.008 −55.04 −33.78 ]
11 0.83 39.8 108 [ 0.281 −0.085 −83.76 −63.07 ]
12 0.55 9.8 160 [ 0.030 −0.194 −30.79 −15.69 ]
13 0.83 31.6 160 [ 0.085 −0.379 −48.14 −24.51 ]
signed such that the closed-loop poles are p1,2 = −ωspζsp ± jωsp
and p3,4 = −ωphζph ± jωph
1 − ζ 2 sp
1 − ζ 2 ph , where ζsp =0.8, ωsp =3.6 rads −1 ,
ζph =0.9andωph =0.1rad s −1 . The subscripts sp and ph denote the shortperiod
and phugoid motion, respectively. The resulting controllers are given
in Table 3.2.
The fuzzy gain-scheduled state feedback controller for which the operating
points are identified through fuzzy clustering is validated in the test flight
conditions and compared with the fixed state feedback controller designed

3.3. Fuzzy Partitioning of the Flight Envelope 39
K1
0.2
0.15
0.1
0.05
180
160
140
120
Dynamic pressur [mbar]
K3
−40
−50
−60
−70
180
160
140
120
Dynamic pressur [mbar]
0.6
0.6
K2
0.3
0.2
0.1
0
180
0.8
160
0.7
140
120
Mach number [−] Dynamic pressur [mbar]
K4
−20
−30
−40
−50
180
0.8
160
0.7
140
Mach number [−] Dynamic pressur [mbar]
120
0.6
0.6
0.7
0.8
Mach number [−]
0.7
0.8
Mach number [−]
Figure 3.10: The four gains of the scheduled state feedback matrix K =
[K1 K2 K3 K4] as a function of Mach number and dynamic pressure.
in FC9. In Figure 3.9 the closed-loop poles in the test flight conditions
are illustrated, both using the fixed (Fig 3.9a) as well as the fuzzy gainscheduled
state feedback controller (Fig 3.9b). The spread of the closed-loop
poles using the fixed state feedback controller is much larger than of the
closed-loop using the scheduled state feedback controller. It should be noted
that using the fuzzy gain-scheduled state feedback controller, the damping
of the closed-loop poles is in most test flight conditions better (higher) than
in the design flight conditions. The reason for this phenomenon is not clear,
but it cannot be expected that this is generally the case.
For the fuzzy gain-scheduled state feedback controllers, the rule-base to
compute the state feedback matrix as a function of Mach number and dynamic
pressure becomes as in Equation 3.11 for Nr = 4. The state feedback
matrix is computed in the same way as illustrated by Equations 3.7 to 3.9.
The four gains of the state feedback matrix are illustrated in Figure 3.10
as a function of Mach number and dynamic pressure.
In Figure 3.11 the time histories of the pitch rate resulting from the initial
condition response of the closed-loop system are illustrated. The light
gray area denotes the simulations using the fixed controller. The black area
denotes the simulations using the gain-scheduled controller which uses the
flight conditions FC10 to FC13 as operating points. The dark gray area
denotes the simulations using the gain-scheduled controller for which the

40 Chapter 3. Fuzzy Clustering for Partitioning of the Flight Envelope
Pitch rate [deg ⋅ sec −1 ]
1.2
1
0.8
0.6
0.4
0.2
0
Fixed controller
Scheduled Controller (square)
Scheduled controller (fuzzy)
−0.2
0 1 2 3 4 5
Time [s]
Figure 3.11: Time histories of the initial condition response in pitch rate of the
closed-loop system. The gray area denotes the simulations using the fixed controller,
the black area denotes the simulations using the scheduled controller and the dark gray
area denotes the simulations using the scheduled controller for which the operating
points are identified through fuzzy clustering.
operating points are identified through fuzzy clustering. As expected, the
simulations using the scheduled state feedback controllers are much more
consistent than the simulations using the fixed state feedback controller.
Moreover, the scheduled state feedback controller using the operating points
obtained through fuzzy clustering outperforms the scheduled state feedback
controller with manual selected operating points. The difference between
the two gain-scheduled controllers is not big, which is partly due to the
fact that the arbitrarily selected operating points are close to the operating
points identified through fuzzy clustering.
3.4 Conclusions
Fuzzy clustering is applied to a data set that is selected based on the flight control
law design objectives and the handling quality requirements. Subsequently the parameters
of the clustering algorithm are selected and the obtained fuzzy partition
is used to determine the operating points and membership functions. The scheduling
variables are selected based on the orientation of the fuzzy clusters.
A simple example demonstrates the effectiveness of the proposed methodology. It
can be concluded that the scheduled state feedback controller using the operating
points identified through fuzzy clustering slightly outperforms the scheduled state
feedback controller with arbitrarily selected operating points. However, the main
objective of the automated approach based on fuzzy clustering is not to improve

3.4. Conclusions 41
the performance, but is focussed towards reducing the design effort with respect
to locating the operating points.
The intended reduction of the design effort can be achieved by the automated
identification of the operating points, circumventing a time-consuming, iterative
procedure for the identification of the operating points and the tuning of the
FCL parameters. Moreover, this model-based approach is likely to result in fewer
operating points, which further reduces the design effort.
As can be seen in Figure 3.7, three regimes can be found as a function of Mach
number, namely a regime for M 0.46, a regime for 0.46 M 0.70 and a regime
for M 0.70. It turns out that the aerodynamic model is structured in exactly the
same way. All the operating points are more or less in the center of each of these
regimes. When an operating point is placed on, or close to, the boundary between
two regimes, the corresponding operating regime would be relatively small because
of the nonlinear dynamics in these parts of the flight envelope and would therefore
result in an unnecessary high number of operating points. This can be prevented
by using a model-based approach to identify the operating points.
The transparency of the resulting scheduler, using the same operating points
and scheduling variables for all longitudinal FCL parameters, makes it easier to
predict the effect of small adjustments made to the FCL parameters in the later
stages of the design process. This will also contribute to the reduction of the design
effort.

42 Chapter 3. Fuzzy Clustering for Partitioning of the Flight Envelope

4
Scheduled Classical Control
The problem addressed in this chapter is the design of a parameter scheduling
scheme for the longitudinal stability and control augmentation system of the
SCA model. The baseline flight control laws that are available within the synthetic
environment are used to illustrate the fuzzy gain scheduling approach.
The six most relevant gains of the longitudinal controller are scheduled using
this approach, while for all the other gains, including those of the lateral
controller, the baseline scheduling scheme is preserved.
This chapter is organized as follows: In Section 4.1 the longitudinal stability and
control augmentation system is described and the gain scheduling problem is addressed.
The partitioning of the flight envelope in operating regimes is discussed
in Section 4.2. Section 4.3 describes the design procedure that is used to tune the
local flight control law parameters. The resulting rule-base and implementation
of the fuzzy gain scheduled controller is discussed in Section 4.4. In Section 4.5
the validation of the gain-scheduled flight control laws is described, which includes
both linear analysis as well as pilot-in-the-loop simulation of the complete nonlinear
aircraft model. Concluding remarks are given in Section 4.6. In Appendix A a
brief description the longitudinal equations of motion and the longitudinal aerodynamic
model of the SCA model is given.
4.1 Stability and Control Augmentation System
The simplified structure of the “classical” longitudinal FCLs of the SCA model is
illustrated in Figure 4.1. Refer to Appendix A for a brief description of the longitudinal
flight dynamics of the SCA model. The functionality of the longitudinal
FCLs can be divided into the following parts:
Pitch damper. Proportional feedback of the pitch rate q is the most basic
approach for augmentation of Cmq , which is the aerodynamic coefficient that
represents the change in the moment around the Y-axis m due to a change in
q. This aerodynamic coefficient is directly related to the natural short-period
damping ζsp of the aircraft. The ideal value for the gain GQ is given by the
43

44 Chapter 4. Scheduled Classical Control
Figure 4.1: Simplified representation of the longitudinal FCLs.
desired augmentation of Cmq , i.e. ∆Cmq , and the elevator effectiveness Cmδ :
GQ = ∆Cmq
, (4.1)
Cmδ
where Cmδ is the aerodynamic coefficient that represents the change in the
moment around the Y-axis m due to a change in δe.
Feedback path. Proportional feedback of the angle-of-attack α is the most
basic approach for augmentation of Cmα , which is the aerodynamic coefficient
that represents the change in the moment around the Y-axis m due
to a change in α. This aerodynamic coefficient is directly related to stiffness
and the frequency of the short-period motion ωsp. The ideal feedback gain
Gα is given by the desired augmentation of Cmα , i.e. ∆Cmα , and the elevator
effectiveness:
Gα = ∆Cmα
. (4.2)
Cmδ
Since the AoA signal is not available for control, the normal acceleration
and pitch rate are used with appropriate scheduling. The short-period approximation
transfer function from normal acceleration to angle-of-attack α,
ignoring the effect of the elevator deflection, is described by the following
expression:
α mg
nz
=
1
2 ρV 2 T SCNα
, (4.3)

4.1. Stability and Control Augmentation System 45
Figure 4.2: Definition of the membership degrees µδc and µVC .Left:µδcas a function
of the absolute value of the column deflection |δc| in degrees. Right: µVC as a function
of the calibrated airspeed VC in knots.
where m is the mass of the aircraft, g the gravity acceleration, ρ the air
density, VT the true airspeed, S the wing reference area, and CNα is the
aerodynamic coefficient that represents the change in normal force N due to a
change in AoA. The short-period approximation transfer function from pitch
rate to normal acceleration, ignoring the effect of the elevator deflection, is
described by the following expression:
nz
q
= VT
g
1
, (4.4)
τs+1
where τ = U0
in the trimmed condition. The latter equation is reconstructed
Nα
in the feedback path illustrated in Figure 4.1. The derivation of the relations
given in Equations 4.3 and 4.4 is given in Appendix B.2.
The normal acceleration that is fed back is a blending of the normal acceleration
measurement and the normal acceleration signal obtained through
the pitch rate measurement (see Equation 4.4). The blending is a function
of the column deflection δc and the calibrated airspeed VC:
nz,fb = wnz +(1− w)
VT
g
1
τs+1
q, (4.5)
where nz,fb denotes the resulting feedback signal. The weight w is computed
as follows:
w = µδc
µVC , (4.6)
where µδc is the membership degree computed from the column deflection
and µVC is the membership degree computed from the calibrated airspeed,
see also Figure 4.2. With the column centered (w = 0), only the pitch rate
signal is contributing to the feedback signal. The same holds for large stick
deflections at low calibrated airspeed. The normal acceleration signal is contributing
to the feedback signal for large stick deflections at medium and
high calibrated airspeed (w >0).
The ideal proportional gain is obtained as follows:
mg
, (4.7)
KP = Gα
1
2 ρV 2 T SCNα

46 Chapter 4. Scheduled Classical Control
where the ideal value for Gα is given in Equation 4.2. The integral action
KI
s is added in the feedback path in order to reduce the steady-state error
of the achieved pitch rate compared to the commanded pitch rate and to
increase the robustness of the controller with respect to the performance in
off-design flight conditions and disturbances.
Feedforward path. The feedforward path feeds the shaped pilot command,
multiplied by the feedforward gain, directly to the elevator. This augments
the control of the aircraft without compromising its stability. The shortperiod
approximation transfer function from the elevator deflection to the
normal acceleration nz is:
nz(s)
δe(s)
= VT
g
Kq
( s
ωsp )2 +2ζsp( s
ωsp )+1.
With the feedforward gain this relation is modified as follows:
nz(s)
δe(s)
= GFF
VT
g
Kq
( s
ωsp )2 +2ζsp( s
ωsp )+1.
(4.8)
(4.9)
Command shaping filter. This lead-lag filter shapes the pilot command
such that the desired attitude (and flight path) response is obtained (Gibson
1999). The short-period approximation transfer function from elevator deflection
δe to the pitch rate q is as follows:
q(s)
δe(s) =
Kq(Tθ2 s +1)
( s
ωsp )2 +2ζsp( s
ωsp )+1.
(4.10)
The command shaping is achieved by cancelling the (stable) zero corresponding
to the short-period motion and place a new (stable) zero at the desired
location:
Kq(Tθ2,news +1)
( s
ωsp )2 +2ζsp( s
where τLAG = Tθ2 and τLEAD = Tθ2,new .
4.2 Partition
ωsp )+1 = τLEADs +1
τLAGs +1
Kq(Tθ2 s +1)
( s
ωsp )2 +2ζsp( s
ωsp )+1,
(4.11)
The procedure to obtain a fuzzy partition of the flight envelope using fuzzy clustering
is described in more detail in Chapter 3. The fuzzy partition that is derived
in Chapter 3 is used in this chapter to design the gain-scheduled classical FCLs.
The fuzzy partition computed by using the modified membership functions (see
Figure 3.5b) is illustrated in Figure 4.3.
4.3 Automatic Tuning Procedure
Usually, the tuning of the FCLs is performed using different objectives for different
operating points. The operating points represent the centers of the clusters, see

4.3. Automatic Tuning Procedure 47
Figure 4.3: Decomposition of the flight envelope into eight operating regimes. The
dots denote the operating points, while the asterisks denote the test flight conditions.
Each operating point and test flight condition is labelled with a number.
Section 3.3. For instance, an operating point for the landing phase flight condition
requires high precision attitude control, while in an operating point representing
the cruise flight condition the requirements are less restrictive.
The tuning of the gains of the classical FCLs, the default FCLs in the SE, is a
time-consuming procedure. In order to be able to evaluate the performance of the
scheduler in an early stage of the design process, before implementation in the
nonlinear FCLs, the simplified FCLs (see Figure 4.1) are used in combination with
an automatic tuning procedure. This procedure makes use of a fourth order linear
longitudinal model. When the gains are set to zero, except for the feedforward gain
which is set to one, the closed-loop system is equal to the bare aircraft model. In
each iteration in the automatic tuning procedure, the Matlab/Simulink TM model
is trimmed and linearized and the damping and frequency of the short-period and
phugoid motion are evaluated. The automatic tuning strategy, which is developed
in cooperation with experienced flight control engineers from industry, consists of
four steps:
1. Tuning GQ to obtain the required short-period damping.
The main effect of increasing GQ is the increase of the short-period damping
ζsp. The pitch damper gain GQ is set such that the short-period damping is
ζsp =0.80, using an optimization algorithm.

48 Chapter 4. Scheduled Classical Control
Table 4.1: The operating points and their corresponding FCL parameters resulting
from the automated tuning procedure on the simplified longitudinal FCL structure.
FC Mach Alt. Dyn. GQ KP KI GFF τLEAD τLAG
nr. [kft] press.
[-] [mbar]
1 0.83 38.0 118 1.09 0.22 0.59 0.16 0.42 1.49
2 0.59 32.3 72 0.90 0.63 1.74 0.14 0.58 1.68
3 0.65 11.0 220 0.53 0.22 0.63 0.07 0.29 0.67
4 0.39 5.9 90 0.63 0.66 2.06 0.12 0.48 1.09
5 0.33 15.4 44 0.61 1.87 3.60 0.21 0.88 3.28
6 0.59 23.2 108 0.77 0.42 1.34 0.12 0.44 1.21
7 0.55 9.8 160 0.56 0.29 0.92 0.08 0.35 0.81
8 0.79 25.8 185 0.78 0.28 0.97 0.09 0.30 0.81
2. Tuning KI and KP to obtain the required phugoid damping and
maintain the required short-period damping.
The main effect of increasing KI is the increase of the phugoid damping
ζph. However, an undesirable side-effect is the decrease of the short-period
damping. Although the main effect of increasing KP is the increase of the
short-period frequency (see Section 4.1), it also results in an increase of the
short-period damping. However, an undesirable side-effect is the decrease
of the phugoid damping. The tuning KP and KI is therefore an iterative
process. The objective is to tune KI and KP such that the phugoid damping
is equal to ζph =0.90, while the short-period damping stays at ζsp =0.80.
3. Tuning GFF to provide optimal direct input to the elevator.
The dynamics of the aircraft are augmented and fixed at this point. The
purpose of the feedforward path is to improve handling performance. The
feedforward gain GFF should be such that the direct input to the elevators
brings the aircraft close to the commanded set point. The pilot input is in
fact a commanded normal acceleration. The feedforward gain is set equal
to the inverse of the gain of the short-period approximation of the transfer
function from elevator deflection δe to normal acceleration nz.
4. Tuning the lead-lag filter to obtain zero dropback.
The lead-lag filter denominator time constant is chosen to cancel the zero of
the short-period motion Tθ2 . The lead-lag filter numerator time constant is
chosen to obtain zero dropback in pitch attitude.
The operating points and their corresponding FCL parameters resulting from the
automatic tuning procedure are given in Table 4.1.
4.4 The Scheduler
The scheduler for the gains of the FCLs is implemented as a TS fuzzy model. The
rule-base of the TS fuzzy model is equivalent to the rule-base given in Equation 3.11

4.4. The Scheduler 49
Figure 4.4: Hierarchical structure of the scheduler.
Table 4.2: Aircraft configurations.
Config. Flaps Slats Remarks
[deg] [deg]
1 0 0 clean configuration
2 0 25
3 12 25
4 20 25 take-off configuration
5 40 25 landing configuration
and the string of local FCL parameter is defined as Ki =[GQi ,KPi ,KIi ,GFFi ,
τLEADi ,τLAGi ].
The gain scheduler that is described above is designed for clean configuration.
However, during take-off and landing the configuration of the aircraft is typically
modified to optimize its performance with respect to the corresponding flight task.
This is achieved by deploying the flaps and slats that are mounted on the trailing
and leading edge of the wing, respectively. The five configurations that can be
selected in the SCA model, in terms of flaps and slats deflection, are described
in Table 4.2. Since the aircraft dynamics change significantly with aircraft configuration,
the FCL parameters need to be scheduled accordingly. The landing
gear is not taken into account because it has only minor effects on the aircraft
dynamics when it is extended. Besides the gain scheduler that is described above,
a second gain scheduler is designed for the landing configuration using the same
method. Since the flight envelope for landing configuration is much smaller, fewer
operating points are needed compared to the clean configuration. The number of
operating points for the landing configuration in this case is two. The rule-base
for the landing configuration therefore has two rules. The schedulers for Clean
Configuration (CC) and Landing Configuration (LC) run in parallel and their respective
outputs, KCC and KLC, are scheduled as a function of the flap deflection:
R1 : If δfl = SMALL then K = KCC
R2 : If δfl = BIG then K = KLC

50 Chapter 4. Scheduled Classical Control
G Q
τ LEAD
K I
1
K P
0.5
2
0.8
0.6
0.4
200
150
100
50
0.8
0.6
0.4
200
150
100
50
Mach number [−]
Dynamic pressure [mbar]
Mach number [−]
Dynamic pressure [mbar]
15
10
5
G FF
0.8
0.6
0.4
200
150
100
50
0.8
0.6
0.4
200
150
100
50
Mach number [−]
Dynamic pressure [mbar]
Mach number [−]
Dynamic pressure [mbar]
0.35
0.3
0.2
0.15
τ LAG
0.8
0.6
0.4
200
150
100
50
0.8
0.6
0.4
200
150
100
50
Mach number [−]
Dynamic pressure [mbar]
Mach number [−]
Dynamic pressure [mbar]
Figure 4.5: The gain scheduled FCL parameters GQ, KP , KI, GFF, τLEAD, and
τLAG as a function of Mach number and dynamic pressure (clean configuration).
Configurations 1 through 3 in Table 4.2 correspond to small flap deflections, while
configurations 4 and 5 correspond to big flap deflections. A schematic interpretation
of the resulting scheduler is given in Figure 4.4. The rule-bases for clean and
landing configuration could be combined in a single rule-base with flap deflection
as a third antecedent or scheduling variable. However, this would result in more
rules and it would obscure the underlying physical structure.
4.5 Evaluation
As mentioned before, the FCL parameters given in Table 4.1, which are obtained
through the automated tuning procedure, are only used to evaluate the scheduler
in the early stages of the design. The FCL parameters that are implemented in
the full nonlinear FCLs, see Table 4.3, are taken from the FCLs that serve as the
default of the SE. Figure 4.5 shows the corresponding FCL parameters as a function
of Mach number and dynamic pressure.
In this section the closed-loop system is evaluated in a number of test flight conditions
selected by flight control engineers from industry. This evaluation considers
straight and level flight conditions in clean configuration. The GS FCLs are eval-
8
6
10
5
1
0.5

4.5. Evaluation 51
Table 4.3: The operating points and their corresponding FCL parameters resulting
from manual tuning using the full longitudinal FCL structure.
FC Mach Alt. Dyn. GQ KP KI GFF τLEAD τLAG
nr. [kft] press.
[-] [mbar]
1 0.83 38.0 118 1.13 0.14 0.58 0.16 0.42 1.49
2 0.59 32.3 72 0.90 0.62 1.74 0.14 0.58 1.68
3 0.65 11.0 220 0.57 0.11 0.62 0.07 0.29 0.67
4 0.39 5.9 90 0.64 0.62 2.06 0.12 0.48 1.09
5 0.33 15.4 44 0.59 1.99 3.60 0.21 0.88 3.28
6 0.59 23.2 108 0.79 0.36 1.35 0.12 0.44 1.21
7 0.55 9.8 160 0.58 0.20 0.92 0.08 0.35 0.81
8 0.79 25.8 185 0.82 0.18 0.96 0.09 0.30 0.81
Table 4.4: Test flight conditions.
FC Mach Altitude Dynamic
number [kft] pressure
[-] [mbar]
14 0.35 15.0 51
15 0.30 5.0 54
16 0.70 35.0 92
17 0.60 25.0 104
18 0.50 5.0 157
19 0.70 15.0 221
20 0.75 40.0 85
uated by linear analysis and nonlinear simulation. The test flight conditions are
given in Table 4.4 and illustrated in Figure 4.3.
4.5.1 Linear off-line evaluation
The linear analysis consists of evaluating the gain margin GM, the phase margin
PM, the short-period frequency ωsp and the short-period damping ζsp. Ideally the
GM exceeds 12 dB and the PM exceeds 60 ◦ . It can be seen in Table 4.5 that these
criteria are amply met in all the test flight conditions. Furthermore a number of
handling qualities are evaluated: the Control Anticipation Parameter (CAP) criterion,
the pitch rate time history criteria (PR), the pitch axis equivalent time delay
criterion (ETD), and the dropback criterion (DB). These handling qualities are
considered to be the most significant (Gibson 1999, Alony et al. 1998). In order
to comply with the design requirements for normal operation the CAP should be
between 0.085 and 3.6, the pitch rate transient peak ratio should be less than 0.05,
TriseVT should be between 5.33 and 296.2, the ETD should be less than 0.10 sec,
and for the DB the point ( qpeak
qss
, ∆θ
qss
) should be below the line that runs through

52 Chapter 4. Scheduled Classical Control
q [deg⋅s −1 ]
n z [−]
θ [deg]
δ e [deg]
4
2
0
−2
0 5 10 15
2
1.5
1
0.5
0 5 10 15
15
10
5
0
0 5 10 15
0
−5
−10
0 5 10 15
Time [s]
Figure 4.6: Linear simulation results with fuzzy gain scheduling in the cruise flight
condition: pitch rate q, normal acceleration nz, attitude θ, and elevator deflection δe
as a function of time.
the points (0, 3) and (1, 2.35). These criteria are met in all the test flight conditions,
see Table 4.5.
In Figure 4.6 the results of a linear simulation example with fuzzy gain scheduling
are illustrated for test condition FC20. This flight condition corresponds to the
cruise flight condition. During the simulation a block-shaped pilot input is used,
which starts at t = 1 sec and terminates at t = 7 sec. It can be seen from the time
history of θ that the dropback in pitch attitude is indeed negligible.
Table 4.5: Evaluation results in the test flight conditions: gain margin GM, phase margin
PM, short-period frequency ωsp, short-period damping ζsp, control anticipation
parameter criterion (CAP), pitch rate time history criteria (PR), pitch axis equivalent
time delay criterion (ETD), and the dropback criterion (DB).
FC GM
[dB]
PM
[deg]
ζsp
[-]
ωsp
[rad s −1 ]
CAP PR
∆q2
∆q1
PR
TriseVT
ETD DP
( DB
qss
, qmax
qss )
14 23.6 147.7 0.78 2.22 0.44 0.020 50.6 0.03 (0.19,1.19)
15 23.3 137.2 0.81 2.31 0.44 0.012 42.6 0.03 (0.18,1.18)
16 26.4 179.9 0.80 3.22 0.37 0.015 65.6 0.03 (0.13,1.18)
17 26.8 138.9 0.79 3.57 0.43 0.019 54.0 0.03 (0.12,1.18)
18 28.5 124.5 0.82 4.51 0.48 0.011 38.7 0.03 (0.09,1.17)
19 30.4 134.0 0.77 5.53 0.47 0.022 45.0 0.03 (0.08,1.19)
20 26.5 179.5 0.79 3.03 0.37 0.017 74.3 0.03 (0.14,1.18)

4.5. Evaluation 53
Mach number [−]
Dynamic pressure [mbar]
Flap deflection [deg]
0.4
0.3
0.2
0 20 40 60
100
50
0
0 20 40 60
40
30
20
10
0 20 40 60
Time [s]
(a) Inputs.
G Q
K I
τ LEAD
1.5
1
0.5
0
CGS
FGS
0
15
20 40 60
10
5
0 20 40 60
0.6
0.4
0.2
0
0 20 40 60
Time [s]
K P
G FF
τ LAG
20
10
0
0
20
20 40 60
15
10
5
0
0.7
20 40 60
0.65
0.6
(b) Outputs.
0.55
0 20 40 60
Figure 4.7: (a) Inputs of the Fuzzy Gain Scheduler. (b) Outputs of the conventional
(CGS) and fuzzy (FGS) gain scheduler. The solid line denotes the conventional gain
scheduler (CGS), while the dash-dotted line denotes the fuzzy gain scheduler (FGS).
4.5.2 Pilot-in-the-loop evaluation
Pilot-in-the-loop tests were performed in the Research Flight Simulator (RFS) of
the National Aerospace Laboratory (NLR) in the Netherlands. In Figure 4.7 the
results of a flight simulator test with the Fuzzy Gain Scheduler (FGS) are illustrated.
This specific test is to verify the transient behavior due to the transition
from landing configuration to clean configuration while the aircraft is accelerating.
During this test the aircraft crossed four operating regimes, namely two for
landing configuration and two for clean configuration. The inputs to the FGS are
shown in Figure 4.7a. The Mach number and the dynamic pressure are continuously
increasing (which indicates the acceleration), while the flaps are retracted
from 40 degrees to 0 degrees. The latter is performed in three stages according
to the following settings: flaps/slats 40/25 (landing configuration), 20/25 (take-off
configuration), 12/25 (intermediate configuration) and 00/00 (clean configuration)
degrees. The transition from intermediate to clean configuration is not shown in
Figure 4.7. The parameters of the Conventional Gain Scheduler (CGS) are added
using the same input data in order to give an indication of the similarities and
differences. The CGS parameters are therefore not resulting directly from pilotin-the-loop
simulation. The differences between the parameters of the FGS and
those of the CGS are due to the differences in the scheduling mechanism. In the
operating points of the FGS the parameters are the same for both controllers. As
can be seen in Figure 4.4, the scheduling as a function of flap deflection takes place
between 12 and 20 degrees. This transition occurs at about 46 sec. Comparing the
SCAS parameters from the CGS with those of the FGS it is clear that the gains
of the FGS are much smoother, especially during the transition from flaps/slats
Time [s]

54 Chapter 4. Scheduled Classical Control
20/25 to 12/25 degrees. In the CGS the scheduling as a function of flap deflection
is a switch at 15 degrees, which explains the discontinuous behavior of the CGS
gains in this region. During this flight simulator test of the FGS the pilot commented
that he could feel no transients due to flaps/slats retraction.
During the evaluation of the FGS, covering about 80% of the flight envelope, the
two test pilots agreed that the FGS performed as good as the CGS. The FGS was
not evaluated in the upper left corner of the flight envelope, see Figure 4.3, due to
the lack of available simulator time and because this region is uninteresting from an
operational point of view. The fact that the FGS is designed in a more automated
fashion indicates the improvement of this approach. Although it should be taken
into account that the FGS approach is implemented only for the most relevant
parameters of the longitudinal SCAS, we conclude that the flight simulator test
results are very encouraging and that they serve as a proof-of-principle of the
proposed fuzzy gain scheduling design approach.
4.6 Conclusions
The presented procedure shows good potential for automated design of gain scheduled
flight control laws. The objective was to reduce the design effort while at
least maintaining performance. The results were evaluated and discussed extensively
with the industrial partners of the ADFCS project. The experienced test
pilots could not detect any significant difference between the conventional FCLs
and those implemented with FGS for the SCAS of the longitudinal axis. Even
though the FGS used fewer operating points than in the conventional approach, it
can be concluded that its performance is comparable. The reduction of the design
effort seems evident because of the more automated approach resulting in fewer
operating points, but this has not been evaluated in detail.
When the flight control engineer designs the gain scheduler for a classical controller,
this is typically performed separately for each FCL parameter that requires
scheduling. Moreover, each FCL parameter is not necessarily scheduled with the
same scheduling variable(s). In other words, each FCL parameter has its own set
of operating points, which makes the structure of the gain scheduler opaque. This
iterative, single-loop approach of tuning the (scheduled) FCL parameters is timeconsuming
and contributes therefore significantly to the total design cost. Due to
this opaque structure, the mutual dependencies of the FCL parameters are hard
to identify. This makes it difficult to understand what needs to be changed when it
turns out that the performance is not as expected in a certain flight condition. By
using the same scheduling variables and operating points for all related FCL parameters
that require scheduling, the mutual dependencies of the FCL parameters
are clearer and corrections are easier to make. However, this does not necessarily
mean that the best performance is achieved by scheduling all FCL parameters in
exactly the same way. To get the same closed-loop dynamics over the entire operating
range, each FCL parameter should have a dedicated scheduling mechanism.
Moreover, it makes sense to use different operating points for the longitudinal and
lateral FCL parameters, since they correspond to different aircraft dynamics.

5
Scheduled Robust Multivariable
Control
The main contribution of this chapter is the combination of the gain scheduling
technique based on fuzzy clustering, introduced in Chapter 3, and an H∞ design
approach. The design objective is to have the closed-loop system match a predefined
reference model, taking into account disturbances and uncertainties. The
H∞ controllers are designed locally in a number of operating points and the
interpolation between them (scheduling) takes place through fuzzy membership
functions. The H∞ design approach requires less design effort than the classical
design approach, but it does not necessarily result in a better controller in
terms of stability and performance. The performance using the gain-scheduled
H∞ controller is significantly improved compared to the performance using a
fixed H∞ controller.
This chapter is organized as follows: An overview of scheduled robust multivariable
control is given in Section 5.2. In Section 5.3, the H∞ design problem is
described. Section 5.4 outlines the design process of the local H∞ controller, while
in Section 5.5 the partition of the flight envelope and the design points are briefly
described. The parameter scheduling approach for H∞ controllers is described in
Section 5.6. Concluding remarks are given in Section 5.7.
5.1 Introduction
The main advantage of using robust multivariable (MV) control techniques over
classical control techniques is that they replace the single-loop design approach
by a multi-loop design approach. This reduces the required design effort, while
simultaneously taking into account model uncertainties and disturbances. In the
case of an aircraft, examples of model uncertainties are:
1. Variation in the aircraft dynamics as a function of flight condition.
2. Variations in the aircraft dynamics as a function of the weight and the centerof-gravity.
55

56 Chapter 5. Scheduled Robust Multivariable Control
3. Aerodynamic uncertainties, e.g., due to variations in the production or model
mismatch.
4. Unmodelled higher-order dynamics.
The latter is not considered in this thesis, since the controller validation is performed
using the same model as used for the controller design. The SCA model
does facilitate the option to modify the aerodynamic characteristics.
It is unlikely that the design specifications can be met over the entire operating
range using a single robust multivariable controller. Typically a compromise has to
be found between performance and robustness. Two types of robustness are considered,
namely robust stability and robust performance. Robust stability implies
that the closed-loop system is stable for all possible plants as described by the uncertainty
description. Robust performance implies that the performance objective
is achieved under all possible plants as described by the uncertainty description.
Expanding the uncertainty, i.e. expanding the operating regime, reduces the probability
that robust stability and robust performance can be achieved. Scheduling
is a common approach to overcome this trade-off, however, scheduling of multivariable
controllers is a difficult task.
5.2 Overview of Scheduled Robust MV Control
One of the difficulties with implementing gain scheduled multivariable control laws
is the complexity of such control laws. For a controller of order n with ni inputs
and no outputs, there are n(1+nino)+nino controller parameters to schedule (Ly
et al. 1985, Nichols et al. 1993, Hyde and Glover 1993). Two categories of gainscheduling
robust multivariable controllers can be found in the literature:
1. Scheduling of the controller output matrix.
2. Scheduling of both the controller dynamics and the controller output matrix.
An example of the first category can be found in (Garg 1997). A nominal controller
is designed that gives a stable closed-loop system in the entire operating range.
The parameters of the output matrix are optimized such that the closed-loop system
at the off-design points closely matches the closed-loop system in the design
point. The controller eigenvalues are not affected. It is possible that this simplified
scheduling scheme is not sufficient to account for significant variations in the plant
poles and zeros.
Many examples of the second category can be found in the literature, showing a
wide variety of design approaches. However, in all approaches an attempt is made
to reduce the order of the controller and/or to impose a certain structure on the
controller in order to simplify the scheduling problem.
In (Nichols et al. 1993) a gain-scheduled robust multivariable controller is designed
for the autopilot function of a missile. The system has two inputs and one
output. The order of each controller is reduced from seven to four. Moreover, after
the order reduction, two poles and two zeros that do not vary significantly from
one operating condition to the next are replaced by their average values. This

5.3. General Description of the Robust Control Problem 57
modification yields fourth-order linearized controller transfer functions containing
an identical second-order factor at each operating point. This significantly reduces
the number of parameters to be scheduled. Scheduling takes place directly on the
gains, poles and zeros of the fourth order controllers. This is possible if the poles
and zeros are identifiable or recognizable, i.e. if it is possible to identify the pole(s)
and/or zero(s) of a certain mode in each local controller. Scheduling is performed
through linear interpolation as a function of Mach number and angle-of-attack. A
similar approach is taken in (Lin and Khammash 2001). Moreover, in this paper
the parameters are modified such that they monotonically increase/decrease with
the indicated airspeed.
The scheduling of H∞ controllers would be simplified if it is possible to write
the controller as a plant observer plus state feedback:
˙ˆx = Aˆx + H(y − C ˆx)+Bu
u = F ˆx.
The clear structure lends itself to gain scheduling of the F and H matrices. In
general, it is not clear that H∞ controllers can be written as exact plant state
observers as there will be a worst disturbance term entering the observer state
equation as shown in (Doyle et al. 1989). However, in (Hyde and Glover 1993)
it is shown that the normalized coprime factor robust stabilization approach produces
a controller which can be written as a plant observer plus state feedback.
The application describes the design of a controller for a Very Short Take-Off and
Landing (VSTOL) aircraft, which has significant variations in airspeed. The scheduler
is based on linear interpolation of the matrices F and H as a function of true
airspeed and angle-of-attack. A similar design approach can be found in (Pellanda
et al. 2000).
An alternative to the approaches described above is to design all the controllers
simultaneously, inherently solving the stability problems that can occur when
scheduling multivariable controllers. This can be done by using Linear Matrix Inequalities
(LMIs) and is described in (Apkarian et al. 1995, Apkarian and Gahinet
1995, Apkarian et al. 2000). The nonlinear model of the plant is transformed
into the Linear Parameter-Varying (LPV) format, where the parameters vary as
a function of the scheduling variables. The resulting controller is also described
in the LPV format. By incorporating the scheduling variables, the controller adjusts
to the variations in the plant dynamics in order to maintain stability and
high performance along all trajectories of the scheduling variables. This approach
is valid only when a single Lyapunov function is used over the entire operating
range. The design approach is demonstrated on a second order LPV model of a
missile (Apkarian et al. 1995).
5.3 General Description of the Robust Control Problem
Figure 5.1 illustrates the robust control problem (Zhou et al. 1995). The control
objective is to design the dynamic controller K such that the closed-loop system

58 Chapter 5. Scheduled Robust Multivariable Control
Figure 5.1: Robust Control Problem.
meets the design requirements, while taking into account certain disturbances
d and model uncertainties. The block ∆ is a matrix of which the elements δi,j
vary between -1 and 1, δi,j ∈ [−1, 1]. The model uncertainty can be defined by
appropriately choosing the vector signal z and the structure of the matrix ∆. An
illustrative example can be found in (Balas et al. 1991). For a detailed description
of the robust control problem, the reader is referred to (Zhou et al. 1995).
5.3.1 The generalized plant
The generalized plant is a linear model of the nominal open-loop system (plant
model, actuator dynamics, sensor model, anti-aliasing filters, etc), including the
reference model and weighting filters for the design of the controller. Since the
generalized plant is a linear approximation of a nonlinear system, it will vary with
the operating point. The three outputs of the generalized plant are the vectors z,
e and y:
z inputs of the uncertainty block ∆ (variations from the nominal model).
e performance measures: tracking error (e1) and control activity penalty (e2).
y feedback signals that are used as input to the controller K.
The three inputs of the generalized plant are the vectors w, d and u:
w outputs of the uncertainty block ∆.
d disturbances, to be separated in the command signal (d1), sensor noise (d2)
and disturbances (d3).
y control signal (output of the controller).
The reference model that is used for the design of the robust multivariable controller
represents the desired closed-loop system dynamics. The reference model
should be chosen such that it fulfills the stability and performance requirements.

5.4. Robust Multivariable Flight Control Design 59
Each measurement is corrupted with sensor noise which becomes more severe with
increasing frequency. This is phenomenon is realized through the noise filter. The
tracking and actuator filters are (dynamic) weighting filters that need to be tuned.
5.3.2 Model uncertainties
The model uncertainties can be divided into two categories:
1. Model uncertainties due to scheduling.
2. Model uncertainties due to changes in variables which are not scheduling
variables, due to parameter uncertainties and due to unmodelled (higherorder)
dynamics.
The first category of model uncertainties are due to nonlinearities in the aircraft
model. In order to effectively deal with the nonlinearities, a scheduling mechanism
for the controller is introduced. However, in general, the scheduled controller at
off-design flight conditions is not equal to the controller designed for that specific
flight condition (Babuˇska and Oosterom 2003). This mismatch between model and
controller can be interpreted as a model uncertainty. Examples of uncertainties
of the second category are uncertainties due to variations of the weight and the
position of the center-of-gravity, parameter uncertainties, and uncertainties due to
the (false) assumption of a rigid body. It should be noted that the second category
of uncertainties are present both in the design flight condition as well as in the
off-design flight conditions.
5.3.3 Controller
The controller needs to be robust against the model uncertainties and the disturbances,
meaning that it should perform within the specifications under all uncertainties
represented by the block ∆ and the disturbances represented by the vector
d. There are several techniques to design the controller, such as the µ-synthesis
approach or the LMI approach. In this chapter the controller results from design
process using LMIs (Doyle et al. 1989). This is discussed in more detail in
Section 5.4 and Appendix F.
5.4 Robust Multivariable Flight Control Design
The objective is to design a fuzzy gain-scheduled H∞ controller for the full flight
envelope of the SCA model. Furthermore, the FGS H∞ controller should maintain
robust stability and robust performance under the above described uncertainties.
The generalized plant illustrated in Figure 5.2, has seven inputs and six outputs.
The seven inputs are (the dimensions are given in brackets):
• w[1]: scalar output from the uncertainty block (representing model uncertainties)

60 Chapter 5. Scheduled Robust Multivariable Control
2
d1: pilot
command
Column Dynamics
Fs [lb] Col [deg]
4
y1: q_c
5
y2: q_fb
6
y3: nz_fb
K
7
u1: control
signal
G2
G1
Col [deg] q [deg/s]
Reference Model
1
z1
∆
1
w1
tau.s+1
tau.s
5
d4: u_gust
6
d5: w_gust
de
de_c
de_dot
Elevator
Actuator
u_gust
w_gust sens em
de
Linear AC
de_dot perf_de
Control Activity
Weight
q [deg/s]
q VT/g
theta [deg]
nz [g]
nz_comp
VT [m/s]
Anti−aliasing Filters and
Computation
q_noise
3
d2: q_noise
nz_noise 4
d3: nz_noise
Noise Filter
Figure 5.2: The generalized plant with input uncertainty.
Tracking
Weight
num(s)
den(s)
2
e1: perf_q
3
e2: perf_de
• d[5]: the pitch reference signal (d1), two sensor noise sources (d2 and d3),
and two atmospheric turbulence sources (d4 and d5)
• u[1]: the control input (commanded elevator deflection).
while the six outputs are:
• z[1]: scalar input of the uncertainty block
• e[2]: the pitch rate error signal (e1) and the control activity signal (e2)
• y[3]: the command path signal (y1, commanded pitch rate) and two feedback
signals (y2 and y3, pitch rate and normal acceleration ).
The controller is a multi-input, single-output system with the input vector y and
with the output scalar u. The pitch rate error signal penalizes the difference between
the true pitch rate q and the output of the reference model qref. The control
activity signal penalizes the time derivative of the elevator deflection, ˙ δe.
5.4.1 Reference model
The following longitudinal handling quality requirements are used as a guideline for
the reference model: CAP criterion, pitch attitude bandwidth criterion, dropback
criterion and the pitch rate time history criteria (Hodgkinson 1999, Gibson 1999).
The controller is of the pitch rate command/attitude hold type, which implies
that the commanded pitch rate is a function of the column deflection and that
the pitch attitude is kept constant at the current value when the column is at the
center position. The required attitude hold capability implies that the phugoid
motion should be cancelled out and for this reason the phugoid motion is not
included in the reference model. The reference model therefore represents only the

5.4. Robust Multivariable Flight Control Design 61
short-period dynamics. For the clean configuration the reference model is:
q(s)
δe(s) =
Kq(Tθ2 s +1)
( s
ωsp )2 +2ζsp( s
ωsp )+1,
where Kq =7.84, Tθ2 =0.56, ζsp =0.9 andωsp =3.2. The short-period damping
ζsp and natural frequency ωsp are chosen such that they comply with the
handling qualities for normal operation. The time-constant Tθ2 is selected such
that zero dropback is achieved (Gibson 1999). The gain Kq is determined based
on the pilot comments concerning the required stick force to maneuver the aircraft.
To further improve the attitude hold performance, the output of the controller is
, see Figure 5.2.
fed through a low-frequency integrator H(s) = τs+1
τs
5.4.2 Model uncertainty
Two types of model uncertainty can be manipulated in the synthetic environment,
namely uncertainty in the weight and balance and uncertainty in the aerodynamic
model. The ranges of these uncertainties are described below.
Weight and Balance Envelope
In the SCA model the Aircraft Inertia Matrix (AIM) depends on the aircraft weight
and the position of the Center-of-Gravity (CG) along the X-axis. The position of
the CG along the Y -axis and Z-axis has no impact on the AIM and they are therefore
not taken into account. The weight and balance envelope, or centogramme,
is illustrated in Figure 5.3. The centogramme is defined in cooperation with the
manufacturer of the aircraft on which the SCA model is based.
Aerodynamic Uncertainties
One can select from four pre-defined aerodynamic models (see Appendix A for
more details on the aerodynamic coefficients):
1. Nominal model (default).
2. Increased control effectiveness (15% increase of ∆Clδe ,∆Cmδe ,∆Clδs ,∆Cmδs ,
∆CYδr ,∆CRδa ,∆CRδr ,∆CNδa and ∆CNδr ).
3. Decreased control effectiveness (15% decrease of ∆Clδe ,∆Cmδe ,∆Clδs ,∆Cmδs ,
∆CYδr ,∆CRδa ,∆CRδr ,∆CNδa and ∆CNδr ).
4. Reduced stability (30% decrease of ∆Cmq and ∆CRp and 10% decrease of
∆Cm, ∆Cn and ∆CNr ).
Furthermore it is possible to modify the (longitudinal) aerodynamic derivatives
Cmα , Cmq , Cmδe , Cmδs , Clδe ,andClδs . Only the four pre-defined aerodynamic
models are considered.
In this chapter the model uncertainties with respect to the weight and balance and
the aerodynamic model are not implicitly described in the uncertainty block ∆.
These uncertainties are only used for the validation of the controller. The model

62 Chapter 5. Scheduled Robust Multivariable Control
weight [10 3 lbs]
45
43
41
39
37
35
33
31
29
6
7
2 3
maximum take−off weight
1
maximum flight weight
maximum landing weight
27
minimum weight
25
18 20 22 24
5
26 28 30 32
X [% mac]
CG
34 36 38
4
40
Figure 5.3: The centogramme of the SCA model as a function of aircraft weight and
the position of the center-of-gravity along the X-axis.
uncertainty description that is used for the design of the local H∞ controllers is
an uncertainty on the input gain, see Figure 5.2. The gains G1 and G2 determine
the level of the input uncertainty. The block ∆ is in this case a scalar δ and varies
between -1 and 1. When the input u should vary between cminu and cmaxu, the
gains G1 and G2 have the following values:
G1 = cmax + cmin
2
, G2 = cmax − cmin
.
2
Taking into account the uncertainty scalar δ, the uncertain input becomes:
5.4.3 Weight functions
uunc =(G1 + δG2) u.
In the generalized plant, as presented in Figure 5.2, there are two weight functions
that need to be tuned, the tracking weight function and the control activity weight
function. The proportion between these weights determines the relative importance
of the following two objectives:
1. Design the controller such that the closed-loop system matches the reference
model.
2. Limit the required actuator control activity.
These weights can either be constant or frequency dependent. These weights influence
the achievable γ, which is defined as ||Tzw||∞ (see also Figure 5.1).

5.4. Robust Multivariable Flight Control Design 63
Gain [dB]
30
20
10
0
−10
−20
−30
−40
Reference model
Tracking weight
Control Activity Weight
Local Model (Mach = 0.55, Alt = 12 kft)
10 −2
10 0
Frequency [rad ⋅ sec −1 ]
Figure 5.4: Gain of the reference model, tracking weight and control activity weight
as a function of the frequency.
The tracking weight is frequency dependent:
Wtrack =
432 (s +1)
.
(s +0.1)(s + 12) 2
The tracking weight has a high DC-gain of 30 dB to suppress steady state errors
on the pitch rate. In the frequency range of interest with respect to the reference
model, between 0.01 and 10 rad sec −1 , the gain should be lower. For this reason
the pole p1 = −0.1 is in the denominator of the reference model. To prevent the
gain of the tracking weight to become too small in the frequency range of interest
of the reference model, the zero z1 = −1 is added to the numerator. Finally the
two poles p2,3 = −12 are added to the denominator to make sure the emphasis is
on the DC-gain and the frequency range of interest of the reference model. The
gain of the reference model, the tracking weight, the control activity weight are
illustrated in Figure 5.4 as a function of frequency. The gain of the open-loop longitudinal
model in one of design flight conditions (FC27) as a function of frequency
is added as an example, see also Section 5.5.
The weight corresponding to the actuator control activity is frequency independent
and is set equal to one. In this way the control activity weight is inferior
to the tracking weight in the frequency range of interest with respect to the reference
model, see also Figure 5.4. At the same time the control activity weight
is superior to the tracking weight in the high end of the frequency range of the
elevator actuator. The maximum speed of the elevator is 50 rad/sec.
5.4.4 Controller Design
The output-feedback H∞ controller is designed in continuous-time using the LMI
approach, see Appendix F for a brief description. This approach is further moti-
10 2

64 Chapter 5. Scheduled Robust Multivariable Control
Figure 5.5: Possible ways to tackle the robust multivariable control problem.
vated below.
The objective is to design a low-order discrete-time H∞ controller. There are a
number of arguments for low-order controllers, e.g., to reduce the required computational
power or the required storage space. However, in this case the main reason
to reduce the order of the controller is to minimize the number of parameters that
need to be scheduled, which greatly simplifies the scheduling problem for robust
multivariable controllers.
The order of the controller is equal to the order of the generalized plant for the
LMI approach (Boyd et al. 1994, Apkarian et al. 1995) or equal to the order of the
generalized plant plus twice the order of the scaling filter D for the µ-synthesis
approach (Zhou et al. 1995). The LMI approach is selected over the µ-synthesis
approach because the resulting controller is of significant lower order. Additional
order reduction is required to further simplify the scheduling problem.
There are a number of approaches to the design of a low-order, discretetime
controller through the LMI approach (see Figure 5.5). The design of the
H∞ controller can be performed in continuous-time and in discrete-time. In this
case it is chosen to perform the design of the controller in continuous-time, using
the full-order generalized plant. Experiments show that order reduction on
the generalized plant has a negative impact on the performance and robustness
of the resulting controller. Performing order reduction on the controller is much
less sensitive in this respect. For this reason it was chosen not to perform order
reduction on the generalized plant. The main argument to perform the design in
continuous-time is that in Matlab TM there are tools readily available for order reduction
of continuous-time systems. Since the order reduction is performed after
the design of the controller, this means that the design itself should be performed
in continuous-time. The order reduction and discretization are discussed in more
detail in the following paragraphs.

5.4. Robust Multivariable Flight Control Design 65
5.4.5 Order reduction
As mentioned before, the order of the controller is equal to the order of the generalized
plant for the LMI approach. This typically leads to a controller of higher
order than is strictly necessary to comply with the design specifications. Consequently,
the excess of poles and zeros are pushed to the high frequency range in
order to minimize their (unwanted) influence. A low-order controller is obtained
in two steps, namely:
1. Hankel order reduction
The minimum realization of the generalized plant, see Figure 5.2, is of 18 th
order. The controller is of the same order. Using Hankel order reduction
the order of the controller is reduced to eleven, while the Bode diagram
does not change in the frequency range of interest ([0.01, 10] rad sec −1 ). The
frequency range of interest is such that it includes both the phugoid motion
as well as the short-period motion frequency, see also Figure 5.4. However,
the Hankel order reduction still leaves some high frequency modes.
2. Remove high-frequency modes
The 11 th order controller has modes with a frequency higher than the Nyquist
frequency, ωN = fs · π, where fs denotes the sample frequency. The sample
frequency of the flight control system of the SCA model is equal to fs =
50 Hz and the Nyquist frequency is therefore equal to ωN = 157.1 radsec −1 .
Removing these modes results in a ninth order controller.
5.4.6 Discretization
Since the controller is to be used in a digital system, it needs to be discretized.
A straightforward Tustin discretization is used to obtain the discrete-time controller,
since this method results in a discrete-time controller that best matched
the continuous-time controller in terms of the frequency response.
5.4.7 Transformation to the δ-operator form
First attempts showed that parameter scheduling using the discrete-time transfer
function form resulted in unstable controllers in a number of off-design flight conditions.
A method approach to reduce the parameter-pole sensitivity is to transform
the controller to the δ-operator form (Aström and Wittenmark 1997):
⎡
⎤
a1 1 0 ... 0
⎡ ⎤
b1
⎢
.
a2 ⎢ 1 1
..
.
⎥
. ⎥ ⎢b2⎥
⎥ ⎢ ⎥
⎢
A = ⎢
.
⎢a3
0 1
..
⎥ ⎢ . ⎥
0⎥
, B = ⎢
⎥ ⎢ . ⎥ , C =
⎢ .
⎣ .
.
.
. .
.
.. . ⎥ ⎢ . ⎥
..
1⎦
⎣ . ⎦
an 0 ... 0 1 bn
1 0 ... 0 , D = d .
In this form the parameter-scheduled controller turned out to be stable for all flight
conditions. An example is illustrated in Figure 5.6. The poles of the scheduled

66 Chapter 5. Scheduled Robust Multivariable Control
Imaginary Axis
2
1
0
−1
−2
x 10 −3
Transfer function
0.998 0.999
Real Axis
1 1.001
Imaginary Axis
2
1
0
−1
−2
x 10 −3
δ−operator
0.998 0.999
Real Axis
1 1.001
Figure 5.6: Coefficient-pole sensitivity. Comparison between the discrete-time transfer
function form (left) and the δ-operator form (right). The black x-marks denote the
poles of the five controllers. The grey x-marks denote the poles of scheduled controllers.
controllers using the transfer function form are scattered, moreover, they cross
the unit circle. The poles of the scheduled controllers using the δ-operator move
around in a much more predictable manner. It should be noted that during the
transformation from discrete-time transfer function form to the δ-operator form,
the poles do not stay exactly in the same position due to the finite computer
accuracy (compare the black x-marks in the left and right plot of Figure 5.6).
5.5 Partition of the Flight Envelope
The partitioning of the flight envelope is obtained by fuzzy clustering of the relevant
aerodynamic derivatives. The approach is the same as in Section 3.3, however,
due to some modifications in the aerodynamic model the resulting partition of the
flight envelope is not equivalent to the partition presented in Chapter 4. Based on
validation and performance criteria, the optimal number of clusters is found to be
nine. Since in this case the aerodynamic database is smoother, due to a discontinuity
in the aerodynamic database that has been fixed, the number of clusters has
increased. The resulting partition of the flight envelope is illustrated in Figure 5.7
as a function of Mach number and altitude. Note that the membership functions
are defined in terms of Mach number and dynamic pressure (see also Chapter 3).
The nine design points are denoted by black dots, see Table 5.1 for more details.
The light areas around the design points denote large membership degrees, i.e., areas
where one controller is dominating, the dark areas denote the flight conditions
where the controller parameters of neighboring design points are interpolated. Interpolation
takes place through the membership functions, which are defined as
functions of the two scheduling variables: Mach number and dynamic pressure.
The membership functions are shown in Figure 5.8.

5.6. Parameter Scheduled Robust Multivariable Control 67
Figure 5.7: Partitioning of the flight envelope of the aircraft for clean configuration
as a function of Mach number and altitude.
5.6 Parameter Scheduled Robust Multivariable Control
The concept of the parameter scheduling approach is illustrated in Figure 5.9.
There is a single linear dynamic controller and the scheduling is performed directly
on the parameters of this controller. In contrast to output scheduling, the parameter
scheduling concept imposes a number of restrictions on the local controllers. In
order to make sure that the scheduling takes place amongst like parameters, the
local controllers need to be of the same order and need to have the same structure,
e.g., the observable canonical form. In order to design local H∞ controllers that
Table 5.1: The nine design points resulting from fuzzy clustering, denoted by FC26
to FC34.
FC Mach Altitude Dynamic
number [kft] pressure
[-] [mbar]
26 0.60 7.2 214
27 0.55 12.0 147
28 0.77 24.1 188
29 0.56 20.5 108
30 0.53 30.2 63
31 0.24 1.7 39
32 0.83 28.0 188
33 0.75 35.0 108
34 0.34 8.3 62

68 Chapter 5. Scheduled Robust Multivariable Control
Membership degree
Membership degree
1
0.5
0
0.2 0.3 0.4 0.5 0.6 0.7 0.8
Mach number [−]
1
0.5
0
50 100 150 200 250
Dynamic pressure [mbar]
Figure 5.8: Membership functions for the flight envelope in clean configuration.
Figure 5.9: Schematic representation of the parameter scheduling concept.
are suitable for gain scheduling, the following three issues are of importance:
1. The controllers should be stable (preferably also with stable zero dynamics).
2. The controllers should be of low-order (designed through low-order generalized
plant and order reduction on the H∞ controller, see Section 5.4).
3. The controllers should have equivalent structures (the same number of poles
and zeros).
It is known that a necessary and sufficient condition of the existence of a stable
stabilizing controller (strong stabilization) is the so-called parity interlacing
property (Youla et al. 1974). Since the H∞ controller is in general not unique, it is
reasonable to expect that even if the H∞ central controller is unstable, there might
still be a stable controller that could satisfy the H∞ norm bound. In (Zeren and
Özbay 1999, Cao and Lam 2000) an approach for designing such high order stable
H∞ controller has been suggested based on the parametrization of all suboptimal
H∞ controllers. This approach conservatively converts the stable H∞ controller
design problem into another 2-block standard H∞ problem. In addition, the order
of the controller may be twice as high as the order of the central controller.

5.6. Parameter Scheduled Robust Multivariable Control 69
Altitude [kft]
45
40
35
30
25
20
V C = 150 Kts
FC30
FC29
FC33
M = 0.85
MAX
FC36
FC28
FC32
15
10
FC34
FC27
FC26
V = 375 Kts
C
5
FC37
0
0.1 0.2
FC31
0.3 0.4 0.5 0.6
Mach number [−]
0.7 0.8
Figure 5.10: The black dashed continuous line denotes the flight envelope for clean
configuration (FC26 → FC34). The black dots denote the design flight conditions
for clean configuration, which are obtained through fuzzy clustering. The light gray
dashed line denotes the flight envelope for landing configuration. The light gray dot
denotes the design flight condition for landing configuration (FC37). The black asterisk
denotes a test flight condition. The dark gray continuous line denotes in which parts
of the flight envelope the evaluation of the parameter scheduling concept has taken
place.
In (Campos-Delgado and Zhou 2001) a weighting function is introduced to alleviate
the conservativeness of this approach.
Fortunately the central controller is stable for this application, which is also
influenced by the reference model and weight functions, and the above mentioned
method is therefore not applied in this thesis.
5.6.1 Scheduling of the H∞ controllers
The ideal situation would be to schedule the poles and zeros of the local controllers
directly. However, this only makes sense when there is an unambiguous
relation between the poles and zeros of each of the H∞ controllers. If this relation
is not present, the parameters of the controllers need to be scheduled instead. In
that case the local controllers should be written in a format with low coefficientpole
sensitivity in order to improve their schedulability. It is well known that the
coefficient-pole sensitivity increases with the order of the system (Aström and
Wittenmark 1997). For this reason the order of the H∞ controller should be as
low as possible. Rewriting the local controllers in the δ-operator form further reduces
the coefficient-pole sensitivity.

70 Chapter 5. Scheduled Robust Multivariable Control
Figure 5.11: Hierarchical structure of the parameter scheduler.
The design procedure as described in Section 5.4 is performed for 10 flight conditions,
see Figure 5.10. One design point represents the landing configuration and
nine design points represent the clean configuration. The order of all 10 controllers
is reduced to nine.
For the parameter scheduling the controller is rewritten from the discrete-time
description into the observable canonical δ-operator form. Since the basic controller
is a three input single output system, the controller is defined as follows:
⎡
⎤
where
K =
⎢
⎣
A On×n On×n B1 On×1 On×1
On×n A On×n On×1 B2 On×1
On×n On×n A On×1 On×1 B3
C C C D1 D2 D3
⎡
⎤
a1 1 0 ... 0
⎡ ⎤
bi1
⎢
.
a2 ⎢ 1 1
..
.
⎥
. ⎥
⎢bi2⎥
⎥
⎢ ⎥
⎢
A = ⎢
.
⎢a3
0 1
..
⎥
⎢ .
0⎥
, Bi = ⎢ .
⎥
⎥
⎢ . ⎥ ,
⎢ .
⎣ .
.
.
. .
.
.. . ⎥
⎢ . ⎥
..
1⎦
⎣ . ⎦
an 0 ... 0 1
bin
C = 1 0 ... 0 , Di =
di , i =1, 2, 3.
The number of parameters that are scheduled is equal to (1 + ni) n + ni for the
dynamic controller, where n denotes the order of the controller and ni denotes the
number of inputs, plus the time-constant of the low-frequency integrator. In this
case the controllers are of 9th order, with three inputs and one output. In total
there are 40 parameters subject to scheduling. These parameters are tuned in the
design flight conditions illustrated in Figure 5.10. The flight condition FC37 denotes
the design flight condition for landing configuration and the flight conditions
FC26 through FC34 denote the nine design flight conditions for clean configuration.
The parameter scheduler is illustrated in Figure 5.11. The landing configuration is
covered by a single operating point, and therefore also by a single parameter set for
the controller K = KLC and the time-constant τ = τLC. The clean configuration
⎥
⎦ ,

5.6. Parameter Scheduled Robust Multivariable Control 71
Table 5.2: Gain and phase margin of the closed-loop system in the 10 design flight
conditions and an additional test flight condition (FC36).
FC Mach Alt. Dyn. δfl δsl LG GM PM
nr. [kft] press. [deg] [deg] [0/1] [dB] [deg]
[-] [mbar]
26 0.60 7.2 214 0 0 0 6.5 23.3
27 0.55 12.0 147 0 0 0 7.2 26.4
28 0.77 24.1 188 0 0 0 7.1 26.0
29 0.56 20.5 108 0 0 0 8.0 30.5
30 0.53 30.2 63 0 0 0 9.6 37.2
31 0.27 2.0 49 0 0 0 10.4 55.6
32 0.83 28.0 188 0 0 0 7.5 28.1
33 0.75 35.0 108 0 0 0 8.3 32.4
34 0.34 8.3 62 0 0 0 9.4 37.2
36 0.67 23.0 144 0 0 0 6.7 27.1
37 0.22 2.0 32 40 25 1 12.4 50.3
is covered by nine operating points, and therefore also by nine parameter sets for
the controller K = KCC. The parameters for K = KCC are a function of Mach
number and dynamic pressure and are computed using the following rule-base:
Ri : If M is MNi and q is DPi then KCC = Ki
The membership functions corresponding to M is MNi and M is DPi are illustrated
in Figure 5.8. In all nine operating points for clean configuration, the same
time-constant τ = τCC is used. The controller parameters for landing and clean
configuration are then scheduled as a function of flaps deflection to obtain K and τ:
R1 : If δfl is Extended then K = KLC and τ = τLC
R2 : If δfl is Retracted then K = KCC and τ = τCC
5.6.2 Stability analysis
In Table 5.2, the gain margin and phase margin of the closed-loop system in the
10 design points and an additional test point (FC36) are given. In all cases the
gain margin exceeds 6 dB, however, the 30 degrees phase margin is not always
achieved.
It should be noted that in the aerospace industry often a minimum gain margin
of 12 dB and a minimum phase margin of 60 degrees is taken as a reference. In order
to achieve this, the local designs have to be improved. The stability margins can
be influenced through the reference model, the weight functions, the uncertainty
model description, etc. However, here we have concentrated on the scheduling
aspects of the local controllers and not on the optimization of the local controllers

72 Chapter 5. Scheduled Robust Multivariable Control
Altitude [kft]
45
40
35
30
25
20
15
10
5
V C = 150 Kts
FC31
FC34
FC30
FC29
FC27
FC26
FC33
M MAX = 0.85
FC28
FC32
V C = 375 Kts
0
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Mach number [−]
Figure 5.12: Randomized stability analysis. The gray dots denote the flight conditions
that have been used for stability analysis. The black dots denote the two design flight
conditions, while the black o-marks denote the flight conditions for which the output
gains are tuned. The black x-marks denote the 9 flight conditions for which the gain
margin is smaller that 3 dB and/or the phase margin is smaller than 15 degrees.
to meet industrial standards.
5.6.3 Controller validation
The stability margins have been evaluated at 200 randomly defined flight conditions
for clean configuration. Not only the Mach number and altitude were selected
randomly, but also the aircraft weight and center-of-gravity as well as the aerodynamic
uncertainty. Nine flight conditions resulted in a closed-loop system with
a gain margin lower than 3 dB and/or a phase margin lower than 15 degrees, see
Figure 5.12. All these flight conditions correspond to a low dynamic pressure, lower
than any of the nine design points. This indicates that either the design points
should be closer to the edge of the flight envelope or the robust stability of the
local controllers needs to be improved.
The parameter-scheduled controller was successfully tested during pilot-in-the-loop
simulations in the NLR Research Flight Simulator. The pilot performs longitudinal
maneuvers to evaluate the dynamics and performance of the closed-loop system
at several flight conditions scattered over the flight envelope and rates the system
using the Cooper-Harper (CH) rating scale (Cooper and Harper Jr. 1969), see
Figure 5.13. As can be seen in Figure 5.10, a large part of the flight envelope was
covered. In all flight conditions where the longitudinal dynamics were evaluated
the pilot gave the system CH-1. With respect to the required stick force when
performing longitudinal maneuvers, the pilot gave ratings ranging from CH-2 to

5.6. Parameter Scheduled Robust Multivariable Control 73
Figure 5.13: Cooper-Harper rating scale (Source: (Cooper and Harper Jr. 1969)).
CH-4. There is a clear correlation between the dynamic pressure and the pilot
rating, namely the lower the dynamic pressure, the higher the pilot rating. Note
that a high CH rating means a poor performance. The reason for this is that the
same reference model was used for all nine design flight conditions for clean configuration.
The pilot expects a more responsive system for low dynamic pressure.
This can be easily solved by modifying the reference model accordingly.
Figures 5.14 and 5.15 show the time histories of an approach and landing. In
Figure 5.14 the position of the aircraft in terms of the deviation from the glide-slope
is illustrated as a function of time. In Figure 5.15 the variables that are interesting
with respect to the scheduling mechanism and/or configuration changes are illustrated.
The main interest of this exercise is to evaluate the scheduling mechanism
while going through a number of configuration changes and covering a fair part
of the range in terms of Mach number and dynamic pressure. No anomalies were
found. In Figure 5.16 the time history of a push-pull to pitch attitude is illustrated.
The data are taken from the RFS recordings. The pilot controls the aircraft to -5,
0 and 5 degrees of pitch attitude while evaluating the performance and dynamics.
It can be seen that the control in pitch attitude is fairly accurate.

74 Chapter 5. Scheduled Robust Multivariable Control
Lateral deviation [m]
Altitude [kft]
100
0
−100
−15 −10 −5
Flight simulator data
Glide slope
0
2
1
0
−15 −10 −5
Downrange [km]
0
Figure 5.14: Flight simulator time histories of the final approach and landing. The
lateral deviations are the result of intentional pilot inputs to evaluate the lateral controller.
Mach number [−]
Dynamic pres. [mbar]
Flaps/Slats [deg]
Landing gear [−]
0.25
0.2
0 20 40 60 80 100 120 140 160 180 200 220
60
40
20
0
40
20 40 60 80 100 120 140 160 180 200 220
25
20
12
Flaps deflection
0
Slats deflection
0 20 40 60 80 100 120 140 160 180 200 220
1
0.5
0
0 20 40 60 80 100 120 140 160 180 200 220
Time [s]
Figure 5.15: Flight simulator time histories of the final approach and landing. The
illustrated variables are the scheduling variables and/or denote configuration changes.

5.7. Conclusions 75
Column position [deg]
Pitch attitude [deg]
4
2
0
−2
−4
0 5 10 15 20 25 30
5
0
−5
0 5 10 15
Time [s]
20 25 30
Figure 5.16: Flight simulator time histories of a push/pull maneuver to ± 5 degrees
pitch attitude.
5.7 Conclusions
A gain scheduling concept for multivariable H∞ controllers is presented. The design
points and the scheduling mechanism are obtained through fuzzy clustering
of relevant aerodynamic derivatives. The local H∞ controllers are designed in
continuous-time using LMIs and order reduction is performed before the transformation
to discrete-time.
The gain scheduling concept has been evaluated off-line (linear and nonlinear
simulations, stability analysis) and through pilot-in-the-loop simulations. The results
are satisfactory, although additional tuning is required to further improve
the performance and stability characteristics.
There are still stability problems in the operating regimes between the outer
operating points and the edge of the flight envelope. These problems occurred
in particular in the low dynamic pressure region. It is recommended to force the
outer operating points closer to the edge of the flight envelope, such that these
stability problems no longer occur. Since the fuzzy clustering algorithm uses the
Euclidean distance measure, one possible approach is to weight the distance for
data points close to the edge of the flight envelope more than for data points in
the center of the flight envelope. Another option is to use the operating points
obtained through fuzzy clustering as an initial condition for a similar procedure
as described in (McNichols and Fadali 2003). In this way the operating points and
the parameters of the scheduler are tuned based on the performance of the global
nonlinear closed-loop system.

76 Chapter 5. Scheduled Robust Multivariable Control

Virtual Angle-of-Attack Sensor
An aircraft carries many (redundant) hardware sensors on board, measuring
a wide variety of variables. Due to the relations between the measured signals,
a lot of redundant information is available. This redundant information
can be used to estimate a certain variable through a number of available signals
that represent other variables, i.e. non-like signals. In this chapter, the
design of a virtual angle-of-attack sensor is described. The virtual sensor consists
of a linear parameter varying model, whose parameters are determined by
a Takagi-Sugeno fuzzy model, plus a nonlinear black-box model. The Takagi-
Sugeno fuzzy model is designed using data from linear models. The black-box
model consists of a neural network that is trained to reduce the estimation
error of the linear parameter varying model. The inputs of the neural network
are selected using a genetic search algorithm followed by a backward elimination
procedure. The neural network is designed and trained using data from
nonlinear simulations.
This chapter is organized as follows: A brief introduction is given in Section 6.1
followed by the description of the design requirements for the virtual angle-ofattack
sensor in Section 6.2. The structure of the virtual sensor is addressed in
Section 6.3. In Section 6.4 the design and parameter optimization of the Takagi-
Sugeno fuzzy model and the neural network are discussed. The validation of the
virtual sensor is described in Section 6.5. Concluding remarks and suggestions for
future work are presented in Section 6.6.
6.1 Introduction
Voting/monitoring systems that are based on cross-comparison of like-signals are
unable to isolate a sensor failure when there are two physical sensors in operation.
A virtual sensor can in this case be used as a discriminator, which may also be
used to identify a fault in the last available physical sensor. In terms of reliability
or flight safety, with the virtual sensor it is possible to get more performance out
of the same number of physical sensors. A second option is to replace a physical
sensor by a virtual sensor, which results in fewer hardware components and the
77
6

78 Chapter 6. Virtual Angle-of-Attack Sensor
Altitude [kft]
45
40
35
30
25
20
15
10
5
0
V C = 150 Kts
M MAX = 0.85
V C = 375 Kts
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Mach number [−]
Figure 6.1: Flight envelope of the SCA. The black solid line defines the flight envelope
in clean configuration. The grey line defines the flight envelope in landing configuration.
associated cost benefits. Although the integration of sensor information has its
advantages, it should be noted that it introduces mutual dependencies amongst
non-like signals. This is true for all model-based FDI methods.
In particular for the physical Angle-of-Attack (AoA) sensor, there are more issues
to consider that motivate the use of a virtual sensor. First of all, these sensors
are mounted on the outside of the fuselage. Due to its vicinity to the fuselage, the
physical AoA sensor is not situated in a free airflow and the measured AoA signal
is therefore corrupted and noisy. Moreover, these sensors are also vulnerable
to damage because of their location and harsh operating environment. Because
of these problems that are encountered with the use of physical AoA sensors, a
virtual AoA sensor can make a significant contribution.
As the research described in this chapter serves as a proof-of-principle for the proposed
methodology, only the flight envelope for the landing phase is considered
rather than the entire flight envelope. This is typically the flight regime where the
AoA is high and therefore most critical with respect to potential stall. In view of
the AoA protection system, the potential added value of a virtual AoA sensor is
higher for the landing configuration than for the clean configuration.
6.2 Design Requirements
The estimation error of the virtual AoA sensor has to be less than one degree.
This level of accuracy makes the virtual sensor suitable for monitoring the physical
angle-of-attack sensors. The virtual AoA sensor should be designed such that
this accuracy is met under the following circumstances:

6.3. Structure of the Virtual Angle-of-Attack Sensor 79
Dynamic maneuvering (both longitudinal as well as lateral).
Variations of flight condition within the landing flight envelope.
The flight envelope for the landing configuration is denoted by the grey line
in Figure 6.1. The calibrated airspeed varies between 110 and 195 knots,
while the altitude up to 10 kft is considered.
Variations of aircraft weight and center-of-gravity.
The allowed variation in aircraft weight and the variation in the position of
the CG along the X-axis is illustrated in Figure 5.3.
Aerodynamic uncertainties.
The aerodynamics of each aircraft of the same type are different, for example
due to production variations, but also weather conditions have an impact on
the aerodynamics. In this chapter, the four pre-defined aerodynamic models
that are available in the SE are considered, see Section 5.4.
6.3 Structure of the Virtual Angle-of-Attack Sensor
The main difference of the approach proposed in this section with respect to using,
for example, a Neural Network (NN) or NARX-model to estimate the AoA, is
that in this case only those parts that are unknown and/or (highly) nonlinear are
estimated by a black-box model. The aircraft dynamics are well-known and it does
not make sense to use a (black-box) model with a completely different structure
to represent them.
The proposed virtual AoA sensor consists of two parts, a Linear Parameter Varying
(LPV) model and a neural network model:
ˆα = αLP V + αNN
(6.1)
The NN model is a black-box model, while the LPV model has an interpretable
structure with varying parameters (white-box structure). The latter consists of
the trimmed AoA (α0) plus a linear Short-Period (SP) approximation of the AoA
(αSP):
(6.2)
αLP V = α0 + αSP
The structure of these three elements of the virtual AoA sensor, namely α0, αSP
and αNN are discussed below.
6.3.1 The trimmed angle-of-attack α0
The trimmed angle-of-attack α0 is the AoA of the aircraft in an equilibrium condition.
To estimate the trimmed angle-of-attack, a Takagi-Sugeno fuzzy model is

80 Chapter 6. Virtual Angle-of-Attack Sensor
Figure 6.2: Architecture of the TS fuzzy model to estimate α0.
used (Takagi and Sugeno 1985). The inputs of the TS fuzzy model, which are selected
using a Genetic Algorithm (GA) optimization procedure (see Section 6.4),
are Mach number, dynamic pressure, bank angle, position of the CG along the
X-axis and aircraft weight:
Ri : If M is ZM,j M and q is Zq,j q and φ is Zφ,j φ and XCG is ZX CG,j XCG
and W is ZW,j W then α0 = α0,i for i =1,...,Nr (6.3)
where jM, jq, jφ, jXCG and jW denote the jth membership function of their
corresponding variable. It should be noted that two of the inputs of the TS fuzzy
model, namely the position of the CG along the X-axis and the aircraft weight,
cannot be measured directly. Instead a NN based virtual sensor developed by Idan
et al. (2004) is used to provide an estimate of these variables. The inputs of this
virtual sensor are Mach number (M), dynamic pressure (q), pitch attitude (θ),
flight path angle (γ) and elevator deflection (δe).
The correlation between the aircraft weight and the angle-of-attack is more
or less linear. The aircraft weight is directly related to the required lift, which is
mainly dependent on the lift coefficient CL and the dynamic pressure. Not taking
into account configuration changes, the lift coefficient is mainly a function of the
angle-of-attack. Because of this direct relation between AoA and aircraft weight,
the TS fuzzy model to estimate α0 consists of two stages. The first stage consists
of a number of TS fuzzy models that each estimate the α0 for a specific aircraft
weight, see Figure 6.2. The inputs of these TS fuzzy models are Mach number,
dynamic pressure, bank angle and the position of the CG along the X-axis. The
corresponding rule-base is as follows:
Ri : If M is ZM,j M and q is Zq,j q and φ is Zφ,j φ and XCG is ZX CG,j XCG
then α0,W 1 = α0,W 1i for i =1,...,Nr (6.4)
In the second stage the outputs of these TS fuzzy models are weighted as a function
of the aircraft weight using linear interpolation.

6.3. Structure of the Virtual Angle-of-Attack Sensor 81
6.3.2 The short-period approximation of the angle-of-attack αSP
The starting point for the derivation of the expression for αSP is the linear shortperiod
approximation (see also Appendix B):
˙w
=
˙q
Zw Zq
˜Mw
˜Mq
w
+
q
Zδe
˜Mδe
δe
(6.5)
where the variables w, q and δe denote the variation of the corresponding variables
with respect to the trimmed condition. Using the formula:
G(s) =C (sI − A) −1 B (6.6)
where in this case the matrix C is an identity matrix of dimension two, the following
two transfer functions can be derived from Equation 6.5:
w(s)
δe(s) =
q(s)
δe(s) =
Zδes +(Zq ˜ Mδe − ˜Mq) Zδe
s2 − (Zw + ˜ Mq)s +(Zw ˜ Mq − Zq ˜ Mw)
˜Mδe s ˜Mw +(Zδe − Zw ˜ Mδe )
s2 − (Zw + ˜ Mq)s +(Zw ˜ Mq − Zq ˜ Mw)
(6.7)
(6.8)
The transfer function from the pitch-rate to the downward velocity can be obtained
by dividing Equation 6.7 by Equation 6.8:
w(s)
q(s) = Zδes +(Zq ˜ Mδe − ˜Mq) Zδe
˜Mδe s ˜Mw +(Zδe − Zw ˜ Mδe )
(6.9)
Using the relation α = w/U0, Equation 6.9 can be rewritten to the transfer function
from pitch-rate to angle-of-attack:
α(s)
q(s)
= 1
U0
Equation 6.10 can be simplified to:
Zδes +(Zq ˜ Mδe − ˜Mq) Zδe
˜Mδe s ˜Mw +(Zδe − Zw ˜ Mδe )
α(s)
q(s) = b1s + b0
a1s + a0
where b1 = 1
U0 Zδe , b0 = 1
U0 (Zq ˜ Mδe − Zδe
˜Mq), a1 = ˜ Mδe and a0 =(Zδe
(6.10)
(6.11)
˜Mw −
Zw ˜ Mδe ). The parameters b1, b0, a1 and a0 are obtained from the linearized aircraft
model. Since the virtual sensor is implemented in the digital flight control system,
the model must be discretized. The Tustin discretization is used to obtain the
discrete-time short-period approximation. This method results in a discrete-time
model that matches best the continuous-time model in terms of the frequency
response. The discrete time transfer function becomes:
α(k) = ˆb1σ + ˆb0 q(k) (6.12)
σ +â0

82 Chapter 6. Virtual Angle-of-Attack Sensor
Figure 6.3: Architecture of the LPV model αLP V .
where σ denotes the shift operator. Nonlinear simulation experiments have shown
that keeping the parameters ˆ b0 and ˆ b1 fixed at the center values of their ranges,
instead of varying them as a function of flight condition, has no effect on the
performance of the virtual sensor. For this reason these parameters are kept constant
and only â0 is varying as a function of flight condition. Besides the TS fuzzy
model to estimate α0, see Equation 6.3, a second TS fuzzy model is designed to
approximate â0. The complete LPV part of the virtual sensor is as follows:
αLP V (k) =α0(x(k)) + ˆb1σ + ˆb0 q(k)
σ +â0(x(k))
(6.13)
where x =[M,q, φ, XCG,W]. In order to get the correct pitch rate during lateral
maneuvers, the pitch rate measurement needs to be modified as follows:
qφ = q − g
tan(φ)sin(φ) (6.14)
VT
The term g
tan(φ)sin(φ) describes the approximation of the output of the pitch
VT
rate sensor during turns with constant pitch rate, constant bank angle and constant
turn rate (McLean 1990). This signal is therefore not related to the pitch rate in
terms of the rotational velocity of the aircraft around the Y-axis. Substitution of
q(k) byqφ(k) in Equation 6.13 results in:
where
αLP V = α0(x)+H(σ;â0(x(k))) qφ(z) (6.15)
H(σ;â0(x(k))) = ˆb1σ + ˆb0 . (6.16)
σ +â0(x(k))
In the first simulation experiments it turned out that, although the static estimation
of α0 and â0 is good, during dynamic nonlinear simulation the performance
of the LPV model was poor. The TS fuzzy model is designed based on data of
the linearized SCA model in trimmed condition. However, during maneuvering the
aircraft is not trimmed, which means that the estimated trimmed angle-of-attack
is leading the true angle-of-attack. This effect is compensated for by a low-pass
filter, which lags the estimated α0 during maneuvering. The resulting structure of
αLP V is illustrated in Figure 6.3.

6.4. Design of the TS Fuzzy Model and the NN Model 83
Figure 6.4: Architecture of the virtual AoA sensor.
6.3.3 The neural network αNN
Neural networks have been trained to perform complex functions in various fields
of application including pattern recognition, identification, classification, speech,
vision and control systems. In general a neural network is used when the exact
nature of the relationship between inputs and outputs is not known. If this relationship
is known, it should be modelled directly. The other key feature of neural
networks is that they learn the input/output relationship through training. Neural
networks are discussed in more detail in Appendix D.3, where also a number of
useful references are given.
As mentioned above, neural networks are used when the exact nature of the
relationship between inputs and output is not known. In this case the objective is
to use a neural network to reduce the estimation error ∆α of the LPV model:
∆α = α − αLP V . (6.17)
In other words, the neural networks is used to account for that part of the AoA
signal that is not accounted for using the LPV model.
The structure of the entire virtual AoA sensor is illustrated in Figure 6.4 and is
written as follows:
ˆα(k) =α0(x(k)) + H(σ;â0(x(k))) qφ(k)+αNN(xNN(k)) (6.18)
where αNN denotes the output of the NN model and xNN =[θ nz, nz, qφ, cos φ]
results from a nonlinear input selection procedure (see Section 6.4).
6.4 Design of the TS Fuzzy Model and the NN Model
The parameters of the LPV model are computed through a TS fuzzy model. The
TS fuzzy model is designed using data from linear models in trimmed condition.

84 Chapter 6. Virtual Angle-of-Attack Sensor
On the other hand, the NN model is trained using data from (dynamic) nonlinear
simulations.
6.4.1 TS fuzzy model design
The TS fuzzy model is designed using data obtained from the nonlinear SCA
model. The SCA model is trimmed and linearized for a large number of flight
conditions within the flight regime of interest, which in this case is the flight envelope
for landing configuration. From each trimmed flight condition, the trimmed
angle-of-attack is extracted and put in a data set together with the corresponding
input values, i.e. [M,q, φ, XCG] (see also Figure 6.2).
The design of the Takagi-Sugeno fuzzy model is performed in two steps. First the
optimal structure of the TS fuzzy model is determined, i.e. the inputs of the model
and the number of membership functions for each input. This is done using a GA
optimization approach (Goldberg 1989, Michalewicz 1996), see Appendix E. The
consequent part of the TS fuzzy model is computed by Least Squares (LS) optimization,
with the objective to minimize the Root Mean-Square Error (RMSE),
see also Appendix C.1, between the data set and the TS fuzzy model output. The
objective function J for the GA optimization is:
J = 1
N
N
k=1
(α0,k − ˆα0,k) 2 Nr
· e
0.5( 64 ) (6.19)
where α0,k denotes the kth sample of the data set, ˆα0,k denotes the kth output
of the TS fuzzy model, N denotes the number of data points and Nr denotes the
number of rules. The fitness of each chromosome in the population is evaluated
using the 10-fold cross validation method, see Appendix C.2. The secondary objective
is to keep the model transparent, for a TS fuzzy model this means to keep the
number of rules low. The RMSE of each chromosome is therefore multiplied by the
Nr
factor e
0.5( 64 ) , where the numbers 0.5 and 64 are the result of an iterative tuning
process. The more complicated the TS fuzzy models, i.e. the more rules describe
the TS fuzzy model, the larger the additional penalty. The resulting inputs of the
TS fuzzy model are given in Section 6.3. It should be noted that the structure of
the TS fuzzy model is determined for fixed aircraft weight, see also Figure 6.2.
In the second step, once the inputs and the number of membership functions
per input are determined, the parameters of the membership functions are optimized
taking again a GA approach, in combination with the 10-fold cross validation
method, and the same data set. Again the objective function in Equation 6.19 is
Nr
used, except for the penalty term e
0.5( 64 ) .
6.4.2 NN model design
The most common type of artificial neural network consists of three groups, or
layers, of units: a layer of input units is connected to a layer of hidden units, which
is connected to a layer of output units. This type of neural network is also used

6.4. Design of the TS Fuzzy Model and the NN Model 85
in this thesis. The activity of the input units represents the raw information that
is fed into the network. The activity of each hidden unit is determined by the
activities of the input units and the weights on the connections between the input
and the hidden units. The behavior of the output units depends on the activity of
the hidden units and the weights between the hidden and output units.
The NN model is designed such that it reduces the estimation error (∆α) of
the LPV model, see Equation 6.17. In order to obtain data for the design of the
NN model, the LPV model is evaluated through a large number of nonlinear simulations
(450). The initial condition and the pilot input, in terms of input shape
and input force, for each simulation are selected randomly from a set of predefined
samples. The initial condition is determined by the Mach number, altitude,
aircraft weight, position of the CG along the X-axis and the aerodynamic model.
The predefined pilot input shapes are the so-called 3-2-1-1 input in column and
in wheel, the block-shaped input in column and in wheel, the pull-up and the
push-down maneuver manoeuvre , the wind-up turn maneuver and the combined
wind-up turn and pull-up maneuver. From the data that are generated, 60% is
used for the training and 40% is used for the validation of the NN model.
The NN model is designed in two steps. First the inputs of the NN model are
determined through a nonlinear input selection procedure, then the number of
neurons in the hidden layer is determined. The objective is to find a compromise
between good performance and low complexity (few inputs and few neurons in the
hidden layer).
The inputs for the NN are determined in two steps. First a set of candidate inputs
is selected using a GA based search algorithm. From this set the inputs of
the NN are selected through a backward elimination procedure. In the first step
a pre-selection is made on the basis of second order polynomial models. In the
second step the final selection is made on the basis of NN models.
The GA based search algorithm (Maertens et al. 2004) is used to generate candidate
inputs for the NN model. The genetic algorithm evolves “in parallel” a large
number (e.g. 100) of different structures of the polynomial model. For each given
structure, the parameters are determined by the LS method and the fitness value
for that model is calculated from the corresponding RMSE. If some of the regressor
variables are not present in any of the polynomial terms, they are deleted from the
regressor set and the procedure is repeated with a smaller number of regressors
until a desired (user-defined) number of variables is reached. Due to the random
nature of the GA, the search has to be performed multiple times (e.g. 50) in order
to get a statistically sound result. The six most promising variables are selected
as candidate variables for the NN model, namely: θ nz, nz, cosφ, qφ, qφ VT and
qφ/VT.
First the performance of the NN model with these six candidate variables as
inputs is evaluated. The neural three-layer feed-forward backpropagation network
is trained using the training data set. Both the input layer and the hidden layer
have as many neurons as the number of inputs of the NN (in this case six). The output
layer has one neuron. The network is trained using the Levenberg-Marquardt

86 Chapter 6. Virtual Angle-of-Attack Sensor
RMSE
0.17
0.16
0.15
0.14
0.13
0.12
0.11
0.1
6 5 4 3 2
Number of inputs
Figure 6.5: Backward elimination procedure, the RMSE as a function of the number
of inputs of the NN model.
algorithm (Bishop 1995, Shepherd 1997). In order to determine the least valuable
candidate variable, six neural networks are trained and validated with five inputs,
leaving each time one of the candidate variables out. Again the input layer and
hidden layer both have as many neurons as the number of inputs of the NN (in
this case five) and the output layer has one neuron. The candidate variable that is
left out of the NN model with the best performance of the six NN models, is the
least valuable and therefore removed from the set of candidate variables. Typically
the performance if this NN model is less than that of the NN model with all six
candidate variables as inputs. This procedure, the so-called backward elimination
procedure, is terminated when the removal of the latest input results in a relatively
large reduction in the performance. This is also illustrated in Figure 6.5,
where it can be seen that there is a relatively large reduction in the performance
of the NN model when removing the fourth input variable. The number of inputs
for the NN model is therefore set to four and these inputs are: θ nz,nz,cosφ and qφ.
The optimal number of hidden neurons is determined based on a number of NN
models with increasing numbers of neurons. Each extra neuron should have a
significant contribution in the performance of the NN model (on the training and
on the validation data set). The number of hidden neurons is set to eight.
6.5 Validation of the Virtual AoA Sensor
In this section all the design requirements, as described in Section 6.2, are taken
into account, except for the variation in aircraft weight. The aircraft weight varies
between W = 31850 and W = 35750 [lbs], which represents 20% of the total range
in aircraft weight. Due to the modular structure of the virtual AoA sensor, the
total range in aircraft weight can be covered by adding more LPV models designed
for different aircraft weights.

6.5. Validation of the Virtual AoA Sensor 87
Table 6.1: Performance of the TS fuzzy models.
W = 31850 [lbs] W = 35750 [lbs]
α0 [deg] â0 [-] α0 [deg] â0 [-]
RMSE 0.1267 2.03 · 10 −4 0.0479 2.07 · 10 −4
VAF [%] 99.86 99.43 99.98 99.23
First the TS fuzzy model and LPV model is validated, before the complete virtual
AoA sensor is validated. This will enable us to appreciate the added value each
element of the virtual sensor. Bank angles larger than 45 degrees are not taken
into account, while the bank angle for the estimation of α0 and â0 is limited
to 30 degrees. As mentioned in Section 6.3, the parameters ˆ b1 and ˆ b0 are kept
constant at their center values. For the compensation of the pitch rate signal, see
Equation 6.14, the bank angle is limited to 33 degrees.
6.5.1 Validation of the TS fuzzy models
Separate TS fuzzy models have been designed for the estimation of α0 and â0. The
TS fuzzy model to estimate α0 has seven membership functions, namely two for
the Mach number, three for the dynamic pressure and two for the bank angle. The
TS fuzzy model uses ten rules, which means that not all the possible 12 combinations
of the MFs are used. The two combinations of the MFs that are not used,
covered spaces that do not contain data, i.e. the aerodynamic model is not defined
there, which means that you can not fly there. The corresponding rule-base is as
follows:
R1 : If M is ZM,1 and q is Zq,1 and φ is Zφ,1 then α0 = α0,1
R2 :
.
If M is ZM,1 and q is Zq,1 and φ is Zφ,2 then α0 = α0,2
R10 : If M is ZM,2 and q is Zq,3 and φ is Zφ,2 then α0 = α0,10
where α0i = c0,i + cM,iM + cq,iq + cφ,iφ + cCG,iXCG. The TS fuzzy model to
estimate â0 has nine membership functions, namely two for the Mach number, the
dynamic pressure and the position of the center-of-gravity along the X-axis and
three for the bank angle. The TS fuzzy model uses 16 rules out of the 24 possible
combinations of the MFs. The performance of the TS fuzzy models to estimate α0
and â0 for each aircraft weight is summarized in Table 6.1 in terms of RMSE and
Variance Accounted For (VAF). These performance measures are defined in Appendix
C.1. Performances of the TS fuzzy models are good, although it should be
noted that the TS fuzzy model to estimate α0 for aircraft weight W = 35750 [lbs]
outperforms the TS fuzzy model to estimate α0 for aircraft weight W = 31850 [lbs].

88 Chapter 6. Virtual Angle-of-Attack Sensor
Table 6.2: Performance of the LPV model in estimating the angle-of-attack in nonlinear
simulation.
W = 31850 W = 35750 31850 < W < 35750
[lbs] [lbs] [lbs]
RMSE [deg] 0.2748 0.2317 0.2323
VAF [%] 99.38 99.52 99.53
∆ α [deg]
α [deg]
15
10
5
0
0 0.5 1 1.5 2 2.5 3
x 10 4
−5
Sample
1
0.5
0
−0.5
−1
0 0.5 1 1.5 2 2.5 3
x 10 4
Sample
Figure 6.6: Performance of the LPV model in nonlinear simulation.
6.5.2 Validation of the LPV model
The performance of the LPV model during nonlinear simulation is evaluated in
terms of RMSE and VAF with respect to the true angle-of-attack signal, see Table
6.2. In this table the performance is given for W = 31850 and W = 35750 [lbs],
the two aircraft weights for which the TS fuzzy models are designed, and for the
range from W = 31850 to W = 35750 [lbs]. Since the performance of the latter is
of the same order as for the fixed aircraft weight cases, it can be concluded that the
linear interpolation as a function of aircraft weight works well. The RMSE value
is below 0.24 degrees while the VAF value is above the 99%. The corresponding
time history is shown in Figure 6.6.
6.5.3 Validation of the virtual AoA sensor
The virtual sensor is validated using the validation data set that has not been used
in any stage of the design. The performance of the virtual sensor is given in Ta-

6.5. Validation of the Virtual AoA Sensor 89
Table 6.3: Performance of the virtual sensor in estimating the angle-of-attack in
nonlinear simulation.
∆ α [deg]
α [deg]
15
10
5
0
Training Validation
RMSE [deg] 0.1088 0.1125
VAF [%] 99.90 99.90
max |∆α| [deg] 0.7314 0.7781
0 0.5 1 1.5 2
x 10 4
−5
0.8
Sample
0.4
0
−0.4
0 0.5 1 1.5 2
x 10 4
−0.8
Sample
Figure 6.7: Performance of the virtual sensor in nonlinear simulation on the training
data set.
ble 6.3. The RMSE values are below 0.12 degrees while the VAF values are again
above the 99%. It can be seen that the difference in performance on the training
and validation data set is in balance, suggesting that the complexity of the NN
is about right. The corresponding simulation results are illustrated in Figures 6.7
and 6.8. In terms of the RMSE, the improvement in the performance of the virtual
sensor is significant compared to the LPV model. The absolute estimation error
remains below the 0.8 degrees for both the training and the validation data set.
In this section the variation in aircraft weight is 3900 lbs. The total allowable
variation for the SCA model is 17400 lbs, see also Figure 5.3, which means that
five local LPV models would be needed to cover the entire centogramme. In case
the performance requirements of the virtual AoA sensor cannot be met with a
single NN for the entire range of the aircraft weight, a separate NN model should
be designed for each aircraft weight regime (W 1 → W 2, W 2 → W 3, etc). This
option will result in more accuracy at the price of a more complicated virtual AoA
sensor.

90 Chapter 6. Virtual Angle-of-Attack Sensor
∆ α [deg]
α [deg]
15
10
5
0
−5
0
0.8
2000 4000 6000
Sample
8000 10000
0.4
0
−0.4
−0.8
0 2000 4000 6000 8000 10000
Sample
Figure 6.8: Performance of the virtual sensor in nonlinear simulation on the validation
data set.
6.6 Conclusions
The virtual AoA sensor described in this chapter consists of two parts, namely a
LPV model and a NN model. In the LPV model part the TS fuzzy model estimates
the trimmed angle-of-attack α0 as well as the parameter in the denominator
of αSP, while in the (nonlinear) black-box part the NN is designed to fit the estimation
error remaining from the LPV model.
The design example in this chapter serves as a proof-of-principle for the proposed
methodology. The virtual sensor performed well, both for the training and the
validation data set. The design requirement of estimation errors less than 1 degree
was met. The RMSE was kept well below 0.12 degrees while the VAF was kept
well above 99%. It should be noted that the automatically generated data does
not reach the maximum angle-of-attack of 16 degrees, because this is prevented by
the envelope protection system. Around this angle-of-attack the aircraft model is
more nonlinear and it can therefore be expected that improving the data set will
result in slightly more complex TS fuzzy models (and neural network) in order
to achieve the same performance. It should also be noted that in the simulation
results the true aircraft weight or the true position of the center-of-gravity along
the X-axis have been used as inputs to the TS fuzzy model and NN. These should
in fact be estimates of the virtual sensor developed by IIT (Idan et al. 2004). It is
expected that the corresponding decrease in performance of the virtual sensor is
negligible, since these variables can be estimated accurately.

7
Soft Sensor Management and
Virtual Sensors for FDIR
A sensor management system based on soft computing techniques has been
developed and implemented in the flight control system of the SCA model.
Unlike in the conventional sensor management system, the signals from sensors
are assigned weights based on fuzzy membership functions and the consolidated
signal is computed as a weighted average. This approach improves the quality
of the consolidated signal and reduces transients due to sensor failures. The soft
voting scheme serves as a basis for soft flight control law reconfiguration. In
addition, it is illustrated how a virtual sensor can serve as an arbitrator which
enables the isolation of the failed sensor in the duplex mode and the detection of
a sensor failure in the simplex mode. The effectiveness of the proposed methods
is demonstrated by using the SCA model, taking into account sensor failures in
pitch rate and normal acceleration. The properties of the conventional sensor
management system have been retained, with the additional advantage that
the quality of the consolidated signal is improved, failure-induced transients
are reduced and the consolidated signal remains available up to the last valid
sensor.
This chapter is organized as follows: In Section 7.1 a short introduction of the
sensor management problem is given. The conventional sensor management system
and the flight control law reconfiguration are discussed in Section 7.2. Section 7.3
introduces the sensor management system and flight control law reconfiguration
based on soft computing, which is extended to include virtual sensors in Section 7.4.
Concluding remarks and future research are discussed in Section 7.5.
7.1 Introduction
Sensor management based on majority voting and point consolidation of like signals
is a proven technology in modern fly-by-wire flight control systems (Rosenberg
1998). The assumption is that the majority of like signals represent the truth and
that any single dissimilar signal is the result of a failure. Such a signal must be
91

92 Chapter 7. Soft Sensor Management and Virtual Sensors for FDIR
disconnected as soon as the failure is detected. The probability of multiple simultaneous
failures is considered to be extremely remote. In the conventional approach,
the decision whether a sensor has failed or not is crisp. In order to reduce the sensitivity
of this decision to uncertainties like quantization and measurement noise,
a properly adjusted threshold is used. This threshold is a compromise between
two goals: the absence of false alarms and the ability to detect all possible failures
within a certain time frame. This inevitably leads to a transient response during
which the consolidated signal temporarily differs from the true value.
The soft sensor management system introduced in this chapter maintains the key
properties of the conventional sensor management system (majority voting) while
improving its performance by applying fuzzy logic techniques.
Using fuzzy logic, the decision whether a sensor has failed or not is no longer
crisp. In the soft sensor management system the signals are assigned weights based
on a cross-comparison of like signals by using fuzzy membership functions. The
consolidated signal is then computed as a weighted average. Compared to the
conventional management system, the soft management system intervenes at an
earlier stage by reducing the weight of the suspected faulty sensor signal, while the
failure declaration occurs at a later stage. This approach improves the quality of the
consolidated signal with respect to its difference from the true value due to sensor
failures. An additional attractive feature of this approach is the reduction of the
transients due to sensor failures. Although FL techniques have been implemented
in other application domains, such as the process industry (Frank and Marcu 1999,
Schneider and Frank 1996), their application for FDI in flight control systems has
not been extensively investigated yet.
Furthermore, a virtual sensor is introduced in order to be able to identify
failed sensors in the duplex mode and to detect a sensor failure in the simplex
mode (which is not possible with the current sensor management systems). In the
literature, many applications of analytical redundancy for fault detection and fault
isolation in flight control systems have been reported (Patton et al. 1989, Patton
and Chen 1992, Isermann 1984), however, the use of virtual sensors in aerospace
applications is novel.
7.2 Conventional Sensor Management and FCL Reconfiguration
Each signal is measured by a number of independent sensors. The sensor management
system has two tasks, namely the computation of a consolidated signal from
these measurements (voting) and the validation of each of the sensors (monitoring).
The consolidated (or voted) signal is fed to the flight control computer and
at the same time serves as a reference for sensor validation.

7.2. Conventional Sensor Management and FCL Reconfiguration 93
Figure 7.1: Conventional triplex sensor management system (Source: (Rosenberg
1998)).
7.2.1 Conventional voting/monitoring scheme
The redundancy level for each signal, i.e. the number of redundant sensors in
normal mode, is related to the system architecture, the failure probability of the
sensor and the consequence of losing the corresponding signal. For example, the
consequence of losing the pitch rate signal is a catastrophic failure. The probability
of a catastrophic failure must be less than 10 −9 per hour of flight. This level
of reliability can not be achieved with a single pitch rate sensor and therefore
redundant pitch rate sensors need to be installed. It should be noted that a sensor
failure in the duplex mode (two sensor signals available) results in losing the signal,
since the conventional voting/monitoring scheme is not able to identify the failed
sensor in this case.
In order to keep the presentation simple, the triplex sensor system (three
sensor signals available) will be used to explain the conventional voting/monitoring
philosophy (see Figure 7.1). The three sensor signals are first sorted from the
largest value to the smallest one. The mid-value signal is taken as a reference and
the two extreme-value signals are limited in their deviation from the mid-value
signal. When the limits are not invoked, the voted (consolidated) signal is given
by:
Svoted = S2 +0.25 (S1 − S2)+0.25 (S3 − S2) =0.25 (S1 +2S2 + S3).
For two valid signals, a duplex voter is used, and the voted signal is a simple average.
The monitor compares each of the three sensor signals Si with the consolidated
signal Svoted. If the absolute difference is smaller than a predefined threshold ∆,
the monitor count is decreased by one, otherwise it is increased by two:

94 Chapter 7. Soft Sensor Management and Virtual Sensors for FDIR
If |Si − Svoted| ≤∆ then (count rate) i = −1
If |Si − Svoted| > ∆ then (count rate) i =+2.
The updated count value is bounded between zero and the failure declaration
value. If the count value has reached the failure declaration value, a failure is declared
and the signal is latched (see Figure 7.1). Since in this case only two sensor
signals are still available, automatically the sensor management system reconverts
to duplex mode.
The logic in the conventional sensor management system is such that a failed sensor
output continues to contribute to the voted signal until it is latched. This implies
that when signal is latched, its contribution to the consolidated signal is instantly
removed. This discontinuity in the consolidated signal results in a transient in the
aircraft motion, which is illustrated by nonlinear, closed-loop simulation examples
using the SCA model in Matlab/Simulink TM .
7.2.2 Simulation examples
The functionality and performance of the sensor management systems is demonstrated
using pitch rate and normal acceleration sensor failures during longitudinal
maneuvering. The closed-loop transients due to sensor failures are not of the same
order for each signal. From Figure 4.1 it can be seen that the pitch rate is fed back
through a proportional gain in the pitch damper path and through a proportional
and an integral gain in the feedback path. In the feedback path the normal acceleration
is fed back in a similar way. However, discontinuities in the voted normal
acceleration signal are suppressed by the low-pass filter in the normal acceleration
feedback path (Figure 4.1). Closed-loop transients are therefore less evident. For
this reason the pitch rate signal is used to demonstrate this additional benefit of
the sensor management system based on soft computing techniques. The normal
acceleration signal is used only to demonstrate the functionality of the conventional
management system in the case of a sensor failure in duplex mode, since
this requires a reconfiguration mode which is not available for the loss of the pitch
rate signal in the SCA model. As mentioned in Section 7.2.1, the loss of the pitch
rate signal is a catastrophic failure.
A simplified representation of the longitudinal flight control laws is illustrated
in Figure 4.1. In this figure it can be seen that the feedback signal consists of a
blending of the pitch rate and the normal acceleration signal. In order to compare
the performance of the conventional sensor management system to that of the
soft management system, sensor failures are considered in these two signals. The
blending function is described in Section 4.1.
Two nonlinear simulations are used to illustrated the functionality of the conventional
sensor management system. For both simulations, the initial condition
is a straight and level flight at a Mach number of M =0.75 and an altitude of
h = 40 kft, which is the cruise flight condition for the SCA model. In this flight
condition the calibrated airspeed is equal to VC = 225 knots. The pilot input
is a block-shaped input of the maximum positive column deflection starting at

7.2. Conventional Sensor Management and FCL Reconfiguration 95
a) q meas [deg s −1 ]
b) ∆ q voted [deg s −1 ]
c) count
d) valid
6
4
2
q
1
q
2
q
3
0
0
0.5
1 2 3 4 5
0
−0.5
0
20
1
q
1
2 3 4 5
10
q
2
q
3
0
0 1 2 3 4 5
3
2
1
0
0 1 2 3 4 5
Time [s]
e) δ e [deg]
f) n z [g]
g) q [deg s −1 ]
−5.2
−5.4
−5.6
−5.8
−6 conv
no fault
−6.2
3.2 3.4 3.6 3.8 4 4.2
1.65
1.6
1.55
3.2
5.5
3.4 3.6 3.8
conv
no fault
4
conv
4.2
5
no fault
4.5
4
3.5
3.2 3.4 3.6 3.8 4 4.2
Time [s]
Figure 7.2: Conventional sensor management: drift failure of a pitch rate sensor.
Figures e-g are zoomed in on the failure-induced transients.
t =1sec and lasting for 6 sec. During this maneuver, the normal acceleration signal
is partly contributing to the feedback signal. This is necessary in order to be
able to evaluated failure-induced transients when considering failures in the normal
acceleration sensors.
The time histories corresponding to the first simulation example are illustrated
in Figure 7.2. At t =1sec, a drift failure of 1 deg sec −2 occurs in one of the pitch
rate sensors (Figure 7.2a). Figure 7.2b shows the difference ∆qvoted between the
voted signal and the true pitch rate. Due to the drift failure, the voted signal diverges
from the true pitch rate until the contribution of the failed sensor output
is limited and the mismatch remains constant. When the difference between the
failed sensor (q1) and the voted signal (qvoted) exceeds the monitor threshold, the
monitor count rate increases from −1 to +2 (denoted by the first vertical dashdotted
line). When the monitor count (Figure 7.2c) reaches the failure declaration
value (denoted by the second dash-dotted vertical line) the signal is latched and
the number of valid signals reduces to two (Figure 7.2d). The faulty contribution
of q1 is omitted instantaneously, which results in an undesirable discontinuity in
the consolidated signal (Figure 7.2b). The resulting transients in the elevator deflection
(Figure 7.2e), the normal acceleration (Figure 7.2f), and the true pitch
rate (Figure 7.2g) signals are evident (solid line) compared to the fault free case
(dash-dotted line). The second simulation example illustrates a cut-off sensor fail-

96 Chapter 7. Soft Sensor Management and Virtual Sensors for FDIR
a) q meas [deg s −1 ]
b) ∆ q voted [deg s −1 ]
c) count
d) valid
6
4
2
q
1
q
2
q
3
0
0
0.5
1 2 3 4 5
0
−0.5
0
20
1
q
1
2 3 4 5
10
q
2
q
3
0
0 1 2 3 4 5
3
2
1
0
0 1 2 3 4 5
Time [s]
e) δ e [deg]
f) n z [g]
g) q [deg s −1 ]
−4.5
−5
−5.5
−6
1.6
1.55
1.5
1.45
conv
no fault
1.4
2.5 3 3.5
5.6
5.4
5.2
5
4.8
conv
no fault
2.5 3 3.5
conv
no fault
4.6
2.5 3
Time [s]
3.5
Figure 7.3: Conventional sensor management: cut-off failure of a pitch rate sensor.
Figures e-g are zoomed in on the failure-induced transients.
ure which occurs at t =3sec in one of the pitch rate sensors (Figure 7.3a). Due
to the abrupt nature of the sensor failure, discontinuities in the voted signal occur
both when the failure is inserted and when the corresponding signal is latched
(Figure 7.3b). Again the behavior of the voted signal is undesirable since it is by
no means representing the behavior of the true value. The transients in the elevator
deflection (Figure 7.3e), the normal acceleration (Figure 7.3f) and the pitch
rate (Figure 7.3g) are evident (solid line), especially when compared to the fault
free case (dash-dotted line).
7.2.3 Flight control law reconfiguration
The voted signal in the duplex mode is computed as the average of the two sensor
signals. If a sensor fails in the duplex mode, the majority voting principle can no
longer be used to identify the failed sensor. As soon as the difference between these
two signals exceeds a certain threshold, both sensors are declared invalid and the
FCS reconfigures to not using this particular signal. In Figure 7.4 a drift failure of
the second normal acceleration sensor is simulated (Figure 7.4a). The voted signal
is the average of the two valid signals. The initial condition is a straight and level
flight at a Mach number of M =0.70 and an altitude of h = 25 kft. This flight
condition is selected to increase the contribution of the normal acceleration signal

7.3. Sensor Management and FCL Reconfiguration Based on Soft Computing 97
a) n z,meas [g]
b) ∆ n z,voted [g]
2.5
2
1.5
1
n z,2
n z,3
5
0.3
6 7 8 9 10
0.2
0.1
0
−0.1
5 6 7 8 9 10
Time [s]
c) valid
d) blend
3
2.5
2
1.5
1
0.5
0
5 6 7 8 9 10
1
0.8
0.6
0.4
0.2
0
5 6 7 8 9 10
Time [s]
Figure 7.4: Conventional sensor management: drift failure of the second normal acceleration
sensor.
in the feedback path. The pilot input is a block-shaped input of maximum positive
column deflection starting at t =6sec and lasting for 6 sec. When the difference
between the two sensor signals exceeds the threshold, the monitor count of both
sensor signals is set to the failure declaration value instantaneously and both input
signals are latched (Figure 7.4c). At this point the consolidated signal is no longer
available and the FCS reconfigures to not using this signal (Figure 7.4d). This
implies that only the pitch rate signal is used in the feedback path for the entire
range of the admissible column deflection and calibrated airspeed. The blending
of the pitch rate signal and the normal acceleration signal in the feedback path is
explained in more detail in Section 4.1.
7.3 Sensor Management and FCL Reconfiguration Based on
Soft Computing
The simulation examples in Section 7.2.2 and 7.2.3 have demonstrated the main
shortcomings of the conventional sensor management approach, namely the failureinduced
discontinuities in the consolidated signal and the inability to identify sensor
failures in duplex mode and/or to detect a sensor failure in simplex mode. A
fuzzy logic approach can be used to improve the conventional sensor management
system without changing the basic concept of majority voting. In this section we
focus on reducing, or even eliminating, the transients in the voted signal due to
sensor failures.

98 Chapter 7. Soft Sensor Management and Virtual Sensors for FDIR
Figure 7.5: Soft voting in the triplex mode. The current value of each sensor signal
forms the center of its corresponding membership function, which is used to determine
the membership degree of this sensor signal.
7.3.1 Soft voting/monitoring scheme
The soft voter is different from the conventional voting scheme in the sense that
each input signal is assigned a weight, and the consolidated signal is the weighted
average of the input signals:
n
Svoted = wiSi, (7.1)
where wi denotes the weight assigned to the ith input signal Si and n denotes the
number of valid sensors. The weight wi is the normalized membership degree µi:
wi =
i=1
µi
n j=1 µj
, (7.2)
where 0 ≤ µi ≤ 1. The computation of the membership degree µi is illustrated in
Figure 7.5. The current value of each signal forms the center of its corresponding
membership function. The membership degree of the signal is the largest membership
degree of the remaining valid signals according to this membership function:
µi = max
i=j (µi(qj)). (7.3)
In Figure 7.5a the membership function of q3 is illustrated. With respect to this
membership function, q1 has a membership degree of µ3(q1) =0.35 and q2 has a
membership degree of µ3(q2) =0.65. This implies that q3 has a membership degree
of:
µ3 = max(µ3(q1),µ3(q2)) = 0.65 .

7.3. Sensor Management and FCL Reconfiguration Based on Soft Computing 99
Figure 7.6: Soft triplex sensor management system.
Clearly, the majority voting concept of the conventional sensor management system
is also used in the soft sensor management system. The signal q3 is not in
agreement with the signals q1 and q2, and therefore its weight in the voted signal
is reduced. In Figure 7.5b the discrepancy between signal q3 and the signals q1
and q2 is further increased. The corresponding membership degree is now reduced
to µ3 =0.
Both the conventional and the soft voting scheme are based on majority voting.
The major difference is the way the like sensor signals contribute to the consolidated
signal. In the conventional voting scheme, the contribution of a faulty signal
is limited, while in the soft voting scheme its weight is reduced. The implementation
of the soft voting scheme is illustrated in Figure 7.6 for a triplex sensor
system. The vector of like signals is split and sorted. The membership degrees are
computed according to Equation 7.3, put back in the original order and combined
again in a vector. The voted signal is then computed according to Equations 7.1
and 7.2.
In the monitor part, the count rate of the ith signal is the following function
of its corresponding membership degree µi:
If µi =1then (count rate) i = −1
If 0

100 Chapter 7. Soft Sensor Management and Virtual Sensors for FDIR
a) q meas [deg s −1 ]
b) ∆ q voted [deg s −1 ]
c) weight
d) count
6
4
2
q
1
q
2
q
3
0
0
0.5
1 2 3 4 5
0
−0.5
0 1 2 3 4 5
1
q
1
0.5
q
2
q
3
0
0 1 2 3 4 5
20
10
q
1
q
2
q
3
0
0 1 2 3 4 5
Time [s]
e) δ e [deg]
f) n z [g]
g) q [deg s −1 ]
−4
−4.5
−5
−5.5
−6
1.6
1.5
1.4
1.3
5.5
5
4.5
4
soft
conv
no fault
2.5 3 3.5 4
soft
conv
no fault
2.5 3 3.5 4
soft
conv
no fault
2.5 3
Time [s]
3.5 4
Figure 7.7: Soft sensor management: drift failure of a pitch rate sensor. Figures e-g
are zoomed in on the failure-induced transients.
corresponding signal is latched. This is illustrated with the help of two closed-loop
simulation examples.
7.3.2 Simulation examples
The setting is identical to the simulation examples discussed in Section 7.2.2 except
for the sensor management system.
The time histories of the first simulation example are given in Figure 7.7. At
t =1sec, a drift failure of 1 deg sec −2 occurs (Figure 7.7a). Figure 7.7b shows the
difference between the voted signal and the true pitch rate ∆qvoted. Due to the
drift failure the voted signal diverges from the true pitch rate until the weight of
the failed sensor output is reduced to zero (Figure 7.7c) and the voted signal is
again equal to the true value (not taking into account uncertainties such as quantization,
sensor noise, etc.). One can see that the voted signal is smoother than in
the conventional sensor management case. By this time the monitor count rate is
increased from −1 (µ1 =1)to0(0

7.3. Sensor Management and FCL Reconfiguration Based on Soft Computing 101
a) q meas [deg s −1 ]
b) ∆ q voted [deg s −1 ]
c) weight
d) count
6
4
2
−0.2
q
1
q
2
q
3
0
0 1 2 3 4 5
0.2
0
0 1 2 3 4 5
1
0.5
q
1
q
2
q
3
0
0 1 2 3 4 5
20
10
q
1
q
2
q
3
0
0 1 2 3 4 5
Time [s]
e) δ e [deg]
f) n z [g]
g) q [deg s −1 ]
−5
−5.5
−6
1.65
1.6
1.55
1.5
1.45
5.5
5
4.5
4
soft
conv
no fault
3 3.5 4
soft
conv
no fault
3 3.5 4
soft
conv
no fault
3 3.5
Time [s]
4
Figure 7.8: Soft sensor management: cut-off failure of a pitch rate sensor. Figures e-g
are zoomed in on the failure-induced transients.
the elevator deflection (Figure 7.7e), normal acceleration (Figure 7.7f), and the
true pitch rate (Figure 7.7g). For comparison, the time histories of the simulations
of the fault free case (dash-dotted line) and conventional voting/monitoring case
(dotted line) are also included in Figures 7.7e-g.
The second simulation example is illustrated in Figure 7.8. At t =3sec, a cutoff
sensor failure occurs (Figure 7.8a). Due to the abrupt nature of the sensor failure,
the weight of the failed signal output becomes zero immediately (Figure 7.8c)
and therefore there are no transients in the elevator deflection (Figure 7.8e), the
normal acceleration (Figure 7.8f), and the pitch rate (Figure 7.8g) signals.
7.3.3 Flight control law reconfiguration
The soft voting logic is extended to soft flight control law reconfiguration. Also here
the voted signal is computed as the average of the two sensor signals. Both normal
acceleration sensor signals automatically have the same membership degree, and
are therefore equally weighted in the consolidated signal. However, their mutual
membership degree is multiplied with the contribution of the normal acceleration
signal in the feedback path as well. The blending between the pitch rate and
the normal acceleration signals in the feedback path, see also Section 4.1, is now

102 Chapter 7. Soft Sensor Management and Virtual Sensors for FDIR
a) n z,meas [g]
b) ∆ n z,voted [g]
2.5
2
1.5
1
n
z,2
n
z,3
5
0.2
6 7 8 9 10
0.1
0
−0.1
5 6 7 8 9 10
Time [s]
c) weight
d) blend
1
0.8
0.6
0.4
0.2
n
z,2
n
z,3
0
5
1
6 7 8 9 10
0.8
0.6
0.4
0.2
0
5 6 7 8 9 10
Time [s]
Figure 7.9: Soft sensor management: drift failure of the second normal acceleration
sensor.
a function of the column deflection δc and the calibrated airspeed pressure VC
multiplied by the maximum membership degree of the normal acceleration signals.
Equation 4.6 then becomes:
w =(µδc µVC ) µnz , (7.4)
where µnz is equal to the maximum membership degree of all nz signals:
µnz = max(µi). (7.5)
When the difference between the two signals is such that their mutual membership
degree becomes equal to zero, the flight control laws are already reconfigured to
not using the normal acceleration signal in the feedback path.
This is illustrated in Figure 7.9, where the time histories of a simulation of a
drift failure of the second normal acceleration sensor are illustrated (Figure 7.9a).
The voted signal is the average of the two input signals. During the maneuver, the
signal in the feedback path is for 90% derived from the normal acceleration signal
(Figure 7.9d). This is reduced to zero due to the growing discrepancy between
the two valid normal acceleration sensor signals. By the time the nz signal is no
longer available (Figure 7.9b), the FCLs are reconfigured to not using this signal
(Figure 7.9d).
7.3.4 Discussion
In principle the conventional and the soft sensor management system are much
alike. The soft sensor management system is a weighted implementation of the

7.3. Sensor Management and FCL Reconfiguration Based on Soft Computing 103
conventional sensor system, retaining all the benefits of this system.
The conventional voting/monitoring system has two separate crisp thresholds, one
to limit the contribution of a suspected faulty sensor signal and one for the failure
declaration. The selected thresholds are a compromise between two goals: the absence
of false alarms and the ability to detect all possible failures within a short
time frame. The latter is important to minimize the effect of a sensor failure. This
inevitably leads to a transient response during which the consolidated signal temporarily
differs from the true value. Although it is possible to reduce transients
by introducing filters, the soft sensor management is a more direct solution to
this problem. In the soft sensor management system, the compromise between
false alarms and the ability to detect sensor failures within a certain time frame is
avoided by introducing a soft threshold. Through the soft threshold, the objectives
of no false alarms and the minimization of the transient effects are well separated.
When the failure declaration procedure is activated, the weight of the corresponding
sensor signal is equal to zero and the negative impact of the suspected faulty
sensor is avoided. As transients are reduced or even removed, the tuning of the
membership function parameters is only driven by the sensor characteristics. For
example, expensive sensors are more accurate and may allow for more narrow
membership functions than other sensors. The additional computation due to the
soft sensor management system is considered to be negligible. The thresholds used
in the simulation examples were selected such that the characteristics of both
voting/monitoring systems become clear. The crisp threshold in the conventional
voting system is in between the upper and lower bound of the soft threshold of
the soft voting/monitoring system.
The thresholds of the soft sensor management system can always be designed such
that the performance is equal to or better than that of the conventional sensor
management system, without increasing the probability of false alarms. Performance
is expressed in the accuracy of the consolidated, or voted, signal Svoted and
in the magnitude of the discontinuities in the voted signal. In Figure 7.10 three
different locations of the soft threshold (solid line) are illustrated compared to the
crisp threshold (dashed line). The simulation results of a slow drift failure shown
in Figure 7.11 correspond to these three locations of the soft threshold. The signal
∆Svoted is a measure of the inaccuracy of the voted signal. In case (c), where the
left hand side of the soft threshold coincides with the crisp threshold, the maximum
deviation when using the soft threshold is equal to that when using the crisp
threshold. Typically case (b) will be applied in practise. An advantage of the soft
sensor management system is that the deviation of the voted signal from the true
value is reduced when the deviation of the failed sensor signal continues to grow.
When using the conventional sensor management system, it remains constant until
the corresponding sensor signal is latched.
The benefits of the soft sensor management system are most evident for cutoff
failures. The worst case sensor failure for the soft sensor management system
is a step-like sensor failure that does not result in a membership degree of the
corresponding signal that is equal to zero. Only in this case a discontinuity in

104 Chapter 7. Soft Sensor Management and Virtual Sensors for FDIR
Figure 7.10: Various locations for the soft threshold compared to the crisp threshold.
Crisp thresholds (dashed line) versus soft thresholds (solid line). The gray area denotes
the range covered by the soft threshold.
∆ S voted
0.3
0.25
0.2
0.15
0.1
0.05
crisp
soft − a
soft − b
soft − c
0
0 0.5 1
∆ S
1
1.5 2
Figure 7.11: Comparison of the deviation of the voted signal from the true value using
a crisp threshold and three using a soft threshold, each with a different location with
respect to the crisp threshold (see also Figure 7.10). The simulations are performed in
triplex mode.
the consolidated signal occurs. The magnitude of the discontinuity will be always
be equal or smaller than in the conventional case, since the weight of the corresponding
signal will always be equal or smaller to one. In the conventional case
the weight of each sensor signal in the consolidated signal is equal to one until the
signal is latched.
Information on the membership degree can be used for maintenance purposes. If
a sensor has regularly a membership degree lower than one, this is an indication
that something is wrong and that the sensor need to be replaced.
7.4 Virtual Sensor for FDIR
The conventional sensor management system works well down to two signals, where
any discrepancy can no longer be related to a majority. In this instance, the system
will either reject both signals and reconfigure to not using this information, or,
for essential data, a simple average will be used as the best compromise. However,

7.4. Virtual Sensor for FDIR 105
Figure 7.12: Soft voting in the duplex mode. The virtual sensor output forms the
center of a membership function, which is used to determine the membership degree
of the hardware sensor outputs.
there is additional information available that can be used to identify the failed
sensor in the duplex mode and to detect a failure in the simplex mode. This
additional information can be used to estimate the signal of interest by including
a virtual sensor, see also Chapter 6. Monitoring of the hardware sensor(s) in the
duplex and simplex mode is then performed by comparison with the virtual sensor
output. In this section a virtual normal acceleration sensor is used that is designed
using similar techniques as for the LPV part of the virtual AoA sensor described
in Chapter 6.
7.4.1 Dynamic thresholds
When using virtual sensors, the residuals resulting from the cross-comparison are
more sensitive to uncertainties than in the case when only physical sensors are
used, especially with respect to unmodelled dynamics. Dynamic thresholds have
been implemented to optimize the performance of the voter/monitor based on the
accuracy of the virtual sensor without risking false alarms. Typically the estimation
error of the virtual normal acceleration sensor is small during steady-state
flight and increases during (aggressive) maneuvering. For this reason, the support
of the membership functions widens during maneuvering. In this way, a dynamic
threshold is realized (see also Figure 7.12b). For the normal acceleration, the parameters
of the membership function are adjusted as follows:
bl = min(bl,max,bl,min + Cb,l nz,voted)
bu = min(bu,max,bu,min + Cb,u nz,voted)

106 Chapter 7. Soft Sensor Management and Virtual Sensors for FDIR
a) n z,meas [g]
b) ∆ n z,voted [g]
2.5
2
1.5
1
n
z,2
n
z,3
n
z,virt
5
0.2
6 7 8 9 10
0.1
0
−0.1
5 6 7 8 9 10
Time [s]
c) weight
1
0.8
0.6
0.4
0.2
n
z,2
n
z,3
0
5 6 7 8 9 10
d) threshold
0.6
0.4
0.2
∆ n z,2
∆ n z,3
mf lb
mf ub
0
5 6 7 8 9 10
Time [s]
Figure 7.13: Soft sensor management: second drift failure of a normal acceleration
sensor.
where bl and bu denote the lower and upper bound of the soft threshold, respectively
(see also Figure 7.12a) and Cb,l and Cb,u are scaling factors. The dynamic
thresholds have minimum (bl,min, bu,min) and maximum (bl,max, bu,max) values,
where the maximum values are typically reached during maneuvering.
The dynamic thresholds can be used in order to make maximum use of the
virtual normal acceleration sensor. In the areas in which the virtual sensor signal
is known to be accurate, the soft thresholds can be held tight, while in the areas
where the virtual sensor is less accurate but can still be used as an arbitrator,
the soft threshold is widened to avoid false failure declarations. In the case of the
virtual normal acceleration sensor the accuracy is known to reduce for extreme
normal accelerations (extreme for commercial standards). In Figure 7.13 it can be
seen that the soft threshold reaches its maximum size when the normal acceleration
reaches 1.8 g. The dynamic thresholds are enabled in duplex and simplex
mode, when the sensor management system is using the virtual sensor signal.
7.4.2 Simulation examples
The virtual sensor enables the sensor management system to identify the failed
sensor in duplex mode for a drift failure of a second normal acceleration sensor,
see Figure 7.13. While signal nz,3 and nz,virtual are in agreement, nz,2 starts to
diverge from nz,virtual (Figure 7.13a). In Figure 7.13b the difference between the
voted signal and the true normal acceleration ∆nz,voted is shown. Due to the drift,
the voted signal diverges from the true normal acceleration until the weight of the
failed sensor output is reduced to zero (Figure 7.13c). The absolute differences

7.4. Virtual Sensor for FDIR 107
a) n z,meas [g]
b) ∆ n z,voted [g]
2.5
2
1.5
1
n
z,2
n
z,3
n
z,virt
5
0.2
6 7 8 9 10
0.1
0
−0.1
5 6 7 8 9 10
Time [s]
c) weight
1
0.8
0.6
0.4
0.2
n
z,2
n
z,3
0
5 6 7 8 9 10
d) threshold
0.6
0.4
0.2
∆ n z,2
∆ n z,3
mf lb
mf ub
0
5 6 7 8 9 10
Time [s]
Figure 7.14: Soft sensor management: second drift failure of a normal acceleration
sensor including sensor noise and severe turbulence.
between the sensor signals and the virtual sensor output (∆nz,2 and ∆nz,3) are
illustrated in Figure 7.13d together with the dynamic lower and upper bounds
of the membership function connected to the virtual sensor signal. Here it is also
illustrated that the dynamic thresholds indeed correlate with the estimation errors
during maneuvering.
This scenario is repeated with sensor noise on all signals, including the inputs
signals of the virtual sensor, and severe atmospheric turbulence, see Figure 7.14.
The soft sensor management system still performs well.
Using the virtual sensor, the sensor management system is even capable of identifying
a failure of the last available sensor, which is illustrated in Figure 7.15 for
a drift failure of the third normal acceleration sensor (Figure 7.15a). The monitor
count is disengaged during simplex mode. As soon as the membership degree of
nz,3 becomes equal to zero, the monitor count reaches the failure declaration value
immediately (Figure 7.15d) and the signal is no longer available. In Figure 7.15c
it is shown how the feedback path of the FCLs smoothly reconfigures to not using
the normal acceleration signal.
It should be noted that even when a sensor failure is detected during simplex
mode, it could be both the last available hardware sensor or the virtual sensor. In
both cases the best strategy is to reconfigure to not using the normal acceleration
signal. The implementation of the virtual sensor is not limited to the soft sensor
management system, and could also be implemented in the conventional sensor
management system.

108 Chapter 7. Soft Sensor Management and Virtual Sensors for FDIR
a) n z [g]
b) threshold
2.5
2
1.5
1
0.6
0.4
0.2
n z,3
n z,virt
n z
10 11 12 13 14 15
∆ n z,3
mf lb
mf ub
0
10 11 12 13 14 15
Time [s]
c) weight
d) blend
1
0.8
0.6
0.4
0.2
0
n
z,3
10
1
11 12 13 14 15
0.8
0.6
0.4
0.2
0
10 11 12 13 14 15
Time [s]
Figure 7.15: Soft sensor management: third drift failure of a normal acceleration
sensor.
7.4.3 Flight simulator results
The soft sensor management system, including the virtual sensor, has been successfully
evaluated during pilot-in-the-loop simulations in the Research Flight Simulator
of the NLR. Figure 7.16 illustrates the result of one particular flight simulator
test. During the simulation, the data was recorded in batches containing mainly
the insertion of the sensor failures and the more aggressive maneuvers performed
by the pilot. The task of the test pilot was to try to find problems in the system
and the virtual sensor, in the latter case by exiting the normal acceleration.
Failures in two of the normal acceleration sensors were introduced at t =95sec
and t = 300 sec. The test pilot took his job very seriously, which can be concluded
from the fact that the normal acceleration exceeded the maximum allowed
value of nz =2.5 g (Figure 7.16a) and the bank angle reached a maximum value
of φ =87deg (Figure 7.16b). Note that in normal flight the maximum bank angle
is limited to φ =66deg when the pilot is controlling the bank angle and to
φ =33deg when the wheel is centered. Even in these extreme situations, the soft
sensor management system performed as expected.
7.4.4 Discussion
In the conventional sensor management system the consolidated signal is no longer
available after a sensor failure in the duplex mode, even if one of the sensors is
still healthy. The implementation of a virtual sensor makes it possible to monitor
the last available hardware sensor. Since the virtual sensor is implemented in the

7.5. Conclusions 109
a) n z,meas [g]
b) φ [deg]
c) n z,voted [g]
d) ∆ n z,virt [g]
e) valid
3
2
1
30
0
−30
−60
−90
3
2
1
0.1
0
−0.1
2
n z,1
n z,2
n z,3
0
0 50 100 150 200
Time [s]
250 300 350
Figure 7.16: Flight simulator test results. After 300 sec the aircraft is flying on the
last available normal acceleration sensor that is monitored by the virtual normal acceleration
sensor.
software, the same safety level can be accomplished with less hardware. The cost
reduction is more than the cost of the sensor itself, since it requires less supporting
equipment and maintenance. Of course the development cost will increase because
of the design of the virtual sensor. Virtual sensors can be implemented to increase
the capability of the available hardware sensors, or to be able to reduce the number
of hardware sensors without compromising the availability of the FCS.
7.5 Conclusions
Fuzzy logic techniques have been applied in the sensor management system, FCL
reconfiguration, and the virtual sensor for normal acceleration. The improvement
with respect to the conventional sensor management system is in the quality of the
consolidated signal and results in a reduction of transients due to sensor failures.
Furthermore, the virtual sensor increases the capability of the available hardware
sensors, since it adds the ability to identify the failed sensor in the duplex mode
and to detect a sensor failure in the simplex mode. This has been demonstrated
by means of closed-loop simulation examples using a realistic aircraft model.
Final evaluation of the soft sensor management system and the TS fuzzy model

110 Chapter 7. Soft Sensor Management and Virtual Sensors for FDIR
based virtual sensor has taken place during pilot-in-the-loop simulations in the
RFS of the NLR.

8
Conclusions
The thesis described two major challenges in the further development of digital
fly-by-wire flight control systems: cost reduction by improving the architecture
and design efficiency of the flight control system and the enhancement of flight
safety. Fuzzy logic techniques and neural networks are used throughout the
thesis to investigate novel applications and more automated design procedures
that can provide a substantial contribution to solving these problems.
The contributions of this thesis are summarized in Section 8.1. In Section 8.2
an answer is given to the research question whether soft computing techniques
can be used to improve the efficiency of the design process. In Section 8.3 an
answer is given to the research question whether soft computing can contribute to
increasing flight safety. Recommendations and suggestions for future research are
given in Section 8.4.
8.1 Thesis Contributions
In Chapter 3, a novel automated procedure is proposed for the identification of
the operating points for which the local flight control law parameters need to be
tuned. This procedure is based on the application of fuzzy clustering to a set
of aerodynamic derivatives that are relevant with respect to the dominant aircraft
dynamics. The resulting cluster centers serve as the operating points, while
the scheduling variables should capture the nonlinearities of the system to be
controlled. A singleton TS fuzzy model structure provides the interpolation mechanism
for the Flight Control Law (FCL) parameters in order to obtain a global
nonlinear controller.
Chapter 4 describes the application of the automated design procedure to the
classical FCLs that are available in the synthetic environment. The scheduling
mechanism of the six most relevant controller parameters is replaced by the
scheduling mechanism designed by using the fuzzy clustering approach. The selected
scheduling variables are the Mach number and the dynamic pressure. Eight
operating points are identified for the flight envelope in clean configuration and
111

112 Chapter 8. Conclusions
two for the flight envelope in landing configuration. Pilot-in-the-loop simulations
demonstrated that the performance of the FCLs designed by using the automated
design procedure is equivalent to the performance of the default FCLs. The proposed
approach uses fewer operating points and requires less design effort.
The proposed methodology can be used in any application domain, provided
that sufficient information of the system to be controlled is available. This information
can either be in the form of a (detailed) nonlinear model or in the form of
recorded time histories of relevant system variables.
Few examples of an automated approach to the identification of operating
points can be found in literature. In these examples an iterative procedure of
scheduling, controller design and closed-loop evaluation is described. The model is
used to evaluate the closed-loop system and can only be used in combination with
an automatic controller design procedure.
The controller design procedure described in Chapter 5 goes one step further.
Here not only the identification of the operating points and the design of the
scheduling mechanism are automated by using the approach of Chapter 3, but also
the local controllers are designed using robust MV control techniques. The latter
removes the need for the time consuming single-loop iterative tuning of the local
flight control law parameters. The problem with robust MV control techniques is
the opaque structure of the resulting dynamic controller, which complicates the
gain scheduling of the FCL parameters.
A design procedure for a gain-scheduled robust MV controller is proposed.
The local H∞ controllers are designed in continuous-time using Linear Matrix Inequalities
(LMIs) and order reduction is performed before the transformation to
the δ-operator form. The gain-scheduled H∞ controller has been evaluated offline
(linear and nonlinear simulations, stability analysis) and through pilot-inthe-loop
simulations. With respect to the aircraft dynamics, the gain-scheduled
H∞ controller was given a Cooper-Harper (CH) rating of 1 in all flight. However,
due to the high control force needed to maneuver the aircraft, especially in the
low dynamic pressure region, the overall CH ratings were between 2 and 3. The
results are satisfactory, however, additional tuning is required to further improve
the performance and stability characteristics.
Although many examples of scheduled robust multivariable control can be
found in literature, in these examples an attempt is made to reduce the order of the
controller and/or to impose a certain structure on the controller in order to simplify
the scheduling problem. This leads to the undesirable modification of the original
controller. In this thesis however, the order reduction of the H∞ controller is done
in such a way that its performance is not affected. The use of the δ-operator form
for parameter scheduling is proposed to further improve the schedulability of the
H∞ controller without modifying their performance. Parameter-scheduled robust
multivariable control can in principle be used in any application domain. However,
it has been shown that the parameter scheduling of higher-order H∞ controllers becomes
problematic. In that case either an alternative scheduling procedure should
be chosen, e.g. scheduling of the output matrix only, or an alternative controller
design methodology should be selected.

8.1. Thesis Contributions 113
The design procedure of the virtual angle-of-attack sensor is described in detail
in Chapter 6. The philosophy of the design procedure is to use as much as possible
the well know relations of the linearized aircraft dynamics. The linear, white-box
part of the virtual sensor consists of the approximation of the trimmed angle-ofattack
plus the linear approximation of the angle-of-attack due to longitudinal
maneuvers. The trimmed AoA as well as the parameters for the approximation of
the dynamic AoA change as a function of the flight condition. These parameters
are estimated by a TS fuzzy model that uses Mach number, dynamic pressure,
bank angle, position of the CG along the X-axis and aircraft weight as inputs. The
remaining error is estimated by a nonlinear, black-box model. For this a neural
network is used. The inputs of the neural network are determined using a nonlinear
input selection approach.
The performance of the virtual sensor is demonstrated by a large number of
nonlinear simulations for which the flight conditions and maneuvers are selected
randomly. The performance of the virtual sensor is good, with maximum estimation
errors for the angle-of-attack of less than 0.8 degrees.
Virtual sensors are used in many application domains, for example chemical
industry, nuclear power plants, and medicine. Most virtual sensors are used to
estimate variables that can not be measured directly, either because such a sensor
does not exist or, if such a sensor does exist, because it can not survive the
harsh environments it needs to operate in. In most cases a black-box modelling
approach is adopted, making us of artificial neural networks. In the literature,
many applications of analytical redundancy for fault detection and fault isolation
in flight control systems have been reported, however, the use of virtual sensors in
aerospace applications is novel. The virtual angle-of-attack sensor design approach
proposed in this thesis distinguishes itself from other approaches due to the combination
of white-box and black-box modelling.
Sensor management is typically performed by the comparison of like sensor signals.
The sensor signal that does not coincide with the other like sensor signals is
assumed to be faulty. Crisp thresholds are used to determine whether a sensor signal
is faulty or not. An alternative sensor management approach is introduced
in Chapter 7. This approach distinguishes itself from the conventional sensor
management approach in three ways, namely the integration of the voting and
monitoring function, the application of soft thresholds and the use of a virtual
sensor for sensor management. This results in a more reliable consolidated signal,
while the transients induced by sensor failures are reduced significantly. Moreover,
it is demonstrated how a virtual sensor can be used to identify the faulty sensor
in the case there is a discrepancy between two like sensor signals. When the last
sensor fails, the signal is no longer available and the FCLs reconfigure to not using
this signal. The FCL reconfiguration is smooth as a direct result of the soft sensor
management strategy. The soft sensor management system, including a normal
acceleration virtual sensor, has been demonstrated by means of closed-loop simulation
examples using the synthetic environment and by means of pilot-in-the-loop
simulations.
The proposed sensor management system is an extension to the conventional
sensor management system of cross comparison of sensor signals. Although soft

114 Chapter 8. Conclusions
computing techniques have been implemented in other application domains, such
as the process industry, their application for FDI in flight control systems has
not been extensively investigated yet. The proposed soft sensor management system
can be used in any application domain where hardware redundancy is implemented.
8.2 Efficiency Improvement of the Design of the System
Fuzzy clustering has been applied to partition the flight envelope into operating
regimes in an attempt to improve the efficiency of flight control law design.
When the flight control engineer designs the gain scheduler for a classical controller,
this is typically performed separately for each FCL parameter that requires
scheduling. Moreover, each FCL parameter is not necessarily scheduled with the
same scheduling variable(s). In other words, each FCL parameter has its own set
of operating points, which makes the structure of the gain scheduler opaque. This
iterative, single-loop approach of tuning the (scheduled) FCL parameters is timeconsuming
and therefore contributes significantly to the total design cost. Due to
this opaque structure, the mutual dependencies of the FCL parameters are hard
to identify. This makes it difficult to understand what needs to be changed when
the performance is not as expected in a certain flight condition. In Chapter 3 it is
shown that applying fuzzy clustering to relevant aerodynamic derivatives results
in a partition of the flight envelope that makes sense to a flight control engineer.
The main advantage of this approach is that the operating points are identified
simultaneously, compared to the iterative trial-and-error approach that is carried
out by the flight control engineer. Besides the reduction in the design effort with
respect to identifying the operating points, the fuzzy clustering approach leads to
fewer operating points. Clearly this results in less design effort for tuning the FCL
parameters. By using the same scheduling variables and operating points for all
related FCL parameters that require scheduling, the mutual dependencies of the
FCL parameters are clearer and corrections are easier to make. However, this does
not necessarily mean that the best performance is achieved by scheduling all FCL
parameters in exactly the same way. In order to get the same closed-loop dynamics
over the entire operating range, each FCL parameter should have a dedicated
scheduling mechanism. Moreover, it makes sense to use different operating points
for the longitudinal and lateral FCL parameters, since they correspond to different
aircraft dynamics. This has not been investigated further in this thesis.
In Chapter 4 it is illustrated that the automated design approach for the identification
of the operating points works well on the classical FCLs that serve as the
default in the SE. However, this approach can potentially be used for any linear
design approach.
In Chapter 5 the automated design approach is used in combination with a
robust MV design approach to design the local FCL parameters, which further
reduces the required effort for the design of gain-scheduled FCLs.
In conclusion, the contribution of the fuzzy clustering approach to improving the

8.3. Enhancement of Flight Safety 115
efficiency of the design of flight control laws is significant. It is a global approach,
all the operating points are identified simultaneously, that results in a transparent
scheduling scheme with fewer operating points. Moreover, it is a model-based
approach that uses the nonlinear dynamics in the design phase and not only in
the evaluation phase. In combination with modern MV control techniques for the
design of the local controllers, the reduction in the design effort for the design of
the flight control laws is even more evident.
8.3 Enhancement of Flight Safety
Two approaches have been proposed to enhance the flight safety by means of soft
computing. The first approach is the introduction of a virtual sensor that can be
used as an arbitrator or as an additional sensor. The second approach is to apply
soft computing for the signal consolidation and sensor monitoring.
In Chapter 6, a design procedure for a virtual angle-of-attack sensor is proposed,
which strongly relies on soft computing techniques. With a virtual sensor it is
possible to identify the faulty signal when a discrepancy is detected between the
signals from two like sensors. Moreover, it is possible to detect a failure on the last
remaining physical sensor. The virtual sensor increases the integrity of the FCS
and therefore contributes to flight safety.
In the conventional approach, signal consolidation and signal monitoring are
performed separately. Moreover, both applications make use of crisp thresholds. In
Chapter 7, signal consolidation and sensor monitoring is integrated with the use
of soft thresholds. This results in a more accurate consolidated signal and reduces
the failure induced transients, which contributes not only to flight safety, but also
to passenger comfort.
In conclusion, the contribution of the soft computing to increase flight safety is
significant. Clearly virtual sensors can be designed using other techniques, however,
the advantage of fuzzy logic techniques is that it is possible to use linear
techniques for nonlinear systems. This allows for an easy interpretable structure
in combination with accuracy and reliability of the virtual sensor. The contribution
of soft sensor management to flight safety is less significant than for the virtual
sensor, but it does improve the system at no extra cost.
8.4 Recommendations for Further Research
With respect to the partitioning of the flight envelope using fuzzy clustering, both
the cluster centers as well as the membership functions are used for the design of
the gain scheduled flight control laws in this thesis. An alternative approach would
be to tune the FCL parameter in the operating points that are identified by fuzzy
clustering and then design a scheduler that optimizes the performance over the
entire flight envelope. One could for example design a (non)linear polynomial that
depends on a predefined set of scheduling variables. In this way improved performance
can be achieved in off-design flight conditions. It should be noted that the

116 Chapter 8. Conclusions
scheduling does not have to be the same for all FCL parameters.
The use of more advanced clustering algorithms, i.e. cluster algorithms that
are more flexible with respect to the size, orientation and volume of the clusters
to be identified in the data set, such as the fuzzy maximum likelihood estimates
clustering, may result in an improved partition of the flight envelope.
As shown in Chapter 5, there are still stability problems in the operating regimes
between the outer operating points and the edge of the flight envelope. These problems
occurred in particular in the low dynamic pressure region. It is recommended
to force the outer operating points closer to the edge of the flight envelope, such
that these stability problems no longer occur. Since the fuzzy clustering algorithm
uses the Euclidean distance measure, one possible approach is to weight the distance
for data points close to the edge of the flight envelope more than for data
points in the center of the flight envelope.
Although the results obtained with the application of scheduled robust MV control
described in this thesis are promising, it is a relatively new subject that requires
much additional research. First of all, the design of the local H∞ controllers is performed
in a relatively crude manner in the sense of the tuning of the input uncertainty
and weighting filters. Most of the design effort for the local H∞ controllers
was spent on the tuning of the reference model based on the comments of the
test pilots. It is believed that much can be gained by investigating the uncertainty
description and weighting filters more closely. This might further improve the robustness
of the local H∞ controllers and remove some of the instability problems
that were found in the low dynamic pressure region of the flight envelope.
The parameters in the robust MV controllers have no apparent physical meaning
and are therefore hard to interpret. This is a serious drawback, not only with
respect to gain scheduling, but especially with respect to FCL reconfiguration.
FCL reconfiguration is typically performed because a sensor signal or actuator is
no longer available or, in the case of military aircraft, there is structural damage.
It is not expected that the a controller designed for nominal mode has the same
inputs and outputs as a controller designed for degraded mode. Therefore these
controllers will not have similar structures, which rules out the option of FCL reconfiguration
through gain scheduling. Further research is required to investigate
this problem. Output scheduling could work, but it means that for each anticipated
failure a separate dynamic controller is running in parallel. This will put a
heavy burden on the available computing power and storage capacity.
Ultimately it would be interesting to investigate the capabilities of a set of
scheduled FCLs entirely based on robust MV control techniques, both for the longitudinal
and lateral-directional aircraft motion. With respect to interpretability
and/or schedulability it is expected that the best possible approach is to design the
controllers for the longitudinal and lateral-directional aircraft motion separately
and integrate them afterwards. This is how it is done with classical flight control
laws as well and is accepted as best practice in the aeronautical world. This would
enable the introduction of a highly automated design approach for the primary
flight control laws.

8.4. Recommendations for Further Research 117
With respect to virtual sensors, another topic of interest is how they could best
be used in the (soft) sensor management system. When there is a whole set of
non-like virtual sensors available, they are all largely depending on the same signals
(except for the signal they are designed to estimate). This means that there
is a high degree of integration and mutual dependency among the physical and
virtual sensors. It should be investigated what the most fault tolerant architecture
is for the case of multiple non-like virtual sensors, possibly in combination with
alternative FDI methodologies.

118 Chapter 8. Conclusions

A
Synthetic Environment and
Real-Time Code Generation
The Synthetic Environment (SE), developed within the ADFCS project, is a
software tool aimed to perform detailed six degrees-of-freedom simulation of a
DFBW aircraft. The SE is developed in the Matlab/Simulink TM environment
and includes models of sensors, actuators, digital flight control computers, etc.
It is defined as an ultra-high fidelity simulation tool that is structurally representative
of a practical implementation (Rosenberg 2001). The SE allows to
identify potential problems at an early stage by means of extensive and detailed
simulation sessions, such that problems can be identified and solved prior to
manufacture. In this way time and cost penalties associated with rework later
in the life-cycle are avoided. The SE allows for evaluation of new concepts in
a realistic simulation environment. Moreover, the SE can be run in real-time
and can therefore be used for hardware-in-the-loop testing and demonstration.
The content of this appendix is for a large part taken from (Ciniglio 2002).
This appendix is organized as follows: In Section A.1 the architecture of the synthetic
environment is discussed, together with a description of the separate modules.
This section provides background information for readers who are familiar
with the notation used in aerospace engineering. Interested readers who are unfamiliar
with the notation that is used in this section, are referred to (McLean 1990).
The procedure to generate real-time code starting from the synthetic environment
is described in Section A.2.
A.1 Synthetic Environment
A.1.1 Synthetic Environment Architecture
The simulation model of the entire augmented SCA model has been structured
in several separate blocks, see Figure A.1. Some of the modules in the SE are
available with different levels of complexity. The lower the level of complexity, the
lower the required computing power to simulate the corresponding module. This
119

120 Appendix A. Synthetic Environment and Real-Time Code Generation
Figure A.1: Synthetic Environment Architecture (Ciniglio 2002).
allows the user to modify the SE such that it meets the requirements of each specific
simulation test, while minimizing the total required computing power. Each of the
blocks illustrated in Figure A.1 is described in more detail below.
A.1.2 Bare Airframe
As illustrated in Figure A.1, the bare airframe is defined by the six Degree-Of-
Freedom (DOF) equations of motion, gravity model, aerodynamic model, hinge
moment model and undercarriage model together with the air data and the engines.
Equations of motion
The six DOF equations of motions are:
˙u = FX g
W
˙v = FY g
W
˙w = FZ g
W
− qw + rv (A.1)
− ru + pw (A.2)
− pv + qu (A.3)

A.1. Synthetic Environment 121
˙p =
˙q =
˙r =
IZ
IXIZ − I 2 XZ
1
+
IXZ
IXIZ − I 2 XZ
[L +(IY − IZ)qr + IXZpq]
[N +(IX − IY )pq − IXZqr] (A.4)
[M +(IZ − IX)pr − IXZ(p
IY
2 − r 2 )] (A.5)
IXZ
IXIZ − I 2 XZ
+
IXZ
IXIZ − I 2 XZ
[L +(IY − IZ)qr + IXZpq]
[N +(IX − IY )pq − IXZqr]. (A.6)
The forces and moments (aerodynamics, engine and gravity) defined in the body
axes are:
where
FX = qSCX + FXE − W sin(θ) (A.7)
FY = qSCY + FYE + W cos(θ)sin(φ) (A.8)
FZ = qSCZ + FZE + W cos(θ)cos(φ) (A.9)
L = qSCRb + FZE (YE − YCG) − FYE (−ZE + ZCG) (A.10)
M = qSCMc + FXE (−ZE + ZCG) − FZE (−XE + XCG) (A.11)
N = qSCNb + FYE (−XE + XCG) − FXE (YE − YCG), (A.12)
CX = CL sin(α) − CD cos(α)cos(β)
CZ = −CL cos(α) − CD sin(α)cos(β)
FXE = T cos(ψT )cos(θT )
FYE = T sin(ψT )
FZE = −T cos(ψT )sin(θT ).
Note that Equations A.7 to A.12 hold for the right engine. For the left engine one
should substitute FYE = T sin(ψT )byFYE = −T sin(ψT ) and substitute YE by
−YE. The aerodynamic coefficients CL, CD, CY , CR, CM and CN are described
in more detail later in this section.
The SE provides the capability to trim and linearize the nonlinear aircraft model
for the given flight condition and aircraft configuration. The linear longitudinal
and lateral aircraft models are given below, respectively:
⎡ ⎤ ⎡
⎤ ⎡ ⎤ ⎡ ⎤
˙u
˙w ⎥
˙q ⎦
˙θ
=
Xu Xw Xq Xθ u
⎢ Zu Zw Zq Zθ ⎥ ⎢
⎢
⎥ ⎢w
⎥
⎣ ˜Mu
˜Mw
˜Mq
˜Mθ ⎦ ⎣ q ⎦
0 0 1 0 θ
+
Xδe
⎢
⎣
Xδs
Zδe
Zδs
⎥
˜Mδe
˜Mδs
⎦
0 0
⎢
⎣
δe
δs
(A.13)

122 Appendix A. Synthetic Environment and Real-Time Code Generation
⎡ ⎤
˙v
⎢
˙p ⎥
⎣ ˙r ⎦
˙φ
=
⎡
⎤ ⎡ ⎤
Yv Yp Yr Yφ v
⎢ ˜Lv
˜Lp
˜Lr
˜Lφ ⎥ ⎢
⎢
⎥ ⎢p
⎥
⎣Ñv
Ñp Ñr Ñφ
⎦ ⎣r
⎦
0 1 0 0 φ
+
⎡ ⎤
Yδa
Yδr
⎢ ˜Lδa
˜Lδr
⎥
⎢ ⎥
⎣ ⎦
Ñδa
Ñδr
0 0
Aerodynamic model - Longitudinal coefficients
δa
δr
. (A.14)
CD = CD0 +∆CD(α, δe,δs,δfl)+CDBIAS (M)+CDSB (α, δsp,M)
+CDLG (CL,δfl,δsl) (A.15)
CL = CL0 +∆CL(α, δe,δs,δfl)+CLSB (α, δsp,M)
+(CLqq + CL
c
˙α) ˙α
2VT
(A.16)
CM = CM0 +∆CM (α, δe,δs,δfl)+CMSB (α, δsp,M)
+(CMqq + CM
c
˙α) ˙α
2VT
+ CX(ZCG − ZREF )
−CZ(XCG − XREF )+CMLG (CL,δfl,δsl). (A.17)
Aerodynamic model - Lateral coefficients
Gravity model
CY = ∆CY (α, β, δfl)+∆CY (M)β +∆CY (α, δa,δr,δfl)
b
+[CYp (M)p + CYr (M)r]
2VT
(A.18)
CR = ∆CR(α, β, δfl)+∆CR(M)β +∆CR(α, δa,δr,δfl)
b
+[CRp (M)p + CRr (M)r]
2VT
−CY (ZCG − ZREF ) c c
− CZYCG
b b
(A.19)
CN = ∆CN (α, β, δfl)+∆CN (M)β +∆CN (α, δa,δr,δfl)
b
+[CNp (M)p + CNr (M)r]
2VT
−CY (XCG − XREF ) c c
+ CXYCG .
b b
(A.20)
The gravity model is the standard International Standard Atmosphere (ISA) gravity
model (Ruijgrok 1990).
Hinge moment model
The hinge moments of elevator, aileron and rudder are given below in mbar m 3 :
HMELEV = CELEV (CHαtail (M) αtail + CHELEV (M) δe) q (A.21)
HMAIL = CAIL CHAIL(α, δa,M) q (A.22)
HMRUD = CRUD CHRUD(δr,β) q, (A.23)

A.1. Synthetic Environment 123
where
Undercarriage model
αtail = α [1 − ɛα(δfl)] − ɛ0(δfl)+δs. (A.24)
When the undercarriage is retracted, an additional drag and corresponding moment
is modelled. The undercarriage has no influence on the aircraft inertia matrix.
Air data model
The air data probe and the corresponding Air Data Computer (ADC) are considered
to be part of the bare airframe. The air data probe measures static pressure,
impact pressure and temperature. Based on these measurements, the ADC computes
pressure altitude (Hp) [ft], rate-of-climb (c) [ft/min], calibrated airspeed (VC)
[kts], Mach number (M), true airspeed (VT ) [kts], dynamic pressure (q) [mbar]the
corrected Angle-of-Attack (AoA) [deg].
Engine model
The thrust T is a function of throttle position δth, altitude h and Mach number
M:
T = f(δth,h,M).
The thrust dynamics are implemented as a simple first order lag.
A.1.3 Outside World
The atmosphere model is the standard ISA atmosphere model (Ruijgrok 1990).
Furthermore the SE includes two turbulence models. One turbulence model is
based on the Dryden theory (Etkin 1972). This turbulence model can be implemented
with three levels of intensity (light, medium and severe). A second turbulence
model can be used instead of the Dryden model in order to better reproduce
the off-line environment that will be used during the simulator sessions, since this
turbulence model is implemented in the flight simulator. Finally a windshear model
with arbitrary altitude profile has been implemented.
A.1.4 Flight Control Computer
The flight control computer model includes the cross-channel data link, the voter/
monitor for sensor/actuator fault detection and identification, the flight control
laws to improve stability and control, the autopilot and the flight envelope protection
modes.
A.1.5 Actuation
The main components of the actuation model are the models of the hydraulic actuators
together with the Actuator Control Electronics (ACE) channels, with related

124 Appendix A. Synthetic Environment and Real-Time Code Generation
LVDT sensors. The most complex ACE card model includes sensors, demodulation
and actuator loop closure.
A.1.6 Sensors, Switches and Controls
The airframe sensor module is implemented in the non-linear dynamic simulation
model of the whole DFBW suite. The sensor models allow for modelling of sensor
failures (bias, change in the scale factor, hold-on model, etc.).
A.1.7 Cockpit
This cockpit model includes generic blocks for pilot command signals generation
(or the pilot commands recorded during flight simulator sessions). No displays are
available in the SE.
A.2 Real-Time Code Generation
The software environment of the NLR RFS is controlled by the Programme and
Real-time Operations SIMulation support tool (PROSIM). Configuring the RFS
with the FGS design was ultimately performed by the software tool MOSAIC
(Model-Oriented Software Automatic Interface Converter) developed at the NLR
(Lammen et al. 1999). The MOSAIC tool automatically transfers the model from
Matlab/Simulink TM to PROSIM. The tool takes as input the model source code
that has been generated by Real-Time Workshop of Simulink 3.0 and delivers as
output the model source code that can run in PROSIM. This automated process
of generating real-time code for the flight simulator is only possible if the Simulink
model complies with defined software development guidelines (Heesbeen 2001). A
more detailed description of this process is given in (Smaili 2001).

B
Short-Period Approximation
In this appendix the short-period approximation is described. This approximation
is a useful tool for longitudinal flight control design, since it greatly
simplifies the equations of motion while maintaining the dominant longitudinal
aircraft dynamics.
The short-period motion is derived from the linear longitudinal aircraft model
in Section B.1. In Section B.2 a few relations are derived from the short-period
approximation which are used in Chapter 4 of this thesis.
B.1 Derivation of the Short-Period Approximation
The linear longitudinal aircraft model is derived in Appendix A and is shown again
below: ⎡ ⎤
˙u
⎢
˙w ⎥
⎣ ˙q ⎦
˙θ
=
⎡
Xu
⎢ Zu ⎢
⎣ ˜Mu
0
Xw
Zw
˜Mw
0
Xq
Zq
˜Mq
1
⎤ ⎡ ⎤
Xθ u
Zθ ⎥ ⎢
⎥ ⎢w
⎥
˜Mθ ⎦ ⎣ q ⎦
0 θ
+
⎡ ⎤
Xδe
⎢ Zδe
⎥
⎢ ⎥
⎣ ˜Mδe
⎦
0
δe. (B.1)
The longitudinal aircraft motion is divided into two oscillatory eigenmotions,
namely the short-period motion and phugoid motion. The phugoid motion has
a low eigenfrequency and is typically poorly damped. During the phugoid motion
the aircraft exchanges kinetic energy for potential energy and vice versa, which
results in variations in airspeed and altitude.
The short-period motion describes the aircraft dynamics along the Y -axis in terms
of the pitch rate and the angle-of-attack (or downward speed):
˙w
=
˙q
Zw Zq
˜Mw
˜Mq
w
+
q
Zδe
˜Mδe
δe. (B.2)
The short-period approximation follows from the assumption that the forward
speed remains constant, i.e. u = 0 (fixed speed assumption). In other words, the
125

126 Appendix B. Short-Period Approximation
short-period approximation assumes that short-period transients are of sufficiently
short duration that variations which arise in the forward speed as a result of aircraft
motion, control surface deflections and atmospheric turbulence are negligible.
B.2 Derivation of Short-Period Related Equations
B.2.1 Pitch rate to angle-of-attack
From Equation B.2 it can be seen that:
sw= Zww + Zqq + Zδe δe, (B.3)
where s denotes the Laplace operator. Assuming the elevator deflection to be equal
to zero (δe = 0) and with the substitution of w = U0α, where U0 is the (constant)
trimmed forward speed, this equation can be rewritten to:
sα= Zwα + Zq
q. (B.4)
Since Zw = − Nα
U0 and Zq = U0 (McLean 1990) it follows that:
where the aerodynamic derivative Nα is defined as:
U0
sα= − Nα
α + q, (B.5)
U0
Nα = ρU 2 0 SCNα
. (B.6)
2m
The transfer function from pitch rate to angle-of-attack then becomes:
which can be rewritten to:
α
q
α
q =
= U0
Nα
1
s + Nα
U0
1
U0 s +1.
Nα (B.8)
B.2.2 Normal acceleration to angle-of-attack
, (B.7)
From kinematics it follows that during longitudinal motion the normal acceleration
can be described by the following relation:
nz = 1
g (U0q − sw). (B.9)
Substitution of w = U0α results in:
nz = U0
(q − sα). (B.10)
g

B.2. Derivation of Short-Period Related Equations 127
Substitution of Equation B.5 in Equation B.10 results in:
nz = Nα
α. (B.11)
g
The transfer function from the angle-of-attack to the normal acceleration becomes:
nz
α
Nα
= . (B.12)
g
The transfer function from the normal acceleration to the angle-of-attack can
therefore be written as:
α
nz
= g
.
Nα
(B.13)
B.2.3 Pitch rate to normal acceleration
The transfer function from the pitch rate to the normal acceleration can be constructed
as follows:
nz α nz
= . (B.14)
q q α
Substitution of Equation B.8 and Equation B.12 in Equation B.14 results in:
which can be simplified to:
nz
q
= U0
Nα
nz
q
Introducing the parameter τ = U0
Nα
in:
nz
q
= U0
g
Nα
g
1
U0 s +1,
Nα (B.15)
1
U0 s +1.
Nα (B.16)
and substitution of τ in Equation B.16 results
= U0
g
1
. (B.17)
τs+1

128 Appendix B. Short-Period Approximation

C
Performance Measures and Cross
Validation
The performance measures described in this appendix provide a measure of
the (mis)match between a system and its model. In order to evaluate the predictive
capability of a model, these performance measures should be used in
combination with a cross validation procedure.
This appendix is organized as follows: In Section C.1 the performance measures
that are used throughout the thesis are defined. The cross validation technique,
which is used in Chapter 6, is described in Section C.2.
C.1 Performance Measures
The Variance Accounted For (VAF) index is computed by:
VAF =
1 −
var(y − ˆy)
var(y)
100%, (C.1)
where ‘var’ denotes variance, y denotes the data vector and ˆy denotes the model
output vector. The VAF of 100% means a perfect model prediction (disregarding
a constant offset), the VAF of 0% is obtained when the “model” is the mean of
the data. Negative VAF values indicate even worse models.
The Root-Mean-Squared Error (RMSE) index is computed by:
RMSE = 1
N
(yk − ˆyk)
N
2 . (C.2)
The RMSE of 0 means a perfect model prediction. Worse models will have larger
RMSE values.
k=1
129

130 Appendix C. Performance Measures and Cross Validation
The advantage of the VAF index is its insensitivity to the scale of the data. 1 This
makes it possible to compare results obtained for different variants of one data
set (e.g., original and scaled data). The VAF index, however, is not sensitive to a
constant offset. The RMSE index is sensitive both to the scale of the data and to
offsets. Therefore, the two indices are sometimes used in parallel.
C.2 Cross Validation
Cross validation is a technique that is often used for estimation of the prediction
error of a classification or regression function. Estimating prediction error on the
same data used for model estimation tends to give underestimates, because the
parameter estimates are “fine-tuned” to the peculiarities of the sample. For very
flexible methods, e.g. neural networks or tree based models, the error on the training
sample can usually be made close to zero. The true error of such a model will
usually be much higher however: the model has “overfitted” the training sample.
An alternative is to divide the available data into a training sample and a test
sample. If the available sample is rather small, this method is not preferred because
the test sample may not be used for model estimation in this scenario. By
using cross validation all data points are used for training as well as testing. The
general V-fold cross validation procedure works as follows:
The data set D is randomly split into V mutually exclusive subsets (the
folds) D1,D2, ..., DV of approximately equal size. The model is trained and
tested V times; each time v ∈ 1, 2, ..., V , it is trained on D \ Dv and tested
on Dv.
The performance measures presented in the previous section can be used in cross
validation by defining ˆy as follows:
ˆy =
ˆy v . (C.3)
v=1,2,..,V
1 Note, however, that modelling methods usually are sensitive to the scale of the data.

D
Soft Computing Techniques
Soft computing is a generic term for techniques like fuzzy clustering, fuzzy
reasoning, neural networks, etc. For example, in fuzzy clustering a certain data
point can belong partly to several clusters, hence the term soft. In contrast
to crisp clustering, where a certain data point belongs entirely to a certain
cluster (hard). Soft computing is used in a wide range of application areas
and is especially useful for piecewise linear modelling and when dealing with
uncertainties. In this appendix a number of soft computing techniques that are
used in this thesis are described.
This appendix is organized as follows: In Section D.1 fuzzy clustering is described
in more detail, including the Gustafson-Kessel algorithm. The application of fuzzy
clustering for system identification is addressed in Section D.2. Some background
information on neural networks is given in Section D.3.
D.1 Fuzzy Clustering
An effective approach to the identification of complex nonlinear systems is to
partition the available data into subsets and approximate each subset by a linear
model. Fuzzy clustering can be used as a tool to obtain a partition of data where
the transitions between the subsets are gradual rather than abrupt. The concept of
fuzzy partitioning is essential for cluster analysis and consequently also for fuzzy
modelling and identification techniques that are based on fuzzy clustering. Fuzzy
partitions can be seen as a generalization of hard partitions, which is formulated
in terms of classical subsets.
D.1.1 Preliminaries
Clustering techniques are unsupervised methods that can be used to organize data
into groups based on similarities among the individual data items. Most clustering
algorithms do not rely on assumptions common to conventional statistical methods,
such as the underlying statistical distribution of data, and therefore they are
131

132 Appendix D. Soft Computing Techniques
useful in situations where little prior knowledge exists.
Clustering techniques can be applied to data that are quantitative (numerical),
qualitative (categorical), or a mixture of both. The data are typically observations
of some physical process. Each observation consists of n measured variables,
grouped into an n-dimensional column vector zk =[z1k ...znk ], zk ∈ IR n .Asetof
N observations is denoted by Z = {zk|k =1, 2,...,N} and is represented as an
n × N pattern or data matrix:
⎡
⎢
Z= ⎢
⎣
z11 z12 ... z1N
z21 z22 ... z2N
.
.
.
zn1 zn2 ... znN
Various definitions of a cluster can be formulated, depending on the objective
of clustering. Generally, one may accept the view that a cluster is a group
of objects that are more similar to one another than to members of other clusters
(Bezdek 1981, Dave 1991). The term “similarity” should be understood as
mathematical similarity, measured in some well-defined sense. In metric spaces,
similarity is often defined by means of a distance norm. Distance can be measured
among the data vectors themselves, or as a distance from a data vector to some
prototypical object (prototype) of the cluster.
Fuzzy clustering methods allow the objects to belong to several clusters simultaneously,
with different degrees of membership. Objects on the boundaries
between several classes are not forced to fully belong to one of the classes, but
rather are assigned membership degrees between zero and one indicating their
partial membership.
The concept of fuzzy partitioning is essential for cluster analysis and consequently
also for the identification techniques that are based on fuzzy clustering.
Conditions for the fuzzy partition are given by (Ruspini 1970):
⎤
⎥
⎦ .
µik ∈ [0, 1], 1 ≤ i ≤ c, 1 ≤ k ≤ N (D.1)
0 <
c
µik =1, 1 ≤ k ≤ N (D.2)
i=1
c
µik < N, 1 ≤ i ≤ c, (D.3)
i=1
where µik denotes the membership degree of the kth data point with respect to
the ith cluster, c denotes the number of clusters or fuzzy subsets and N denotes
the number of data points. The ith row of the fuzzy partition matrix U contains
values of the ith membership function of the fuzzy subset Ai of Z:
⎡
⎤
µ11 µ12 ... µ1N
⎢
⎥
⎢
U= ⎢
⎣
µ21 µ22 ... µ2N
.
.
.
µc1 µc2 ... µcN
⎥
⎦ .

D.1. Fuzzy Clustering 133
Equation D.1 implies that the membership degree of zk with respect to subset Ai
can adopt any value from zero to one. The total membership of each zk in Z equals
one, which is defined in Equation D.2. Equation D.3 implies that the subset Ai is
never empty and never contains all the data. The fuzzy partition space for Z is
therefore the set:
Mfc = {U ∈ R c×N |µik ∈ [0, 1], ∀i, k;
c
µik =1, ∀k; 0 <
i=1
D.1.2 Fuzzy clustering algorithms
c
µik < N, ∀i}. (D.4)
In this section we focus on fuzzy clustering algorithms with an objective function.
These methods are relatively well understood and mathematical results are available
concerning the convergence properties and fuzzy validity assessment.
A large family of fuzzy clustering algorithms is based on minimization of the
objective function formulated as:
where
J(Z; U, V, {Ai} =
is a fuzzy partition matrix of Z;
c
N
i=1 k=1
U=[µik] ∈ Mfc
(µik) m D 2 ikAi
V=[v1, v2,...,vc], vi ∈ R n
i=1
, (D.5)
(D.6)
(D.7)
is a vector of cluster prototypes (operating points) which have to be determined;
D 2 ikAi = zk − vi 2 =(zk − vi) T Ai(zk − vi) (D.8)
is a squared inner-product distance norm and:
m ∈ [1, ∞) (D.9)
is a parameter which determines the fuzziness of the resulting clusters. The value
of the cost function defined in Equation D.5 can be seen as a measure of the total
variance of zk from vi.
The minimization of the objective function represents a nonlinear optimization
problem that can be solved by using a variety of methods, including iterative
minimization or genetic algorithms. A number of parameters must be specified beforehand:
the number of clusters, c, the ’fuzziness’ exponent, m, the termination
tolerance, ɛ and the norm-inducing matrices, Ai. Moreover, the fuzzy partition
matrix, U, must be initialized.
The number of clusters c is the most important parameter, in the sense that
the remaining parameters have less influence on the resulting partition. Two main
approaches to determine the appropriate number of clusters in data can be distinguished:

134 Appendix D. Soft Computing Techniques
• Validity measures are scalar indices that assess the goodness of the obtained
partition. Clustering algorithms generally aim at locating well-separated and
compact clusters. Cluster validity analysis is performed by running the cluster
algorithm for different values of c and usually also several times for each
c with a different initialization. The validity measure is calculated for each
run and the number of clusters which minimizes (maximizes) the validity
measure is chosen as the ”correct” number of clusters in the data. Many
validity measures have been introduced in the literature (Bezdek 1981, Gath
and Geva 1989, Backer 1995, Pal and Bezdek 1995).
• The basic idea of cluster merging is to start with a sufficiently large number
of clusters and successively reduce this number by merging clusters that are
(compatible) with respect to some well-defined criteria (Krishnapuram and
Freg 1992, Kaymak and Babuka 1995, Setnes and Kaymak 1998).
The fuzziness parameter m is a rather important parameter as well, because it significantly
influences the fuzziness of the resulting partition. As m approaches one
from above, the partition becomes hard and vi are ordinary means of the clusters.
As m →∞, the partition becomes completely fuzzy and the cluster means are all
equal to the mean of Z. These limit properties are independent of the optimization
method used. Once the number of clusters is selected, the fuzziness parameter m
is chosen by running the cluster algorithm for different values of m and evaluating
the resulting partition. The fuzziness parameter m is usually selected between 1.5
and 2.5.
The clustering algorithm stops iterating when the norm of the difference between
U in two successive iterations is smaller than the termination parameter ɛ.
For the maximum norm maxik(|µ (l)
ik − µ(l−1)
ik |) the usual choice is ɛ =10−3 ,even
though ɛ =10−2works well in most cases, while drastically reducing the computing
time.
The shape of the clusters is determined by the choice of the norm-inducing
matrices Ai in the distance measure. The norm-inducing matrices can be fixed
beforehand, but they can also be subject to optimization themselves. A common
choice is Ai = I, which gives the standard Euclidean norm:
D 2 ik =(zk − vi) T (zk − vi). (D.10)
The norm metric influences the clustering criterion by changing the measure of
dissimilarity. The Euclidean norm induces hyper-spherical clusters (surfaces of
constant membership are hyper-spheres). Norm-inducing matrices non-equal to
the identity matrix generates hyper-ellipsoidal clusters.
A common limitation of clustering algorithms based on a fixed distance norm
is that such a norm forces the objective function to prefer clusters of a certain
shape even if they are not present in the data. Generally, different matrices Ai are
required, but there is no guideline as to how to choose them a priori. Therefore,
clustering with adaptive distance measure is often used, such as in the Gustafson-
Kessel algorithm presented below.

D.2. Identification via Fuzzy Clustering 135
D.1.3 Gustafson-Kessel algorithm
The Gustafson-Kessel clustering algorithm described in this section taken from (Gustafson
and Kessel 1979).
Given the data set Z, choose the number of clusters 1 0. Initialize the partition matrix
randomly, such that U (0) ∈ Mf .
Repeat for l =1, 2,...
1. Compute the cluster prototypes (means).
v (l)
i =
n 2. Compute the cluster covariance matrices.
k=1 (µ(l−1)
ik ) mzk n k=1 (µ(l−1)
, 1 ≤ i ≤ c. (D.11)
ik ) m
N k=1
Fi =
(µ(l−1)
ik ) m (zk − v (l)
i )(zk − v (l)
i )T
N k=1 (µ(l−1)
ik ) m
3. Compute the distances.
4. Update the partition matrix.
if
otherwise
µ (l)
ik
, 1 ≤ i ≤ c. (D.12)
D 2 ikAi =(zk − vi) T [det(Fi) 1
n F −1
i ](zk − vi). (D.13)
DikAi
> 0, 1 ≤ i ≤ c, 1 ≤ k ≤ N (D.14)
µ (l)
ik =
c
=0 ifDikAi > 0 and µ(l)
ik
until U (l) − U (l−1)

136 Appendix D. Soft Computing Techniques
Figure D.1: Extraction of Takagi-Sugeno rules by fuzzy clustering.
The Takagi-Sugeno (TS) model (Takagi and Sugeno 1985) is used to smoothly
combine the linear submodels. The TS rules have the following form:
Ri : If x1 is Zi1 and ... and xp is Zip
then yi = a T i x + bi, i =1, 2,...,K. (D.17)
This is a MISO model with inputs and output y. Zij are linguistic terms (like low,
medium, high, etc.), represented by membership functions. Finally, ai and bi are
real-valued consequent parameters.
TS rules are extracted from data by clustering in the product space of the inputs
and outputs. By applying clustering algorithms that are capable of detecting linear
substructures in data, a nonlinear regression problem is automatically decomposed
it into several local linear subproblems. Each obtained cluster is represented by one
rule in the TS fuzzy model. The antecedent membership functions are obtained
by projection (Figure D.1) and the consequent parameters can be estimated by
various Least Squares (LS) methods (Babuˇska 1998).
Illustrative example: Consider a nonlinear function y = f(x) defined
piecewise by:
y = 0.25x, for x ≤ 3
y = (x − 3) 2 +0.75, for 3 6.
(D.18)
Figure D.2 shows a plot of this function evaluated in 50 samples uniformly
distributed over x ∈ [0, 10]. Zero-mean, uniformly distributed noise with
amplitude 0.1 was added to y.
The data set {(xi,yi)|i =1, 2,...,50} was clustered into four clusters. Figure
D.3 shows the local linear models obtained through clustering, the bot-

D.2. Identification via Fuzzy Clustering 137
y
12
10
8
6
4
2
y = 0.25 ⋅ x
y = 0.25 ⋅ x + 8.25
y = (x−3) 2 + 0.75
0
0 2 4 6 8 10
x
Figure D.2: The nonlinear function D.18 with zero-mean, uniformly distributed noise.
y
Membership degree
15
10
y 4 = 0.26⋅ x + 8.19
5
y = 4.45⋅ x − 17.40
3
y = 0.25⋅ x + 0.05
1
0
0 2
1
4
y = 1.42⋅ x − 3.76
2
6
x
8 10
0.5
Z 1
Z 2
0
0 2 4 6 8 10
x
Figure D.3: Cluster prototypes and the corresponding fuzzy sets.
tom plot shows the corresponding membership functions. In terms of TS
rules, the fuzzy model is expressed as:
R1 : If x is Z1 then y =0.25 x +0.05
R2 : If x is Z2 then y =1.42 x − 3.76
R3 : If x is Z3 then y =4.45 x − 17.40
R4 : If x is Z4 then y =0.26 x +8.19.
Note that the consequents of R1 and R4 correspond almost exactly to the
first and third equation D.18. Consequents of R2 and R3 are approximate
local linear models of the parabola defined by the second equation of D.18.
Z 3
Z 4

138 Appendix D. Soft Computing Techniques
D.3 Neural Networks
This section provides a brief introduction into neural networks, their architecture
and their training algorithms. For a more detailed description, the interested reader
is referred to (Fausett 1994, Gurney 1997, Haykin 1999).
D.3.1 Introduction to neural networks
An Artificial Neural Network (ANN) is an information processing paradigm that
is inspired by the way biological nervous systems, such as the brain, process information.
The key element of this paradigm is the structure of the information
processing system. It is composed of a large number of highly interconnected processing
elements (neurones) working together to solve specific problems. ANNs,
like people, learn by example. Learning in biological systems involves adjustments
to the synaptic connections that exist between the neurones. This is true of ANNs
as well.
Neural networks are applicable in virtually every situation in which a relationship
between the predictor variables (independents, inputs) and predicted variables
(dependents, outputs) exists, even when that relationship is very complex and not
easy to articulate in the usual terms of correlations or differences between groups.A
few representative examples of problems to which neural network analysis has been
applied successfully are detection of medical phenomena, stock market prediction,
credit assignment, monitoring the condition of machinery and engine management.
D.3.2 Architecture of neural networks
To capture the essence of biological neural systems, an artificial neuron is defined
as follows:
• It receives a number of inputs (either from original data, or from the output
of other neurons in the neural network). Each input comes via a connection
that has a strength (or weight); these weights correspond to synaptic efficacy
in a biological neuron. Each neuron also has a single threshold value.
The weighted sum of the inputs is formed, and the threshold subtracted, to
compose the activation of the neuron.
• The activation signal is passed through an activation function (also known
as a transfer function) to produce the output of the neuron.
If the step activation function is used then the neuron acts just like the biological
neuron described earlier, see Figure D.4. Actually, the step function is rarely used
in artificial neural networks, as will be discussed later in this section. Note also
that weights can be negative, which implies that the synapse has an inhibitory
rather than excitatory effect on the neuron: inhibitory neurons are found in the
brain.
If a network of neurons is to be of any use, there must be inputs (which carry
the values of variables of interest in the outside world) and outputs (which form

D.3. Neural Networks 139
Figure D.4: Example of a neuron with step activation.
predictions, or control signals). Inputs and outputs correspond to sensory and motor
nerves such as those coming from the eyes and leading to the hands. However,
there also can be hidden neurons that play an internal role in the network. The
input, hidden and output neurons need to be connected together. A simple network
has a feedforward structure: signals flow from inputs, forwards through any
hidden units, eventually reaching the output units. Such a structure has stable behavior.
However, if the network is recurrent (contains connections back from later
to earlier neurons) it can be unstable and has very complex dynamics. Recurrent
networks are very interesting to researchers in neural networks, but so far it is the
feedforward structures that have proved most useful in solving real problems.
A typical feedforward network has neurons arranged in a distinct layered topology,
see Figure D.5. The input layer is not really neural at all: these units simply
serve to introduce the values of the input variables. The hidden and output layer
neurons are each connected to all of the units in the preceding layer. It is possible
to define networks that are partially-connected to only some units in the preceding
layer; however, for most applications fully-connected networks are better.
When the network is executed, the input variable values are placed in the
input units and then the hidden and output layer units are progressively executed.
Each of them calculates its activation value by taking the weighted sum of the
outputs of the units in the preceding layer and subtracting the threshold. The
activation value is passed through the activation function to produce the output
of the neuron. When the entire network has been executed, the outputs of the
output layer act as the output of the entire network.
D.3.3 Training of the neural network
The best-known example of a neural network training algorithm is back propagation
(Fausett 1994, Patterson 1996, Haykin 1999). Modern second-order algorithms
such as conjugate gradient descent and Levenberg-Marquardt (Bishop 1995,
Shepherd 1997) are substantially faster for many problems, but back propagation
still has advantages in some circumstances and is the easiest algorithm to understand.
In the back propagation algorithm, the gradient vector of the error surface is cal-

140 Appendix D. Soft Computing Techniques
Figure D.5: Feedforward neural network.
culated while propagating backwards through the network (from output layer to
input layer). This vector points along the line of steepest descent from the current
point. Moving a short distance along this line will decrease the error. A sequence
of such moves (slowing when approaching the minimum) will eventually find a
minimum of some sort. The difficult part is to decide how large the steps should
be. In practice, the step size is proportional to the slope (so that the algorithms
settles down in a minimum) and to a special constant: the learning rate. The correct
setting for the learning rate is application-dependent, and is typically chosen
by experiment; it may also be time-varying, getting smaller as the algorithm progresses.
The algorithm is also usually modified by inclusion of a momentum term: this
encourages movement in a fixed direction, so that if several steps are taken in
the same direction, the algorithm “picks up speed”, which gives it the ability to
(sometimes) escape local minimum, and also to move rapidly over flat spots and
plateaus.
The algorithm progresses iteratively through a number of epochs. On each
epoch the training cases are each submitted in turn to the network, the target
and actual outputs are compared and the error is calculated. This error, together
with the error surface gradient, is used to adjust the weights and then the process
repeats. The initial network configuration is random and training stops when a
given number of epochs elapses, or when the error reaches an acceptable level, or
when the error stops improving (to be selected by the user).
The Levenberg-Marquardt algorithm is typically the fastest of the training algorithms.
However, it has some important limitations such as: it can only be used
on single output networks, it can only be used with the sum squared error function
and it has memory requirements proportional to W 2 (where W is the number

D.3. Neural Networks 141
of weights in the network). This makes the Levenberg-Marquardt algorithm impractical
for reasonably big networks. Conjugate gradient descent algorithms are
nearly as good and do not suffer from these restrictions.
The second-order training algorithms seem to be prone to stick in local minima
in the early phases. To overcome this problem, one could start with a short
burst using the back propagation algorithm, before switching to a second-order
algorithm.

142 Appendix D. Soft Computing Techniques

E
Genetic Algorithms
Genetic algorithms are search algorithms that imitate the principles of natural
evolution for optimization problems. They combine survival of the fittest
among a population of potential solutions with a structured yet randomized
information exchange.
This appendix is organized as follows: A brief introduction into genetic algorithms
is given in Section E.1. In Section E.2 the basic principles on how genetic algorithms
work are described on the basis of an example. The theoretical foundation and
application areas of genetic algorithms are discussed in Sections E.3 and E.4,
respectively.
E.1 Introduction
Genetic Algorithms (GAs) are randomized optimization algorithms that are based
on the principle of survival of the fittest. The population of potential solutions
is mani- pulated repeatedly to form a new generation. Each potential solution is
defined in a string structure, referred to as a chromosome. A new generation is
formed by deleting weak chromosomes from the population, while using strong
chromosomes to produce new chromosomes through genetic operators. A more detailed
description of how GAs work is offered in Section E.2.
Genetic algorithms are especially useful for non-convex optimization problems
with large search spaces. For convex optimization problems the use of a gradientbased
optimization algorithm is more efficient. For non-convex optimization problems
with small search spaces classical exhaustive search methods usually suffice.
For larger search spaces a more intelligent search technique must be employed,
such as a GA.
E.2 How Do They Work?
Genetic algorithms can be divided into two main groups, namely binary-coded
and real-coded GAs. The first group expresses the value of the parameters to be
143

144 Appendix E. Genetic Algorithms
Figure E.1: Flowchart of a genetic algorithm.
optimized in a binary code, while the latter group expresses them by their true
value. In principle these two implementations are equivalent, however, it has been
shown that the real-coded genetic algorithm is faster, more consistent from run to
run and provides a higher accuracy (Goldberg 1990, Michalewicz 1996).
In this section, the basic elements that are present in all genetic algorithms are
discussed, see also Figure E.1. The starting point of the genetic algorithm is an
initial population of chromosomes. Each chromosome is evaluated with respect to
the fitness function. The selection procedure randomly selects two sets of chromosomes,
namely a set of chromosomes to be removed from the population and a
set of chromosomes to be used to create new chromosomes. Weaker chromosomes
have a higher probability to be removed from the population than stronger chromosomes
and stronger chromosomes have a higher probability to be used to create
new chromosomes than weaker chromosomes (survival of the fittest). New chromosomes
are created using genetic operators (crossover and mutation operators).
The fitness of these new chromosomes is evaluated and the selection procedure
starts again. The algorithm stops when a certain termination criterion is met.
In the remainder of this section, the genetic algorithm that is used in this
thesis is explained with a simple example. For this specific genetic algorithm there
are five parameters that need to be set by the designer, namely:
N number of generations
Nc number of chromosomes in the population
Nop number of genetic operations per generation
percentage of crossover operations
pco

E.2. How Do They Work? 145
y
0
−20
−40
−60
−80
1
0.5
0
x 2
−0.5
−1
−1
Figure E.2: The function y =(x1 − 1) 5 +(x2 − 1) 5 over the domain x1 ∈ [−1, 1]
and x2 ∈ [−1, 1].
md
differentiation parameter.
For this example, these parameters are set to be: N = 100, Nc = 20, Nop = 10,
pco =0.80 and md =4.
−0.5
E.2.1 Description of the optimization problem
The goal is to find the Takagi-Sugeno model that fits the surface illustrated in
Figure E.2. The data set that is used for optimizing the TS model is obtained
by evaluating y = f(x1,x2) at a number of grid points. The grid is defined by
x1 =[−1, −0.98,...,0.98, 1] and x2 =[−1, −0.98,...,0.98, 1].
In this example the structure of the TS fuzzy model is fixed. There are two
fuzzy membership functions for each antecedent variable and four rules. The corresponding
rule-base is as follows:
R1 : If x1 is Z11 and x2 is Z21 then ˆy = c01 + c11x1 + c21x2
R2 : If x1 is Z11 and x2 is Z22 then ˆy = c02 + c12x1 + c22x2 (E.1)
R3 : If x1 is Z12 and x2 is Z21 then ˆy = c03 + c13x1 + c23x2
R4 : If x1 is Z12 and x2 is z22 then ˆy = c04 + c14x1 + c24x2.
For each antecedent variable, the two membership functions are defined by two
parameters, see Figure E.3. In total this TS fuzzy model requires 16 parameters,
namely four for the membership functions in x1 and x2 and 12 for the local linear
models. Each chromosome in the genetic algorithm consists of a string of four realvalued
numbers for the membership functions. Once the membership functions are
defined, the 12 parameters of the local linear models are determined through least
squares optimization.
A single chromosome is denoted as vt i , where the subscript i denotes the
x 1
0
0.5
1

146 Appendix E. Genetic Algorithms
Membership degree
1
0.5
Z 11
Z 12
0
−1 −0.5 d 0 d 0.5 1
1 x 2
1
Figure E.3: The membership functions Z11 and Z12, defined by the two parameters
d1 and d2.
position of the chromosome in the population and the superscript t denotes the
generation. A single element of this chromosome is denoted by vt ij , where the subscript
j denotes the position of the element in the chromosome vt i . The constraints
) ∈ [−1, 1].
for the parameters of the ith chromosome are (v t i1 ,...,vt i4
E.2.2 Initial population
The initial population is the result of an initialization process. In this case the
initial population is selected randomly, taking into account the constraints, and
consists of 20 chromosomes:
v 0 1
v 0 2
.
v 0 19
v 0 20
=
=
=
=
−0.0462 −0.4774 −0.9921 −0.1692
0.5846 0.4012 0.6715 0.0017
0.6643 0.0037 −0.5748 0.2749
−0.2857 0.0560 −0.4451 −0.5821 .
E.2.3 Evaluation of the fitness of the chromosomes
Each chromosome represents two sets of membership functions, namely for x1
and for x2. To evaluate the fitness of the ith chromosome, a TS fuzzy model is
created using the corresponding membership functions. The parameters of the
consequent part of the rule-base are computed through LS optimization (see also
Equation E.1). For each data point the output ˆyj of the corresponding TS fuzzy
model is evaluated. The corresponding fitness value is the root mean-squared error
over the K data points:
F (v t i)=
K
j=1 (yj − ˆyj) 2
, (E.2)
K
where y and ˆy are the true output and the output of the TS fuzzy model using
the parameters of chromosome vt i , respectively.
Each chromosome of the initial population is evaluated using the fitness func-

E.2. How Do They Work? 147
tion. In this case this results in:
F (v 0 1) = 1.5121
F (v 0 2)
.
= 1.5534
F (v 0 19) = 4.4841
F (v 0 20) = 5.0600.
The chromosomes are sorted according to their fitness. Since this is a minimization
problem, the stronger the chromosome the lower its fitness value.
E.2.4 Selection procedure
Two sets of chromosomes need to be selected. The first set consists of the chromosomes
that will be used for reproduction and the second set consists of the
chromosomes that will be replaced by the new chromosomes. The basic principle
is that chromosomes with a strong fitness value have a higher probability to be
selected for reproduction than chromosomes with a weak fitness value. For the
selection of chromosomes for deletion this is the other way around.
The selection procedure is as follows:
1. Divide the fitness value of the strongest chromosome by the fitness value of
all the chromosomes in the population and raise the result to the power md.
Pop(i) =
t F (v1) F (vt i )
md , i =1,...,Nc. (E.3)
2. Normalize the result, such that the sum of the vector is equal to one.
Pop(i) :=
Pop(i)
Nc
j=1 Pop(j) , i =1,...,Nc. (E.4)
3. The vector for deletion is the flipped vector for genetic operation, i.e. reproduction.
Pdel(i) =Pop(Nc +1− i), i =1,...,Nc. (E.5)
4. Compute the cumulative sum for both vectors.
Pop(i) :=
Pdel(i) :=
i
Pop(j), i =1,...,Nc (E.6)
j=1
i
Pdel(j), i =1,...,Nc. (E.7)
j=1

148 Appendix E. Genetic Algorithms
Figure E.4: Roulette wheel.
The higher the value for md, the more the probability is differentiated, i.e. weak
chromosomes are less likely to be selected for operation. The vectors Pop and Pdel
are shown below:
Pop(1) = 0.1385 Pdel(1) = 0.0011
Pop(2)
.
= 0.2629 Pdel(2)
.
= 0.0029
Pop(19) = 0.9989 Pdel(19) = 0.8615
Pop(20) = 1 Pdel(20) = 1.
These two vectors are used to select the chromosomes for operation and deletion
respectively. This process is explained using the vector for operation:
1. Generate a random number r ∼ U[0, 1].
2. If r ≤ Pop(1),
then select the first chromosome (v t 1)
else select the ith chromosome such that Pop(i − 1)

E.2. How Do They Work? 149
E.2.5 Genetic operators
Two kinds of genetic operators are used, namely crossover and mutation. The
probability that a crossover operator is selected is determined by the parameter
pco. The probability that a mutation operator is selected is equal to 1 − pco. For
each genetic operation, three versions are used. When a chromosome is selected
for crossover (or mutation) one of the used crossover (or mutation) operators are
applied with equal probability.
Crossover operators
For crossover operations, the chromosomes are treated in pairs (v t i , vt j ).
1. Simple arithmetic crossover.
vt i and vt j arecrossedoveratthekth position, where k is a random integer
between 1 and m. The resulting offsprings are:
v t+1
i = v t i,1,...,v t i,k|v t j,k+1,...,v t
j,m
(E.8)
v t+1
j = v t j,1,...,v t j,k|v t i,k+1,...,v t
i,m . (E.9)
2. Whole arithmetic crossover.
A linear combination of vt i and vt j resulting in:
where the parameter r ∼ U[0, 1].
3. Heuristic crossover.
vt i and vt j are combined such that:
v t+1
i = r v t i +(1− r) v t j (E.10)
v t+1
j = (1− r) v t i + r v t j, (E.11)
v t+1
i = v t i + r1 (v t i − v t j) (E.12)
v t+1
j = v t j + r2 (v t j − v t i), (E.13)
where the parameters r1 ∼ U[0, 1] and r2 ∼ U[0, 1].
Mutation operators
For mutation operations, single chromosomes are selected.
1. Uniform mutation.
A randomly selected element vt i,k , k ∈{1, 2,...,m} is replaced by v′ k ,which
is a random number in the range vmin i,k ,vmax
i,k . The resulting chromosome
is:
v t+1
i =(vt i,1,...,v t i,k−1,v ′ k,v t i,k+1,...,v t i,m). (E.14)

150 Appendix E. Genetic Algorithms
Root Mean−Squared Error
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0 20 40 60 80 100
Generation
Figure E.5: Root mean-squared error as function of the generation.
2. Multiple uniform mutation.
Uniform mutation of n elements in the chromosome vt i , where n is a randomly
selected integer between 1 and m. The position of each of the n elements in
the chromosome is again determined by a randomly selected integer between
1andm.
3. Gaussian mutation.
All elements of a chromosome are mutated such that:
v t+1
i =(v′ 1,...,v ′ k,...,v ′ m), (E.15)
where v ′ k = vk + fk, k =1, 2,...,m. Here fk is a random number drawn
from a Gaussian distribution with zero mean and an adaptive variance
(T − t) (v
t
max
k
− vmin
k )
σk = , (E.16)
3
where T denotes the predefined maximum number of generations. The parameter
tuning performed by this operator becomes finer as the generation
counter t increases.
For an overview of a wide variety of genetic algorithm implementations, selection
procedures, genetic operators, etc., the reader is referred to (Bäck et al. 2000a,Bäck
et al. 2000b).
In this example, it is chosen to let the parameters defining the membership functions
be subject of the GA optimization. The parameters of the local linear models
are determined through LS optimization (once the membership functions are defined).
One could for example also choose to optimize both the parameters defining
the membership functions and the parameters of the local linear models using the

E.3. Theoretical Foundation 151
Membership degree
Membership degree
1
0.5
Z 11
Z 12
0
−1 −0.5 0
x
1
0.5 1
1
0.5
Z 21
Z 22
0
−1 −0.5 0
x
2
0.5 1
Figure E.6: The optimal membership functions Z11, Z12, Z21 and Z22.
GA. However, this increases the number of parameters to be optimized by the
GA from 4 to 16 and overcomplicates the optimization process by introducing
additional degrees of freedom.
E.2.6 Result
Figure E.5 illustrates the convergence of the GA towards the optimal result. The
optimal fitness value is equal to root mean-squared error of RMSE = 0.6479. Due
to the small number of parameters to be optimized, the optimal value is reached
in the 56th generation. The optimal membership functions are illustrated in Figure
E.6 and the corresponding rule-base is as follows:
R1 : If x1 is Z11 and x2 is Z21 then ˆy = 4.30 + 32.9x1 +32.9x2
R2 : If x1 is Z11 and x2 is Z22 then ˆy = 1.02 + 32.9x1 +1.63x2
R3 : If x1 is Z12 and x2 is Z21 then ˆy = 1.02 + 1.63x1 +32.9x2
R4 : If x1 is Z12 and x2 is Z22 then ˆy = −2.26 + 1.63x1 +1.63x2.
The output of the optimal TS fuzzy model is illustrated in Figure E.7a, while the
modelling error with respect to the original data (see Figure E.2) is illustrated in
Figure E.7b.
E.3 Theoretical Foundation
Although GAs are widely applied, there is a profound lack of hard theory accompanying
the empirical results provided by GA researchers (White and Flockton 1995).
The description of GAs as schema processing algorithms was first set out
by Holland (1975) and forms the foundation of many of the theoretical results.
Other methods based on Walsh functions (Goldberg and Rudnick 1990, Forrest

152 Appendix E. Genetic Algorithms
y
0
−50
−100
1
0
x 2
−1
(a) Output of the optimal TS fuzzy model.
−1
x 1
0
1
y
2
0
−2
−4
1
0
x 2
−1
(b) Modelling error of the optimal TS fuzzy
model.
Figure E.7: Output (left) and modelling error (right) of the optimal TS fuzzy model.
See Figure E.2 for the original data.
and Mitchell 1992, Field 1994), Markov chains (Goldberg and Segrest 1987, Davis
and Principe 1993), statistical mechanics (Prügel-Bennett and Shapiro 1994) and
simulated annealing-like convergence analysis (Goldberg 1987) have since been developed
and have attempted, in their turn, to provide a quantitative explanation
of the dynamics of the GA.
For the class of Elitist GAs it is proven that it converges to the optimum of
the fitness function (Jong 1975, Suzuki 1995). However, despite of the vast amount
of effort spent, it cannot be proven for other (simple) GAs (Lozano et al. 1999).
Other proofs of convergence involve simulated annealing-like genetic algorithms.
E.4 Application Areas
Genetic algorithms are widely applied in planning (routing, scheduling, packing),
design, simulation and identification, control and classification (Bäck et al. 2000a).
Optimization problems in these areas are typically constrained, non-convex
optimization problems that involve both continuous as well as binary or integer
parameters, i.e. hybrid optimization problems. Unlike gradient-based optimization
techniques, GAs are well suited for these kind of optimization problems. The main
disadvantage of GAs is the computational effort needed to solve an optimization
problem.
−1
x 1
0
1

F
Linear Matrix Inequalities for
Control
In this appendix it is briefly explained how the robust control problem is defined
through linear matrix inequalities. More specifically, the output-feedback
robust control problem is considered.
This appendix is organized as follows: The output-feedback H∞ control problem
is explained in Section F.1. In Section F.2 it is described how the output-feedback
H∞ control problem is transformed into a Linear Matrix Inequality (LMI). How
to transform constraints on the location of the controller poles into LMIs is briefly
discussed in Section F.3.
F.1 Output-feedback H∞ Control Problem
Suppose a linear time-invariant open-loop system is described by:
⎡ ⎤ ⎡
˙x A
⎣z⎦
= ⎣ C1
B1
D11
B2
D12
⎤ ⎡ ⎤
x
⎦ ⎣w⎦
,
y
u
C2 D21 D22
with state x, exogenous inputs w, control inputs u, controlled outputs z and measured
outputs y. The assumptions on the plant parameters are:
1. (A, B2,C2) is stabilizable and detectable.
2. D22 =0.
The first assumption is necessary and sufficient to allow for plant stabilization by
dynamic output feedback. The second assumption is considerably simplifying the
equations, while it incurs no loss of generality.
The output-feedback controller is a finite dimensional linear time-invariant system
described as:
153

154 Appendix F. Linear Matrix Inequalities for Control
Figure F.1: Robust control design problem.
˙xK
=
u
AK BK
CK DK
xK
,
y
where xK is the state of the controller. The closed-loop system therefore becomes
(see also Figure F.1):
⎡
˙ξ
= ⎣
z
=
A + BDKC BCK B1 + BDKF
BKC AK BKF
C1 + EDKC ECK D1 + EDKF
A B
C D
ξ
w
,
⎤
⎦
ξ
w
where ξ = T x xK . The suboptimal H∞ control problem of parameter γ consists
of finding a controller K such that (Gahinet and Apkarian 1994):
1. The closed-loop system is internally stable.
2. The H∞ norm of Tzw(s) =D + C(sI −A) −1 B (the maximum gain from w
to z) is strictly less than γ, i.e.
F.2 LMI Approach
||Tzw||∞

F.2. LMI Approach 155
to be solved. These nonlinearities can be eliminated by an appropriate change of
controller variables. This change of variables was introduced in (Gahinet 1996) and
is implicitly defined in terms of the (unknown) Lyapunov matrix X. Specifically,
partition X and its inverse as:
X =
R M
M T U
, X −1 =
The new controller variables are defined as:
BK := NBK + SB2DK
S N
N T
. (F.3)
V
CK := CKM T + DKC2R (F.4)
AK := NAKM T + NBKC2R + SB2CKM T + S(A + B2DKC2)R.
The identity XX −1 = I together with Equation F.3 gives:
MN T = I − RS. (F.5)
Thus, M and N have full row rank when I − RS is invertible. The invertibility of
I − RS can be assumed without loss of generality, see Lemma 4.2 in Chilali and
Gahinet (1996).
Theorem F.1
The output-feedback H∞ control problem is solvable if and only if the following
system of LMIs is feasible:
R
I
I
> 0
S
⎡
AR + B2
⎢
⎣
(F.6)
ĈK +(∗)
(B1 + B2DKD21)
∗ ∗ ∗
T ÂK +(A + B2DKC2)
−γI ∗ ∗
T SB1 + ˆ BKD21 SA + ˆ C1R + D12
BKC2 +(∗) ∗
ĈK D11 + D12DKD21 C1 + D12DKC2
⎤
⎥
⎦ < 0,
−γI
(F.7)
where “ ∗ ” denotes the complex-conjugate transpose. See (Chilali and Gahinet
1996) for the proof of this theorem.
Given any solution of this LMI system:
• Compute via singular value decomposition a full-rank factorization MN T =
I − RS of the matrix I − RS (M and N are then square and invertible).
• Solve the system of linear Equations F.4 for BK, CK and AK (in this order)
• Set K(s) :=DK + CK(sI − AK) −1 BK.
Then K(s) isannth order controller such that ||Tzw(s)||∞

156 Appendix F. Linear Matrix Inequalities for Control
Figure F.2: Region S(α, r, θ). Source: (Chilali and Apkarian, 1996).
The LMI formulation of constrained H∞ optimization is appealing from a practical
standpoint. LMIs can be solved by efficient interior-point optimization algorithms
such as those described in for example (Nesterov and Nemirovskii 1993).
F.3 Pole Placement
The transient response of a linear system is related to the location of its poles.
By constraining the poles to lie in a prescribed region, a satisfactory transient
response can be ensured. In addition, fast controller dynamics can be prevented
by prohibiting large closed-loop poles, which is often desirable for digital implementation.
One way of simultaneously tuning the H∞ performance and transient
behavior is therefore to combine the H∞ and pole placement objectives.
Regions of interest include α-stability regions Re(s) ≤ −α, vertical strips,
disks and conic sectors. Another interesting region for control purposes is the set
S(α, r, θ) of complex numbers x + jy such that:
x

F.3. Pole Placement 157
matrix α =[αkl] ∈ R m×m and a matrix β =[βkl] ∈ R m×m such that:
with
D = {z ∈ C : fD(z) < 0} (F.9)
fD(z) :=α + zβ + zβ =[αkl + βklz + βlkz] 1≤k,l≤m . (F.10)
Note that the characteristic function fD takes values in the space of m × m Hermitian
matrices and that “< 0” stands for negative definite.
In other words, an LMI region is a subset of the complex plane that is representable
by an LMI in z and z, or equivalently, an LMI in x = Re(z) andy = Im(z). As a
result, LMI regions are convex. Moreover, LMI regions are symmetric with respect
to the real axis since for any z ∈D, fD(z) =fD(z) < 0. For more details, the
reader is referred to (Chilali and Gahinet 1996).

158 Appendix F. Linear Matrix Inequalities for Control

Acknowledgements
In the first place I would like to thank Henk Verbruggen and Gerard Schram for
giving me the opportunity to work for the European project named “Affordable
Fly-By-Wire Flight Control Systems for Small Commercial Aircraft” (ADFCS).
Their preparation work and supervision during the first year made it possible for
me to get started immediately and to quickly catch up with the project. I would
also like to thank Robert Babuˇska, who became my supervisor after one year and
later also my promotor. He gave me the freedom to find my own way, but he was
always available for a technical discussion or to provide me with whatever I needed
to make my (professional) life easier. I also acknowledge the help of the supporting
staff of the department.
I would like to thank all the people who worked for ADFCS for their numerous
constructive comments and suggestions and for the useful discussions. I valued the
personal contact and the atmosphere during the meetings and the flight simulator
sessions, both from a professional as well as personal point of view.
During my stay in Örebro, Sweden, in the beginning of 2002 I have worked on
scheduled robust multivariable control together with Dimiter Driankov and Pontus
Bergsten. The cooperation and discussions greatly improved my understanding on
the subject and I would like to thank them for that.
Furthermore, I would like to acknowledge the European Union and Marie Curie
Foundation for their financial support.
Regarding the non-scientific aspects, I am grateful for the social activities at
the department, mainly organized by De Biercommissie and friends. I have enjoyed
the many (at random) bierages and all the other activities that have taken place.
Because of them it was possible for me to find a good balance between working
and having fun.
I take this opportunity to thank my parents, brothers and other friends for
their social support. Although they have not been directly involved in my life as a
scientist, they are important to me as a person. In particular I would like to thank
Daniela for her love and for withstanding the tension and frustration that haunted
the house from time to time during the writing of this thesis.
Finally I would like to thank my paranimphen Domenico Bellomo and Hans
Roubos for the pleasant working environment when we shared an office and for
assisting me during the defence of this thesis.
Marcel Oosterom Rotterdam, May 2005
159

160 Acknowledgements

Curriculum Vitae
Marcel Oosterom was born on March 1, 1974 in Deurne, The Netherlands. In June
1998 he received his M.Sc. degree from the Delft University of Technology, Faculty
of Aeronautical Engineering, Department of Control & Simulation. The thesis
was entitled “Flight control during final approach in windshear - The MBPC approach”.
From September 1998 he worked almost six years for the European project
“Affordable Digital Fly-By-Wire Flight Control Systems for Small Commercial
Aircraft” on intelligent flight control at the Control Laboratory of the Faculty of
Electrical Engineering (now part of the Delft Center for Systems and Control).
During this period he spent three months at the Örebro University, Center for
Applied Autonomous Sensor Systems, Sweden.
Since February 2005 he is employed at Huisman-Itrec, Schiedam, The Netherlands,
working in the field of control system engineering.
161

162 Curriculum Vitae

Chapters in books
List of Publications
1. Schram G., M. Oosterom, and H.B. Verbruggen, “Fuzzy Modeling and Control
in Avionics”. In “Fuzzy Logic Control - Advances in Applications”, Verbruggen
and Babuˇska (Eds.), World Scientific Series in Robotics and Intelligent
Systems, Vol. 23, pp. 275-291, 1999.
2. Babuˇska R. and M. Oosterom, “Fuzzy Clustering for Multiple-Model Approaches
in System Identification and Control”. In “Granular Computing -
An Emerging Paradigm”, Physica-Verlag, pp. 306-323, 2001.
Scientific journal publications
3. Oosterom M., R. Babuˇska and H.B. Verbruggen, “Soft Computing Applications
in Aircraft Sensor Management and Flight Control Law Reconfiguration”.
IEEE Transactions on Systems, Man and Cybernetics - Part C:
Applications and Reviews, Vol. 32, No. 2, May 2002.
4. Oosterom M. and R. Babuˇska, “Design of a Gain-Scheduling Mechanism for
Flight Control Laws using Fuzzy Clustering”. Accepted for publication in
the Control Engineering Practice Journal.
5. Oosterom M. and R. Babuˇska, “Virtual Angle-of-Attack Sensor”. Submitted
to the Journal of Aircraft.
6. Oosterom M. and R. Babuˇska, “Scheduled Robust Multivariable Control”.
Submitted to the Control Engineering Practice Journal.
Conference publications
7. Oosterom M., G. Schram, H.B. Verbruggen and R. Babuˇska, “Automated
Procedure for Gain Scheduled Flight Control Law Design”. In: Proceedings
of the AIAA Guidance Navigation and Control Conference, AIAA-2000-4253,
Denver, CO, USA, 2000.
8. Oosterom M. and R. Babuˇska, “Virtual Sensor for Fault Detection and Isolation
in Flight Control Systems - Fuzzy Modeling Approach”. In: Proceedings
of the IEEE Conference on Decision and Control, Sydney, Australia, pp.
2645-2650, 2000.
163

164 Publications
9. Oosterom M. and R. Babuˇska, “Fuzzy Logic Applications in Aircraft Sensor
Management Systems”. In: Proceedings of the 40th Israel Annual Conference
on Aerospace Sciences, Tel Aviv/Haifa, Israel, pp. 160-167, 2001.
10. Oosterom M. and R. Babuˇska, “Aircraft Sensor Management and Flight
Control Law Reconfiguration - Fuzzy Logic Approach”. In: Proceedings of
the AIAA Guidance Navigation and Control Conference, AIAA-2001-4358,
Montreal, Canada, 2001.
11. Oosterom M. and R. Babuˇska, “Fuzzy Gain Scheduling for Flight Control
Laws”. In: Proceedings of the FUZZ’IEEE Conference, Melbourne, Australia,
pp. 716-719, 2001.
12. Abonji J., J.A. Roubos, M. Oosterom and F. Szeifert, “Compact TS-Fuzzy
Models through Clustering and OLS plus FIS Model Reduction”. In: Proceedings
of the FUZZ’IEEE Conference, Melbourne, Australia, pp. 1-4, 2001.
13. Oosterom M. and R. Babuˇska, “Fuzzy Gain-Scheduled H∞ Flight Control
Law Design”. In: Proceedings of the AIAA Guidance Navigation and Control
Conference, AIAA-2002-4847, Monterey, CA, USA, 2002.
14. Babuˇska R. and M. Oosterom, “Design of Optimal Membership Functions
for Fuzzy Gain-Scheduled Control”. In: Proceedings of the FUZZ’IEEE Conference,
St. Louis, MO, pp. 476-481, 2003.

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