Introduction Overview of GKYP Lemma-Based Integrated Servo ...

*Corresponding author: Chee Khiang Pang
2012 ASME-ISPS /JSME-IIP Joint International Conference on
Micromechatronics for Information and Precision Equipment
MIPE2012
June 18-20, 2012, Santa Clara, California, USA
INTEGRATED SERVO-MECHANICAL DESIGN OF HIGH PERFORMANCE
MECHANICAL SYSTEMS USING GENERALIZED KYP LEMMA
Yan Zhi Tan 1 , Chee Khiang Pang 2 *, Fan Hong 3 , Tong Heng Lee 1,2
1 NUS Graduate School for Integrative Sciences and Engineering, Singapore 117456
E-mail: yanzhi@nus.edu.sg, eleleeth@nus.edu.sg
2 Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576
E-mail: justinpang@nus.edu.sg
3 Data Storage Institute, A*STAR, 5 Engineering Drive 1 (Off Kent Ridge Crescent, NUS), Singapore 117608
E-mail: HONG_Fan@dsi.a-star.edu.sg
Introduction
Traditional R&D process for servo-mechanical design is a
cyclical process that begins with the mechanical structure
design, followed by the manufacturing and evaluation of
the prototype before the prototype is passed to control
engineers for servo system design and analysis [3]. The
two main problems associated with the traditional servomechanical
design approach are first, the overall servomechanical
system is not necessarily optimal as a result
of designing the plant and the controller independently.
Second, the performance specifications might not be met
by the same controller for different batches of the same
mechanical system. As such, it is generally difficult to
satisfy the demanding performance specifications of a
high-performance mechatronic control system solely
through the design of an inexpensive and simple low
order controller [4].
In [1], an integrated system design by separation
approach is proposed. Rather than optimizing the plant
and the controller according to some objectives, this
approach focuses primarily on designing a “good” plant
with a finite frequency positive real (FFPR) inputdifferentiated
output transfer characteristic by using the
Generalized Kalman-Yakubovic-Popov (GKYP) Lemma.
In this paper, a novel GKYP Lemma-based integrated
servo-mechanical design algorithm is proposed for
frequency domain design of a “good” plant according to a
pre-designed low order controller and the performance
specifications required of the overall servo-mechanical
system. The proposed algorithm does not impose the
FFPR condition and is carried out in discrete-time to
prevent deterioration in performance with discretization
of a continuous-time system.
The following notations are used. I denotes an identity
matrix of appropriate dimensions. For a matrix M, its
376
transpose and complex conjugate transpose are denoted
by M T and M * , respectively, where M T and M * are
matrices of appropriate dimensions.
Overview of GKYP Lemma-Based Integrated
Servo-Mechanical Design
A typical closed-loop configuration to regulate the output
y of plant P(z) subjected to output disturbance w is shown
in Fig. 1. C(z) represents the controller and u represents
the plant’s input. The transfer function from w to z
corresponds to the output sensitivity function.
The GKYP Lemma-based algorithm for plant design with
a pre-designed controller makes use of the sensitivity
function because disturbance attenuation capabilities can
be considered and well-known time domain determinants
of the quality of a step response can be easily transformed
into frequency domain specifications of the output
w
z Nominal plant P0 *
Plant P(z)
+
y u
+
C*(z)
C(z)
y
+ -
0
v
C*(z) Estimator
K
Q(z)
Fig.1: Output regulation configuration with Youla
parameterization for plant design.
e
ˆx
C c
D c
+ +
û
+
-

sensitivity function [5]. The following lemma shows the
application of the GKYP Lemma to the loop shaping of
the discrete-time sensitivity function [2].
Lemma 1: Given sensitivity function S(z) = CS(zI-AS) -1 BS
+DS with AS being stable. |S(e j )| r over a finite
frequency range for a given scalar r > 0 if
and only if there exist Hermitian matrices P and V 0
such that
* *
A B 0 0
S S A B S S C S
*
I 0
I 0
0 I
D 0,
S
C D S S
I
where = -r 2 , = -1, and
(1)
P V
*
V P2cosRV .
(2)
Applying Schur’s complement to (1), = -r and = -r,
and r becomes a variable to be optimized. In (2), the
values of scalars R and depend on the frequency range
considered and are shown in Table 1.
Table 1: R and for Different Frequency Range
l
h
l h
R l ( )/ 2
h l
h
1 j ( l h)/2
e
However, the plant design parameters are to be restricted
to CS and DS of (1). To overcome this, Youla
parameterization is required. The Youla parameterization
for plant design is shown in Fig. 1. Its details will be
explained along with the presentation of the steps for the
proposed integrated servo-mechanical design algorithm in
the next section.
Design Algorithm
Step 1: Design a low order controller C(z) with state
space matrices denoted as (Ac, Bc, Cc, Dc) to compensate
for the desired low frequency characteristics of the plant.
Step 2: Compute the gain matrix L of C*(z) Estimator
and feedback gain matrix K using MATLAB commands
given by L = Acdlqe(Ac, Bc, Cc, I, I) and K = dlqr(Ac, Bc,
Cc T Cc, I), respectively. C*(z) Estimator is an estimate of
C*(z) with w set to zero.
Step 3: Formulate the equations given by
-1
377
x A B
Kx B B w
k1c c k c c k
,
xˆ
LC A
xˆ
LD B
v
k1 c E k c c k (3)
z 0 1 1
k Kxk
wk ,
e
C C
xˆ
D 0
v
k c c k c k (4)
where AE = Ac – BcK - LCc, and xk refers to the value of x
at discrete time k. (3-4) denote the state space
representation for the combination of C*(z) with the
nominal plant P0*(z), and the inclusion of e and v.
Step 4: Design Q(z) as an m th order Finite Impulse
Response (FIR) filter given by
0 A q
0 I 0
m1
, B ,
q
0
1
(5)
C q q ... q , D q , (6)
q m m1 1 q 0
where q is a vector of design parameters. Set q0 to zero so
that the designed plant will not have direct feedthrough.
Step 5: Let Twz represent the transfer function from w to z
and obtain the state space matrices of the sensitivity
function S = Twz + TvzQTwe given by
Awz
A 0 S
0
0
At BC q t
0 Bwz
0
, B B ,
S t
(7)
A BD q q t
CS Cwz D C q t C , q
DS D D
D , (8)
wz q t
where (Awz, Bwz, Cwz, Dwz) and (At, Bt, Ct, Dt) denote the
state space matrices of Twz and TvzTwe , respectively.
Step 6: Specify i constraints on the sensitivity function
using i LMIs in the form of (1) to obtain values for the
vector q.
Step 7: Obtain state space matrices of the plant given by
A
p
A LD K B C LDC
E c c q c q
BDKBCABDC ,
(9)
q c q c q q c q
T
B , , .
p L B C K
q C D D
p q (10)
p q
Simulation Example
In this section, the proposed integrated servo-mechanical
design algorithm is applied to the design of a highperformance
mechatronic system.
Lead (Pb) Zirconate (Zr) Titanate (Ti) is a piezoelectric
material commonly known as PZT. The frequency
response of a PZT microactuator from a commercial 3.5”

Magnitude (dB)
Phase (deg)
Magnitude (dB)
80
60
40
20
0
-20
180
90
0
-90
10 1
-180
10
0
-10
-20
-30
-40
10 1
-50
10 2
10 2
Frequency (Hz)
10 3
10 3
Frequency (Hz)
Original Plant
Nominal Plant
Designed Plant
Lag Compensator
Open Loop
Fig.2: Bode plots of original plant, nominal plant,
designed plant, lag compensator and open loop.
Closed-loop w ith Nominal Plant
Closed-loop w ith Designed Plant
Fig.3: Comparison of sensitivity functions.
dual-stage Hard Disk Drive (HDD) is measured and
modeled as having two main high frequency resonant
modes at 4.70 kHz and 13.5 kHz. A discrete-time first
order lag compensator with a sampling frequency Fs of 40
kHz is designed to produce an open loop bandwidth of 2
kHz with the original plant. The Bode plots of the original
plant model and the lag compensator are shown in Fig. 2.
As the closed-loop system is unstable due to the 4.7 kHz
resonant mode, the proposed algorithm is used to redesign
the frequency response characteristics of the PZT
microactuator. m is chosen to be 6, and the constraints are
a) |S(f1)| -22 dB, 4.68 kHz f1 4.72 kHz,
b) |S(f2)| -3 dB, 12.8 kHz f2 14.2 kHz,
c) |S(f3)| r3, f3 2.6 kHz,
d) |S(f4)| r4, 9.00 kHz f4 9.20 kHz, and
e) |S(f5)| r5, 17.0 kHz f5 20.0 kHz,
10 4
10 4
378
where fi = iFs/2, and r3,4,5 are variables to be minimized.
Constraints (a-b) shape the plant’s resonant modes so that
noise and disturbances in those regions are attenuated by
the closed-loop system, constraint (c) changes the closedloop
bandwidth, and constraints (d-e) limit the sensitivity
function’s gain beyond its 0 dB cross-over frequency.
The sensitivity function of the closed-loop system with
the designed plant is shown in Fig. 3. It is shaped from
the sensitivity function of the closed-loop system with the
nominal plant according to the constraints specified.
The Bode plots of the first-order nominal plant and
seventh-order designed plant are shown in Fig. 2. A high
order plant is not an issue unlike a high order controller.
The first resonant mode of the designed plant at 4.70 kHz
has a phase of 3.46º while the second resonant mode at
13.5 kHz has a phase of -363.67º. The Bode plot of the
resulting open loop is also shown in Fig. 2. The open loop
bandwidth is 1.5 kHz and its phase margin is 94º. The
drop in bandwidth is due to zeros in that region. Finally,
the resonant modes are phase stabilized [4] by the lag
compensator despite being out-of-phase.
Conclusion
In this paper, a GKYP Lemma-based integrated servomechanical
design algorithm has been proposed. The
algorithm yields a plant design that can meet the
performance specifications of the overall servomechanical
system with a low order controller through
loop shaping of the sensitivity function. Our future work
includes the consideration of plant parameter pertubations
in the design algorithm. With such a modification, a set of
plant designs that can meet the required closed-loop
performance specifications with a low order controller
will be obtained.
References
[1] T. Iwasaki, 1999, “Integrated System Design by
Separation,” in Proceedings of the IEEE International
Conference on Control Applications, 97-102.
[2] T. Iwasaki and S. Hara, 2005, “Generalized KYP
Lemma: Unified Frequency Domain Inequalities with
Design Applications,” IEEE Trans. on Automatic
Control, 50:41-59.
[3] C. K. Pang, F. L. Lewis, T. H. Lee, and Z. Y. Dong,
2011, Intelligent Diagnosis and Prognosis of Industrial
Networked Systems, CRC Press, Taylor and Francis
Group, Boca Raton, FL, USA.
[4] T. Yamaguchi, M. Hirata, and C. K. Pang (eds.),
2011, High-Speed Precision Motion Control. CRC press,
Taylor and Francis Group, Boca Raton, FL, USA.
[5] K. Zhou, J. C. Doyle, 1996, Essentials of Robust
Control. Prentice Hall, New Jersey, USA.