ANALYSIS AND CONTROL OF LINEAR
PARAMETER-VARYING SYSTEMS
a dissertation
submitted to the department of aeronautics & astronautics
and the committee on graduate studies
of stanford university
in partial fulfillment of the requirements
for the degree of
doctor of philosophy
By
Sungyung Lim
September 1998
c 1998 by Sungyung Lim
Copyright All Rights Reserved.
ii
I certify that I have read this dissertation and that in my opinion it is fully adequate,
in scope and quality, as a dissertation for the degree of Doctor of Philosophy.
Jonathan P. How
Department of Aeronautics and Astronautics
(Principal Adviser)
I certify that I have read this dissertation and that in my opinion it is fully adequate,
in scope and quality, as a dissertation for the degree of Doctor of Philosophy.
Stephen P. Boyd
Department of Electrical Engineering
I certify that I have read this dissertation and that in my opinion it is fully adequate,
in scope and quality, as a dissertation for the degree of Doctor of Philosophy.
Stephen M. Rock
Department of Aeronautics and Astronautics
I certify that I have read this dissertation and that in my opinion it is fully adequate,
in scope and quality, as a dissertation for the degree of Doctor of Philosophy.
Andrew Packard
Department of Mechanical Engineering
University of California, Berkeley
Approved for the University Committee on Graduate Studies:
iii
To my wife, Yunhee and my daughter, Ellen
iv
Abstract
The area of analysis and control of linear parameter-varying (LPV) systems has received
much recent attention because of its importance in developing systematic techniques for
gain-scheduling. An LPV system resembles a linear system that nonlinearly depends on
one or more time-varying parameters. Nonlinear systems are often modeled in the LPV
system via the parameterized Jacobian linearization.
Typical approaches for the analysis and control of LPV systems are the scaled smallgain approach and the dissipative systems framework using smooth parameter-dependent
Lyapunov functions (PDLFs). The dissipative systems framework is the more desirable of
the two techniques because it can directly treat time-varying parameters and yield an LPVtype controller. Furthermore, the dissipative systems framework attractively formulates
analysis and synthesis problems as convex optimization problems involving linear matrix
inequalities (LMIs), which are now very eciently solved by computer. However, the current
dissipative systems framework has two major potential drawbacks: (1) diculty in selecting
an optimal PDLF in order to reduce conservatism of the dissipative systems approach; (2)
diculty in solving exactly convex optimization problems involving an innite number of
LMIs.
The thesis presents new analysis and control design techniques to avoid these potential drawbacks of the smooth dissipative systems framework. The thesis focuses on a
piecewise-ane parameter-dependent linear parameter-varying (PALPV) system which is
a new class of LPV systems. Associated with the PALPV system is a piecewise-ane
parameter-dependent Lyapunov function (PAL). To address the non-dierential nature of
both the PALPV system and the PAL, the thesis develops a nonsmooth dissipative systems
framework. Then, the thesis fully characterizes several interesting analysis and synthesis problems, such as L2 -gain, L1-gain, H2 -norm, passivity, and robust counterparts, of
PALPV system with the developed nonsmooth dissipative systems framework.
v
The new approach is shown to yield a less conservative, reliable result than previously
published LPV approaches. The improvement is a direct result of (1) using a more accurate
model in the analysis and control, and (2) using a very general class of PDLFs. The
derived analysis and synthesis formulations are also nite-dimensional convex optimization
problems which can be solved extremely eciently by computer. Furthermore, the new
approach enables us a trade-o between conservatism and computational eort of the design
technique.
Several benchmark problems including a missile autopilot design problem are used to
demonstrate the usefulness, reliability, and feasibility of the proposed new approach.
vi
Acknowledgments
Thanks to all.
vii
Contents
Abstract
v
Acknowledgments
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List of Tables
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List of Figures
xii
List of Symbols
xiv
List of Acronyms
xvi
1 Introduction
1
1.1 Previous Research : : : : : : : :
1.1.1 Linearization Methods :
1.1.2 Analysis : : : : : : : : :
1.1.3 Synthesis : : : : : : : : :
1.2 Thesis Objectives and Overview
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2 Mathematical Preliminary
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2.1 Norms and Normed Spaces of Signals : : : :
2.2 Stability : : : : : : : : : : : : : : : : : : : :
2.2.1 Lyapunov Stability : : : : : : : : : :
2.2.2 Input-Output Stability : : : : : : : :
2.3 Small-Gain Theorem and Passivity Theorem
2.4 Lebesgue Integral Theorem : : : : : : : : : :
2.5 Ordinary Dierential Equations : : : : : : :
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3
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2.6 Linear Matrix Inequality : : : : : : : : : : : : :
2.6.1 Linear Matrix Inequality Problems : : :
2.6.2 Numerical Algorithms for LMI Problems
2.6.3 Miscellaneous Results on LMIs : : : : : :
2.7 \Convexifying" Techniques : : : : : : : : : : : :
2.8 Bilinear Matrix Inequality : : : : : : : : : : : :
3 Dissipative Systems Framework
3.1 Denition of Dissipative Systems : : : : : : : : :
3.2 \Dini-Dierential" Dissipation Inequality : : : :
3.3 \Dissipation" implies : : : : : : : : : : : : : : :
3.3.1 Lyapunov Stability : : : : : : : : : : : :
3.3.2 L2 -Gain : : : : : : : : : : : : : : : : : :
3.3.3 L1 -Gain : : : : : : : : : : : : : : : : : :
3.3.4 H2 -Norm : : : : : : : : : : : : : : : : : :
3.3.5 Passivity : : : : : : : : : : : : : : : : : :
3.4 Dissipation of Feedback Systems : : : : : : : : :
3.5 System with Structured Dynamics Uncertainties
4 Analysis
4.1 PALPV System : : : : : : : : : : :
4.2 PAL : : : : : : : : : : : : : : : : : :
4.3 Analysis Formulations : : : : : : : :
4.3.1 Lyapunov Stability : : : : :
4.3.2 L2 -Gain : : : : : : : : : : :
4.3.3 L1 -Gain : : : : : : : : : : :
4.3.4 H2 -Norm : : : : : : : : : : :
4.3.5 Passivity : : : : : : : : : : :
4.3.6 Robust L2 -Gain and Others
4.4 Generalized Results : : : : : : : : :
4.4.1 PALPV system : : : : : : :
4.4.2 PAL with s 3 : : : : : : :
4.4.3 Analysis formulations : : : :
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5 Synthesis
5.1 QPALPV System : : : : : :
5.1.1 PALPV System : : :
5.1.2 LPV Controller : : :
5.2 QPAL : : : : : : : : : : : : :
5.3 Synthesis Formulations : : :
5.3.1 L2 -Gain : : : : : : :
5.3.2 L1 -Gain : : : : : : :
5.3.3 H2 -Norm : : : : : : :
5.3.4 Passivity : : : : : : :
5.3.5 Robust L2 -Gain : : :
5.4 Generalized Results : : : : :
5.4.1 QPALPV system : :
5.4.2 QPAL with s 3 : :
5.4.3 Synthesis formulations
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6 Numerical Studies
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6.1 Stability Margin and L2 -Gain Problems : : : : :
6.2 L2 -Gain Synthesis : : : : : : : : : : : : : : : : :
6.3 Autopilot Design : : : : : : : : : : : : : : : : : :
6.3.1 Missile Model and Performance Objective
6.3.2 PALPV Modeling : : : : : : : : : : : : :
6.3.3 Comparison of LPV control techniques :
6.3.4 Autopilot Design and Simulations : : : :
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7 Conclusions
168
Bibliography
173
7.1 Summary : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 168
7.2 Conclusions and Contributions : : : : : : : : : : : : : : : : : : : : : : : : : 170
7.3 Recommendations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 171
x
List of Tables
6.1
6.2
6.3
6.4
Relative RMS of Ek (M; ) in (%) : : : : : : : : : : : : : : : : : : : : : : :
Features of LPV control techniques : : : : : : : : : : : : : : : : : : : : : :
Maximum eigenvalues from the synthesis and the post-analysis : : : : : : :
Characteristics of design techniques and their results. Note that the NGS
technique does not provide any guaranteed L2 -gain. : : : : : : : : : : : :
xi
153
156
157
159
List of Figures
2.1
3.1
3.2
4.1
4.2
5.1
6.1
6.2
6.3
6.4
6.5
6.6
Feedback system : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
Feedback system : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
Dynamic system with two uncertainty blocks in a feedback loop : : : : : :
Partitioned parameter subspaces (s = 2) with m1 = m2 = 2 : : : : : : : :
Example of a continuous, piecewise-ane P () with s = 2. For simplicity,
the parameter space is only split into 2 2 regions. : : : : : : : : : : : : :
Example of a continuous, piecewise-ane X () with s = 2. Y () is also
similarly dened with Yij 's. For simplicity, each parameter space is only
split into 2 regions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
max vs. _max from the gridding technique, multi-convexity, S -procedure,
and our PAL with a number of dierent partitions (N ). The multi-convexity
approach is equivalent to our PAL with N = 1. : : : : : : : : : : : : : : :
P () from our PAL with N = 5 at dierent values of _max's. : : : : : : : :
L2-gain vs. _max from the gridding technique, multi-convexity, S -procedure
and our PAL with a number of dierent partitions (N ). The multi-convexity
approach is equivalent to our PAL with N = 1. : : : : : : : : : : : : : :
Block diagram of a benchmark problem : : : : : : : : : : : : : : : : : : :
L2-gain vs. _max from the gridding technique, S -procedure, multi-convexity
approach and our QPAL. The multi-convexity approach is equivalent to our
QPAL with N = 1. PCS means the Popov controller for _max = 0. The dotline indicates the minimum L2 -gain that can be obtained by the pointwise
H1 control for _max = 0. : : : : : : : : : : : : : : : : : : : : : : : : : : :
L2-gain vs. _max from the same synthesis techniques as in Fig. 6.5 except
that Y () is constrained to be constant. All the labels are same as Fig. 6.5.
xii
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146
6.7
6.8
6.9
6.10
6.11
6.12
6.13
6.14
6.15
6.16
6.17
6.18
6.19
H1-norm vs. of the closed-loop system for _max = 0, the controller of which
is designed by pointwise-H1 , PCS, and QPAL approach with N = f1; 2; 3g,
respectively. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
Weighted open-loop interconnection of the missile plant : : : : : : : : : :
f2(M; ) and its approximation error (E2 (M; )) with a number of dierent
partitions (N = 1; 3; 5) : : : : : : : : : : : : : : : : : : : : : : : : : : : :
L2-gain v.s. N (number of partitions) from GRID, GRID1, QPAL and
QPAL1. Black 2 indicates the result is veried by the post-analysis, while
white means the result is not. : : : : : : : : : : : : : : : : : : : : : : : :
Mach number prole for Case I : : : : : : : : : : : : : : : : : : : : : : : :
Normal acceleration (t) from NGS, C-, GRID, and QPAL for Case I :
Mach number prole for Case II without noise : : : : : : : : : : : : : : :
Normal acceleration (t) from C-, GRID, and QPAL for Case II without
noise : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
Mach number prole for Case II with and without noise : : : : : : : : : :
Normal acceleration (t) from the QPAL approach for Case II with and
without noise : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
Response of tail deection rate (_ (t)) from the QPAL approach for Case II
with v(t) = 0 and v(t) 2 [;0:05; 0:05] : : : : : : : : : : : : : : : : : : : : :
Response of tail deection rate (_ (t)) from the QPAL approach for Case II
with v(t) 2 [;0:15; 0:15] and v(t) 2 [;0:25; 0:25] : : : : : : : : : : : : : :
Angle-of-attack () and its estimated value (~) from the QPAL approach
for Case II with v(t) = 0 and v(t) 2 [;0:25; 0:25] : : : : : : : : : : : : : :
xiii
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163
164
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167
List of Symbols
R
R+
Rn
I
D
Dcl
W
U
Z
Y
Fs
C1
K
kxk
kxkp
h; i
B(c)
set of real numbers
set of nonnegative numbers
set of real n-vectors
[t0 ; 1)
set of states
set of closed-loop system states
set of performance inputs
set of control inputs
set of performance outputs
set
n of outputs
o
2 C 1(I ; Rs ) : (t) 2 P ; _(t) 2 ; 8t 2 I ; where
P = [1; 1] [s; s] and = [;1 ; 1] [;s; s]
class of continuously dierentiable functions
class of continuous, strictly increasing functions with f (0) = 0
"X
n
Zi=1
jxij2
1
t0
#1=2
and the corresponding normed space l2
kx(t)kp dt
1=p
and the corresponding normed space Lp
inner product
fx 2 D : kx ; ck g
I
identity matrix
diag(X1 ; ; Xn ) diagni=1 (Xi ) i.e., block-diagonal matrix with X1 ; ; Xn
X?
orthogonal complement of X , i.e., X T X? = 0 and
[X X? ] is of maximum rank
xiv
TrX
XT
X ;1
X>0
X ( ) > 0
Co[X1 ; ; Xr ]
trace of X
transpose of a matrix X
inverse of a matrix X
positive denite, i.e., X = X T and xT Xx > 0 8x 2 Rn
positive denite for all 2 P
convex hull of the set fX1 ; ; Xr g
uncertainty block
S
set of uncertainties
1 [V (x(t + h); (t + h)) ; V (x(t); (t))]
D+V (x(t); (t)) lim sup
h!0+ h
1 [V (x + hu; ) ; V (x; )]
D+V (x; )(u; 0) lim sup
h!0+ h
1 [V (x + hu; ) ; V (x; )]
D+V (x; )(u; 0) lim
inf
+
h!0 h
Ex
The expected value of the random variable x
l
ii is
1(s)
1| {z 1} with 1(0) = ;
[A()]ij
2
8
2
s
Aij (ij )
\belongs to"
\for all"
end of the proof
xv
List of Acronyms
ALF
ALPV
BMI
C-
GRID
IQC
LDI
LFT
LMI
LPV
LTI
MIMO
NGS
PAL
PALPV
PDLF
PJL
QAL
QLPV
QPAL
QPALPV
Ane parameter-dependent Lyapunov Function
Ane parameter-dependent Linear Parameter-Varying
Bilinear Matrix Inequality
Complex synthesis
Gridding approach in the dissipative systems framework
Integral Quadratic Constraint
Linear Dierential Inclusion
Linear Fraction Transformation
Linear Matrix Inequality
Linear Parameter-Varying
Linear Time-Invariant
Multi-Input Multi-Output
Naive Gain-Scheduling
Piecewise Ane parameter-dependent Lyapunov function
Piecewise Ane parameter-dependent Linear Parameter-Varying
Parameter-Dependent Lyapunov Function
Parameterized Jacobian Linearization
Quasi-Ane parameter-dependent Lyapunov function
Quasi Linear Parameter-Varying
Quasi-Piecewise-Ane parameter-dependent Lyapunov function
Quasi-Piecewise-Ane parameter-dependent Linear Parameter-Varying
xvi
Chapter 1
Introduction
All physical systems are virtually nonlinear and time-varying in nature. Examples of nonlinearities we are often confronted with range from simple nonlinearity, such as saturation,
rate limiters, and backlash, to the inherently nonlinear behavior of physical systems such
as robotic manipulators, aircraft, and chemical process plants. Nevertheless, it is often
possible to describe the operation of the physical system by a linear system, under the circumstances that the real operation of the physical system does not deviate too much from
the \nominal" operating (or equilibrium) condition. Therefore, the analysis and synthesis
of linear systems have occupied an important place in systems theory. Consequently, many
(computationally tractable) analysis and synthesis techniques have been developed.
We often encounter situations where the linearized model around a nominal operating
condition is inadequate or inaccurate. For example, the missile dynamics involves a wide
range of variation in system dynamics over the operation range so that it can not be represented by a linearized model. In this case, a linear controller from linear systems theory
may perform well on the linearized model but may not even be stable when implemented
on the real physical system. To play with \inaccuracies," linear systems theory has been
extended to linear robust systems theory that takes into account these inherent inaccuracies
as uncertainties and then provides systematic analysis and design techniques in the face of
these \uncertainties." However, introducing a uncertainty in the design process leads to
degradation in the performance of a designed controller (Note that the performance degradation is often alternatively expressed by conservatism in the literature). Therefore, one
1
CHAPTER 1. INTRODUCTION
2
of the important issues in linear robust systems theory is how to eciently play with uncertainties, i.e., to reduce conservatism of analysis and synthesis techniques exploiting the
nature of uncertainties, such as \structured," \real" and \time-invariant."
The \inaccuracy" can also be reduced via sophisticated linearization methods, such as
the parameterized Jacobian linearization method (PJL). PJL linearizes a nonlinear physical
system around parameterized operating conditions rather than a single operating condition.
The control of the linearized model from PJL is traditionally done by the gain-scheduling
technique. The gain-scheduling technique designs linear controllers at several operating
conditions and then interpolates these designed controllers along the user-dened parameter trajectory. The gain scheduling method does enjoy widespread usage in a variety of
applications, such as aircraft control, missile autopilot, jet-engine control, and process control. However, it remains an ad hoc methodology because robustness, performance, or even
nominal stability of a gain-scheduling controller are not addressed explicitly in the design
process. Rather, such properties are inferred from extensive simulations.
The demand for systematic, theoretically rigorous techniques for the gain-scheduling
method has stimulated a great deal of research on linear parameter-varying (LPV) systems,
where system matrices are matrix functions of time-varying parameters. Associated with
LPV systems are roughly two main analysis and design approaches: linear robust systems
theory and dissipative systems theory using various parameter-dependent Lyapunov functions (PDLFs). The dissipative systems framework is the more desirable of the two techniques because it can directly treat real time-varying parameters and also yield an LPV-type
controller. Furthermore, the dissipative systems framework attractively formulates analysis
and synthesis problems as convex optimization problems involving linear matrix inequalities
(LMIs) which are now very eciently solved by computer. However, two major important
issues still remain unsolved: (1) diculty in selecting an optimal PDLF in order to reduce
conservatism of the dissipative systems approach; (2) diculty in solving exactly convex optimization problems involving an innite number of LMIs when using the PDLF. Note that
these issues imply a zero-sum game between conservatism and computational complexity of
the dissipative systems framework.
In this thesis, we investigate a new approach to avoid these potential issues associated
with the dissipative systems framework. In other words, the devised new approach should
automatically select an optimal PDLF during the analysis and synthesis optimization process and then yield nite-dimensional convex optimization problems. Furthermore, the new
CHAPTER 1. INTRODUCTION
3
approach should enable us an explicit trade-o between conservatism and computational
complexity of the dissipative systems framework.
1.1 Previous Research
This section is intended to review some of the key approaches most relevant to this research
in order to place the present work in context. This section mainly reviews some linearization
methods for a nonlinear system, linear robust systems theory and LPV systems theory.
We should clarify the dierence between a parametric uncertainty in linear robust systems theory and a parameter in LPV systems theory. Both are assumed to be unknown but
constrained a priori to lie in some known, bounded real set. However, the parameter is even
further assumed to be measurable in real time. Therefore, both are not distinguishable in
the analysis, while they are distinguishable in the synthesis. In other words, we can exploit
a parameter-dependent controller in the LPV framework.
1.1.1 Linearization Methods
Some direct linearization methods for nonlinear systems could be roughly categorized into
three types: (I) linearization about an equilibrium, (II) linearization about a (parameterized) state trajectory, and (III) global linearization. Method I is used to represent a
nonlinear system with a linear time-invariant (LTI) system around an equilibrium condition [Vid92]. The system representation in Method I is the simplest of three dierent
approaches and so are the related analysis and synthesis techniques. However, this approach is limited to characterizing only the local properties of a nonlinear system around
an equilibrium condition [Vid92].
Method II is used where the nonlinear system follows prescribed trajectories from repeated maneuvers and the outcome of some trajectory optimization [BH75]. Method II is
also used where the nonlinear system can be approximated by a family of linearizations or
the parameterized linearization [Rug91, SA90]. In particular, this case motivates an LPV
system representation which will be intensively studied here. Since the model from Method
II is valid around a state trajectory rather than a single equilibrium, Method II can represent
a nonlinear system in a wider range of operating conditions than Method I.
The last approach is used to represent a nonlinear system with a set of linear timevarying (LTV) systems [Liu68, Lsl69], which is often called linear dierential inclusions
CHAPTER 1. INTRODUCTION
4
(LDIs) in the literature [BGFB94]. Since Method III approximates the set of trajectories of
a nonlinear system with trajectories of LDIs, it can represent a nonlinear system in the entire
operating range. However, Method III can be very conservative because there may be many
trajectories of the LDI that are not actual trajectories of the nonlinear system [BGFB94].
Method II lies between Method I and Method III in every category of comparison. The
main dierence between Method II and Method III is whether or not a nonlinear system
can be parameterized by index or parameters or whether a nonlinear system is represented
by an LTI system with parametric uncertainties or an LPV system.
Note that the above linearization methods are a direct way to derive a linear system,
while the feedback linearization method [Isi89] in nonlinear controls is an indirect way in
the sense that a nonlinear system is linearized by a feedback loop for further analysis.
1.1.2 Analysis
Analysis techniques for LPV systems fall roughly into one of two categories: the scaled smallgain framework and the dissipative systems framework. These approaches overlap with
counterparts for uncertain systems because an LPV system can be treated as a uncertain
system with parametric uncertainties. Note that as shown in the literature [How93,Vid92],
these two approaches for uncertain systems are closely related to each other.
The small-gain framework originates from the work done by Zames [Zam66], which
provided an exact robust stability test for an LTI system with unstructured dynamic uncertainty. This approach led to the use of singular values as an important tool in robust
control [DFT92]. However, the small-gain framework provides only sucient conditions for
an LTI system with structured parametric or dynamic uncertainty and thus may be very
conservative. Hereby, the small-gain framework have been modied to exploit the structure
and type of a uncertainty. The most remarkable results are the structured singular value
() [Doy82] and the multivariable stability margin (Km ) [Saf82] for linear fractional transformation (LFT) systems [Red60]. However, these quantitative measures are very dicult
to calculate exactly, so that they are often estimated by both the (computationally feasible)
upper and lower bounds. Therefore, one of the main issues associated with these approaches
is how to derive a tight upper and lower bound. The problem of deriving a tight upper
bound is focused on in particular, because an upper bound condition is often convex and
provides a sucient condition for robust analysis.
CHAPTER 1. INTRODUCTION
5
A number of researchers have considerably derived tighter upper and lower bounds
with scaling techniques [FTD91, Hel95, PD93, Pag96] and developing ecient computation
schemes for these bounds [BDGPS91,Hel95,Yo93]. The scaling matrix for the upper bound
is originally frequency-dependent for static parametric uncertainties [FTD91,Hel95] or LTI
uncertainties [PD93]. The scaling matrix is constrained to be constant for quickly varying parametric uncertainties or structured nonlinear (dynamic) uncertainties [AG95,Hel95,
Dan96,Pac94,SG,Sha94]. Furthermore, the constant scaling is extended to the Popov multiplier which eectively takes into account the \real" uncertainty. However, the constant
scaling matrix can be very conservative for the analysis of LPV systems with slowly varying
parameters.
In a parallel approach, tight upper bounds have been also derived by the passivity
framework [BHPD94, Hel95] and integral quadratic constraints (IQC) approach [Jon96,
Meg93] using various multipliers. As the IQC approach describes the uncertainty in terms
of integral quadratic constraints, it generalizes the standard passivity approach [DV75].
One of the attractive properties of the IQC approach is its ability to build a new multiplier
as a convex combination of known basic multipliers. These multiplier-based approaches can
analyze less conservatively the uncertain system with slowly varying uncertainties using the
swapping lemma [Mor80]. An explicit result is shown in [Jon96] using the Popov multiplier,
which is the simplest type of multiplier. However, further study is required to investigate
the desired type of multiplier which leads to a tight upper bound. Furthermore, the use of a
general (frequency-dependent) multiplier makes the synthesis problem very computationally
dicult.
The dissipative systems framework was originally developed to dene \dissipativeness"
(a generalized concept of passivity) in terms of an inequality involving the storage function (or Lyapunov function)and the supply rate [Wil72]. This framework has been extended to formulate many sucient conditions for performance analysis problems of LTI
systems [BGFB94, Iwa93, SGC97], LTV systems [HM76], and nonlinear systems [Scha96].
One of the attractive characteristics of the dissipative systems framework is its ability to
formulate analysis problems as convex optimization problems involving linear matrix inequalities (LMIs) [BGFB94], which are now very eciently solved by computer.
The appearance of LMIs in the control community started with the Lyapunov stability
theory. The important role of LMIs in control theory was already recognized in the early
1960's, especially by Yakubovich [Yak67]. Since the late 1980's, the LMI approach in control
CHAPTER 1. INTRODUCTION
6
theory have been revived because of the development of computationally ecient interiorpoint algorithms for LMIs [NG94,NN94,VB96]. Several software packages now exist which
allow users to represent LMI problems with a high-level language and to interface with
MATLAB (LMILAB [GNLC95] and SDPSOL [WB96] are examples of this software).
In the dissipative systems framework, a supply rate is uniquely related to a performance
analysis problem. Therefore, one of the important issues associated with the dissipative
systems framework is to determine for a given uncertain system an optimal Lyapunov
function which leads to a less conservative analysis result (Note that the dissipative systems framework often provides only sucient conditions). So far, various types of Lyapunov functions have been proposed, from the typical (smooth) quadratic Lyapunov function [BGFB94, Iwa93, SGC97] to nonsmooth Lyapunov functions [AC84, SS95]. However,
one of these Lyapunov functions is selected such that the Lyapunov function should have
the same uncertainty-dependence as the uncertain system. A famous example is the LurePostnikov Lyapunov function for the analysis of Lure systems [NT73]. This key idea has
been also applied to the selection of parameter-dependent Lyapunov functions (PDLFs) for
the analysis of LPV systems [AA97,AWU97,Beck95,FAG95,GAC96,YS95]. For example, associated with an ane parameter-dependent LPV system is an ane parameter-dependent
Lyapunov function.
The use of a PDLF allows the dissipative systems framework to directly analyze an LPV
system with slowly varying parameters [Wu95]. Hereby, using the PDLF implies that the
dissipative systems framework provides a less conservative analysis tool for the analysis of
LPV systems than the scaled small-gain framework. However, using the PDLF introduces
two major potential issues: diculty in selecting an optimal PDLF vs. given time variation
of parameters and solving exactly an innite number of LMIs. The rst issue can be
addressed by an example. Consider an ane LPV system, where the parameter (p) has
broad range of time variation, p_ 2 [0; 1]. Based on the result of the Lure system [NT73], an
ane PDLF is associated with the ane LPV system with p_ = 0. However, a (parameterindependent) quadratic Lyapunov function is associated with the ane LPV system with
p_ = 1 (See [GAC96]). Thus, the current rule for selecting PDLFs may not be eective
when applied to an LPV system with time-varying parameters. This potential problem
spurs a new study on the selection of an optimal PDLF.
The second issue is one of the main diculties in the analysis of uncertain and LPV
systems. Due to the nature of these systems, the analysis technique should play with the
CHAPTER 1. INTRODUCTION
7
(convex) compact set of uncertainties and parameters. Therefore, the derived formulations actually represent convex problems involving an innite number of LMIs. As these
systems are constrained to a couple of special cases, such as LDIs or polynomial (parameterdependent) LPV systems, the number of LMIs can, however, be reduced to a nite number
of LMIs using \convexity" [AA97, AWU97, FAG95, GAC96, YS95]. Note that associated
with these systems are the PDLFs with the same parameter-dependency. However, LPV
systems and PDLFs that are more general than these special cases are desired to develop
a less conservative, guaranteed analysis technique for a broad class of LPV systems. Unfortunately, the corresponding analysis formulations are an innite number of LMIs, so the
guaranteed result cannot be found by computer. In this case, nite basis methods and
heuristic gridding techniques [Wu95] are typically used. However, these approaches do not
guarantee the analysis result and furthermore need further study on the selection of the
best basis functions.
The dissipative systems approach using a nonsmooth Lyapunov function can be found
in the analysis problems for the switched, hybrid and other nonlinear controls [Bran94,
HelZ97, MBA96, JR96, RJ97, WP94]. The stability analysis of switched or hybrid systems [Bran94, MBA96, WP94] is attempted by a piecewise continuous Lyapunov function.
While this Lyapunov function reduces conservatism of the stability analysis, it is limited
to the stability analysis problem. As using continuous, piecewise quadratic Lyapunov functions, LMI formulations have been developed for other performance analysis problems of
the switched system [JR96,RJ97]. However, these Lyapunov functions may be conservative
for the analysis of LPV systems because they cannot directly treat time-varying parameters
in LPV systems. Furthermore, extensions of these approaches to performance synthesis
problems are not often straightforward.
1.1.3 Synthesis
The synthesis is a more complicated problem than the analysis: while an analysis problem nds only the optimal scaling matrix or PDLF, a synthesis problem nds an optimal
scaling matrix or PDLF and unknown controllers. Thus, it involves more technical steps
and assumptions than the analysis. However, most synthesis techniques stem from the
analysis techniques so synthesis techniques can be categorized as analysis techniques: the
scaled small-gain framework and the dissipative systems framework. Therefore, most of the
discussions and references made in the analysis are still eective here.
CHAPTER 1. INTRODUCTION
8
It should be mentioned that the LPV control is obviously dierent from the robust
control. The dierence lies in the assumption that while a parametric uncertainty in an
uncertain system is not measurable, a parameter in an LPV system is measurable in real
time. Therefore, the LPV control can design an LPV-type controller rather than an LTI
controller.
Before we discuss synthesis techniques for LPV systems, we briey address the typical gain-scheduling approach. The gain-scheduling technique is based on the matured
linear systems theory and user-dened scheduling scheme [AW89]. In detail, local linear
controllers are designed for linearized models of a nonlinear system at several dierent
operating conditions. A global nonlinear controller for the nonlinear system is then obtained by interpolating, or scheduling, the gains of the local linear controllers. Due to
simplicity, the gain-scheduling approach has been successfully applied to many interesting
problems [AW89, NRR93, Ste80, WP94]. However, it remains an ad hoc methodology. For
example, the robustness, performance, or even nominal stability properties of the global
gain-scheduled controller are not addressed explicitly in the design process [SA92].
The lack of guaranteed properties stems from the fact that there is no theoretic tool to
verify the interpolation process. The theoretic lack for the interpolation has been compensated with a couple of useful guidelines, such as \the scheduling variable should capture the
plant's nonlinearities" and \the scheduling variable should vary slowly" [SA92]. However,
these guidelines may not be applicable when applied to advanced missile autopilot designs.
For example, some parameters, such as the angle-of-attack, are arbitrarily quickly varying.
Furthermore, these guidelines may lead to a very complicated scheduling scheme because
missile dynamics is a highly nonlinear MIMO system [NRR93]. A theoretic improvement
of gain-scheduling is obviously to include the scheduling process in the controller design
problem. This idea spurs the studies on LPV systems.
The scaled small-gain approach for the LPV control is a special case of the counterpart
for the robust control except that the scaling matrix in the LPV control is constrained to
be constant and a block full matrix associated with a block-repeated uncertainty structure.
The rst property is obvious because parameters of LPV systems are time-varying. Note
that the constant scaling matrix includes the Popov multiplier which eectively takes into
account the \real" parameter. The second property is due to the fact that the controller is
assumed to be the same as parameter-dependency as the LPV system, i.e., parameters and
their copies for the controller appear in the uncertainty block. This property contributes
CHAPTER 1. INTRODUCTION
9
LMI formulations for the synthesis [AG95, Hel95, Pac94, SG] rather than bilinear matrix
inequality (BMI) formulations [Hel95, PD93, Yo93], which are typical in the robust control
synthesis.
Due to the nature of the scaled small-gain approach, most researches have focused
on the (robust) L2 -gain synthesis problem. However, a systematic approach using full
block scaling, which is a simplied IQC approach with constant multiplier, was recently
devised [Sche96, SW96]. This approach can be applied to other performance synthesis
problems and also improve the results from the scaled small-gain approach because of the
richness of scaling matrices [Sche96, SW96]. Note that the integral quadratic constraint
is originally shown in the dissipative systems framework [BGFB94], so versatility of the
IQC approach is as much as the dissipative systems approach. The advantage of the scaled
small-gain approach makes full use of the matured scaled small-gain approach in the robust
control. In other words, any result in the robust control could be directly extended to the
LPV control. Furthermore, it could be even simplied due to the structure of the uncertainty
block. However, the benet is obviously limited by conservatism of the synthesis tool
because the constant scaling matrix cannot eciently account for the slowly time-varying
parameter.
The synthesis procedure of the dissipative systems framework originates from the standard three steps for the H1 synthesis of an LTI system [GA94]: (1) to derive the analysis
formulation for the closed-loop system; (2) to eliminate the unknown controller dynamics
from the analysis formulation and then solve the remaining formulation; (3) to construct the
unknown controller dynamics from the results of (2). This procedure has been rened and
applied to many other performance synthesis problems of an LTI system [Iwa93, SGC97].
Including the \convexifying" step at (2), this standard step has also been applied to the
L2-gain synthesis problem of an LPV system [AA98, BP94, KJS96, Wu95, Woo95, YS95].
However, the nature of the dissipative systems framework enables us to derive many other
performance synthesis problems for the LPV system with the same format of the L2 -gain
synthesis. As discussed in the analysis, the crucial factor associated with the dissipative
systems framework is selecting an optimal PDLF and \convexifying" an innite number of
LMIs.
Note that we observe results similar to the scaled small-gain approach. Consider the
parametric robust control for a Lur'e system subject to parametric uncertainties [Ban97,
CHAPTER 1. INTRODUCTION
10
FAG95]. The corresponding Lyapunov function, V = xTcl Pcl (p)xcl , is
2
3 0
2
31
X
(
p
)
Z
(
p
)
Y
(
p
)
5 @Pcl;1(p) = 4
5A ;
Pcl (p) = 4
Z (p)T E (p)
where X (p) is as same uncertainty dependency as the Lur'e system and Z (p) and E (p) are
constant. The reason for Z (p) and E (p) being constant is that the feedback controller should
be uncertainty-independent (Note that these Z (p) and E (p) are related to the controller
dynamics [Ban97]). As shown in [Ban97, FAG95], the parametric robust control problem
then yields a nonconvex optimization involving BMIs. Next, consider the LPV control for
an ane LPV system. Since the controller can be parameter-dependent, consider the PDLF
such that X (p) and Y (p) is as same parameter-dependency as the ane LPV system, i.e.,
ane in p [AA98]. In this case, the LPV control leads a convex optimization problem
involving LMIs of X (p) and Y (p). Furthermore, it yields a less conservative result than the
robust control because of the richness of PDLFs.
1.2 Thesis Objectives and Overview
The primary goal of this thesis is to develop analysis and synthesis tools to avoid two potential issues associated with the dissipative systems framework: (1) diculty in selecting an
optimal PDLF in order to reduce conservatism of the dissipative systems approach; (2) diculty in solving exactly convex optimization problems involving an innite number of LMIs.
In other words, the devised new approach should automatically select an optimal PDLF
during the analysis and synthesis optimization process and then yield nite-dimensional
convex optimization problems. The approach is also desired to provide an explicit trade-o
between conservatism and computational complexity of the design technique. Furthermore,
the approach can derive several performance analysis and synthesis formulations for LPV
systems without any diculty.
This thesis attempts to consider both a piecewise-ane parameter-dependent linear
parameter-varying (PALPV) system and a continuous, (quasi-) piecewise-ane parameterdependent Lyapunov function (PAL) for analysis and synthesis. From the survey of previous works, both an ane LPV system and ane PDLF turns out to be the simplest
pair that leads to a nite convex optimization problem. Therefore, a generalization for
reducing conservatism should be made without destroying the attractive property of \ane
CHAPTER 1. INTRODUCTION
11
parameter-dependency." It intuitively leads to the idea that \piecewise-ane parameterdependency" may provide one solution for the potential issues associated with the dissipative
systems framework (Similar approaches can be found in many other elds, such as hybrid
control [RJ97]). However, the concept of \piecewise-ane parameter-dependency" needs
a \nonsmooth" dissipative systems framework rather than the typical smooth dissipative
systems framework. As a result, this thesis focuses on the development of a nonsmooth
dissipative systems framework and applications on interesting performance analysis and
synthesis problems of PALPV systems.
The results of this thesis address the usefulness of the concept of \piecewise-ane
parameter-dependency." The new approach allows a trade-o between conservatism and
computational eort: tuning the number of piecewise terms, the new approach produces
various results ranging from the existing result (based on ane parameter-dependency) to
improved new results. As a supporting tool, the developed nonsmooth dissipative systems
framework turns out to be very similar to the current smooth dissipative systems framework. Furthermore, this nonsmooth dissipative systems framework contributes to deriving
many analysis and synthesis formulations for PALPV systems.
Chapter 2 of the thesis outlines some of the mathematical preliminaries for this work. It
presents the overview of several important topics, such as stability of systems, Lebesgue integral theorem, ordinary dierential equations (or inequalities) and linear matrix inequality.
In particular, the overview of Lebesgue integral theorem and ordinary dierential inequalities are presented because of the essential roles that they play in the later developments.
Through this chapter, some extensions of the existing mathematical results have been also
presented for the later developments.
Chapter 3 builds a nonsmooth dissipative systems framework using results. This new
framework is developed with the Lebesgue integral theorem. The derived results are extensions of the results for LTI systems [BGFB94,Iwa93,SGC97] to LPV systems. This framework is a simplication of general nonsmooth dissipative systems frameworks [AC84,SS95]
to eectively support our idea in mathematics. This chapter explicitly demonstrates that
many interesting analysis problems for LPV systems, such as Lyapunov stability, L2 -gain,
L1-gain, H2-norm, passivity and robust counterparts, can be formulated in terms of a
Lipschitz Lyapunov function and the supply rate.
Chapter 4 derives several interesting analysis formulations for PALPV systems using the
developed dissipative systems framework with the continuous, Lipschitz PAL. It includes
CHAPTER 1. INTRODUCTION
12
Lyapunov stability, L2 -gain, L1-gain, H2 -norm, passivity analysis problems and robust
counterparts for PALPV systems. A construction method for the PAL is also explicitly
presented. The promising property of the dissipative systems framework allows us to systematically derive these analysis problems. The derived formulations are nite-dimensional
convex optimization problems involving LMIs.
Chapter 5 continues the work of Chapter 4 to address the synthesis. The PAL is extended
to be a continuous, quasi-piecewise-ane parameter-dependent Lyapunov function (QPAL)
for the synthesis problem. This chapter then presents several control design problems of
PALPV systems. It includes L2 -gain, L1 -gain, H2 -norm, passivity and robust L2 -gain
synthesis problems for PALPV system. The derived formulations are nite-dimensional
convex optimization problems involving LMIs.
Chapter 6 demonstrates the eectiveness of the proposed approach in the analysis and
synthesis. Several benchmark problems are used to demonstrate conservatism of the analysis
and synthesis techniques. A realistic benchmark problem of a missile autopilot design is
also used to address some important issues: the impact of the model used in the controller
design on performance and reliability of the designed controller; and the richness and its
conservatism of PDLFs used for the LPV control.
Chapter 2
Mathematical Preliminary
This chapter is comprised of some basic denitions and elementary results in linear algebra, system theory, Lebesgue integral theorem, ordinary dierential equation and convex
optimization. While the treatment of this material is by no means exhaustive, it should be
sucient as a reference for this work.
2.1 Norms and Normed Spaces of Signals
We dene some standard norms and normed spaces for signals. Various norms for engineering problems are introduced in [BGFB94,DV75,Vid92] and references therein.
Denition 2.1 Norms on the linear vector space Rn are as follows:
n
X
kxk1 = jxij and the corresponding normed space l1.
i=1
kxkp =
"X
n
i=1
jxijp
#1=p
and the corresponding normed space lp .
kxk1 = 1max
jx j and the corresponding normed space l1.
in i
Note that for simplicity, we dene k k as the l2 -norm on Rn .
Denition 2.2 Let I = [t0 ; 1) and E = ff : I ! Rn j f locally Lebesgue integrable g.
Norms on appropriate subsets of E are as follows:
kxk1 =
Z1
t0
kx(t)kdt and the corresponding normed space L1(I ; Rn).
13
CHAPTER 2. MATHEMATICAL PRELIMINARY
kxkp =
Z 1
t0
1=p
kx(t)kp dt
14
and the corresponding normed space Lp (I ; Rn ).
kxk1 = ess sup kx(t)k and the corresponding normed space L1(I ; Rn).
t2I
Denition 2.3 Suppose f : I ! Rn. Then for each T 2 I , the function fT : I ! Rn is
dened by
8
< f (t) t0 t T
fT (t) = :
0
t>T
and is called the truncation of f to the interval [t0 ; T ].
Denition 2.4 The normed space Lpe consists of all Lebesgue integrable functions f :
I ! Rn with property that fT 2 Lp for all nite T , and is called the extension of Lp or the
extended Lp-space.
Denition 2.5 Let H : Lpe ! Lpe be causal (non-anticipative) if
(Hf )T = (HfT )T ; 8 T 2 I and f 2 Lpe:
Another popular norm is the inner product, denoted by < ; >.
Denition 2.6 Let E L2(I ; Rn). The inner product h; i : E E ! R+ is dened as
follow:
hx; yi =
Z1
t0
x(t)T y(t)dt for x; y 2 E
hx; xi = kxk22 for x 2 E
2.2 Stability
We briey discuss some denitions of stability: Lyapunov stability, input-output stability
and input-state stability. The rst two denitions [Kh96, Vid92] have been widely used in
the control community. However, the last denition has been recently recognized as a useful
tool for the Lyapunov nonlinear controls such as backstepping [Son89, KKK95]. We will
review only the rst two denitions here.
CHAPTER 2. MATHEMATICAL PRELIMINARY
15
2.2.1 Lyapunov Stability
We introduce uniform (asymptotic) stability and exponential stability of a nonlinear timevarying system. The \uniformity" is necessary to characterize time-varying systems whose
behavior has a certain consistency for dierent values of initial time t0 (Refer to [Kh96,
Vid92] for details).
We dene a nonlinear time-varying system:
x_ = f (x; t); x(t0 ) = x0 :
This system is assumed to have at least a solution x() on I . It is also assumed that the
origin x = 0 is an equilibrium point for this system, i.e.,
f (0; t) = 0; 8 t 2 I :
Note that if the equilibrium under study is not the origin, we can always redene the coordinates on Rn in such a way that the equilibrium of interest becomes the new origin [Vid92].
Denition 2.7 The equilibrium point x = 0 is locally uniformly stable if for any R > 0,
there exists a positive scalar r = r(R) such that
kx(t0 )k < r =) kx(t)k < R 8 t t0:
Denition 2.8 The equilibrium point x = 0 is locally uniformly asymptotically stable if for
any R0 > 0, there exist positive scalars R1 ; R2 and T (R1 ; R2 ) > 0 such that 0 < R2 < R1 <
R0 and 8 t 0,
kx(t0 )k < R1 =) kx(t)k < R2 8 t t0 + T (R1 ; R2 ):
Denition 2.9 The equilibrium point x = 0 is locally exponentially stable if there exist two
positive numbers, and , such that for suciently small x(t0 ),
kx(t)k kx(t0 )ke;(t;t0 ) 8 t t0:
CHAPTER 2. MATHEMATICAL PRELIMINARY
16
Note that exponential stability implies uniformly asymptotic stability but the converse
is not necessarily true.
2.2.2 Input-Output Stability
We briey state the basic denitions of input-output stability (Refer to [Kh96, Vid92] for
details).
Suppose H : X ! X is a mapping. H denes a binary relation R on X , i.e.,
R = f(x; Hx) : x 2 X g:
Denition 2.10 Suppose R is a binary relation on Lpe. Then R is said to be Lp-stable if
(x; y) 2 R; x 2 Lp =) y 2 Lp :
R is Lp-stable with nite gain if it is Lp -stable, and in addition there exist nite constant
p and p such that
(x; y) 2 R; x 2 Lp =) kykp p kxkp + p :
(2.1)
Furthermore, when p = 0, R is Lp -stable with nite gain and zero bias.
Denition 2.11 Suppose R is a binary relation on Lpe. If R is Lp-stable with nite gain,
then the Lp -gain of R is dened as p = inf fp : 9 p 0 such that Eq. 2.1 holds g:
Denition 2.12 Suppose H : Lpe ! Lpe. Then the map H is said to be Lp-stable if and
only if the corresponding binary relation R on Lpe is Lp-stable.
An alternative approach to the L2 -stability is the passivity approach using the inner
product.
Denition 2.13 Suppose H : L2e ! L2e. Then the map H is said to be passive if there
exists constant such that
hHx; xiT :
CHAPTER 2. MATHEMATICAL PRELIMINARY
v1
17
w1
H1
z1
z2
H2
w2
-
v2
Fig. 2.1: Feedback system
Furthermore, the map H is said to be strictly passive if there exists constant such that
hHx; xiT > :
(2.2)
where hHx; xiT = h(Hx)T ; xT i :
Note that there exist several dierent formulations on Eq. 2.2 [TGPS96].
Eq. 2.2 () h(Hx)T ; xT i kxT k22 + input strictly passive
() h(Hx)T ; xT i k(Hx)T k22 + output strictly passive
() h(Hx)T ; xT i (kxT k22 + k(Hx)T k22 ) + input/output strictly passive
2.3 Small-Gain Theorem and Passivity Theorem
The general frameworks to study input-output stability of complex systems such as a feedback connected system are the small-gain theorem and passivity theorem for the L2 -stability.
Theorem 2.1 [DV75] Consider the feedback system in Fig. 2.1. Let H1; H2 : Lpe !
Lpe; p 2 [1; 1], be causal and Lp-stable operators with nite gains 1; 2 and associated
constants 1 ; 2 . If
1 2 < 1;
then the feedback system is Lp-stable, i.e., w1T , w2T , z1T and z2T have bounded Lp-norm's
for v1 ; v2 2 Lpe.
CHAPTER 2. MATHEMATICAL PRELIMINARY
18
Theorem 2.2 [DV75] Consider the feedback system in Fig. 2.1. Suppose there exist constants i ; i ; i = 1; 2, such that
hHix; xiT ikxT k22 + i k(Hix)T k22 ; 8 T 0; 8 x 2 L2e; i = 1; 2:
Then the feedback system is L2 -stable with nite gain and zero bias if 1 +2 > 0; 2 +1 > 0:
Theorem 2.2 implies the following corollary.
Corollary 2.1 Consider the feedback system in Fig. 2.1. Then the feedback system is L2stable with nite gain and zero bias if
H1 and H2 are input strictly passive or
H1 and H2 are output strictly passive or
H1 is passive and H2 is input and output strictly passive or reversely.
2.4 Lebesgue Integral Theorem
The reconstruction of a function from its derivative plays an important role in the dissipative
systems framework discussed later. This problem, which shows the connection between
dierentiation and integration, is also a fundamental problem in real analysis [KF70].
Theorem 2.3 [KF70] Suppose that f : I ;! Rn is absolutely continuous on the interval
I ; that is, given any > 0, there is a > 0 such that
n
X
kf (bk ) ; f (ak )k < k=1
for every nite system of pairwise disjoint subintervals, (ak ; bk ) I , of total length
n
X
(bk ; ak ) < :
k=1
Then, the derivative f_ is summable or integrable on I and
f (t2) = f (t1 ) +
Z t2
t1
f_( )d
CHAPTER 2. MATHEMATICAL PRELIMINARY
19
for any t1 ; t2 2 I :
Note that f_ exists almost everywhere (f_ exists except for the measure zero set). Therefore, f_ is not dened on the measure zero set. However, a Lipschitz function has a nice
property that the integration can be constructed by the Dini-derivative which is well-dened
everywhere.
Corollary 2.2 If f : I ;! Rn is Lipschitz on the interval I ; that is, there exists a such
that for any t1 ; t2 2 I , kf (t2 ) ; f (t1 )k kt2 ; t1 k:
Then the Dini-derivative D+ f (t) exists everywhere on I and
Z t2
f (t2) = f (t1 ) +
t1
D+f ( )d
1
for any t1 ; t2 2 I : Here, D+ f (t) = lim sup [f (t + h) ; f (t)]:
h
Proof: The proof is based on the facts [KF70, RHL77]; the Dini-derivative is equal to the
derivative if the derivative exists; the Dini-derivative of a Lipschitz function f (t) is dened
everywhere on I ; and the behavior of f_(t) on the measure zero set does not aect the
integral. Since a Lipschitz function is also absolutely continuous, the following equation is
then immediately obtained:
h!0+
f (t2) = f (t1 ) +
Z t2
t1
f_( )d = f (t1) +
Z t2
t1
D+ f ( )d
for any t1 ; t2 2 I .
2
When f is a multi-variable function, the following chain-rule of the Dini-derivative is
useful. Note that this lemma is an extension of the result [AC84,Yos66].
Lemma 2.1 Let x 2 D C 1(I ; Rn ) and 2 F C 1 (I ; Rs); that is, x and are continuously dierentiable on the interval I . Suppose that a continuous f : DF ;! Rn satises
the following conditions:
f is continuous, Lipschitz on x for each xed ; that is, there is a such that
kf (x2 ; ) ; f (x1; )k kx2 ; x1k
for each 2 F and x1 ; x2 2 D.
CHAPTER 2. MATHEMATICAL PRELIMINARY
20
f is continuous, Lipschitz on for each xed x; that is, there is a such that
kf (x; 2 ) ; f (x; 1)k k2 ; 1 k
for each x 2 D and 1 ; 2 2 F .
Then
D+f (x; )(x;_ 0) + D+ f (x; )(0; _) D+f (x(t); (t)) D+f (x; )(x;_ 0) + D+f (x; )(0; _):
(2.3)
Here, the \partial" Dini-derivative is dened:
1 [f (x + hu; ) ; f (x; )];
D+ f (x; )(u; 0) = lim sup
h
+
h!0
1 [f (x + hu; ) ; f (x; )]:
D+ f (x; )(u; 0) = lim
inf
+
h!0 h
Proof: The proof is based on the result [AC84, Yos66] and the well known properties of
`lim sup' and `lim inf.' According to the property of `lim sup,' D+ f is bounded above:
1 [f (x(t + h); (t + h)) ; f (x(t); (t))]
D+f (x(t); (t)) = lim sup
h
+
h!o
1 [f (x(t + h); (t + h)) ; f (x(t); (t + h))]
lim sup
h!o+ h
1 [f (x(t); (t + h)) ; f (x(t); (t))]:
+ lim sup
+ h
h!o
(2.4)
(2.5)
We consider the rst term (Eq. 2.4). Since x is smooth and f is continuous, Eq. 2.4 becomes
1 [f (x(t) + x_ (t)h + (t; x; h)h; (t)) ; f (x(t); (t))]:
lim sup
+ h
h!o
(2.6)
Furthermore, f (x; ) is Lipschitz on x so Eq. 2.6 becomes
1 [f (x(t) + x_ (t)h; (t)) ; f (x(t); (t))]:
lim sup
+ h
h!o
Therefore,
Eq. 2.4 = D+ f (x; )(x;_ 0):
(2.7)
CHAPTER 2. MATHEMATICAL PRELIMINARY
Similarly,
21
Eq. 2.5 = D+ f (x; )(0; _):
As a result,
D+ f (x(t); (t)) D+f (x; )(x;_ 0) + D+ f (x; )(0; _):
(2.8)
According to the property of `lim sup' and `lim inf,' D+ f is bounded below:
1 [f (x(t + h); (t + h)) ; f (x(t); (t))]
D+f (x(t); (t)) = lim sup
h!o+ h
1 [f (x(t + h); (t + h)) ; f (x(t); (t + h))]
lim sup
h!o+ h
1 [f (x(t); (t + h)) ; f (x(t); (t))]:
+ lim
inf
+
h!o h
(2.9)
(2.10)
As in the derivation of upper bound, we can derive
D+f (x(t); (t)) D+ f (x; )(x;_ 0) + D+ f (x; )(0; _):
(2.11)
2
Eqs. 2.8 and 2.11 implies Eq. 2.3.
2.5 Ordinary Dierential Equations
When dealing with continuous-time systems, it is necessary to have a good understanding
of the basic facts regarding initial-value problems of dierential equations. Consider
x_ = f (t; x); with x(t0 ) = x0 ;
(2.12)
for x 2 D Rn . Such equations result when control and parameter trajectories are
substituted in the right-hand side of a nonlinear parameter-varying system:
x_ = f (x; (t); w(t)):
We state the main result on existence and uniqueness associated with the initial-value
problem.
Theorem 2.4 [Son90] Assume that f : I D ;! Y , where D Rn is open, Y Rn is
open and I is an interval, satises the following conditions:
CHAPTER 2. MATHEMATICAL PRELIMINARY
22
f is continuous, locally Lipschitz on x for each xed t; that is, there are for each
x0 2 D a real number > 0 and a such that the ball B (x0 ) of radius centered at
x0 is contained in D and
kf (t; x) ; f (t; y)k kx ; yk
for each t 2 I and x; y 2 B (x0 ).
f is measurable, locally integrable on t for each xed x0; that is for each xed x0 there
is a such that
kf (t; x0)k for all t.
Then, for each pair (t0 ; x0 ) 2 I D there is some nonempty subinterval J I and
there exists the unique continuous, Lipschitz solution x of Eq. 2.12 on J .
Note that the interval J could be arbitrary small. However, the interval J can be
extended to J = [t0 ; +1) in the following two cases.
Corollary 2.3 [Son90] Let D = Rn. Suppose that f satises the assumptions of Theorem 2.4 except that the function can be chosen independently of x0 with = 1. Then
J = [t0; +1).
Corollary 2.4 [Kh96] Let W be a compact subset of D Rn. Suppose that f satises the
assumptions of Theorem 2.4 and for x0 2 W it is known that every solution of Eq. 2.12 lies
entirely in W . Then, J = [t0 ; +1).
We discuss some properties of the rst-order \dierential" inequality which is frequently
shown in the input-output stability or input-to-state stability [KKK95,Son89].
Lemma 2.2 [Son90] Assume given an interval I , a constant c 0, and two functions,
; : I ;! R+ such that is locally integrable and is continuous. Suppose further that
for some t0 2 I it holds that
Zt
(t) c + ( )( )d
t0
CHAPTER 2. MATHEMATICAL PRELIMINARY
for all t t0 . Then, it must hold that
R t ( )d
(t) ce t0
23
:
Note that this lemma is called the Bellman-Gronwall.
Lemma 2.3 Assume constants b; c 0, and two functions v; w : I ! R+ such that
D+v(t) ;cv(t) + bw(t)2 ; with v(t0 ) 0
then
(2.13)
Zt
;
c
(
t
;
t
)
0
v(t) v(t0 )e
+ be;c(t; ) w( )2 d:
t0
If, in addition, w 2 L2(I ; R+ ), then v 2 L1(I ; R+ ) and
kvk1 1c (v(t0 ) + bkwk22 ):
If, in addition, w 2 L2(I ; R+ ), then v 2 L1(I ; R+ ) and
v(t) v(t0 )e;c(t;t0 ) + bkwk22 :
If, in addition, w 2 L1(I ; R+), then v 2 L1(I ; R+) and
v(t) v(t0 )e;c(t;t0 ) + bc kwk21 :
Proof: We show that Eq. 2.13 implies the rst result. Since v(t) is Lipschitz and thus
absolutely continuous, Eq. 2.13 implies
v_ (t) ;cv(t) + bw(t)2 for almost all t:
Multiplying this inequality by ect , it becomes
d (v(t)ect ) bw(t)2 ect for almost all t:
dt
CHAPTER 2. MATHEMATICAL PRELIMINARY
24
Since v(t)ect is absolutely continuous, Theorem 2.3 implies that
v(t)ect ; v(t0 )ect0
Dividing both sides by e;ct , we derive
v(t) v(t0 )e;c(t;t0 ) +
Zt
t0
Zt
t0
bw( )2 ec d:
be;c(t; ) w( )2 d:
The other results can be derived by the same approach as [KKK95].
2
2.6 Linear Matrix Inequality
2.6.1 Linear Matrix Inequality Problems
Many problems in the LPV systems theory as well as robust linear systems theory can be
formulated as convex optimization problems involving linear matrix inequalities (LMIs).
Detailed references on this topic can be found in [BGFB94, GNLC95, VB96]. An LMI has
the form
m
X
F (x) = F0 + xi Fi > 0
(2.14)
i=1
Fi = FiT 2 Rnn,
where the symmetric matrices
i = 0; 1; ; m are given and x 2 Rm is
variable. The inequality in Eq. 2.14 means that F (x) is positive- denite, i.e., uT F (x)u > 0
for all u 2 Rn , u 6= 0 or the smallest eigenvalue of F (x) is positive. However, most LMI
problems include matrices as variables. For
Texample, the Lyapunov stability problem is
formulated: 9P > 0 such that F (P ) = ; A P + PA > 0: In this case, we must check
whether or not F (P ) can be converted to an LMI. A couple of known facts are useful
to investigate the property of F (P ): if matrix variables (P ) anely enter into the matrix
inequality and are not coupled with other matrix variables, the matrix inequality is an LMI;
even a matrix inequality including coupled matrix variables can be an (enlarged) LMI, when
these coupled terms are related to the schur complement.
CHAPTER 2. MATHEMATICAL PRELIMINARY
25
2
3
F
(
P
)
F
(
P
)
11
12
5 and where F11 (P ) is square.
Lemma 2.4 [BGFB94] Let F (P ) = 4
F21 (P ) F22 (P )
Then F (P ) > 0 if and only if
F11 (P ) > 0 and F22 (P ) ; F12 (P )F11;1 (P )F21 (P ) > 0:
Here, F22 (P ) ; F12 (P )F11;1 (P )F21 (P ), is called schur complement of F (P ).
Note that an LMI (Eq. 2.14) can be nonlinear and nonsmooth on x but still a convex
constraint on x. A set of LMIs can be also combined into an (enlarged) LMI using methods
such as the diagonalization or S -procedure discussed below.
Lemma 2.5 [BGFB94] Let F0 ; F1 ; ; Fp be quadratic functions of the variable 2 Rn:
Fi() = T Ti + 2uTi + vi; i = 0; 1; ; p;
where Ti = TiT . Suppose that there exist 1 ; ; p 0 such that for all ,
F 0 ( ) ;
p
X
i=1
i Fi( ) 0:
Then F0 ( ) 0 for all such that Fi ( ) 0, i = 1; ; p.
Since an LMI denes a convex constraint on the variable x, optimization problems
involving the minimization (or maximization) of a convex performance function f : F ! R
with F = fx j F (x) > 0g belong to the class of convex optimization problems and thus can
employ the full power of convex optimization theory. Associated with the study of LMIs
are three generic problems:
Feasibility problem: whether or not F is an empty set
n
o
Optimization problem: infx cT x j F (x) > 0
Generalized eigenvalue problem: infx f j F (x) ; G(x) > 0; F (x) > 0; G(x) > 0 g
2.6.2 Numerical Algorithms for LMI Problems
The three typical LMI problems can be solved in a numerically ecient way, i.e., polynomial time algorithms such as the cutting plane, the ellipsoid method and very ecient
CHAPTER 2. MATHEMATICAL PRELIMINARY
26
interior-point methods (Refer to [BGFB94] and references therein for details). In practice, the interior-point algorithms are much more ecient than the rst two methods.
Among interior-point methods, the most ecient methods today appear to be primal-dual
methods and projective methods [NN94]. Furthermore, these primal-dual and projective
methods have been extended to exploit the special (Lyapunov) structure of LMI problems [NG94,VB95].
We will now discuss some basic ideas of interior-point methods (Detailed references on
this topic can be found in [BGFB94,GNLC95,VB96]). Consider an optimization problem:
n
T
opt = inf
x c x j F (x) > 0
o
(2.15)
A simple approach, called the Method of Centers, converts this problem to a -feasibility
problem:
n
o
min such that F = xjF (x) > 0; ( ; cT x) > 0 6= ;:
It denes a barrier function such that
is smooth and strictly convex on the interior of the feasibility set F .
approaches innity along each sequence of points xn in the interior of F that
converge to a boundary point of F .
One candidate of is
8
< log det F (x);1 + log( ; cT x);1 if x 2 F
=:
1
otherwise:
The Method of Centers then solves the problem with the following algorithm:
Repeat x 2 F , > cT x, 0 < < 1
Inner Loop
nd the path of center x( ),
;1
T ;1
x( ) = arg min
x log det F (x) + log( ; c x)
by the (iterative) Newton method starting at x
CHAPTER 2. MATHEMATICAL PRELIMINARY
27
Until (New) = (1 ; )cT x + converges to opt
Note that this algorithm is comprised of the inner and outer loop so that it is not ecient
to solve.
A sophisticated approach is the primal-dual optimization method that minimizes to zero
the duality gap, which is dened the dierence between an upper bound and a lower bound
of the optimal value (opt ). Therefore, this approach has some nice properties: the optimal
value of the primal-dual optimization framework is always known as `0'; the algorithm is
supported by the simplied theory; and the interval of the optimal solution (opt ) is always
known so the stopping criteria of the algorithm is directly related to the accuracy of the
optimal solution. A dual problem of Eq. 2.15 is
sup f;TrF0 Z : TrFi Z = ci g :
Z =Z T 0
The primal-dual optimization formulation is then
infT
x;Z =Z
nT
o
c x + TrF0 Z j F (x) > 0; TrFi Z = ci; i = 1; ; m :
0
The performance function is the \duality gap" ( = cT x + TrF0 Z ), which is always nonnegative and specially zero at the optimal condition. Therefore, a barrier function ((x; Z ) =
; log det (F (x)Z )) to nd the path of center (x ; Z ) (for example, see the algorithm
of the Method of Center) has a nice property that (x; Z ) = (x; Z ) ; (x ; Z ) =
n log (=n) + (x; Z ) is always non-negative and specially zero at the path of center. Furp
thermore, an augmented primal-dual potential function '(x; Z ) = n log()+ (x; Z ) can
combine the inner and outer loop of the Method of Centers into one loop.
Repeat given strictly feasible x and Z
nd feasible search direction x and Z by solving a least-square problem
plane search: arg min
p;q ' (x + p x; Z + q Z )
update: x = x + p x and Z = Z + q Z
Until = cT x + TrF0Z CHAPTER 2. MATHEMATICAL PRELIMINARY
28
The most time-consuming step is to solve the least-square problem at each iteration.
This least-square problem can be more eciently solved exploiting the problem structure [VB95]. Note that there exist several software packages which allow users to represent
LMI problems with a high-level language and to interface with MATLAB. Example are
SDPPACK [AHN97], LMITOOL [GNLC95] and SDPSOL [WB96].
2.6.3 Miscellaneous Results on LMIs
We state some results on matrix function elimination and completion. These lemmas form
the backbone of the synthesis in this study. Note that the known properties of a matrix
associated with the matrix elimination and completion are directly extended for a matrix
function, as long as the matrix function is continuous over a compact set.
Lemma 2.6 [ND77] Let 2 P , hyper-rectangle ( Rs). Suppose a symmetric, continuous
matrix function A() > 0 and B () = S ()T A()S (), where S () is a nonsingular continuous matrix function; that is, A() and B () are congruent. Then B () > 0. In other words,
the congruent transformation does not change inequality.
Lemma 2.7 [BGFB94] Let 2 P , hyper-rectangle ( Rs). Given a symmetric, continuous
matrix function G : P ! Rnn and continuous matrix functions U; V : P ! Rnm , let
U?() and V? () be continuous matrices functions whose columns form bases for the kernels
of U () and V (), respectively. There exists a continuous matrix function K : P ! Rmm
satisfying
G() + U ()K ()V ()T + V ()K ()T U ()T < 0; 8 2 P
if and only if U?()T G()U?() < 0 and V?()T G()V?() < 0 for all 2 P .
Lemma 2.8 [Pac94] Let 2 P , hyper-rectangle ( Rs). Given a pair of continuous
positive-denite matrix functions X; Y : P ! Rn+n , there exist continuous matrix functions M; N : P ! Rnr and S; T : P ! Rr+r such that
2
3
2
3
X
(
)
M
(
)
Y
(
)
N
(
)
5 > 0 and Q() = P ();1 = 4
5>0
P () = 4
T
T
M ( )
S ()
N ()
T ( )
CHAPTER 2. MATHEMATICAL PRELIMINARY
if and only if
2
4 X ( )
I
3
02
29
31
I 5
0 and rank @4 X () I 5A n + r:
Y ()
I Y ()
Note that given X (); Y () 2 Rn+n , there exists an integer r such that the dilation can
be completed if and only if X () ; Y ();1 0. If this semi-denite condition holds, then
the rank of X () ; Y ();1 determines the dimension necessary for the dilation. It can be
easily drawn that
I ; X ()Y () = M ()N ()T :
(2.16)
Lemma 2.9 [PZPB91] Suppose r = n (implies X () ; Y ();1 > 0). Then, P () and Q()
can be parameterized as follows:
and
2
X ()
P () = 4
T
M ()
M ( )
5
;
T
M () X () ; Y ();1 ;1 M ()
2
Y ()
Q() = 4
T
N ()
5:
;
T
N () Y () ; X ();1 ;1 N ()
N ( )
3
3
2.7 \Convexifying" Techniques
We will frequently encounter innite-dimensional LMI problems in an LPV systems framework. For example, the stability problem of an LPV system can be formulated: for all
2 [;1; 1] and _ = 0,
A()T P () + P ()A() < 0:
(2.17)
Eq. 2.17 actually represents an innite number of LMIs because it should be checked over
all (innite) 's inside the interval [;1; 1]. Therefore, Eq. 2.17 may be very dicult to
exactly solve. However, for some special cases, Eq. 2.17 can be reduced to a nite number
of LMI conditions using \convexifying" techniques [AA97,FAG95,GAC96,WUFS94,YS95].
We consider two simple \convexifying" techniques for our study.
CHAPTER 2. MATHEMATICAL PRELIMINARY
30
Let 2 P , hyper-rectangle ( Rs ), and be 2s vertices or corners of this hyperrectangle. Consider a matrix quadratic function F : P ! Rnn such that
F (1 ; ; s) = C0 +
s
X
k=1
k Ck +
s kX
;1
X
k=1 p=1
k pCkp +
s
X
k=1
k2 Dk :
(2.18)
The following lemma summarizes the results of [GAC96] for our study. The rst \convexifying" technique is stated.
Lemma s2.10 [GAC96] Suppose that F () is bounded by a multi-convex function, Fub () =
X 2
F () +
k=1
k Mk for Mk > 0, such that
@ 2 Fub () = 2(D + M ) 0; for k = 1; ; s:
k
k
@k2
(2.19)
Then F () (Eq. 2.18) is negative-denite on all 2 P if Fub (w) is negative-denite at all
the corner points w 2 .
Note that Eq. 2.19 is non-strict inequality but still strictly feasible with appropriate Mk > 0.
In fact, to introduce Mk > 0 always makes a non-strict inequality strictly feasible, admittedly with some conservatism. In numerical studies, we use the strictly feasible condition
2
of Eq. 2.19, i.e., @ F@ubk2() = 2(Dk + Mk ) > 0; for k = 1; ; s, rather than the non-strict
inequality.
Another approach is based on the quadratic relation shown in the following lemma and
S -procedure.
Lemma 2.11 [FAG95] Let x; q 2 Rn. There exists 2 R such that q = x and jj if
and only if there exist symmetric matrix S and skew matrix T such that
qT Sq 2 xT Sx and qT Tx = 0:
(2.20)
Note that the skew matrix T is used to account for the \real" quantity of x and q. For
convenience, we dene some matrices:
CHAPTER 2. MATHEMATICAL PRELIMINARY
31
C = [C1 Cs ] T = [T1 Ts] J T = [Inn Inn]
D = diagsk=1Dk S = diagsk=1Sk = diagsk=1 2k Inn
and
2
66 0
6C
; = 66 ..21
64 .
3
0
0 7
0
0 777
..
..
.. 77 :
.
.
. 5
Cs1 Cs2 Cs(s;1) 0
Lemma 2.12 Let jk j k for k = 1; ; s. F () (Eq. 2.18) is negative-denite on all
2 P if there exist Sk = SkT > 0 and Tk = ;TkT , k = 1; ; s such that
2
T
4 C1 0 +T J SJ
2C
1C ; T
2
D;S+;
+T
3
5 < 0:
(2.21)
Proof: Pre-multiply and post-multiply Eq. 2.18 with x. Then,
2
3
s
s kX
;1
s
X
X
X
xT F ()x = xT 4C0 + k Ck +
k pCkp + k2 Dk 5 x < 0:
k=1
k=1 p=1
k=1
(2.22)
Dene a new variable qk = k x and then
Eq. 2.22 = xT C0 xT +
s
X
k=1
qkT Ck x +
s kX
;1
X
k=1 p=1
qkT Ckpqp +
s
X
k=1
qkT Dk qk < 0:
(2.23)
According to Lemma 2.11, the relation between qk and x can be formulated as follows:
qkT Sk qk 2k xT Sk x
qkT Tk x = 0
for Sk = SkT > 0
for Tk = ;TkT :
(2.24)
(2.25)
CHAPTER 2. MATHEMATICAL PRELIMINARY
32
Eqs. 2.24 and 2.25 can be combined into Eq. 2.23 using S -procedure (Lemma 2.5), admittedly with some conservatism. Then,
xT C0xT +
0
s
X
@
k=1
qkT Ck x +
kX
;1
p=1
1
qkT Ckpqp + qkT Dk qk ; qkT Sk qk + 2k xT Sk x + 2qk2 Tk xA < 0:
2
This inequality obviously implies Eq. 2.21.
Remark 2.1 We should properly use these \convexifying" techniques. The technique based
on Lemma 2.10 is less computationally intensive than that based on Lemma 2.12 for a large
problem. For example, the LMI solver based on the projection method [GNLC95] has the
relationship between the number N () of ops needed to compute an -accurate solution
and the problem size:N () M N 3 log(V=); where M is the total row size of the LMI
system, N is the total number of scalar decision variables, and V is a data-dependent scaling
factor. This inequality implies that the LMI problem involving many number of unknown
variables is likely to be more computationally intensive than that involving many number
of LMIs. While the S -procedure approach increases a number of unknown variables, the
multi-convexity approach increases a number of LMIs. Therefore, the technique based on
Lemma 2.10 is less computationally intensive than that based on Lemma 2.12 for a large
problem. For our study, we will use Lemma 2.10 as a \convexifying" technique. The studies
using Lemma 2.12 have been investigated in [LimH97,LimH98].
2.8 Bilinear Matrix Inequality
An optimization problem involving bilinear matrix inequalities (BMIs) is often regarded as
an extension of the LMI problem in the literature [Ban97]. A BMI has the form
F (x; y) = F00 +
m
X
i=1
xi Fi0 +
n
X
j =1
yj F0j +
m X
n
X
i=1 j =1
xi yj Fij > 0:
(2.26)
where Fij = FijT 2 Rpp are given and variables are x 2 Rm and y 2 Rn . For example,
F (Q; T; Y; ) = B + QC T T + (AQ + BY )T C T > 0: Note that the coupling terms such as
QC T T cannot be eliminated by the schur complement because C T is not a positive-denite
CHAPTER 2. MATHEMATICAL PRELIMINARY
33
matrix. The optimization involving BMI constraints is illustrated as
nT
o
T y j F (x; y) 0 :
inf
c
x
+
d
x;y
(2.27)
This optimization often arises in a vast number of robust control synthesis problems such as
the parametric robust controller design, xed order and decentralized controller design. This
optimization is known as an NP-hard problem so that it requires very intensive computation.
However, these many heuristic methods have been developed which can nd only local
solutions. One of these heuristic methods is the alternative algorithm, called D ; K or
V ; K iteration, that utilizes various ecient LMI solvers in order to solve the optimization
involving BMIs: For xed x, nd y to minimize Eq. 2.27 using LMI solvers because Eq. 2.27
is an LMI on y; Similarly, nd x to minimize Eq. 2.27 with xed y. These approaches do
not guarantee the convergence of the algorithm but they seem to converge the local minima
in many practical applications [Ban97].
Chapter 3
Dissipative Systems Framework
We investigate various analysis problems { such as the stability, L2 -gain, passivity and other
performance measures { of a nonlinear parameter-varying system within a nonsmooth dissipative systems framework. In [BGFB94, SW96, Wil72], it is shown that many analysis
problems can be formulated in terms of storage functions and supply rates within the dissipative systems framework. This chapter extends these results to address analysis problems
for a special class of LPV systems which will be discussed later.
3.1 Denition of Dissipative Systems
We now consider a nonlinear parameter-varying system :
x_ = f (x; ; w); x(t0 ) = x0
z = g(x; ; w);
(3.1)
(3.2)
where x : I ;! D Rn is the state, w : I ;! W Rw is the input, z : I ;! Z Rz
is the output, and : I ;! F s is the parameter of the system. Here,
n
o
F s =4 2 C 1(I ; Rs) : (t) 2 P ; _(t) 2 ; 8t 2 I ;
where P = [1 ; 1 ] [s ; s ] and = [;1 ; 1 ] [;s; s ].
Note that with a specic trajectory (), Eq. 3.1 is a nonlinear time-varying system.
Thus, we can use some useful facts of nonlinear time-varying systems to explore the properties of nonlinear parameter-varying systems. However, it should be noted that is a set
34
CHAPTER 3. DISSIPATIVE SYSTEMS FRAMEWORK
35
of nonlinear time-varying systems because is not known in advance except that and _
are bounded by compact sets. Throughout this chapter, we assume
f and g are unbiased in the sense that
f (0; ; 0) = 0; g(0; ; 0) = 0; 8 2 F s :
f and g are continuous, locally Lipschitz on x and w jointly for each xed ; there are
a real number > 0, and such that the ball B (0; 0) of radius centered at (0; 0)
is contained in D W and given 2 F s ,
kf (x; ; w) ; f (y; ; v)k (kx ; yk + kw ; vk)
kg(x; ; w) ; g(y; ; v)k (kx ; yk + kw ; vk)
for (x; w); (y; v) 2 B (0; 0).
f is locally integrable on for each xed x and w; there is a such that given
(x; w) 2 B (0; 0),
kf (x; ; w)k for 2 F s .
every solution of Eq. 3.1 lies entirely in a compact set Dv D that includes x0.
These assumptions imply some properties: the origin x(t) = 0 is an equilibrium for such that f (0; ; 0) = 0; 8 2 F s ; Eq. 3.1 has the unique continuous, Lipschitz solution x()
over I with any x0 2 B (0), 2 F s , and input w 2 B (0) by Theorem 2.4 and Corollary 2.4.
Note that the last assumption simplies the derivation of the dissipative systems framework.
However, this assumption will be shown to be satised by the dissipative systems framework
discussed later; for any locally square integrable w(t), the resulting functions x(t) and z (t)
can be locally square integrable.
Let
r : W Z ;! R
be a mapping and assume for all t0 ; t1 2 R and forZ all input-output pairs (w; z ), the comt1
position function r(w; z ) is locally integrable, i.e.,
jr(w(t); z(t))jdt < 1. The mapping
t0
CHAPTER 3. DISSIPATIVE SYSTEMS FRAMEWORK
36
r will be referred to as the supply rate. Note that since z itself can be a state of , r(w; z )
may also be a function of x.
Denition 3.1 The class K is a set of functions, f : R+ ! R+, that are continuous,
strictly increasing functions with f (0) = 0.
Denition 3.2 is said to be dissipative with respect to the supply rate r for all 2 F s if
there exists a continuous function V : D P ! R+ such that for some functions a; b 2 K,
all t1 2 I , all x0 2 D and all (; w) 2 F s W ,
(I) a(kxk) V (x; ) b(kxk)
(II) V (x(t1 ); (t1 )) V (x(t0 ); (t0)) + Rtt01 r(w( ); z( ))d
where x(t1 ) is the state of at time t1 resulting from the initial condition x0 , () and w().
The supply rate r should be interpreted as the supply delivered to the system.Z This
t1
means that during time interval [t0 ; t1 ] work has been done on the system whenever rd
t0
is positive, while work is done by the system if this integral is negative. The function
V,
called a storage function or Lyapunov function, generalizes the notion of an energy function
for a dissipative system. (II) of Denition 3.2, called the dissipation inequality, formalizes
the intuitive idea that a dissipative system is characterized by the property that the change
of the storage in any time interval [t0 ; t1 ] will never exceed the amount of supply that
ows into the system. Hence, there can be no internal \creation of energy"; only internal
dissipation of energy is possible.
This denition is an extension of [Scha96,SW96,Wil72] to nonlinear parameter-varying
systems. The dierences exist: (I) of Denition 3.2 is included to address \uniformness"
of the dissipation; a Lipschitz, nonsmooth storage function is used rather than a typical
smooth storage function as discussed later. General denitions using nonsmooth storage
functions can be found in [AC84,SS95] in terms of \monotone trajectories" or \nonsmooth
control-Lyapunov functions."
Note that while a storage function is often used in the input-output stability problem,
a Lyapunov function is used in the internal stability problem. However, both stability
problems are equivalent to each other [Kh96,Vid92]. Following the typical notation, we call
V (x; ) a parameter-dependent Lyapunov function (PDLF) rather than a storage function.
CHAPTER 3. DISSIPATIVE SYSTEMS FRAMEWORK
37
3.2 \Dini-Dierential" Dissipation Inequality
Denition 3.2 includes the integral formulation so a dierential version of this denition is
more attractive. However, a continuous function is not necessarily dierentiable so that a
generalized derivative should be used [AC84,Clar83]. In general, the generalized derivative
is often dicult to calculate. Therefore, we consider a special class of nonsmooth PDLFs
(V (x; )) that are continuous, Lipschitz on x and . This type of the PDLF allows us to
use the Dini-derivative which almost preserves the properties of the derivative.
Theorem 3.1 is dissipative with respect to the supply rate r(; ) for all 2 F s if there
exists a continuous, locally Lipschitz PDLF V (x; ) on x and , respectively, such that for
some functions a; b 2 K, all t 2 I , all x0 2 D and all (; w) 2 F s W ,
(I) a(kx(t)k) V (x(t); (t)) b(kx(t)k)
(II) D+V (x(t); (t)) r(w(t); z(t))
where x(t) is the state of at t resulting from the initial condition x0 , () and w().
Proof: The proof is based on Corollary 2.2. First, we show that V (x(t); (t)) is a Lipschitz
function on t 2 I . According to the triangular property of `norm,'
kV (x(t1 ); (t1)) ; V (x(t0 ); (t0 ))k kV (x(t1 ); (t1 )) ; V (x(t0 ); (t1))k
+ kV (x(t0 ); (t1 )) ; V (x(t0 ); (t0 ))k:
Since V is locally Lipschitz on x and x is also Lipschitz on t, Eq. 3.3 becomes
kV (x(t1 ); (t1)) ; V (x(t0 ); (t1 ))k 1 kx(t1 ) ; x(t0 )k 1 2kt1 ; t0k:
Similarly, Eq. 3.4 becomes
kV (x(t0); (t1 )) ; V (x(t0 ); (t0 ))k 3 k(t1) ; (t0)k 3 4kt1 ; t0k:
With = max(1 2 ; 3 4 ),
kV (x(t1 ); (t1 )) ; V (x(t0 ); (t0 ))k kt1 ; t0 k:
(3.3)
(3.4)
CHAPTER 3. DISSIPATIVE SYSTEMS FRAMEWORK
38
It implies that V (x(t); (t)) is Lipschitz function on t 2 I . Therefore,
V (x(t1 ); (t1 )) = V (x(t0); (t0 )) +
Z t1
t0
D+ V (x( ); ( ))d
by Corollary 2.2. Hereby, (II) of Denition 3.2 is equivalent to
Z t1
t0
[D+ V (x( ); ( )) ; r(w( ); z ( ))]d 0:
This condition is always true if D+ V (x(t); (t)) r(w(t); z (t)) for all t 2 [t0 ; t1 ].
2
Note that if V (x; ) is dierentiable and f (x; ; w) is continuous, (II) is equivalent to
the typical dierential dissipation inequality, V_ (x(t); (t)) r(w(t); z (t)). If V is just continuous, the Dini-derivative should be replaced by the upper contingent derivative [AC84].
However, this upper contingent derivative of a multi-variable function may be very dicult
to calculate.
3.3 \Dissipation" implies
The \dissipation" implies many interesting analysis problems { such as uniformly asymptotic stability, L2 -gain, L1-gain and passivity{ for nonlinear parameter-varying systems
using proper supply rates. Most of the interesting supply rates for LTI systems have already been introduced by [BHPD94,SW96,Wil72] and references therein. The results presented here will be extensions of these existing results for analysis problems of nonlinear
parameter-varying systems.
3.3.1 Lyapunov Stability
Lyapunov stability is concerned with the behavior of the trajectories of a system when its
initial state is near an equilibrium. From a practical viewpoint, this issue is very important
because external disturbances, such as noise, wind, and component errors, are always present
in a real system to knock it out of equilibrium.
Consider with no input, i.e., w = 0. We can easily dene several stability for nonlinear parameter-varying systems extending Denitions 2.7 - 2.9. For example, consider
the uniformly asymptotic stability for a parameter-varying systems: the equilibrium point
x = 0 is locally uniformly asymptotically stable for all 2 F s if for any R0 > 0, there exist
CHAPTER 3. DISSIPATIVE SYSTEMS FRAMEWORK
39
positive scalars R1 ; R2 and T (R1 ; R2 ) > 0 such that 0 < R2 < R1 < R0 and
kx(t0 )k < R1 =) kx(t)k < R2 8 t t0 + T (R1; R2 ) and 2 F s:
The dierence is that the stability for the parameter-varying system requires the \uniformness" with respect to the parameter. Other stability denitions can be similarly dened.
Theorem 3.1 provides a sucient condition of the uniformly asymptotic stability with
r(w(t); z(t)) = ;c(kx(t)k) for c 2 K.
Proposition 3.1 Suppose that there exists a continuous, locally Lipschitz PDLF V (x; )
on x and , respectively, such that for some functions a; b; c 2 K, all t 2 I , all x 2 D and
all 2 F s ,
(I) a(kxk) V (x; ) b(kxk)
(II) D+V (x(t); (t)) ;c(kx(t)k).
Then, is uniformly asymptotically stable about the origin x = 0 for all 2 F s .
In particular, if (II)0 D+ V (x(t); (t)) 0, is uniformly stable about the origin x = 0
for all 2 F s .
Proof: The proof follows the standard Lyapunov argument. To begin with, we prove that
conditions (I) and (II) implies the uniform stability for all 2 F s . For any R > 0, there
exists (R) > 0 such that b() < a(R). Let kx(t0 )k < . Condition (II) implies V is
decreasing, i.e.,
V (x(t1 ); (t1 )) V (x(t2 ); (t2 )) for t1 < t2 and 2 F s :
According to condition (I), it becomes
b(R) > b(kx(t1 )k) b(kx(t2 )k) () R > kx(t1 )k kx(t2 )k; 8 2 F s :
Therefore, kx(t)k < R; 8 t t0 and 2 F s : In other words, is uniformly stable about
x = 0 for all 2 F s.
Next, we prove that conditions (I) and (II) mean uniformly asymptotic stability for all
2 F s . Let and such that 0 < < and b() < a(). Also let T = b()=c(). To prove
CHAPTER 3. DISSIPATIVE SYSTEMS FRAMEWORK
40
the uniformly asymptotic stability, we show that for all 2 F s ,
kx(t)k < for some t 2 [t0 ; t0 + T ]:
(3.5)
To prove Eq. 3.5, suppose by way of contradiction that Eq. 3.5 is false, so that for any ,
kx(t)k for all t 2 [t0; t0 + T ]:
Then,
0 < a() V (x(t0 + T ); (t0 + T )) V (x(t0 ); (t0 )) ;
V (x(t0 ); (t0)) ;
Z t0 +T
t0
c()d b() ; T c() = 0:
Z t0 +T
t0
c(kx( )k)d
This is a contraction and so there must exist t1 2 [t0 ; t0 + T ] such that kx(t1 )k < for all
2 F s. Since V is decreasing, it should be satised that kx(t)k < for all t t0 + T and
2 F s . Therefore, is uniformly asymptotically stable about x = 0 for all 2 F s .
2
Note that x(t) never escapes B (0) = fx 2 D : kxk g in a nite time. This result
directly implies that the last assumption of is satised.
A similar result for a nonlinear time-varying system is shown in [RHL77] without proof.
Note that condition (II) is equivalent to D+ V (x(t); (t)) < 0 because c(kxk) plays the role
of expressing a strict inequality in the format of a non-strict inequality. This role of c(kxk)
will be also preserved in other analysis formulations. We now consider a simplied case
which is often used in a linear parameter-varying system.
Corollary 3.1 Suppose that there exists a continuous, locally Lipschitz PDLF V (x; ) on
x and , respectively, such that for ; ; > 0, all t 2 I , all x 2 D and all 2 F s ,
(I) kxk2 V (x; ) kxk2
(II) D+V (x(t); (t)) ;kx(t)k2 .
Then, is exponentially stable (implies uniformly asymptotically stable) about the origin
x = 0 for all 2 F s, i.e., kx(t)k2 e;=(t;t0 ) kx(t0 )k2 ; 8 2 F s .
CHAPTER 3. DISSIPATIVE SYSTEMS FRAMEWORK
41
Proof: From conditions (I) and (II), it can be derived that for all 2 F s,
D+ V (x; ) ;kxk2 ; V (x; ):
Integrate Eq. 3.6 and then,
V (x(t); (t)) V (x(t0 ); (t0 )) +
(3.6)
Zt ; V (x( ); ( ))d:
t0
Applying Lemma 2.2, we can derive
V (x(t); (t)) V (x(t0 ); (t0 ))e;=(t;t0 ):
As once again using condition (I), i.e., kx(t)k2 V (x(t); (t)) and V (x(t0 ); (t0 )) kx(t0 )k2 , it can be derived that
kx(t)k2 e;=(t;t0 ) kx(t0 )k2 :
Therefore, is exponentially stable about the origin x = 0 for all 2 F s .
3.3.2
2
L2-Gain
Let w 2 W L2 (I ; Rw ) and z 2 Z L2 (I ; Rz ). Based on Denition 2.11, we dene the
L2-gain that
(3.7)
2 = sup sups kkwzkk2
w2W 2F
2
with x(t0 ) = 0. The L2 -gain is equivalent to the H1-norm for LTI systems. The L2 -gain
is convenient to express the ratio of the total energy (or RMS value) of the output z with
respect to the total energy (or RMS value) of the input w, and enforce robustness to model
uncertainty.
Theorem 3.1 provides a sucient condition to derive an upper bound of the L2 -gain
using the supply rate
r(w(t); z(t)) = 2 w(t)T w(t) ; z(t)T z(t) ; c(kx(t)k):
CHAPTER 3. DISSIPATIVE SYSTEMS FRAMEWORK
42
Proposition 3.2 Assume there exists a continuous, locally Lipschitz PDLF V (x; ) on x
and , respectively, such that for some functions a; b; c 2 K, all t 2 I , all x 2 D and all
(; w) 2 F s W ,
(I) a(kxk) V (x; ) b(kxk)
(II) D+V (x(t); (t)) 2w(t)T w(t) ; z(t)T z(t) ; c(kx(t)k):
Then, is uniformly asymptotically stable about x = 0 for all 2 F s and its L2 -gain is
less than , i.e., 2 < .
Proof: First, consider a special case of the supply rate with w(t) = 0. Then,
r(w(t); z (t)) = ;z(t)T z (t) ; c(kx(t)k) ;c(kx(t)k):
Therefore, conditions (I) and (II) imply that is uniformly asymptotically stable about
x = 0 for all 2 F s, as shown in Proposition 3.1.
Next, we prove 2 < . For convenience, it is assumed that x(t0 ) = 0. Integrate
condition (II) from t0 to t. Then,
Zt
t0
D+V (x( ); ( ))d 2
Zt
t0
w( )T w( )d ;
Zt
t0
z ( )T z ( )d ;
Zt
t0
c(kx( )k)d: (3.8)
According to Corollary 2.2,
Zt
t0
D+V (x( ); ( ))d = V (x(t); (t)) ; V (x(t0 ); (t0 )):
With x(t0 ) = 0 and V (x(t); (t)) 0, Eq. 3.8 becomes
Zt
t0
z ( )T z( )d 2
Zt
t0
w( )T w( )d ;
Zt
t0
c(kx( )k)d:
Therefore, kkwzkk22 < for all 2 F s and all w 2 W . It implies 2 < .
Note that V (x(t); (t)) is always nite in a nite time because
Zt
V (x(t); (t)) < V (x(t0 ); (t0 )) + 2 w( )T w( ) ; z ( )T z ( )d
t0
2
b(kx(t0 )k) + kwk21 (t ; t0):
2
CHAPTER 3. DISSIPATIVE SYSTEMS FRAMEWORK
43
Therefore, x(t) never escapes a compact set in a nite time. This result implies that the
last assumption of is satised.
As minimizing , we compute the smallest upper bound of the L2 -gain. This smallest
(opt ) is often referred to as the L2-gain.
3.3.3
L1-Gain
Let w 2 W L1(I ; Rw ) and z 2 Z L1 (I ; Rz ). Based on Denition 2.11, we dene the
L1-gain that
1 = sup sups kkwzkk1
w2W 2F
1
with x(t0 ) = 0. This L1-gain is called the peak-to-peak gain and is equivalent to the l1 norm of the impulse response of an LTI system. Therefore, the L1-gain is precisely equal
to the maximum amplication the system exerts on bounded signals.
Theorem 3.1 provides an upper bound of the L1-gain using the supply rate
r(w(t); z (t)) = ;V (x(t); (t)) + w(t)T w(t) ; kx(t)k2 :
This supply rate has been proposed as a way to characterize the L1-gain by [SW96].
Proposition 3.3 Assume there exists a continuous, locally Lipschitz PDLF V (x; ) on x
and , respectively, such that for ; ; ; ; ; > 0, all t 2 I , all x 2 D and all (; w) 2
Fs W,
(I) kxk2 V (x; ) kxk2
(II) D+V (x(t); (t)) ;V (x(t); (t)) + w(t)T w(t) ; kx(t)k2 :
Then, is uniformly asymptotically stable about x = 0 for all 2 F s and its L1-gain is
bounded above, i.e.,
r
1 < 1 +
:
Here, is the smallest Lipschitz constant of Eq. 3.2.
Proof: First, consider a special case of the supply rate with w(t) = 0. Then,
r(w(t); z(t)) = ;V (x(t); (t)) ; kx(t)k2 ;kx(t)k2 :
CHAPTER 3. DISSIPATIVE SYSTEMS FRAMEWORK
44
Therefore, conditions (I) and (II) imply that is uniformly asymptotically stable about
x = 0 for all 2 F s, as shown in Corollary
q 3.1.
Next, we prove 1 < 1 + . According to Lemma 2.3, condition (II) (
+
D V (x(t); (t)) < ;V (x(t); (t)) + w(t)T w(t)) is equivalent to
V (x(t); (t)) < V (x(t0 ); (t0 ))e;(t;t0 ) + kwk21 :
(3.9)
With x(t0 ) = 0 and condition (I), Eq. 3.9 becomes
kx(t)k2
r
2
< kwk1 () kx(t)k < kwk1 :
(3.10)
Eq. 3.2 also lead to
kz(t)k (kx(t)k + kw(t)k) (kx(t)k + kwk1 ):
(3.11)
Plugging Eq. 3.10 into Eq. 3.11, we obtain
r
kz(t)k < 1 + kwk1 8 t t0 and 2 F s :
Therefore,
kzk1 < 1 + r kwk1
q
for all w 2 W and 2 F s . It implies that 1 < 1 + .
Note that Eq. 3.9 implies V (x(t); (t)) is always nite, i.e.,
2
V (x(t); (t)) < V (x(t0 ); (t0 )) + kwk21 :
Therefore, x(t) never escapes a compact set in a nite time. This result implies that the
last assumption associated of is satised.
Note that nding the smallest upper bound of 1 looks quite dicult because an upper
bound is expressed in terms of several variables such as , and . However, we can derive
a simple formulation for a linear parameter-varying system discussed in the next chapter.
CHAPTER 3. DISSIPATIVE SYSTEMS FRAMEWORK
3.3.4
45
H2-Norm
We dene a H2 -norm for the nonlinear parameter-varying system by generalizing the
H2-norm for an LTI system. The H2-norm for an LTI system is convenient to express the
response of the system with respect to white noise, which is accepted as a good model of
many disturbances arising in engineering applications.
Let e1 ; ; ew be an orthonormal basis of the input space W . Given 2 F s , let zi be
the output of with respect to the impulse ei (i.e., an impulse at time t0 in the direction
of the unit vector ei ). Then, the H2 -norm is dened as follows:
kk22 =
sups
w
X
2F q=1
kzq k22 :
Since an impulse is related to a nonzero initial condition, the H2 -norm can be dened
in terms of the maximum output energy given an initial state. Therefore, we consider the
`bound on output energy' problem. The result will serve a basis to derive an upper bound
of the H2 -norm for a linear parameter-varying system. Note that the relationship between
an impulse and an initial state is not clear for nonlinear systems.
We dene the maximum output energy given an initial state with no input that
8Z
9
< 1 T
=
z
(
t
)
satises
Eqs.
3.1
{
3.2
E = sups : z (t) z (t)dt :
t0
with x(t0) = x0 and w() = 0. ;
2F
Theorem 3.1 provides an upper bound of the maximum output energy with the supply
rate r(w(t); z (t)) = ;z (t)T z (t) ; c(kx(t)k).
Proposition 3.4 Suppose that there exists a continuous, locally Lipschitz PDLF V (x; )
on x and , respectively, such that for some functions a; b; c 2 K, all t 2 I , all x 2 D and
all 2 F s ,
(I) a(kxk) V (x; ) b(kxk)
(II) D+V (x(t); (t)) ;z(t)T z(t) ; c(kx(t)k):
Then, is uniformly asymptotically stable about the origin x = 0 for all 2 F s and its
maximum output energy is bounded above, i.e., E < sup V (x(t0 ); (t0 )).
(t0 )2P
CHAPTER 3. DISSIPATIVE SYSTEMS FRAMEWORK
46
Proof: Obviously, r(w(t); z(t)) ;z(t)T z(t) ; c(kx(t)k) ;c(kx(t)k): Therefore, conditions (I) and (II) imply that is uniformly asymptotically stable about x = 0 for all 2 F s ,
as shown in Proposition 3.1.
Next, we prove that E < sup V (x(t0 ); (t0 )). We integrate (II) from t0 to t and then
(t0 )2P
obtain
Zt
Zt
Zt
D+V (x( ); ( ))d ; z ( )T z( )d ; c(kx( )k)d:
(3.12)
t0
t0
t0
According to Corollary 2.2, Eq. 3.12 becomes
Zt
t0
z( )T z ( )d
V (x(t0 ); (t0 )) ; V (x(t); (t)) ;
Since V (x(t); (t)) is always non-negative,
Zt
t0
z( )T z ( )d
V (x(t0 ); (t0 )) ;
Zt
t0
Zt
t0
c(kx( )k)d:
c(kx( )k)d:
Therefore, kz k22 < sup V (x(t0 ); (t0 )):
2
(t0 )2P
Note that x(t) never escapes a compact set in a nite time. This result implies that the
last assumption of is satised.
Remark 3.1 For an LTI system, the H2-norm is equivalently dened:
" ZT
#
w
X
1
2
2
T
kk2 = kzi k2 or Tlim
E
z (t) z(t)dt
!1 T
i=1
0
where zi is the output of the system with respect to an impulse ei and z (t) is the output
of the system with respect to a white noise. However, in the case of nonlinear systems, the
second denition may not be proper because the limit may not exist and the properties of
zero mean and Gaussian are not preserved [Ban97,Sto93].
3.3.5 Passivity
Let w be a locally square integrable, i.e., w 2 W L2 (I ; Rw ). The resulting functions x
and z are locally square integrable. It is also assumed that the number of inputs are equal
to the number of outputs.
CHAPTER 3. DISSIPATIVE SYSTEMS FRAMEWORK
47
We dene the strict passivity which is more convenient in the dissipative systems framework:
hw; ziT kxT k22 + ; 8 2 F s:
This denition is an extension of Denition 2.13. The rst term (kxT k22 ) plays the role of
expressing a strict inequality in the format of a nonstrict inequality. For an LTI system, the
passivity implies a positive-real system which has been widely used in system, circuit, and
control theory. It is also known that the passivity is closely related to the L2 -gain problem
via the Cayley transform.
Theorem 3.1 provides a sucient condition of the strict passivity with the supply rate
r(w(t); z(t)) = 2w(t)T z(t) ; c(kx(t)k).
Proposition 3.5 Suppose that there exists a continuous, locally Lipschitz PDLF V (x; )
on x and , respectively, such that for some functions a; b; c 2 K, all t 2 I , all x 2 D and
all (; w) 2 F s W ,
(I) a(kxk) V (x; ) b(kxk)
(II) D+V (x(t); (t)) 2w(t)T z(t) ; c(kx(t)k):
Then, is uniformly stable about the origin x = 0 and also strictly passive for all 2 F s .
Proof: First, consider a special case of the supply rate with w(t) = 0. Then, r(w(t); z(t)) =
;c(kx(t)k): Therefore, conditions (I) and (II) imply that is uniformly asymptotically
stable about x = 0 for all 2 F s , as shown in Proposition 3.1.
Next, we prove that is strictly passive for all 2 F s . We integrate (II) from t0 to t
and then obtain
Zt
t0
D+V (x( ); ( ))d 2
Zt
t0
According to Corollary 2.2,
V (x(t); (t)) ; V (x(t0 ); (t0)) 2
Since V (x; ) is always non-negative,
2
Zt
t0
w( )T z ( )d Zt
t0
w( )T z( )d ;
Zt
t0
Zt
t0
w( )T z ( )d ;
c(kx( )k)d:
Zt
t0
c(kx( )k)d:
c(kx( )k)d ; V (x(t0 ); (t0 )):
CHAPTER 3. DISSIPATIVE SYSTEMS FRAMEWORK
v1
w1
1
z1
z2
2
w2
-
48
v2
Fig. 3.1: Feedback system
Therefore, is strictly passive for all 2 F s .
2
Note that V (x(t); (t)) is always nite in a nite time because
V (x(t); (t)) < 2
Zt
Zt0t
w( )T z ( )d + V (x(t0 ); (t0 ))
Zt
w( )T w( )d + 2 z ( )T z( )d + V (x(t0 ); (t0 ))
t0
t0
= 2(kwk21 + kz k21 )(t ; t0 ):
2
Therefore, x(t) never escapes a compact set in a nite time. This result implies that the
last assumption of is satised.
3.4 Dissipation of Feedback Systems
So far, we have discussed the dissipation of a nonlinear parameter-varying system. However,
it will be interesting to investigate the dissipation of the feedback system. An earlier result
of [Wil72] showed that the dissipation of the feedback system can be established by the
dissipation of each system. The following proposition shows that the result is still true
in the nonsmooth dissipative systems framework; if there exists a PDLF for each system
that is dissipative with respect to an appropriate supply rate, then these functions can be
combined to form a PDLF for the feedback system.
Proposition 3.6 Consider the feedback system in Fig. 3.1. Furthermore, associated with
these dissipative systems are states x1 , x2 , parameters 1 , 2 , supply rates r1 (w1 ; z1 ),
r2(w2 ; z2 ) and continuous, locally Lipschitz PDLFs V1 (x1 ; 1 ), V2(x2 ; 2 ), respectively. Then
CHAPTER 3. DISSIPATIVE SYSTEMS FRAMEWORK
49
the feedback system is dissipative with respect to r = r1 + r2 for all (1 ; 2 ) 2 F1s1 F2s2 and
V = V1 + V2 is itself a PDLF for the feedback system.
Proof: Each system i is dissipative with respect to ri for all i 2 Fisi . Thus there exist
PDLFs V1 and V2 such that for a1 ; a2 ; b1 ; b2 2 K,
(I) a1 (kx1 k) V1 (x1 ; 1 ) b1 (kx1 k)
a2(kx2 k) V2(x2 ; 2 ) b2 (kx2 k)
+
(II) D V1 (x1 (t); 1 (t)) r1 (w1 (t); z1 (t)) D+ V2 (x2 (t); 2 (t)) r2 (w2 (t); z2 (t)):
Since V = V1 + V2 is continuous and bounded, we can nd some a; b 2 K such that
a(kx1 k + kx2 k) V = V1 + V2 b(kx1 k + kx2 k):
V is also Lipschitz in t so that
D+V = D+ (V1 + V2 ) D+V1 + D+V2 r1 + r2 = r:
Therefore, the feedback system is dissipative with respect to r for all (1 ; 2 ) 2 F1s1 F2s2
and V is itself a PDLF for the feedback system.
2
This proposition implies the passivity theorem or the small-gain theorem with an appropriate supply rate.
Corollary 3.2 Consider the feedback system in Fig. 3.1 with v1 = v2 = 0. Furthermore,
associated with these dissipative systems are states x1 , x2 , parameters 1 , 2 , supply rates
r1(w1 ; z1 ), r2 (w2 ; z2 ) and continuous, locally Lipschitz PDLFs V1(x1 ; 1), V2 (x2 ; 2 ), respectively. Then, the feedback system is uniformly (asymptotically) stable around (x1 ; x2 ) = 0
<)
for all (1 ; 2 ) 2 F1s1 F2s2 if r = r1 + r2 (=
0. Moreover, V = V1 + V2 is itself a PDLF for
the feedback system.
Proof: This corollary is a direct result of Proposition 3.1.
2
Note that the passivity and small-gain theorem are directly related to the condition
r = r1 + r2 (=<) 0, where r1 and r2 are the supply rates for the L2-gain and the passivity.
For example, consider the L2 -gain problem. Let r1 = 2 w1T w1 ; z1T z1 (L2 -gain is ) and
r2 = w2T w2 ; 2 z2T z2 ; c(kx2 k) (L2 -gain is less than ;1 ). From Fig. 3.1 with v1 = v2 = 0,
CHAPTER 3. DISSIPATIVE SYSTEMS FRAMEWORK
50
2
p2
p1
1
q1
w0
q2
z0
Fig. 3.2: Dynamic system with two uncertainty blocks in a feedback loop
w1 = ;z2 and z1 = w2 . Therefore,
r = r1 + r2 = 2 w1T w1 ; z1T z1 + z1T z1 ; 2 w1T w1 ; c(kx2 k) = ;c(kx2 k) < 0:
In other words, the feedback system is uniformly asymptotically stable. This result can be
also immediately obtained by the small-gain theorem because the loop gain is less than 1.
3.5 System with Structured Dynamics Uncertainties
We consider a nonlinear parameter-varying system (Eqs. 3.1 - 3.2) subject to the structured uncertainties depicted in Fig. 3.2. This system is called ^ . Let w 2 W L2 (I ; Rw )
and z 2 Z L2 (I ; Rz ). The input and output of ^ is assumed to be partitioned:
h
i
h
i
wT = w0T pT1 pTr and z T = z0T q1T qrT :
It is also assumed that each uncertainty i (t) is a linear time-varying or nonlinear dynamic
causal operator with the unit L2 -gain, i.e., i 2 Si = fi j ki k2 1g, which implies
kpi k2 kqik2 ; for i = 1; ; r :
CHAPTER 3. DISSIPATIVE SYSTEMS FRAMEWORK
51
Without loss of generality, we assume for notational convenience that every i -block is
square. We also dene
= fdiag[1 r ]ji 2 Si 8i = 1; ; rg
S = fdiag[S1 Sr ]g:
The robust L2 -gain of ^ from w0 to z0 is then dened as follows:
^2 = supw sups sup kkwz0 kk2
w0 2W 0 2F 2S 0 2
with x(t0 ) = 0.
An upper bound of ^2 can be obtained using the supply rate
r(w(t); z(t)) = w(t)T ^ w(t) ; z(t)T z (t) ; c(kx(t)k);
where ^ = diag 2 I; 1 I; ; r I and = diag [I; 1 I; ; r I ] with i 0.
Proposition 3.7 Suppose that there exist ^ ; ( 0) and a continuous, locally Lipschitz
PDLF V (x; ) on x and , respectively, such that for some functions a; b; c 2 K, all t 2 I ,
all x 2 D and all (; w) 2 F s W ,
(I) a(kxk) V (x; ) b(kxk)
(II) D+V (x(t); (t)) w(t)T ^ w(t) ; z(t)T z(t) ; c(kx(t)k).
Then, ^ is uniformly asymptotically stable about the origin x = 0 for all 2 F s and
2 S , and the L2 -gain of ^ from w0 to z0 is less than , i.e., ^2 < .
Proof: First, consider a special case of the supply rate with w(t) = 0. Then,
r(w(t); z (t)) = ;z (t)T z(t) ; c(kx(t)k) ;c(kx(t)k):
Therefore, conditions (I) and (II) imply that ^ is uniformly asymptotically stable about
x = 0 for all 2 F s and 2 S , as shown in Proposition 3.1.
CHAPTER 3. DISSIPATIVE SYSTEMS FRAMEWORK
52
Next, we prove ^2 < . Integrate condition (II) from t0 to T . After several obvious
steps shown in the proof of Proposition 3.2, condition (II) becomes
0 < 2 kw0 T k22 ; kz0 T k22 +
r
X
i=1
i kpi T k22 ; kqi T k22 :
As T ! 1, kpi k2 kqi k2 and then
0 < 2 kw0 k22 ; kz0 k22 +
r
X
i kpi k22 ; kqi k22 < 2 kw0 k22 ; kz0 k22 :
i=1
This inequality means kkwz0 kk2 < for all w0 2 W w0 , 2 F s and 2 S . Therefore, the
0 2
L2-gain of ^ from w0 to z0 is less than , i.e., ^2 < .
2
is equivalent to the constant scaling matrix (D) of the LFT- approach. When
exploring the structure of , can be a block-diagonal matrix which commutes with .
This example shows that the dissipative systems framework can account for the uncertainty
in a similar fashion to the LFT- approach. We refer to [PD93, Pag96, Sha94] for the
relationship between the structure of the scaling matrix and the type of uncertainty.
We consider a special case that each i (t) = i (t)I 2 Si = fi I ji 2 [;1; 1]g, i.e., parametric uncertainty. This parametric uncertainty is a real time-varying operator without
memory. Note that the parameter of ^ can also be considered a parametric uncertainty,
which is the basic idea of the LFT- approach in the LPV systems theory [AG95, Hel95,
Pac94]. However, an obvious dierence exists between them in the LPV control: the parameter is measurable in real time, while the parametric uncertainty is not measurable.
For a parametric uncertainty (pi (t) = i (t)qi (t)) we have the following relation that
pTi ;iqi + qiT ;Ti pi = 0
(3.13)
where each ;i is a skew matrix, i.e., ;i = ;;Ti . In the literature [FTD91, FAG95], the
constraint is well known as a basic constraint for the parametric uncertainty. As discussed
in Section 2.6.3, the constraint (Eq. 3.13) can be extended that for all 2 F s ,
pTi ;i ()qi + qiT ;i()T pi = 0;
(3.14)
CHAPTER 3. DISSIPATIVE SYSTEMS FRAMEWORK
53
where each ;i () is a piecewise-continuous, skew matrix function on 2 F s .
Now, we can derive a tighter upper bound for the robust L2 -gain of the parametervarying system with parametric uncertainties. We dene some piecewise-continuous matrices:
h
i
^ () = diag 2 I; 1 (); ; r ()
() = diag [I; 1 (); ; r ()]
;() = diag [0; ;1 (); ; ;r ()]
with each i () = i ()T > 0 and ;i () = ;;i ()T for all 2 F s .
Proposition 3.8 Suppose that there exist ^ (), () ( 0), ;() and a continuous, locally
Lipschitz PDLF V (x; ) on x and , respectively, such that for some functions a; b; c 2 K,
all t 2 I , all x 2 D and all (; w) 2 F s W ,
(I) a(kxk) V (x; ) b(kxk)
2
3T 2
w
(
t
)
5 4 ^ ()T
(II) D+V (x(t); (t)) 4
z(t)
;()
32
3
;() 5 4 w(t) 5
; c(kx(t)k).
;()
z(t)
Then, ^ is uniformly asymptotically stable about the origin x = 0 for all 2 F s and
2 S , and the L2 -gain of ^ from w0 to z0 is less than , i.e., ^2 < .
Proof: Since a parametric uncertainty is a dynamic uncertainty, we can use the supply
rate of Proposition 3.7 with parameter-dependent scaling matrices:
r0 (w(t); z (t)) = w(t)T ^ ()w(t) ; z (t)T ()z(t) ; c(kx(t)k):
(3.15)
Note that a parametric uncertainty can commute with a parameter-dependent scaling matrix. The right side of condition (II) is equal to r0 (w(t); z (t)) because
X
r(w(t); z (t)) = r0 (w(t); z(t)) +
r
i=1
pi(t)T ;i()qi (t) + qi(t)T ;i()T pi(t) = r0 (w(t); z(t)):
Therefore, it is enough to show that the supply rate r0 (w(t); z (t)) implies the uniformly
asymptotic stability and the L2 -gain of ^ from w0 to z0 being less than .
CHAPTER 3. DISSIPATIVE SYSTEMS FRAMEWORK
54
First, consider a special case of the supply rate with w(t) = 0. Then,
r0(w(t); z (t)) = ;z(t)T ()z(t) ; c(kx(t)k) ;c(kx(t)k):
Therefore, conditions (I) and (II) imply that ^ is uniformly asymptotically stable about
x = 0 for all 2 F s and 2 S , as shown in Proposition 3.1.
Next, we prove ^2 < . Integrate condition (II) from t0 to t. After several obvious steps
shown in the proof of Proposition 3.2, condition (II) becomes
0 < 2
p
Zt
t0
w0T w0 d ;
Zt
t0
z0T z0 d +
p
r Zt
X
(pTi i ()pi ; qiT i ()qi )d:
t
i=1 0
(3.16)
We dene p0i = i ()pi and qi0 = i ()qi . Since i () is nonsingular and commutes with
i (t)I ,
p0i = i qi0 () kp0i(t)k kqi0 (t)k; 8t 2 I :
(3.17)
Therefore, Eq. 3.16 becomes
0 <
2
< 2
Zt
t0
Zt
w0T w0 d
wT w0 d
t0 0
Zt
r Zt
X
T
; z0 z0 d +
(p0iT p0i ; qi0T qi0 )d
t0
i=1 t0
Zt
; z0T z0 d:
t0
This inequality means kkwz0 kk2 < for all w0 2 W w0 , 2 F s and 2 S . Therefore, the
0 2
L2-gain of ^ from w0 to z0 is less than , i.e., ^2 < .
2
As shown in [Hel95], the counterpart of an LTI system is equivalent to the upper bound
of the real- analysis.
Remark 3.2 Other robust performance problems for ^ with parametric uncertainties can
be similarly treated. First, we consider the robust L1-gain problem. The robust L1 -gain is
dened
^1 = sup sups sup kkwz0 kk1
w2W 2F 2S
0 1
CHAPTER 3. DISSIPATIVE SYSTEMS FRAMEWORK
55
with x(t0 ) = 0. A supply rate to derive an upper bound for the robust L1 -gain is
2 3T 2
w
^ ()
r(w; z ) = ;V (x; ) + 4 5 4 1 T
z
;()
32
3
;() 5 4 w 5
; kx(t)k2 ;
;1()
z
where piecewise-continuous ^ 1 () and 1() are dened
^ 1 () = diag [I; 1 (); ; r ()] and 1 () = diag [0; 1 (); ; r ()] :
Second, we dene the maximum output energy given an initial state with no input that
8Z
9
< 1 T
=
^
z
(
t
)
satises
E^ = sups sup : z0(t) z0 (t)dt 0
:
2F 2S t0
with x(t0 ) = x0 and w0(t) = 0. ;
A supply rate to derive an upper bound for E^ is
2 3T 2
w
^ ()
r(w; z ) = 4 5 4 2 T
z
;()
32
3
;() 5 4 w 5
; c(kx(t)k);
;2 ()
z
where piecewise-continuous ^ 2 () and 2 () are dened
^ 2 () = diag [0; 1 (); ; r ()] and 2 () = diag [I; 1 (); ; r ()] :
Chapter 4
Analysis
In the previous section, we have derived sucient conditions for various analysis problems
of a nonlinear parameter-varying system. With these results, we derive new LMI analysis
formulations for a special class of LPV systems, i.e., piecewise-ane parameter-dependent
linear parameter-varying system (PALPV). Associated with the PALPV system is a special
class of continuous, Lipschitz PDLFs, i.e., piecewise-ane parameter-dependent Lyapunov
function (PAL). A pair of the PALPV system and the PAL is recognized as a basic pair
which allows us to derive (computationally tractable) nite-dimensional LMIs for analysis
problems.
4.1 PALPV System
We dene a PALPV system. For simplicity, we consider the case of s = 2, i.e., (t) 2 F 2 .
The general case will be discussed in the last section of this chapter. P is partitioned into
m1 m2 number of the closed rectangles with width 1 2 , where k = (k ; k )=mk :
In this case, each rectangle is represented by Pij with an appropriate index. For each
subspace Pij , we introduce a local coordinate ^ij = [^ij 1 ^ij 2 ]T measured from the center of
Pij (see Fig. 4.1). Here, ^ij is a bound function; that is, ^ij : I ! F^2 such that
F^2 =4 ^ 2 C 1(I ; R2 ) : ^(t) 2 P^ ; ^_ (t) 2 ; 8t 2 I ;
56
CHAPTER 4. ANALYSIS
P22
2
2
P11
^111
^221
^212
^121
^112
P12
^222
^122
2
2
57
P21
^211
1
1
1 1
Fig. 4.1: Partitioned parameter subspaces (s = 2) with m1 = m2 = 2
where P^ = [; 21 ; 21 ] [; 22 ; 22 ] and = [;1 ; 1 ] [;2 ; 2 ]. Note that the relation
between (t) and ^ij (t) is given by
^ij 1(t) = 1 (t) ; 1 ; (2i ; 1) 21
^ij 2 (t) = 2 (t) ; 2 ; (2j ; 1) 22 :
(4.1)
(4.2)
For each Pij , an ane parameter-dependent linear parameter-varying (ALPV) system
is described by a set of nominal dynamics at the center of the parameter subspace and ane
parameter-dependent terms. A PALPV system is then dened as a system that switches
between the m1 m2 number of ALPV systems: for (x; ; w) 2 D F 2 W ,
2 3 m1 m2
2
3 2 3
^) B (^)
X
X
x
_
A
(
4 5=
5 4 x 5;
ij () 4
(4.3)
2
3 2
3 2 2
3
^
^
X
4 A() B () 5 = 4 Aij0 Bij0 5 + ^ijk 4 Aijk Bijk 5 :
^
^
(4.4)
z
where
C ( ) D ( )
i=1 j =1
ij
C (^) D(^)
Cij 0 Dij 0
k=1
ij
w
Cijk Dijk
To avoid ambiguity at the boundary, the switching function ij () is dened as
8
< 1 if (t) 2 Pij ; (P(i+1)j [ Pi(j+1) )
ij () = :
0 otherwise
CHAPTER 4. ANALYSIS
58
4
with Pij =
; for undened Pij . For simplicity, we dene sets of vertices of parameter
subspaces:
k k = (!1 ; !2 ) : !k 2 ; 2 ; 2
denotes the set of 22 vertices of P^ . Similarly,
= f(1 ; 2 ) : k 2 [;k ; k ]g
denotes the set of 22 vertices of .
Remark 4.1 The PALPV system (Eq. 4.3) is locally ane parameter-dependent. A typ-
ical ALPV system description is a special case of the PALPV description. Therefore, the
PALPV system can provide less conservative models for an LPV system than an ALPV
system. Furthermore, the PALPV system can include dynamics that are discontinuous in
(t) on the boundary of Pij , which could provide much less conservative models.
4.2 PAL
To analyze the PALPV system, we use a continuous, piecewise-ane parameter-dependent
Lyapunov function (PAL) which is a Lipschitz PDLF. The PAL, V (x; ) = xT P ()x, is
dened that
m1 X
m2
h i
X
P () =
ij () P^ (^) ij
(4.5)
i=1 j =1
[P^ (^)]ij = Co[Pij ; P(i+1)j ; P(i+1)(j+1) ; Pi(j+1) ]; the vertices of which are positive-denite
matrices dened counterclockwise from the corners of the rectangle Pij (see Figure 4.2).
To ensure that [P^ (^)]ij is ane in ^ij , these matrices must satisfy
Pij + P(i+1)(j +1) = P(i+1)j + Pi(j +1) ;
(4.6)
which is the parallelogram rule.
It will be convenient to express [P^ (^)]ij with the same format of Eq. 4.4:
h^ ^ i
P () ij = Pij 0 +
2
X
^
k=1
ijk Pijk ;
(4.7)
CHAPTER 4. ANALYSIS
59
P32
P ()
P33
P22
P23
P13
P31
P12
P21
P11
P12
P22
P11
2
P21
1
Fig. 4.2: Example of a continuous, piecewise-ane P () with s = 2. For simplicity, the
parameter space is only split into 2 2 regions.
where Pij 0 is the center point and Pijk is the slope of [P^ (^)]ij along the k axis. Since Pij 0
is the center of [P^ (^)]ij ,
Pij0 = 212 Pij + P(i+1)j + P(i+1)(j+1) + Pi(j+1) :
(4.8)
Evaluating Eq. 4.6 with each ij , we can derive the following relation that
Pij = Pi1 + P1j ; P11 :
(4.9)
Plugging Eq. 4.9 into Eq. 4.8, we can derive
Pij0 = 12 (P(i+1)1 + Pi1 + P1(j+1) + P1j ; 2P11 ):
Pijk is the slope of [P^ (^)]ij along the k axis. Thus,
Pij 1 = P(i+1)j ; Pij =1
= P(i+1)1 + P1j ; P11 ; (Pi1 + P1j ; P11 ) =1
(4.10)
CHAPTER 4. ANALYSIS
=
Similarly,
60
P(i+1)1 ; Pi1
1 :
(4.11)
P1(j+1) ; P1j
::
(4.12)
2
h i n
o
Note that for w 2 , P^ (w) ij 2 Pij ; P(i+1)j ; P(i+1)(j +1) ; Pi(j +1) ; where each Pij satises
Eq. 4.9.
Remark 4.2 Each [P^ (^)]ij is dierentiable over the corresponding Pij . P () is also conPij2 =
tinuous over the original parameter space (P ) because of sharing some vertices between
[P^ (^)]ij 's, as shown in Fig. 4.2. Therefore, P () does have the Dini-derivative over P and
hence P () is Lipschitz on [KF70]. Therefore, the PAL is a Lipschitz PDLF.
Remark 4.3 To express the unknown P () in terms of unknown constant matrices allows
us to numerically nd the optimized P () associated with the least conservatism. Furthermore, to dene PAL needs only Pa1 and P1b (a = 1; ; m1 + 1, b = 1; ; m2 + 1). In case
4P.
of s = 1, Eq. 4.7 is still true with Pi1 = Pi2 = = Pi(m2 +1) =
i
4.3 Analysis Formulations
We have dened the PALPV system (Eq. 4.3) and the PAL (Eq. 4.5) in the previous sections.
We now derive (nite-dimensional) LMI formulations for various analysis formulations for
the PALPV system using the results of Chapter 3. Various analysis problems can be similarly derived, as will be shown here. Therefore, we are enough to understand the derivation
of an analysis problem. However, for completion, we include all the derivations in this
thesis. For simplicity, we consider the pair of the PALPV system and the PAL with s = 2.
The results for the general case will be derived at the end of this chapter.
Note that most of the present results are extensions of the results for an LTI system and
a linear dierential inclusion (LDI) [BGFB94, SGC97] to PALPV systems using a general
class of PDLFs, i.e., PALs. Furthermore, these results are generalizations of [GAC96] and
full characterizations to other performance analysis problems using a more general class of
PDLFs than the ane parameter-dependent Lyapunov function.
CHAPTER 4. ANALYSIS
61
4.3.1 Lyapunov Stability
We derive a sucient condition for uniformly asymptotic stability of the PALPV system
(Eq. 4.3) based on Proposition 3.1. For simplicity, we dene
h
i
[L(w; )]ij = A(w)T P^ (w) + P^ (w)A(w) + P^ ( ) ; Pij 0 ij :
(4.13)
Let i = 1; ; m1 , j = 1; ; m2 and k = 1; 2 for all analysis problems.
Proposition 4.1 Assume Aij0 is asymptotically stable and that there exist Mijk (> 0),
positive-denite Pa1 (a = 1; m1 + 1) and P1b (b = 1; ; m2 + 1) such that for all ij ,
[L(w; )]ij +
2
X
k=1
wk2 Mijk < 0
(4.14)
for all (w; ) 2 and
ATijk Pijk + Pijk Aijk + Mijk 0
(4.15)
for all k. Then, the PALPV system (Eq. 4.3) is uniformly asymptotically stable about x = 0
for all 2 F 2 .
Proof: Sucient conditions of the uniformly asymptotic stability are given in Proposi-
tion 3.1. Both the PALPV system and the PAL are used for Proposition 3.1. First, consider
condition (II) of Proposition 3.1. Partition the parameter space (P ) into subspaces (Pij 's).
The interval (I ) is then partitioned:
I=
[
ij
Iij = ftj(t) 2 Pij g :
Consider each Iij . It is then assumed that by a smooth extension, the PALPV system
(Eq. 4.3) and the PAL (Eq. 4.5) are dened over the smallest open set that includes Pij .
This assumption makes x and V (x; ) dierentiable on this sub-domain (possibly admittedly
with some conservatism). According to Lemma 2.1, condition (II) on Iij becomes
D+V (x(t); (t)) D+ V (x; )(x;_ 0) + D+ V (x; )(0; _) < 0:
(4.16)
CHAPTER 4. ANALYSIS
62
Furthermore, the partial Dini-derivative is just the partial derivative because V (x; ) is
dierentiable. Then,
T^ ^ 3
2 T^ ^ @
x P ()x ^_
@
x
P
(
)
x
x
_
+
5
D+V (x(t); (t)) 4
@x
@ ^
ij
= xT A(^)T P^ (^) + P^ (^)A(^) + P^ (^_ ) ; Pij 0 x < 0:
ij}
|
h ^{z^_ i
L(;)
ij
Therefore, condition (II) is satised if for all ij ,
L(^; ^_ ) < 0; 8(^ij ; ^_ ij ) 2 P^ :
ij
(4.17)
Given ij , this inequality, however, leads to an innite number of LMIs. Therefore, we need
to \convexify" Eq. 4.17. To begin with, we regard ^ij and ^_ ij as two independent variables
(admittedly with some conservatism). Since [L(^; ^_ )]ij is ane in ^_ ij , Eq. 4.17 is always
true if
h ^ i
L(; ) ij < 0; 8 ^ij ; 2 P^ :
With xed , [L(^; )]ij consists of ane [A(^)]ij and [P^ (^)]ij . According to Lemma 2.10,
[L(^; )]ij < 0 is always true if
[L(w; )]ij +
and
2
X
k=1
wk2 Mijk < 0; 8(w; ) 2 ATijk Pijk + Pijk Aijk + Mijk 0; k = 1; 2:
(4.18)
(4.19)
As a result, condition (II) is satised if Eqs. 4.18 - 4.19 are satised for all ij .
To complete the proof, consider condition (I) of Proposition 3.1. This condition directly
implies that for all ij ,
h^ ^ i
P () ij > 0; 8^ij 2 P^ :
(4.20)
Given ij , condition (II) implies that
h ^ i h ^T^ ^ ^ ^ ^i
L(; 0) ij = A() P () + P ()A() ij < 0; 8^ij 2 P^ :
(4.21)
CHAPTER 4. ANALYSIS
63
As shown in [GAC96], the classical Lyapunov theory tells that a stable Aij 0 implies the
existence of Pij 0 > 0. Furthermore, [P^ (^)]ij should be nonsingular for all ^ij 2 P^ if Eq. 4.21
is satised for all ^ij 2 P^ . Therefore, it is always satised that [P^ (^)]ij > 0 for all ^ij 2 P^ .
As a result, condition (I) of Proposition 3.1 is redundant.
2
Remark 4.4 Note that we typically use the strictly feasible condition of Eq. 4.15, i.e.,
ATijk Pijk + Pijk Aijk + Mijk > 0 for computation. Introducing Mijk makes the LMI for-
mulations feasible to compute because current LMI solvers can treat the strict inequality [GNLC95]. However, Mijk creates some conservatism so that a minimum size of Mijk
should be used in the \convexifying" step.
4.3.2
L2-Gain
We derive an upper bound of the L2 -gain (2 ) of the PALPV system (Eq. 4.3) based on
Proposition 3.2.
Proposition 4.2 Assume Aij0 is asymptotically stable and that there exist Mijk (> 0),
positive-denite Pa1 (a = 1; m1 + 1) and P1b (b = 1; ; m2 + 1) such that for all ij ,
2
3 2
T C (w)
^ (w)B (w) + C (w)T D(w)
X 2
L
(
w;
)
+
C
(
w
)
P
4
5
+
wk Mijk < 0 (4.22)
B (w)T P^ (w) + D(w)T C (w) ; 2 I + D(w)T D(w)
k
=1
ij
for all (w; ) 2 and
2 T
3
A
P
+
P
A
P
B
ijk
ijk
ijk
ijk
ijk
ijk
4
5 + Mijk 0
T
Bijk Pijk
0
(4.23)
for all k. Then, the PALPV system (Eq. 4.3) is uniformly asymptotically stable about x = 0
for all 2 F 2 , and its L2 -gain (2 ) is less than , i.e., 2 < .
Proof: The proof is similar to that of Proposition 4.1. An upper bound of the L2-gain is
given in Proposition 3.2. Both the PALPV system and the PAL are used for Proposition 3.2.
First, consider condition (II) of Proposition 3.2. As in the proof of Proposition 4.1, condition
(II) over each Iij becomes
D+ V (x; )(x;_ 0) + D+ V (x; )(0; _) ; 2 wT w + zT z < 0:
(4.24)
CHAPTER 4. ANALYSIS
64
Since V (x; ) is dierentiable,
2 T^ ^ 3
@ x P ( )x
@ xT P^ (^)x ^_ 2 T
Eq. 4.24 = 4
x_ +
; w w + zT z5
^
@x
@
ij
= xT L(^; ^_ )x + xT P^ (^)B (^)w + wT B (^)T P^ (^)x ; 2 wT w
T + C (^)x + D(^)w
C (^)x + D(^)w
ij
< 0:
(4.25)
Therefore, condition (II) is satised if for all ij ,
2
3 2
3
_
T
T
^
^
^
^
^
^
^
^
^
C () C () C () D() 5
G(^; ^_ ) = 4 L^(T; ^) ^ P ()B2 () 5 + 4 ^ T ^
< 0 (4.26)
D() C () D(^)T D(^) ij
ij
B () P () ; I ij
for all (^ij ; ^_ ij ) 2 P^ . Furthermore, according to Lemma 2.10, Eq. 4.26 is always true if
for
2
X
[G(w; )]ij + wk2 Mijk < 0; 8(w; ) 2 (4.27)
and
k=1
2 T
3
A
P
+
P
A
P
B
4 ijk ijkT ijk ijk ijk ijk 5 + Mijk 0; k = 1; 2:
(4.28)
Bijk Pijk
0
Note that the coupling terms of the second matrix of Eq. 4.26 can be eliminated by the
schur complement so that they do not have the ^k2 terms. As a result, condition (II) is
satised if Eqs. 4.27 - 4.28 are satised for all ij .
As in the proof of Proposition 4.1, condition (I) of Proposition 3.2 is redundant.
2
As minimizing , we compute the smallest upper bound of the L2 -gain of the PALPV
system (Eq. 4.3). This smallest (opt ) is often used as the L2 -gain.
4.3.3
L1-Gain
We derive an upper bound of the L1-Gain (1 ) of the PALPV system (Eq. 4.3) based on
Proposition 3.3.
CHAPTER 4. ANALYSIS
65
Proposition 4.3 Assume Aij0 is asymptotically stable and that there exist , , Mijk (>
0), positive-denite Pa1 (a = 1; m1 + 1), and P1b (b = 1; ; m2 + 1) such that for all ij ,
2
3
2
^ (w) P^ (w)B (w)
X
L
(
w;
)
+
P
4
5
+
wk2 Mijk < 0;
(4.29)
T
^
B ( w ) P (w )
;I
k
=1
ij
2
3
T (w)
^ (w)
P
0
C
66
7
(4.30)
64 0 ( ; )I DT (w) 775 > 0
C (w )
D (w )
I
for all (w; ) 2 and
ij
2 T
3
A
P
+
P
A
P
B
4 ijk ijkT ijk ijk ijk ijk 5 + Mijk 0
Bijk Pijk
(4.31)
0
for all k. Then, the PALPV system is uniformly asymptotically stable about x = 0 for all
2 F 2 and its L1-gain (1) is less than , i.e., 1 < .
Proof: The proof is similar to that of Proposition 4.1. An upper bound of the L1-gain
can be derived using Proposition 3.3. Both the PALPV system and the PAL are used
for Proposition 3.3. First, consider condition (II) of Proposition 3.3. As in the proof of
Proposition 4.1, condition (II) over each Iij becomes
D+V (x; )(x;_ 0) + D+V (x; )(0; _) + V (x; ) ; wT w < 0:
(4.32)
Since V (x; ) is dierentiable,
2 T^ ^ 3
@ x P ()x
@ xT P^ (^)x ^_
Eq. 4.32 = 4
x_ +
+ xT P^ (^)x ; wT w5
@x
@ ^
_
ij
= xT L(^; ^) + P^ (^) x + 2xT P^ (^)B (^)w ; wT w < 0: (4.33)
ij
Therefore, condition (II) is satised if for all ij ,
2
3
_
^
^
^
^
^
^
^
L
(
;
)
+
P
(
)
P
(
)
B
(
)
5 <0
G(^; ^_ ) = 4
ij
B (^)T P^ (^)
;I ij
(4.34)
CHAPTER 4. ANALYSIS
66
for all (^ij ; ^_ ij ) 2 P^ . Furthermore, according to Lemma 2.10, Eq. 4.34 is always true if
[G(w; )]ij +
and
2
X
k=1
wk2 Mijk < 0; 8(w; ) 2 (4.35)
2 T
3
4 Aijk PijkT + Pijk Aijk Pijk Bijk 5 + Mijk 0; k = 1; 2:
(4.36)
V (x; ) < kwk21
(4.37)
Bijk Pijk
0
As a result, condition (II) is satised if Eqs. 4.35 - 4.36 are satised for all ij .
According to Lemma 2.3, condition (II) also leads to
with x(t0 ) = 0. Based on this relation, we assume a relation to indicate an upper bound of
the L1-gain [SGC97]:
z (t)T z (t) < V (x; ) + ( ; ) kwk21
(4.38)
because plugging Eq. 4.37 into Eq. 4.38,
z(t)T z (t) < 2 kwk21 () kzk21 < 2 kwk21 :
Eq. 4.38 is satised if
(4.39)
z(t)T z (t) < V (x; ) + ( ; ) w(t)T w(t) :
Obviously, this inequality is satised if for all ij ,
xT P^ (^)x + ( ; ) wT w ; ;1 C (^)x + D(^)w
(4.40)
T ^
C ()x + D(^)w > 0: (4.41)
ij
Furthermore, Eq. 4.41 becomes
2
3
^)T
^ (^)
P
0
C
(
66
7
64 0 ( ; )I D(^)T 775 > 0:
C (^)
D(^)
I
ij
(4.42)
CHAPTER 4. ANALYSIS
67
Since this LMI is ane in ^ij , Eq. 4.42 is always true on ^ij 2 P^ if Eq. 4.42 is satised for
all w 2 . As a result, the L1 gain (1 ) is less than , if Eq. 4.42 is satised for all w 2 and all ij .
As in the proof of Proposition 4.1, condition (I) of Proposition 3.3 is redundant.
2
Note that Eq. 4.30 is not an LMI because there exists [P^ (w)]ij . However, Eq. 4.30
becomes an LMI with xed scalar . Therefore, the smallest (opt ) can be found by
minimizing 's at dierent values of 's and selecting the minimum value of . As discussed
in [SGC97], we can nd an upper bound of such that
2
3
2 40; ;2 sup sup Re eig[A(^)]ij 5 :
ij ^ij 2P^
4.3.4
H2-Norm
We derive an upper bound of the H2 -norm of the PALPV system (Eq. 4.3) with [D(^)]ij = 0.
Dene
m1 X
m2
h i
X
B () =
ij () B (^) ij :
i=1 j =1
Given 2 F 2 , let x0q = B ()eq (q = 1; ; w) be a basis for the initial condition where
eq denotes the orthonormal basis of input space W . Let z0q denote the output response
subject to initial condition x0q of the PALPV system (Eq. 4.3) with the 2 F 2 . Then, the
H2-norm is equivalent to
w
X
kk22 = sup2 kz0q k22 :
(4.43)
2F q=1
Therefore, an upper bound of the H2 -norm, while the H2 -norm is dicult to compute, can
be derived based on Proposition 3.4.
Note that the H2 -norm analysis in the stochastic approach is shown in [Wu95].
Proposition 4.4 Assume Aij0 is asymptotically stable and that there exist Q, Mijk , Nijk (>
0), positive-denite Pa1 (a = 1; m1 + 1) and P1b (b = 1; ; m2 + 1) such that for all ij ,
2
h
i X
T
2
L(w; ) + C (w) C (w) ij +
k=1
wk Mijk < 0;
(4.44)
CHAPTER 4. ANALYSIS
2
4
68
3
2
Q
B T (w)P^ (w) 5 X
+ wk2 Nijk > 0
P^ (w)B (w)
P^ (w)
ij k=1
(4.45)
for all (w; ) 2 and
2
4
ATijk Pijk + Pijk Aijk + Mijk 0;
(4.46)
3
T Pijk
0
Bijk
5 + Nijk 0
(4.47)
Pijk Bijk
0
for all k. Then, the PALPV system with [D(^)]ij = 0 is uniformly asymptotically stable
p
about x = 0 for all 2 F 2 and its H2 -norm is less than TrQ, i.e., kk2 < TrQ:
Proof: The proof is similar to that of Proposition 4.1. An upper bound of the H2-norm
can be derived using Proposition 3.4. Both the PALPV system and the PAL are used
for Proposition 3.4. First, consider condition (II) of Proposition 3.4. As in the proof of
Proposition 4.1, condition (II) over each Iij becomes
D+V (x; )(x;_ 0) + D+V (x; )(0; _) + z T z < 0:
(4.48)
Since V (x; ) is dierentiable,
3
2 T^ ^ @ x P ()x
@ xT P^ (^)x ^_ T
Eq. 4.48 = 4
+ z z5
x_ +
@x
@ ^
ij
_
= xT L(^; ^)x + xT C (^)T C (^)x
ij
< 0:
(4.49)
(4.50)
Therefore, condition (II) is satised if for all ij ,
L(^; ^_ ) + C (^)T C (^)
ij
< 0; 8(^ij ; ^_ ij ) 2 P^ :
(4.51)
Furthermore, according to Lemma 2.10, Eq. 4.51 is always true if
h
i
L(w; ) + C (w)T C (w) ij +
2
X
k=1
wk2 Mijk < 0; 8(w; ) 2 (4.52)
CHAPTER 4. ANALYSIS
69
and
ATijk Pijk + Pijk Aijk + Mijk 0; k = 1; 2:
(4.53)
Note that the coupling term (C (^)T C (^)) of Eq. 4.51 can be eliminated by the schur complement. As a result, condition (II) is satised if Eqs. 4.52 - 4.53 are satised for all ij .
With the results of Proposition 3.4 and Eq. 4.43, an upper bound of the H2 -norm is
derived as follows:
kk22 < sup2
w
X
2F q=1
xT0q P (^)x0q = sup2
2F
w
X
TrB ()T P (^)B ()
h ^ T ^ ^ ^q=1
i
< sup sup Tr B () P ()B () ij :
ij ^ij 2P^
(4.54)
Eq. 4.54 immediately implies kk22 < TrQ such that
2
3
T P^ (^)
i
^
Q
B
(
)
5 >0
Q ; B (^)T P^ (^)B (^) ij > 0 () 4 ^ ^ ^
P ()B ()
P^ (^)
ij
h
(4.55)
for all ^ij 2 P^ and all ij . According to Lemma 2.10, Eq. 4.55 is always true if
2
4
3
2
Q
B T (w)P^ (w) 5 X
+
wk2 Nijk > 0; 8w 2 P^ (w)B (w)
P^ (w)
ij k=1
2
4
and
3
T Pijk
0
Bijk
5 + Nijk 0; k = 1; 2:
Pijk Bijk
0
As a result, kk22 < TrQ if Eqs. 4.56 - 4.57 are satised for all ij .
As in the proof of Proposition 4.1, condition (I) of Proposition 3.4 is redundant.
p
(4.56)
(4.57)
2
As minimizing TrQ, we compute the smallest upper bound of the H2 -norm of the
PALPV system (Eq. 4.3).
4.3.5 Passivity
We derive a sucient condition for the strict passivity of the PALPV system (Eq. 4.3) based
on Proposition 3.5.
CHAPTER 4. ANALYSIS
70
Proposition 4.5 Assume Aij0 is asymptotically stable and that there exist Mijk (> 0),
positive-denite Pa1 (a = 1; m1 + 1) and P1b (b = 1; ; m2 + 1) such that for all ij ,
2
3 2
^ (w)B (w) ; C (w)T
X 2
L
(
w;
)
P
4
5
+
w M <0
(4.58)
B (w)T P^ (w) ; C (w) ; D(w)T + D(w) ij k=1 k ijk
for all (w; ) 2 and
2 T
3
A
P
+
P
A
P
B
4 ijk ijkT ijk ijk ijk ijk 5 + Mijk 0
Bijk Pijk
(4.59)
0
for all k. Then, the PALPV system is uniformly asymptotically stable about x = 0 for all
2 F 2 and also strictly passive.
Proof: The proof is similar to that of Proposition 4.1. Sucient conditions of the strict
passivity are given in Proposition 3.5. Both the PALPV system and the PAL are used
for Proposition 3.5. First, consider condition (II) of Proposition 3.5. As in the proof of
Proposition 4.1, condition (II) over each Iij becomes
D+ V (x; )(x;_ 0) + D+ V (x; )(0; _) ; 2wT z < 0:
(4.60)
Since V (x; ) is dierentiable,
2 T^ ^ 3
@ x P ( ) x
@ xT P^ (^)x ^_
Eq. 4.60 = 4
x_ +
; 2w T z 5
@x
@ ^
ij
_
= xT L(^; ^)x + xT P^ (^)B (^)w + wT B (^)T P^ (^)x
;wT
^
^
T ^
^
C ()x + D()w ; C ()x + D()w w < 0:
ij
(4.61)
Therefore, condition (II) is satisef if for all ij ,
2
G(^; ^_ ) = 4
ij
3
L(^; ^_ )
P^ (^)B (^) ; C (^)T 5 < 0
B (^)T P^ (^) ; C (^) ;D(^)T ; D(^) ij
(4.62)
CHAPTER 4. ANALYSIS
71
for all (^ij ; ^_ ij ) 2 P^ . Furthermore, according to Lemma 2.10, Eq. 4.62 is always true if
[G(w; )]ij +
2
X
k=1
wk2 Mijk < 0; 8(w; ) 2 2 T
3
4 Aijk PijkT + Pijk Aijk Pijk Bijk 5 + Mijk 0; k = 1; 2:
and
Bijk Pijk
0
As a result, condition (II) is satised if Eqs. 4.63 - 4.64 are satised for all ij .
As in the proof of Proposition 4.1, condition (I) of Proposition 3.5 is redundant.
(4.63)
(4.64)
2
4.3.6 Robust L2-Gain and Others
Based on Proposition 3.8, we derive an upper bound of the robust L2 -gain (^2 ) of the PALPV
system subject to the structured time-varying parametric uncertainties ((t)) depicted in
Fig. 3.2 (Refer to section 3.5 for details).
For computational simplicity, we consider piecewise constant ^ (), () and ;() such
that
XX
XX
XX
^ () =
ij ()^ ij ; () =
ij ()ij ; ;() =
ij ();ij ;
m1 m2
m1 m2
m1 m2
i=1 j =1
i=1 j =1
i=1 j =1
(4.65)
where ^ ij = ^ Tij , ij = Tij (> 0) and ;ij = ;;Tij .
Proposition 4.6 Assume Aij0 is asymptotically stable and that there exist Mijk , ^ ij , ij
(> 0), ;ij , positive-denite Pa1 (a = 1; m1 + 1) and P1b (b = 1; ; m2 + 1) such that
for all ij ,
2
4 L(w;T )
3 2
32
32
3T 2
5 + 4 C (w)T 0 5 4 ; 5 4 C (w)T 0 5 + X
wk2 Mijk < 0
T
T
T
^
^
B (w) P (w) 0 ij
D(w) I
; ;
D(w) I ij k=1
(4.66)
for all (w; ) 2 and
2 T
3
A
P
+
P
A
P
B
4 ijk ijkT ijk ijk ijk ijk 5 + Mijk 0
Bijk Pijk
0
(4.67)
CHAPTER 4. ANALYSIS
72
for all k. Then, the PALPV system is uniformly asymptotically stable about the origin x = 0
for all 2 F 2 and 2 S , and its robust L2 -gain from w0 to z0 is less than , i.e., ^2 < .
Proof: The proof is similar to that of Proposition 4.1. An upper bound of the robust
L2-gain is given in Proposition 3.8. Both the PALPV system and the PAL are used for
Proposition 3.8. First, consider condition (II) of Proposition 4.1. As in the proof of Proposition 4.1, condition (II) over each Iij becomes
2 3T 2
3 2 3
^
w
;
5 4 w 5 < 0:
D+V (x; )(x;_ 0) + D+ V (x; )(0; _) ; 4 5 4 T
z
; ; ij z
(4.68)
Note that ^ ij , ij and ;ij are assumed to be dened over the smallest open set that includes
Pij . Since V (x; ) is dierentiable,
3 2 3T 2
2 T^ ^ 3 2 3
^ ;
@ x P ()x
@ xT P^ (^)x ^_
w
w5
5
4
5
4
5
4
Eq. 4.68 = 4
x
_
+
;
@x
@ ^
z
;T ; ij z
ij
= xT L(^; ^_ ) + C (^)T C (^) x + xT P^ (^)B (^) + C (^)T D(^) ; C (^)T ;T w
+wT B (^)T P^ (^) + D(^)T C (^) ; ;C (^) x
i
+wT ;^ + D(^)T D(^) ; D(^)T ;T ; ;D(^) w ij < 0:
(4.69)
Therefore, condition (II) is satised if for all ij ,
G(^; ^_ )
ij
2
= 4
3
L(^; ^_ )
P^ (^)B (^) + C (^)T ; 5
B (^)T P^ (^) + ;T C (^) D(^); + ;T D(^) ij
2
3 2
3 2
3
^) D(^) T 0
^) D(^)
C
(
C
(
5 4
5 4
5 < 0 (4.70)
+4
0
I ij 0 ;^ ij 0
I ij
for 8(^ij ; ^_ ij ) 2 P^ . Furthermore, according to Lemma 2.10, Eq. 4.70 is always true if
[G(w; )]ij +
2
X
k=1
wk2 Mijk < 0; 8(w; ) 2 (4.71)
CHAPTER 4. ANALYSIS
73
2 T
3
A
P
+
P
A
P
B
ijk
ijk
ijk
ijk
ijk
ijk
4
5 + Mijk 0; k = 1; 2:
T
and
(4.72)
Bijk Pijk
0
Note that the coupling terms of the second matrix of Eq. 4.70 can be eliminated by the
schur complement so that they do not have the ^k2 terms. As a result, condition (II) is
satised if Eqs. 4.71 - 4.72 are satised for all ij .
As in the proof of Proposition 4.1, condition (I) of Proposition 3.8 is redundant.
2
We can also derive analysis formulation for the robust L1 and the robust H2 -norm of
the PALPV system. For computational simplicity, we use piecewise constant ^ 1 (), 1 ()
such that
m1 X
m2
m1 X
m2
X
X
^ 1 () =
ij ()^ 1ij ; 1 () =
ij ()1ij ;
(4.73)
i=1 j =1
i=1 j =1
where ^ 1ij = ^ T1ij and 1ij = T1ij (> 0).
Proposition 4.7 Assume Aij0 is asymptotically stable and that there exist , , Mijk ,
^ 1ij , 1ij (> 0), ;ij , positive-denite Pa1 (a = 1; m1 + 1), and P1b (b = 1; ; m2 + 1)
such that for all ij ,
20
1 3 2
32
32
3 2
L
(
w;
)+
66 @
A 77 4 C (w)T 0 5 4 1 ; 5 4 C (w)T 0 5T X
+ wk2 Mijk < 0;
64 P^ (w)
75 +
T
T
T
^
D (w ) I
; ;1
D(w) I ij k=1
T^
B (w) P (w)
0 ij
2
3
T (w)
^ (w)
P
0
C
66
7
64 0 ( ; )I DT (w) 775 > 0
C (w )
D (w )
I
for all (w; ) 2 and
(4.75)
ij
2 T
3
A
P
+
P
A
P
B
4 ijk ijkT ijk ijk ijk ijk 5 + Mijk 0
Bijk Pijk
(4.74)
0
(4.76)
for all k. Then, the PALPV system is uniformly asymptotically stable about x = 0 for all
2 F 2 and its robust L1-gain (^1 ) from w0 to z0 is less than , i.e., ^1 < .
Proof: The proof is straightforward from the results of Propositions 4.3 - 4.6.
2
CHAPTER 4. ANALYSIS
74
To clarify the robust H2 -norm problem, partition [B (^)]ij = [Bw0 (^) Bp (^)]ij and
h
i
h
i
X
X
Bw0(^) ij = Bw0ij 0 + ^ijkBw0ijk ; Bp(^) ij = Bpij 0 + ^ijk Bpijk:
2
2
k=1
k=1
We redene the robust H2 -norm of the PALPV system with time-varying parametric uncertainties (called ^ ). Dene
Bw0 () =
m1 X
m2
X
i=1 j =1
h
i
ij () Bw0(^) ij :
Given 2 F 2 , let xw0q = Bw0 ()eq (q = 1; ; w0 ) be a basis for the initial condition where
eq denotes the orthonormal basis of input space W w0 . Let zz0q denote the output response
(z0 ) subject to initial condition xw0q of ^ with the 2 F 2 . Then, the robust H2 -norm is
equivalent to
w0
X
k^ k22 = sup2 sup kzz0q k22 :
(4.77)
2F 2S q=1
For computational simplicity, we use piecewise constant ^ 2 (), 2 () such that
XX
XX
^ 2 () =
ij ()^ 2ij ; 2 () =
ij ()2ij ;
m1 m2
m1 m2
i=1 j =1
i=1 j =1
(4.78)
where ^ 2ij = ^ T2ij and 2ij = T2ij (> 0).
Proposition 4.8 Assume Aij0 is asymptotically stable and that there exist Q, Mijk , Nijk ,
^ 2ij , 2ij (> 0), ;ij , positive-denite Pa1 (a = 1; m1 + 1) and P1b (b = 1; ; m2 + 1)
such that for all ij ,
2
3 2
T 2 C (w) P^ (w)Bp (w) + C (w)T ;
X 2
L
(
w;
)
+
C
(
w
)
4
5
+
wk Mijk < 0;
Bp(w)T P^ (w) + ;T C (w)
;^ 2
k
=1
ij
2
3 2
T (w)P^ (w)
Q
B
w0
4
5 + X wk2 Nijk > 0
P^ (w)Bw0 (w)
P^ (w)
ij
k=1
(4.79)
(4.80)
CHAPTER 4. ANALYSIS
75
for all (w; ) 2 and
2 T
3
A
P
+
P
A
P
B
ijk
ijk
ijk
ijk
pijk
4 ijk T
5 + Mijk 0;
Bpijk Pijk
2
4
0
(4.81)
3
0
BwT 0ijkPijk 5
+ Nijk 0
(4.82)
Pijk Bw0ijk
0
for all k. Then, the PALPV system with [D(^)]ij = 0 is uniformly asymptotically stable
about x = 0 for all 2 F 2 and its robust H2 -norm from w0 to z0 is less than TrQ, i.e.,
k^ k2 < pTrQ:
Proof: The proof is straightforward from the results of Propositions 4.4 - 4.6.
2
4.4 Generalized Results
We extend the results for s = 2. The extension is conceptually straightforward except the
construction of the PAL. We will show that the derived formulations are very similar to
counterparts of the simple case (s = 2).
4 i i . It is
For notational convenience, we make some denitions for index. l =
1
s
4
4
4
also dened that 1(s) = 1| {z 1} for s 1 with 1(0) = ;. For example, Xi1(2) = Xi11 and
4
Xi1(0) =
Xi .
s
4.4.1 PALPV system
We dene a PALPV system for the general case of s 3. The parameter space P is
partitioned into m1 ms number of closed hyper-rectangles with width 1 s, where k = (k ; k )=mk : In this case, each hyper-rectangle is represented by Pl
an appropriate
index. Given each subspace Pl , we introduce a local coordinate ^l =
hwith
i
T
^l1 ^ls (2 F^ s) measured from the center of Pl . Here, ^l is a bound function; that is,
^l : I ! F^s such that
F^ s =4 f^ 2 C 1(I ; Rs ) : ^(t) 2 P^ ; ^_ (t) 2 ; 8t 2 Ig
where P^ = [; 21 ; 21 ] [; 2s ; 2s ] and = [;1 ; 1 ] [;s ; s ].
CHAPTER 4. ANALYSIS
76
Note that the relation between (t) and ^l (t) is then given by
^l1 (t) = 1 (t) ; 1 ; (2i1 ; 1) 21
(4.83)
^ls (t) = s(t) ; s ; (2is ; 1) 2s :
(4.84)
For each Pl , an ALPV system is described by a set of nominal dynamics at the center of
the parameter subspace and ane parameter-dependent terms. A PALPV system is then
dened as a system that switches between ane linear parameter-varying (ALPV) systems:
for (x; ; w) 2 D F s W ,
2 3 m1 ms
2
32 3
^) B (^)
X
X
x
_
A
(
4 5 = l () 4
5 4 x 5;
^
^
(4.85)
2
3 2
3 s 2
3
^) B (^)
X
A
(
A
B
A
B
l
0
l
0
lk
lk
4
5 4
5 + ^lk 4
5:
^
^ =
(4.86)
z
where
i1 =1
is =1
C () D() l
Cl0 Dl0
The switching function l () is dened as
C () D()
k=1
l
w
Clk Dlk
8
<
l () = : 1 if (t) 2 Pl ; (P(i1 +1)is [ [ Pi1 (is +1) )
0 otherwise
4 ; for undened P . For simplicity, we dene sets of vertices of parameter
where Pl =
l
subspaces:
= (!1 ; ; !s ) : !k 2 ; 2k ; 2k
denotes the set of 2s vertices of P^ . Similarly,
= f(1 ; ; s ) : k 2 [;k ; k ]g
denotes the set of 2s vertices of .
CHAPTER 4. ANALYSIS
77
4.4.2 PAL with s 3
We extend the PAL to the general case of s 3. The PAL, V (x; ) = xT P ()x, is dened
such that
m1
ms
h i
X
X
P () = l () P^ (^) l
(4.87)
i1 =1
h^ ^ i
is =1
h
i
P () l = Co Pi1 is ; P(i1 +1)is ; ; P(i1 +1)is ; P(i1 +1)(is+1) ; the vertices of which
|
{zs
}
2
are positive-denite matrices dened counterclockwise from the corners of the rectangle Pl .
To ensure that [P^ (^)]l is ane in ^l , these matrices must satisfy the generalized
parallelogram rule such that
Pij ik + P(ij +1)(ik +1) = P(ij +1)ik + Pij (ik +1);
(4.88)
where ij (ij +1) and ik (ik +1) indicate the j -th and the k-th index, respectively. Note
that except both the j -th and the k-th, the other index is xed.
It will be convenient to express [P^ (^)]l with the same format of Eq. 4.86:
h^ ^ i
P () l = Pl0 +
where
and
s
X
^
k=1
lk Plk
(4.89)
Pl0 = 21 P(i1 +1)1(s;1) + Pi1 1(s;1) + + P1(k;1) (ik +1)1(s;k) + P1(k;1) ik 1(s;k) +
+ P1(s;1) (is +1) + P1(s;1) is ; 2(s ; 1)P1(s)
(4.90)
P1(k;1) (ik +1)1(s;k) ; P1(k;1) ik 1(s;k)
:
(4.91)
k
Therefore, constructing the [P^ (^)]l needs only Pa1(s;1) , , P1(s;1) b (a = 1; ; m1 + 1; ,
b = 1; ; ms + 1).
Plk =
CHAPTER 4. ANALYSIS
78
h
i
n
|
o
}
Note that for w 2 , P^ (w) l 2 Pi1 is ; P(i1 +1)is ; ; P(i1 +1)is ; P(i1 +1)(is +1) ;
where
{zs
2
Pi1 is = Pi1 1(s;1) + + P1(k;1) ik 1(s;k) + + P1(s;1) is ;(s ; 1)P1(s) :
|
{z
}
s
(4.92)
Derivations of Eq. 4.92
Evaluating Eq. 4.88 with each l, we can derive the following relation that
Pij ik = Pij 1 + P1ik ; P11:
(4.93)
Based on this relation, we can derive Eq. 4.92 by an inductive method. With s = 1, Eq. 4.92
is obviously true. Now, we assume that with s = k ; 1,
Pi1 i(k;1) = Pi1 1(k;2) + + P1(k;2) i(k;1) ;(k ; 2)P1(k;1) :
|
{z
}
k;1
Obviously,
Pi1 i(k;1) ik = Pi1 1(k;2) ik + + P1(k;2) i(k;1)ik ;(k ; 2)P1(k;1) ik :
|
Using Eq. 4.93,
{z
}
k;1
Pi1 ik = Pi1 1(k;1) + P1(k;1) ik ; P1(k) + + P1(k;2) i(k;1) 1 + P1(k;1) ik ; P1(k)
;(k ; 2)P1(k;1) ik
= Pi1 1(k;1) + + P1(k;1) ik ; (k ; 1)P1(k) :
Derivations of Eqs. 4.90 - 4.91
Pl0 is the center of [P^ (^)]l and then
i
h
Pl0 = 21s Pi1 is + P(i1 +1)is + + Pi1 (is+1) + P(i1 +1)(is +1) :
|
{zs
2
}
(4.94)
CHAPTER 4. ANALYSIS
79
Using Eq. 4.92, all the terms of Eq. 4.94 become
Pi1 is =
P(i1 +1)is =
=
Pi1 (is +1) =
P(i1 +1)(is +1) =
Pi1 1(s;1) + + P1(s;1) is ; (s ; 1)P1(s)
P(i1 +1)1(s;1) + + P1(s;1) is ; (s ; 1)P1(s)
Pi1 1(s;1) + + P1(s;1) (is +1) ; (s ; 1)P1(s)
P(i1 +1)1(s;1) + + P1(s;1) (is+1) ; (s ; 1)P1(s) :
The right side of Eq. 4.94 is equal to the average of the right sides of the above equations.
The summation of the right sides includes the 2s;1 number of Pi1 1(s;1) , P(i1 +1)1(s;1) , ,
P1(s;1) is and P1(s;1) (is +1) , respectively. It also includes the 2s number of ;(s ; 1)P1(s) . As
a result, we can show that Eq. 4.94 becomes
h
i
Pl0 = 21 Pi1 1(s;1) + P(i1 +1)1(s;1) + + P1(s;1) is + P1(s;1) (is +1) ; 2(s ; 1)P1(s) : (4.95)
Next, we derive Plk . Since Plk is the slope of [P^ (^)]l along the k axis,
Plk = Pi1 (ik +1)is ; Pi1 ik is =k
= P1(k;1) (ik +1)1(s;k) ; P1(k;1) ik 1(s;k) =k
(4.96)
by Eq. 4.92.
4.4.3 Analysis formulations
We can derive various analysis formulations for the generalized PALPV system using the
nonsmooth dissipative systems framework with the generalized PAL. The derivations are
exactly same as those of the sample cases. Thus, we present only the result of the Lyapunov
stability.
Let i1 = 1; ; m1 , , is = 1; ; ms and k = 1; ; s.
Proposition 4.9 Assume Al0 is asymptotically stable and that there exist Mlk (> 0),
positive-denite Pa1(s;1) (a = 1; m1 + 1), , P1(k;1) b1(s;k) (b = 1; ; mk + 1), ,
CHAPTER 4. ANALYSIS
80
P1(s;1) c (c = 1; ; ms + 1) such that for all l,
wk2 Mlk < 0
(4.97)
ATlk Plk + Plk Alk + Mlk 0
(4.98)
[L(w; )]l +
for all (w; ) 2 and
s
X
k=1
for all k. Then, the PALPV system (Eq. 4.85) is uniformly asymptotically stable about
x = 0 for all 2 F s.
Proof: The proof is exactly same as that of Proposition 4.1.
2
Chapter 5
Synthesis
In the previous section, we have derived various analysis formulations of PALPV systems
using the nonsmooth dissipative systems framework. Now, we address various synthesis
problems of PALPV systems. In general, a synthesis problem is considered a direct extension of an analysis problem because it begins with an analysis problem of a closed-loop
system. However, it involves more assumptions and technical steps associated with the
elimination and the construction of a unknown controller dynamics than the analysis problem. Furthermore, the synthesis needs to generalize the results in the previous chapter for a
general class of LPV systems and PDLFs. Hereby, this chapter addresses all the issues from
generalized analysis problems to the construction of a unknown controller dynamics. With
the dissipative systems framework, this chapter also fully characterizes several interesting
synthesis problems with the (almost) same procedure.
Throughout this chapter, we use both a quasi-piecewise-ane parameter-dependent linear parameter-varying (QPALPV) system and a continuous, quasi-piecewise-ane parameterdependent Lyapunov function (QPAL). The pair is recognized as a basic form which enables
us to derive (computationally tractable) nite-dimensional LMIs for synthesis problems of
LPV systems.
Note that most of the present results are extensions of the results for LTI systems and
linear dierential inclusions (LDIs) [BGFB94,SGC97] to PALPV systems using a Lipschitz
PDLF. Furthermore, these results generalize the result of [AA97] and fully characterize other
performance synthesis problems using a more general class of PDLFs than the quasi-ane
parameter-dependent Lyapunov function (QAL).
81
CHAPTER 5. SYNTHESIS
82
5.1 QPALPV System
5.1.1 PALPV System
We dene an open-loop PALPV system similar to that used in section 4.1 (Refer to section 4.1 for details). For simplicity, we consider the case of s = 2, i.e., (t) 2 F 2 . The
general case will be discussed in the last section of this chapter. Let u : I ;! U Ru .
The PALPV system can be dened such that for (x; ; w; u) 2 D F 2 W U ,
2
3 2 3
2 3
^) Bw1 (^) Bw2 Bu (^)
A
(
x
_
66
77 66 x 77
66 77 m m
66 z1 77 = X1 X2 () 66 Cz1(^) 0
0
0 77 66 w1 77
ij 6
77 66 w2 77 ;
66 z2 77
6
0
0
I
C
n
z
2
i
=1
j
=1
u
4
5 4 5
4 5
Cy (^)
y
where
2
^
66 A(^) T
66 Bw1()
66 Bu(^)T
66
4 Cz1(^)
Cy (^)
3 2
77 66 ATij0
77 66 Bw1ij0
T
77 = 66 Buij
0
77 66
5 4 Cz1ij0
ij
Cyij 0
0
Iny
0
3
2
77
66 ATijk
77 X
66 Bw1ijk
2
T
77 + ^ijk 66 Buijk
77 k=1 66
5
4 Cz1ijk
Cyijk
ij
(5.1)
u
3
77
77
77 :
77
5
(5.2)
For convenience, we dene some conventional matrices with wT = [w1T w2T ] and z T =
[z1T z2T ]:
h ^i h ^
i h
iT h
i
Bw () ij = Bw1() Bw2 ij ; Cz (^) ij = Cz1 (^)T CzT2 ij
and
2
3
2 3
2 3
0
0
0
T
5 ; [Dzu]ij = 4 5 ; [Dyw ]ij = 4 0 5 :
[Dzw ]ij = 4
Inu
Iny
[Dzw ]ij = 0 is assumed to simplify the derivation of the controller formula except the
passivity synthesis that requires the nonzero [Dzw ]ij . [Dzu ]ij and [Dyw ]ij are also normalized
to satisfy the rank conditions and simplify the central formula of the controller dynamics.
The normalization can be achieved by norm-preserving transformation using unitary matrices. Furthermore, [Dyu (^)]ij are assumed to be zero. However, the nonzero [Dyu (^)]ij term
can be easily included into the synthesis problem using the loop transformation [DV75].
0 0
CHAPTER 5. SYNTHESIS
83
5.1.2 LPV Controller
Let y : I ;! Y Ry . It is assumed that (t) and _(t) are measurable in real time.
Our problem is then to nd a strictly proper full-order LPV controller of the form that for
(xc ; ; y) 2 D F 2 Y ,
2 3 m1 m2
2
3 2 3
^; ^_ ) Bc(^)
X
X
x
_
A
(
c
c
4 5=
5 4 xc 5 ;
ij () 4
^
u
i=1 j =1
Cc()
0
ij
y
(5.3)
where [Ac (^; ^_ )]ij , [Bc (^)]ij and [Cc (^)]ij are continuous matrix functions on ^ij and ^_ ij .
However, these unknown controller dynamics are allowed to be discontinuous on the boundary of Pij .
Remark 5.1 The class of controllers (Eq. 5.3) could be conned to two special cases; (I)
the controller dynamics are parameter-independent; (II) the controller dynamics are anely
parameter-dependent. The case (I) does not require any measurement of (; _) to implement
the design controller so it leads to the simplest implementable LPV controller. Furthermore, the design controller also works well when the parameters are poorly measurable or
unmeasurable. In the linear robust control, this controller is often called a parametric robust controller [Ban97] when _ = 0. However, it signicantly limits the performance of the
closed-loop system because of the limited class of the controller dynamics and the PDLF used
in the synthesis. Meanwhile, the case (II) has been widely used in the LPV control using
the LFT- approach [AG95, Hel95, Pac94]. It improves the performance of the closed-loop
system with relatively simple implementation. However, the limitation on the performance
still remains compared to the general case such as Eq. 5.3. As it goes on, we observe the
no free lunch theorem that a controller with better performance needs more information
about the system and/or complicated implementation. In our thesis, we separate the implementation issue from the synthesis problem. In the synthesis, we then design the LPV
controller to achieve the best performance of the closed-loop system.
Now, we close the PALPV system (Eq. 5.1) with the LPV controller (Eq. 5.3). In this
thesis, the closed-loop system is called a quasi-piecewise-ane parameter-dependent linear
parameter-varying system (QPALPV). This QPALPV system will be repeatedly used in
various synthesis problems later.
CHAPTER 5. SYNTHESIS
84
X32
X ( )
X33
X23
X13
X31
X22
X12
X21
X11
P12
2
P22
P11
P21
1
Fig. 5.1: Example of a continuous, piecewise-ane X () with s = 2. Y () is also similarly
dened with Yij 's. For simplicity, each parameter space is only split into 2 regions
Let xTcl = [xT xTc ] and Dcl R2n . The QPALPV system is then dened such that for
(xcl ; ; w) 2 Dcl F 2 W ,
2 3 m1 m2
2
3 2 3
_
^
^
^
X
X
x
_
A
(
;
)
B
(
)
cl
4 cl 5 =
5 4 xcl 5 ;
ij () 4 cl ^
z
where
i=1 j =1
Ccl()
0
ij
w
2
3
^
^
^
A
(
)
B
(
)
C
(
)
u
c
5
Acl (^; ^_ ) = 4 ^ ^
_
^
^
ij
Bc()Cy () Ac(; ) ij
2
3
h ^i
^
B () Bw2 5
Bcl() ij = 4 w1
0
Bc(^) ij
2
3
h ^i
^) 0
C
(
z
1
5 :
Ccl() = 4
ij
Cz2
Cc(^)
ij
(5.4)
(5.5)
(5.6)
(5.7)
CHAPTER 5. SYNTHESIS
85
5.2 QPAL
To design the LPV controller (Eq. 5.3), we use a continuous, quasi-piecewise-ane parameterdependent Lyapunov function (QPAL) which is a Lipschitz PDLF. The QPAL, V (xcl ; ) =
xTcl Pcl ()xcl , is dened that
Pcl () =
=
m1 X
m2
X
i=1 j =1
m1 X
m2
X
i=1 j =1
h
i
ij () P^cl (^)
ij
2 ^^
X ()
ij () 4
T
2
X ( )
= 4
T
M ( )
M^ (^)
3
(5.8)
M^ (^) 5
;1
M^ (^)T X^ (^) ; Y^ ;1(^) M^ (^) ij
3
M ( )
5
;
T
M () X () ; Y ;1 () ;1 M ()
Continuous, piecewise-ane X (): over Pij ,
h^ ^i
X () ij = Xij 0 +
2
X
^
k=1
ijk Xijk
(5.9)
with
Xij 0 = 21 (X(i+1)1 + Xi1 + X1(j+1) + X1j ; 2X11 );
X
; Xi1 ; X = X1(j+1) ; X1j :
Xij1 = (i+1)1
ij 2
1
2
Continuous, piecewise-ane Y (): over Pij ,
h^ ^ i
Y () ij = Yij0 +
2
X
^
k=1
ijk Yijk
(5.10)
with
Yij 0 = 21 (Y(i+1)1 + Yi1 + Y1(j +1) + Y1j ; 2Y11 );
Y
; Yi1 ; Y = Y1(j+1) ; Y1j :
Yij1 = (i+1)1
ij 2
1
2
CHAPTER 5. SYNTHESIS
86
Continuous, piecewise-smooth M () such that over Pij ,
h ^ ^ ^ ^ i h ^ ^ ^ ^ Ti
I ; X ()Y () ij = M ()N () ij :
Note that as in section 4.2, [X^ (^)]ij is originally dened as follows:
h
i
[1] X^ (^) ij = Co[Xij ; X(i+1)j ; X(i+1)(j +1) ; Xi(j +1) ]; the vertices of which are positivedenite matrices dened counterclockwise from the corners of the rectangle Pij (see
Figure 5.1).
[2] To ensure that [X^ (^)]ij is ane in ^ij , these matrices must satisfy
Xij + X(i+1)(j +1) = X(i+1)j + Xi(j+1) ;
(5.11)
which is the parallelogram rule.
As in section 4.2, these conditions then lead to Eq. 5.9. [Y^ (^)]ij can be similarly derived.
Note that for w 2 ,
h^
i n
o h
i n
o
X (w) 2 Xij ; X(i+1)j ; X(i+1)(j +1) ; Xi(j+1) ; Y^ (w) 2 Yij ; Y(i+1)j ; Y(i+1)(j +1) ; Yi(j +1) ;
where Xij = Xi1 + X1j ; X11 and Yij = Yi1 + Y1j ; Y11 , respectively.
Remark 5.2 Pcl () is parameterized by X () and Y () as shown in Lemma 2.9. X () and
Y () are (1,1) block elements of Pcl () and Pcl ();1 , respectively. In fact, the synthesis
problem involves only X () and Y () after the elimination step (we will clearly show this
later): when eliminating unknown controller dynamics from the synthesis formulation, (1,2),
(2,1) and (2,2) block elements of Pcl () and Pcl ();1 , which are related to the unknown
controller dynamics, are also eliminated. Therefore, it is enough to assume that X () and
Y () are continuous, piecewise-ane for deriving nite-dimensional LMIs for the synthesis
problem.
Remark 5.3 Each [P^cl (^)]ij is dierentiable. Pcl () is also continuous over the original
parameter space (P ). Therefore, Pcl () does have the Dini-derivative over P and hence
Pcl () is Lipschitz on [KF70]. The QPAL is a Lipschitz PDLF.
CHAPTER 5. SYNTHESIS
87
5.3 Synthesis Formulations
The problem we consider here is to design an LPV controller (Eq. 5.3) that minimizes an
upper bound of the system gain { such as L2 -gain { of the closed-loop system (Eq. 5.4).
The synthesis follows the standard procedure:
[Step I]: Formulate the analysis problem of the QPALPV system. The analysis formulations
consist of sets of (innite-dimensional) LMIs as many as the number of partitions. For
the next steps, consider (innite-dimensional) LMIs over each partitioned subspace.
[Step IIa]: Eliminate unknown controller dynamics from the analysis formulation through
the following steps
Apply the congruence transformation (Lemma 2.6) to the LMIs and express them
in terms of [X^ (^)]ij , [Y^ (^)]ij and intermediate controller variables, i.e., [A^c (^)]ij ,
[B^c (^)]ij and [C^c (^)]ij .
Eliminate these intermediate controller variables using the standard Elimination
Lemma (Lemma 2.7). However, some intermediate controller variables may remain in the formulations. In this case, these remaining variables are assumed to
be piecewise-ane in .
[Step IIb]: \Convexify" the LMIs of [Step IIa] to derive nite-dimensional LMIs and
solve them by LMI solvers. Our approach uses the `multi-convexity' requirement
(Lemma 2.10).
[Step III]: Construct an LPV controller dynamics analytically from the results of [Step
IIb].
This standard procedure is an extension of the procedure for the synthesis of an LTI
system [Gah96, SGC97] to PALPV systems using a Lipschitz PDLFs. Furthermore, this
procedure is a generalization of [GAC96] using a nonsmooth dissipative systems framework.
With the dissipative systems framework, the synthesis step can be applied to various synthesis problems (The following sections explicitly demonstrate this point). Therefore, we
are enough to understand the derivation of the L2 -gain synthesis for all the derivations.
For completion, we, however, include all the details of several synthesis problems in the
following sections.
CHAPTER 5. SYNTHESIS
88
For simplicity, we dene
Lcl (^; ^_ )
ij
= Acl (^; ^_ )T P^cl (^) + P^cl (^)Acl (^; ^_ ) + P^_ cl (^)
ij
ij
LX (A; ^; ^_ )
= X^ (^)A(^) + A(^)T X^ (^) + X^_ (^)
LY (A; ^; ^_ )
= A(^)Y^ (^) + Y^ (^)A(^)T ; Y^_ (^)
ij
ij
ij
LXI (^; ^_ )
LY I (^; ^_ )
ij
= LX (A; ^; ^_ ) + B^c (^)Cy (^) + Cy (^)T B^c (^)T
= LY (A; ^; ^_ ) + Bu(^)C^c (^) + C^c (^)T Bu (^)T
(5.12)
ij
(5.13)
(5.14)
ij
ij
(5.15)
:
(5.16)
For the congruent transformation, we dene nonsingular matrices such that
2
3
h ^i
^ (^) I
Y
S1 () ij = 4 ^ ^ T 5
N ( )
These matrices satisfy
0 ij
2
3
h ^i
^ (^)
I
X
5 :
and S2 () ij = 4
0 M^ (^)T
ij
(5.17)
h^ ^ ^ i h ^ i
Pcl ()S1 () ij = S2 () ij :
Let i = 1; ; m1 , j = 1; ; m2 and k = 1; 2 for all synthesis problems.
5.3.1
L2-Gain
We derive an upper bound of the L2 -gain (2 ) of the QPALPV system (Eq. 5.4) based on
Proposition 3.2.
Lemma 5.1 Suppose there exist dierentiable functions [P^cl (^)]ij 's such that for all ij ,
h^ ^ i
Pcl () ij > 0
and
2
_
66 Lcl(^; ^) P^cl (^)Bcl (^) Ccl (^)T
64 Bcl (^)T P^cl (^)
;I
0
Ccl (^)
0
;I
(5.18)
3
77
75 < 0
ij
(5.19)
for all (^ij ; ^_ ij ) 2 P^ . Then, the QPALPV system (Eq. 5.4) is uniformly asymptotically
stable about x = 0 for all 2 F 2 and its L2 -gain (2 ) is less than , i.e., 2 < .
CHAPTER 5. SYNTHESIS
89
Proof: An upper bound of the L2-gain is given in Proposition 3.2. Both the QPALPV
system and the QPAL are used for Proposition 3.2. First, consider condition (II) of Proposition 3.2. Partition the parameter space (P ) into subspaces (Pij 's). The interval (I ) is
then partitioned:
[
I = Iij = ftj(t) 2 Pij g :
ij
Consider each Iij . It is assumed that by a smooth extension, the QPALPV system (Eq. 5.4)
and the QPAL (Eq. 5.8) are dened over the smallest open set that includes Pij . This
assumption makes xcl and V (xcl ; ) dierentiable on this sub-domain (possibly admittedly
with some conservatism). According to Lemma 2.1, condition (II) over each Iij becomes
D+V (xcl ; )(x_ cl ; 0) + D+V (xcl ; )(0; _) ; 2 wT w + zT z < 0:
(5.20)
Furthermore, the partial Dini-derivative is just the partial derivative because V (xcl ; ) is
dierentiable. Thus,
2 T^ ^ 3
@ xcl Pcl ()xcl
@ xTcl P^cl (^)xcl ^_ 2 T
Eq. 5.20 = 4
x_ cl +
; w w + zT z5
^
@xcl
@
ij
_
= xTcl Lcl (^; ^)xcl + xTcl P^cl (^)Bcl (^)w + wT Bcl (^)T P^cl (^)xcl
i
; 2wT w + xTclCcl (^)T Ccl(^)xcl ij < 0:
Therefore, condition (II) is satised if for all ij ,
2
_
T
66 Lcl (^; ^) P^cl (^)Bcl (^) Ccl (^)
64 Bcl(^)T P^cl (^)
;I
0
^
Ccl ()
0
;I
3
77
75 < 0; 8(^ij ; ^_ ij ) 2 P^ :
(5.21)
(5.22)
ij
Second, condition (I) of Proposition 3.2 is obviously satised if for all ij ,
h^ ^ i
Pcl () ij > 0; 8^ij 2 P^ :
2
CHAPTER 5. SYNTHESIS
90
Since Eq. 5.19 includes unknown controller dynamics, we need to eliminate the controller dynamics from the formulation. The elimination step is comprised of the congruent
transformation (Lemma 2.6) and the elimination lemma (Lemma 2.7). A similar process has
already been applied to the H1 synthesis of the LTI system [Gah96,SGC97] and the LPV
system [AA98]. Using the congruence transformation, Lemma 5.1 is equivalently restated
as follows.
Lemma 5.2 Suppose there exist [X^ (^)]ij , [Y^ (^)]ij , [A^c (^)]ij , [B^c(^)]ij and [C^c(^)]ij 's such
that for all ij ,
2
3
^ (^) I
X
4
5 >0
(5.23)
I Y^ (^) ij
and
2
0
1
0
13
T
^
^
^
^
66 LY I (^; ^_ ) @ A()+T A
@ Y ()CzT()T + A 77
Bw (^)
^
^
66
77
Ac()
C^c(^) Dzu
0
1
66
77
X^ (^)Bw (^)+ A
_
T
66 77 < 0
^
^
^
@
LXI (; )
C z ( )
66
77
B^c(^)Dyw
66 77
;I
0
4
5
;I
ij
(5.24)
for all (^ij ; ^_ ij ) 2 P^ . Then, the QPALPV system (Eq. 5.4) is uniformly asymptotically
stable about x = 0 for all 2 F 2 and its L2 -gain (2 ) is less than , i.e., 2 < .
Proof: Perform the congruence transformation on Eqs. 5.18 { 5.19 for each ij . First,
consider Eq. 5.18. According to Lemma 2.6,
h^ ^ i
h
i
Pcl () ij > 0 , S1(^)T P^cl (^)S1 (^) ij > 0
2
3
^ (^) I
Y
5 > 0:
, 4
I
X^ (^)
ij
CHAPTER 5. SYNTHESIS
91
Next, consider Eq. 5.19. Similarly,
2
3T
2
3
^
^
S
(
)
0
0
S
(
)
0
0
66 1
77
66 1
7
Eq. 5.19 < 0 , 64 0 I 0 75 (Eq. 5.19) 64 0 I 0 775 < 0:
0
0 I ij
0
0 I ij
After some matrix manipulations, we can obtain
2
0
13
T
^
^
^
Y
(
)
C
(
)
+
z
66 T1 (^; ^_ ) T2 (^; ^_ )
@
A 77
Bw (^)
T
^
^)T Dzu
^
66
77
N
(
)
C
(
c
0
1
66
77
T
66 T3 (^; ^_ ) @ X^ (^)Bw (^)+ A
77 < 0
^
C
(
)
z
66
77
M^ (^)Bc(^)Dyw
66 77
;I
0
4
5
;I
ij
where
T1(^; ^_ )
(5.25)
= LY (A; ^; ^_ ) + N^ (^)Cc (^)T Bu (^)T + Bu (^)Cc (^)N^ (^)T
(5.26)
ij
ij
_
^
^
T2(; ) = A(^) ; Y^_ (^)X^ (^) ; N^_ (^)M^ (^)T + Y^ (^)A(^)T X^ (^) + N^ (^)Ac (^; ^_ )T M^ (^)T
ij
T3(^; ^_ )
ij
i
+N^ (^)T Cc(^)T Bu (^)T X^ (^) + Y^ (^)Cy (^)T Bc(^)T M^ (^)T ij
= LX (A; ^; ^_ ) + Cy (^)T Bc(^)T M^ (^)T + M^ (^)Bc (^)Cy (^) :
ij
(5.27)
(5.28)
We dene intermediate controller variables such that
h^ ^i
Ac() ij = ;X^ (^)Y^_ (^) ; M^ (^)N^_ (^)T + X^ (^)A(^)Y^ (^) + X^ (^)Bu (^)C^c(^)
_
+B^c(^)Cy (^)Y^ (^) + M^ (^)Ac (^; ^)N^ (^)T
h^ ^i
h
i
Bc() ij = M^ (^)Bc(^) ij
h^ ^ i
h
i
Cc() ij = Cc(^)N^ (^)T ij :
Then, Eq. 5.25 immediately implies Eq. 5.24 for each ij .
ij
(5.29)
(5.30)
(5.31)
2
CHAPTER 5. SYNTHESIS
92
Some intermediate controller variables of Lemma 5.2 will be even further eliminated
using the elimination lemma.
Lemma 5.3 Suppose there exist [X^ (^)]ij and [Y^ (^)]ij 's such that for all ij ,
2
4 X^ (^)
I
and
3
I 5
> 0;
Y^ (^) ij
(5.32)
2
3
_
T
T
^
^
^
^
^
^
^
^
^
L
(
A;
;
)
;
B
(
)
B
(
)
Y
(
)
C
(
)
B
(
)
u
u
z1
w
66 Y
77
^
^
^
64
Cz1()Y ()
;I
0 75 < 0
Bw (^)T
0
;I ij
2
3
_
T
T
^
^
^
^
^
^
^
~
^
66 LX (A; ; ) ; Cy () Cy () X ()Bw1() Cz () 77
64
Bw1 (^)T X^ (^)
;I
0 75 < 0
Cz (^)
0
;I ij
for all (^ij ; ^_ ij ) 2 P^ , where
h^ ^i
h
A() ij = A(^) ; Bu (^)Cz2
i
and
ij
h~ ^i
h
(5.33)
(5.34)
i
A() ij = A(^) ; Bw2 Cy (^) ij :
Then, the QPALPV system (Eq. 5.4) is uniformly asymptotically stable about x = 0 for
all 2 F 2 and its L2 -gain (2 ) is less than , i.e., 2 < .
Proof: The result is directly obtained by applying Lemma 2.7 (Elimination Lemma) to
Eq. 5.24. For convenience, dene some matrices such that
2
LY (A; ^; ^_ )
A(^)
Bw (^) Y^ (^)Cz (^)T
6
6
_
^T
^ ^_ ^ ^
^
^T
6
G(^; ^) = 66 A(^) T LX (^A;T ^; ^) X ()Bw () Cz ()
64 Bw () Bw () X () ;I
ij
0
^
^
^
^
Cz ()Y ()
Cz ( )
0
;I
2
3
2
3
0 Bu(^)
I
0
66
7 h i 66
h ^i
^)T 777
I
0 77
0
C
(
y
6
6
U () ij = 66
;
77 ; V (^) ij = 66
T 7
7
0
0
0
D
yw
4
5
4
5
0 Dzu
ij
0
0
ij
3
77
77 ;
77
5
ij
CHAPTER 5. SYNTHESIS
93
2
3
^c (^) B^c (^)
A
5 :
= 4 ^ ^
h^ ^i
K () ij
Cc()
0
ij
A simple matrix manipulation shows that
Eq. 5.24 = G(^; ^_ ) + U (^)K^ (^)V (^)T + V (^)K^ (^)T U (^)T
ij
< 0:
According to Lemma 2.7, the above inequality is equivalent to a pair of LMIs that
U?(^)T G(^; ^_ )U?(^) < 0 and V?(^)T G(^; ^_ )V?(^) < 0:
ij
ij
(5.35)
Here, [U?(^)]ij and [V?(^)]ij are orthogonal complements of [U (^)]ij and [V (^)]ij , respectively:
2
66 0
h ^ i 666 0
U?() ij = 66 2 0 3 2
66 I
44 5 4
0
2
3
3
0
0
0
66
7
I
07
0
I
0 77
7
6
77
0
0 77 h
i 666 2 3 2
7
77 :
^
3
0
3 I 777 ; V?() ij = 666 I
7
0
4 5 4
5 0 77
7
6
0
5 05
64 0
75
;Cy (^)
;Bu(^)T
ij
0
0
I
ij
After calculating Eq. 5.35 with these [U?(^)]ij and [V?(^)]ij , we can eliminate unknown
intermediate controller dynamics [A^c (^)]ij , [B^c (^)]ij and [C^c (^)]ij and then derive Eqs. 5.33 5.34.
2
Each of Eqs. 5.32 - 5.34 leads to an innite number of LMIs. Therefore, we need the
\convexifying" step to derive a nite number of LMIs. For notational convenience, we dene
h^ ^i
A() ij = A^ij0 +
2
2
h i
X
X
^ijk A^ijk and A~(^) ij = A~ij 0 + ^ijk A~ijk :
k=1
k=1
(5.36)
CHAPTER 5. SYNTHESIS
94
Proposition 5.1 Suppose there exist Mijk ; Nijk (> 0) (Xa1 ; Ya1 ) (a = 1; ; m1 + 1) and
(X1b ; Y1b ) (b = 1; ; m2 + 1) such that for all ij ,
2
3
^ (w) I
X
4
5 > 0;
(5.37)
I
Y^ (w)
ij
2
3 2
^ w; ) ; Bu (w)Bu (w)T + 1=Bw (w)Bw (w)T Y^ (w)Cz1 (w)T
L
(
A;
Y
4
5 +X wk2 Mijk < 0;
Cz1(w)Y^ (w)
;I
ij k=1
(5.38)
2
3 2
T
T
4 LX (A;~ w; ) ; Cy (w) CyT(w) + 1=Cz (w) Cz (w) X^ (w)Bw1 (w) 5 + X wk2 Nijk < 0
Bw1 (w) X^ (w)
;I
ij k=1
(5.39)
for all (w; ) 2 and
2
3
^Tijk + A^ijk Yijk Yijk CzT1ijk
Y
A
ijk
4
5 + Mijk 0;
(5.40)
2 T
3
~ijk Xijk + Xijk A~ijk Xijk Bw1ijk
A
4
5 + Nijk 0
T
(5.41)
Cz1ijk Yijk
0
Bw1ijk Xijk
0
for all k. Then, the QPALPV system (Eq. 5.4) is uniformly asymptotically stable about
x = 0 for all 2 F 2 and its L2 -gain (2 ) is less than , i.e., 2 < .
Proof: Since Eq. 5.32 is ane in ^ij , it is obvious that
3
2
^ (w ) I
X
5 > 0; 8w 2 :
Eq. 5.32, 8^ij 2 Pij , 4
^
I
Y (w )
ij
Consider Eq. 5.33 (Note that Eq. 5.34 can be similarly treated). Eq. 5.33 is equivalent to
20
1 3
T
2
3
^
^
;
B
(
)
B
(
)
u
u
_
66 @
A 0 77
^ ^; ^_ ) Y^ (^)Cz1 (^)T
L
(
A;
Y
^
^
4
5
^
^)T
G(; ) =
+
6
75 < 0
+1
=B
(
)
B
(
w
w
4
ij
Cz1 (^)Y^ (^)
;I
ij
0
0
ij
(5.42)
CHAPTER 5. SYNTHESIS
95
for all (^ij ; ^_ ij ) 2 Pij . According to Lemma 2.10, Eq. 5.42 is always true on all
(^ij ; ^_ ij ) 2 Pij if
[G(w; )]ij +
2
X
k=1
wk2 Mijk < 0; 8(w; ) 2 (5.43)
2
3
^Tijk + A^ijk Yijk Yijk CzT1ijk
Y
A
ijk
4
5 + Mijk 0; k = 1; 2:
T
and
(5.44)
Cz1ijk Yijk
0
Note that the coupling terms of the second matrix of Eq. 5.42 can be eliminated by the
schur complement so that they do not have the ^k2 terms.
2
As minimizing , we compute the smallest upper bound of L2 -gain of the QPALPV
system (Eq. 5.4). This smallest (opt ) is often used as the L2 -gain. This minimization
problem is a standard convex optimization problem. Furthermore, we can construct X ()
and Y () with solutions of Proposition 5.1.
A central controller formula is constructed in terms of the PALPV system dynamics
(Eq. 5.1) and solutions of Proposition 5.1 ( , X () and Y ()). A similar process has
already been applied to the H1 synthesis of the LTI system [Gah96,SGC97] and the LPV
system [AA98].
Proposition 5.2 Suppose , [X^ (^)]ij and [Y^ (^)]ij are solutions of Proposition 5.1. Then,
a central controller of Eq. 5.3, which makes the L2 -gain of the QPALPV system (Eq. 5.4)
less than , is given as follows: for (t) 2 Pij ,
Ac (^; ^_ )
= M^ (^);1 A^c (^) + X^ (^)Y^_ (^) + M^ (^)N^_ (^)T ; X^ (^)A(^)Y^ (^)
ij
i
;X^ (^)Bu (^)C^c(^) ; B^c(^)Cy (^)Y^ (^) N^ (^);T
(5.45)
ij
h ^i
h ^ ^ ;1 ^ ^ i
Bc() = M () Bc () ij
h ^ iij
h
i
Cc() ij = C^c(^)N^ (^);T ij
h
i h
i
where M^ (^)N^ (^)T ij = I ; X^ (^)Y^ (^) ij and
h^ ^i
h
Ac () ij = ;A(^)T ; ;1 X^ (^)Bw1 (^)Bw1 (^)T + Cy (^)T BwT 2
(5.46)
(5.47)
CHAPTER 5. SYNTHESIS
h^ ^ i
Bc() ij
h^ ^ i
Cc() ij
96
; ;1Cz1(^)T Cz1(^)Y^ (^) + Cz2(^)T Bu(^)T
h
i
= ; CyT (^) + X^ (^)Bw2 ij
h
i
= ; BuT (^) + Cz2 Y^ (^) ij :
i
ij
(5.48)
(5.49)
(5.50)
Proof: It is enough to derive Eqs. 5.48 { 5.50 because Eqs. 5.45 { 5.47 are directly obtained
by Eqs. 5.29 { 5.31. The derivations of Eqs. 5.48 { 5.50 are based on the approach of [Gah96].
We dene some matrices
h ^i
h
i
T
L1 () ij = Bw (^) Y^ (^)Cz (^)T + C^c(^)T Dzu
ij
h ^i
h
i
L2 () ij = X^ (^)Bw (^) + B^c (^)Dyw Cz (^)T ij
2
3
I
0
5:
= 4
0 I
Then,
2
3 2
3
2
3T
_
T
^
^
^
^
^
^
^
L
(
;
)
A
(
)
+
A
(
)
L
(
)
L
(
)
c
5 + 4 1 5 ;1 4 1 5 < 0
Eq. 5.24 = 4 ^ YTI ^ ^
_
^
^
L2 (^) ij
L2(^) ij
A() + Ac () LXI (; ) ij
2
3
^; ^_ ) + L1 (^);1 L1 (^)T A(^) + A^c (^)T + L1 (^);1 L2 (^)T
L
(
Y
I
5 < 0:
= 4
_
;
1
T
^
^
^
^
LXI (; ) + L2 () L2 ()
ij
The idea is to nd [A^c (^)]ij , [B^c (^)]ij , and [C^c (^)]ij such that make (1; 2) = 0, (2; 2) < 0
and (1; 1) < 0 (For example, (1; 2) indicates the (1; 2) block element of the above matrix).
First, consider the condition (2; 2) < 0. With a x and any , let
2 3T
2 3
x
x
(; B^c (^)) = 4 5 (2; 2) 4 5 :
x
x
2 3T 2
_ ) L2 (^) 3 2 x 3
^
^
x
L
(
;
XI
5 4 5<0
(; B^c (^)) = 4 5 4
x
L2 (^)T ; ij x
CHAPTER 5. SYNTHESIS
97
2
3
2
3
2
3
T
D
0
0
yw
7
^c(^)T x 66
^c (^)T x
B
B
7
T ;I
5 6 Dyw
5
= 4
0 75 4
4
x
x
ij
ij
0
0 ;I ij
2
3T
^
C
(
)
x
y
66
77 2 B^c(^)T x 3
5
+ 2 64 Bw (^)T X^ (^)x 75 4
x
ij
Cz (^)x
ij
_
+ xT LX (A; ^; ^) x < 0:
ij
(5.51)
As shown in [Gah96], a [B^c (^)]ij satisfying (; B^c (^)) is the saddle point ([B^c (^)]ij )
that makes the rst gradient of (; B^c (^)) zero:
2
66 0 Dyw
T ;I
64 Dyw
0
0
0
0
;I
3
2
2
3
77 B^c(^)T
66 Cy T(^)
75 4 5 = ; 64 Bw (^) X^ (^)
ij
ij
Cz (^)
3
77
75 :
(5.52)
ij
Note that with = , ( ; B^c (^)) = ( ; B^c (^)). Since ( ; B^c (^)) < 0, ( ; B^c (^)) < 0.
Similarly, we can derive the following equation:
2 3T
2 3
2 3T 2
3 2 3
^; ^_ ) L1 (^)
x
x
x
L
(
Y
I
4 5 (1; 1) 4 5 = 4 5 4
5 4 x 5<0
T
^
x
x
x
L1 ()
; ij x
2
3
T
3T 6 0 0 Dzu
2
77 2 C^c(^)x 3
^c(^)x 6
C
5 6 0 ;I 0 7 4
5
= 4
5
x ij 4
x ij
Dzu 0 ;I ij
2
3
3
^)T x T 2
B
(
u
66
7
^c(^)x
C
7
5
+ 2 64 Bw (^)T x 75 4
x
ij
Cz (^)Y^ (^)x ij
_
+ xT LY (A; ^; ^) x < 0:
ij
(5.53)
CHAPTER 5. SYNTHESIS
98
Similarly, a [C^c (^)]ij satisfying Eq. 5.53 is the saddle point ([C^c (^)]ij ):
3
2
66 0
64 0
2
T
T
0 Dzu
77 2 C^c(^) 3
66 Bu(^) T
;I 0 75 4 5 = ; 64 Bw (^)
ij
Dzu 0 ;I ij
Cz (^)Y^ (^)
3
77
75 :
(5.54)
ij
The condition (1; 2) = 0 immediately implies
h^ ^i
h
i
Ac() ij = ;A(^)T ; L2(^);1 L1 (^)T ij
2
20
33
1T 3T 2
T
^
^
^
^
66
66 @ X ()Bw () A 77 66 0 Bw () 1 7777
T
;
1
^
6
= 6;A() ; 64 +B^c(^)Dyw
:
75 64 @ Cz (^)Y^ (^) A 7577 (5.55)
4
5
^ ^
^
Cz ()
+Dzu Cc ()
ij
After calculating Eqs. 5.52, 5.54 and 5.55, we can obtain Eqs. 5.48 { 5.50.
2
Remark 5.4 Eq. 5.45 includes the ^_ ij (t) (or _(t)). This ^_ ij (t) may be dicult to estimate
in the real world, when ^ij (t) is noisy and quickly varying. To eliminate ^_ ij (t) from the
formulation, several approaches have been proposed such as the BMI formulation [Beck96],
the coordinate transformation [Woo95]. Filtering techniques could be also incorporated to
accurately estimate ^_ ij (t), when ^ij (t) is slowly varying [LeeL97]. One of the most popular
ways is to constraint X () or Y () to constant [AA98], admittedly with some conservatism.
The following relation is useful such that
X^ (^)Y^_ (^) + M^ (^)N^_ (^)T
h
i
h
i ij
_
= ; X^ (^)Y^ (^) + M^_ (^)N^ (^)T
ij
(5.56)
I ; X^ (^)Y^ (^) ij = M^ (^)N^ (^)T ij . Note that this argument can be valid for other synthesis problems which will be discussed later.
5.3.2
L1-Gain
The problem we consider here is to design an LPV controller (Eq. 5.3) that minimizes an
upper bound of the L1 -gain (1 ) of the closed-loop system (Eq. 5.4). In contrast to the
CHAPTER 5. SYNTHESIS
99
L2-gain, it will be assumed that [C^c(^)]ij is ane on ^ij , i.e.,
2
h^ ^ i ^
X
Cc() ij = Ccij 0 + ^ijk C^cijk :
(5.57)
k=1
The synthesis follows the previous standard approach. We refer to section 5.3.1 for detailed
explanation.
Lemma 5.4 Suppose there exist positive , ( < ) and dierentiable functions [P^cl (^)]ij 's
such that for all ij ,
2
3
_
^
^
^
^
^
^
^
L
(
;
)
+
P
(
)
P
(
)
B
(
)
cl
cl
cl
4 cl T
5 <0
^
^
^
Bcl ()Pcl ()
;I
ij
(5.58)
2
3
^cl (^) Ccl (^)T
P
4
5 >0
^
and
Ccl()
I
(5.59)
ij
for all (^ij ; ^_ ij ) 2 P^ . Then, the QPALPV system (Eq. 5.4) is uniformly asymptotically
stable about x = 0 for all 2 F 2 and its L1 -gain (1 ) is less than , i.e., 1 < .
Proof: The proof is similar to that of Lemma 5.1. An upper bound of the L1-gain can
be derived using Proposition 3.3. Both the QPALPV system and the QPAL are used for
Proposition 3.3. Consider condition (II) of Proposition 3.3. As in the proof of Lemma 5.1,
condition (II) over each Iij becomes
D+V (xcl ; )(x_ cl ; 0) + D+V (xcl ; )(0; _) + V (xcl ; ) ; wT w < 0:
Since V (xcl ; ) is dierentiable,
(5.60)
2 T^ ^ 3
@ xcl Pcl ()xcl
@ xTcl P^cl (^)xcl ^_
Eq. 5.60 = 4
x_ cl +
+ xTcl P^cl (^)xcl ; wT w5
@xcl
@ ^
ij
_
= xTcl Lcl (^; ^) + P^cl (^) xcl + 2xTcl P^cl (^)Bcl (^)w ; wT w < 0: (5.61)
ij
CHAPTER 5. SYNTHESIS
100
Therefore, condition (II) is satised if for all ij ,
2
3
^; ^_ ) + P^cl (^) P^cl (^)Bcl (^)
L
(
cl
4
5 < 0; 8(^ij ; ^_ ij ) 2 P^ :
T
Bcl (^) P^cl (^)
;I
ij
(5.62)
As in Proposition 4.3, an upper bound ( ) of the L1-gain can be derived from the
following inequality that
zT z < V (xcl ; ) + ( ; ) wT w :
This inequality is satised if for all ij ,
h
(5.63)
i
xTclP^cl (^)xcl + ( ; ) wT w ; ;1 xTcl Ccl (^)T Ccl (^)x ij > 0:
Furthermore, Eq. 5.64 is equivalent to
2
0
Ccl (^)T
66 P^cl (^)
64 0 ( ; )I 0
Ccl (^)
0
I
3
77
75 > 0:
(5.64)
(5.65)
ij
As a result, the L1 gain (1 ) is less than , if Eq. 5.65 is satised for all ^ij 2 P^ and all
ij .
Condition (I) is satised if [P^ (^)]ij > 0 for all ^ 2 P^ and all ij . However, it is redundant
because Eq. 5.65 implies [P^ (^)]ij > 0.
2
Lemma 5.4 is equivalently restated using the congruence transformation.
Lemma 5.5 Suppose there exist , ( < ), [A^c(^)]ij , [B^c(^)]ij , [C^c(^)]ij , positive-denite
[X^ (^)]ij and [Y^ (^)]ij 's such that for all ij ,
20
1
_
^
^
L
(
;
)
66 @ Y I
A
66 +Y^ (^)
66
66
64
3
0
1
^) + I
A
(
77
@
A
Bw (^)
^c(^)T
+
A
0
1 0
1 777
_
@ LXI (^; ^) A @ X^ (^)Bw (^)+ A 777 < 0
75
B^c (^)Dyw
+X^ (^)
;I
ij
(5.66)
CHAPTER 5. SYNTHESIS
and
101
2
T
T T
66 Y^ (^) I Y^ (^)Cz (^) + CT^c(^) Dzu
64 X^ (^)
Cz (^)
I
3
77
75 > 0
(5.67)
ij
for all (^ij ; ^_ ij ) 2 P^ . Then, the QPALPV system (Eq. 5.4) is uniformly asymptotically
stable about x = 0 for all 2 F 2 and its L1 -gain (1 ) is less than , i.e., 1 < .
Proof: Perform the congruence transformation on Eqs. 5.58 { 5.59.
2
3
2
3
^) 0 T
^) 0
S
(
S
(
1
1
5 (Eq. 5.58) 4
5 < 0:
Eq. 5.58 < 0 , 4
0
I
ij
0
I
ij
(5.68)
After some matrix manipulations, we can obtain
2
3
_
_
^
^
^
^
^
^
^
Bw ()
66 T1 (; ) + Y () T2 (; ) + I 0
1 77
^
^
^
66
X
(
)
B
(
)+
w
A 777 < 0;
T3 (^; ^_ ) + X^ (^) @ ^ ^ ^
66
M ()Bc ()Dyw 75
4
;I
ij
(5.69)
where [T1 (^; ^_ )]ij , [T2 (^; ^_ )]ij and [T3 (^; ^_ )]ij are dened at Eqs. 5.26 { 5.28. With the intermediate controller variables at Eqs. 5.29 - 5.31, Eq. 5.69 is immediately equal to Eq. 5.66.
Similarly,
2
3T
2
3
^
^
S
(
)
0
S
(
)
0
5 (Eq. 5.59) 4 1
5 < 0:
Eq. 5.59 < 0 , 4 1
(5.70)
0 I ij
0 I ij
After simple matrix manipulations, we can obtain Eq. 5.67.
2
Some of intermediate controller variables will be even further eliminated from Lemma 5.5
using the elimination lemma. However, [C^c (^)]ij remains to simplify the construction of
the controller dynamics. Note that [C^c (^)]ij should satisfy both Eq. 5.66 and Eq. 5.67.
Therefore, the construction of [C^c (^)]ij satisfying both inequalities would be dicult.
CHAPTER 5. SYNTHESIS
102
Lemma 5.6 Suppose there exist , ( < ), [C^c(^)]ij , positive-denite [X^ (^)]ij and
[Y^ (^)]ij 's such that for all ij ,
LY I (^; ^_ ) + Y^ (^) + 1=Bw (^)Bw (^)T
ij
< 0;
(5.71)
2
3
~ ^; ^_ ) ; Cy (^)T Cy (^) + X^ (^) X^ (^)Bw1 (^)
L
(
A;
X
4
5 < 0;
BwT 1 (^)X^ (^)
;I
ij
2
3
T
T T
66 Y^ (^) I Y^ (^)Cz (^) + CT^c(^) Dzu 77
64 X^ (^)
75 > 0
Cz (^)
I
ij
(5.72)
(5.73)
for all (^ij ; ^_ ij ) 2 P^ , where [A~(^)]ij = [A(^) ; Bw2Cy (^)]ij . Then, the QPALPV system
(Eq. 5.4) is uniformly asymptotically stable about x = 0 for all 2 F 2 and its L1-gain
(1 ) is less than , i.e., 1 < .
Proof: The result is directly obtained by applying Lemma 2.7 (Elimination Lemma) to
Eq. 5.66. For convenience, dene some matrices such as
2
3
_
^
^
^
^
^
^
L
(
;
)
+
Y
(
)
A
(
)
+
I
B
(
)
w
_
6 YI
77
_
^
^
^
^
^
^
^
G(^; ^) = 664 A(^)T + I
LX (A; ; ) + X () X ()Bw () 75
ij
Bw (^)T
Bw (^)T X^ (^)
;I
ij
2 3
2
3
2
3
66 0 77 h i 66 I 0 T 77 h iT
h ^i
^c (^)T
A
U () ij = 64 I 75 ; V (^) ij = 64 0 Cy (^) 75 ; K^ (^) ij = 4 ^ ^ T 5 :
Bc ()
T
0 ij
0
Dyw
ij
ij
A simple matrix manipulation shows that
Eq. 5.66 = G(^; ^_ ) + U (^)K^ (^)V (^)T + V (^)K^ (^)T U (^)T
ij
< 0:
(5.74)
According to Lemma 2.7, the above inequality is equivalent to a pair of LMIs such that
U?(^)T G(^; ^_ )U?(^) < 0 and V?(^)T G(^; ^_ )V?(^) < 0:
ij
ij
(5.75)
CHAPTER 5. SYNTHESIS
103
Here, [U?(^)]ij and [V?(^)]ij are orthogonal complements of [U (^)]ij and [V (^)]ij , respectively:
2
3
0
0
66
77
2
3
6
77
0
I
I
0
77 h
66
h ^ i 66
i
77 :
U?() ij = 64 0 0 75 ; V?(^) ij = 66 2 3 2
3
77
66 I
0 I ij
0
5 75
44 5 4
0
;Cy (^) ij
After calculating Eq. 5.75 with these [U? (^)]ij and [V?(^)]ij , we can eliminate the unknown
intermediate controller dynamics [A^c (^)]ij and [B^c(^)]ij and then derive Eqs. 5.71 - 5.72. 2
Each of Eqs. 5.71 - 5.73 leads to an innite number of LMIs. Therefore, we need the
\convexifying" technique to derive a nite number of LMIs.
Proposition 5.3 Suppose there exist , ( > ), Lijk; Mijk ; Nijk (> 0), C^cij0, C^cijk,
positive-denite (Xa1 ; Ya1 ) (a = 1; ; m1 + 1) and (X1b ; Y1b ) (b = 1; ; m2 + 1) such that
for all ij ,
h
LY I (w; ) + Y^ (w) + 1=Bw (w)Bw (w)T
i
+
ij
2
X
k=1
wk2 Mijk < 0;
(5.76)
2
3 2
T
~
^
^
X 2
L
(
A;
w;
)
+
X
(
w
)
;
C
(
w
)
C
(
w
)
X
(
w
)
B
(
w
)
y
y
w1
4 X
5
+
wk Nijk < 0; (5.77)
Bw1 (w)T X^ (w)
;I
k
=1
ij
2
3
Y^ (w)
7
66
3
2
7
2
X
66 Cz1(w)Y^ (w)
7
I
7 + w2 4 Lijk 0 5 > 0 (5.78)
66 Cz2Y^ (w) + C^c(w)
0
I
775 k=1 k 0 0
4
Cz1 (w)T CzT2 X^ (w)
I
for all (w; ) 2 and
ij
Aijk Yijk + Buijk C^cijk + (Aijk Yijk + Buijk C^cijk )T + Mijk 0;
2 T
3
~
~
+ Xijk Aijk Xijk Bw1ijk 5
4 Aijk Xijk
+ Nijk 0;
T
Bw1ijk Xijk
0
(5.79)
(5.80)
CHAPTER 5. SYNTHESIS
2
4
104
0
Cz1ijk Yijk
3
Yijk CzT1ijk 5
+ Lijk 0
0
(5.81)
for all k. Here, [A~(^)]ij is dened at Eq. 5.36. Then, the QPALPV system (Eq. 5.4) is
uniformly asymptotically stable about x = 0 for all 2 F 2 and its L1-gain (1 ) is less
than , i.e., 1 < .
Proof: The proof of Eqs. 5.71 - 5.72 is similar to that of Proposition 5.1. Consider Eq. 5.73.
Interchanging the 3-th row and column with the 2-th row and column, Eq. 5.73 is equivalent
to
2
3
^ (^)
Y
6
77
h ^ i 666 Cz1(^)Y^ (^)
77 > 0
I
G() ij = 6
(5.82)
64 Cz2Y^ (^)C^c(^)
0
I
775
I
Cz1 (^)T CzT2 X^ (^) ij
for all ^ij 2 P^ . According Lemma 2.10, this equation is always true on all ^ij 2 P^ if
3
2
2
X
L
0
ijk
5>0
[G(w)]ij + wk2 4
k=1
for all w 2 and
for all k.
2
4
0
Cz1ijk Yijk
0
0
3
Yijk CzT1ijk 5
+ Lijk 0
0
(5.83)
(5.84)
2
Note that as discussed in the analysis, the above formulations are LMIs with xed .
Since is a scalar, we can easily compute the smallest upper bound (opt ) of the L1 -gain of
the QPALPV system (Eq. 5.4) by minimizing 's at dierent values of 's and selecting the
minimum value of these parameterized 's. The smallest (opt ) is often used as the L1 gain. Furthermore, we can construct C^c(), X () and Y () with solutions of Proposition 5.3.
Note that an upper bound of is not analytically given in the synthesis.
A central controller formula is constructed in terms of the PALPV system dynamics
(Eq. 5.1) and solutions of Proposition 5.3.
Proposition 5.4 Suppose , , , [C^c(^)]ij , [X^ (^)]ij and [Y^ (^)]ij are solutions of Proposition 5.3. Then, a central controller of Eq. 5.3, which makes the L1-gain of the QPALPV
CHAPTER 5. SYNTHESIS
105
system (Eq. 5.4) less than , is given as follows: for (t) 2 Pij , Eqs. 5.45 { 5.47 and
h^ ^i
h
i
Ac() ij = ;A(^)T ; I ; 1=X^ (^)Bw1(^)Bw1 (^)T + Cy (^)T BwT 2 ij
h^ ^ i
h
i
Bc() ij = ; Cy (^)T + X^ (^)Bw2 ij :
(5.85)
(5.86)
Proof: The derivations are similar to those of Proposition 5.2. We dene some matrices
h ^i h ^i h ^i h^ ^ ^ ^ ^ i
L1 () ij = Bw () ij ; L2 () ij = X ()Bw () + Bc()Dyw ij :
Eq. 5.66 can be formulated
2
3
2
3 2
3T
_
T
^
^
^
^
^
^
^
^
^
L
(
;
)
+
Y
(
)
A
(
)
+
A
(
)
+
I
L
(
)
L
(
)
c
5 + 1= 4 1 5 4 1 5
Eq. 5.66 = 4 Y I
_
^
^
^
^
L2 (^) ij L2 (^) ij
LXI (; ) + X () ij
20
1 0
13
^) + A^c (^)T + I
^; ^_ ) + Y^ (^)
A
(
L
(
Y
I
66 @
A @
A 77
T
T
^
^
^
^
6
+1
=L
(
)
L
(
)
+1
=L
(
)
L
(
)
1
2
1
1
0
1 777 < 0:
= 66
_
64
@ LXI (^; ^) + X^ (^) A 75
+1=L2 (^)L2 (^)T
ij
The idea is to nd [A^c (^)]ij and [B^c (^)]ij such that make (1; 2) = 0 and (2; 2) < 0, respectively. Consider condition (2; 2) < 0. With a x and any ,
2 3T
2 3
2 3T 2
3 2 3
_
^
^
^
^
^
x
x
x
x 5
L
(
;
)
+
X
(
)
L
(
)
2
4 5 (2; 2) 4 5 = 4 5 4 XI
5
4
<0
x
x
x
L2 (^)T
; ij x
2
3 2
3 2
3
T
^
^c (^)T x T
^
B
0
B
(
)
x
D
yw 5 4 c
5 4 T
5
= 4
x
D
;
I
x
yw
ij
ij
ij
2
3T 2
3
Cy (^)x 5 4 B^c(^)T x 5
+ 24
Bw (^)T X^ (^)x ij
x
ij
_
+ xT LX (A; ^; ^) + X^ (^) x < 0:
ij
(5.87)
CHAPTER 5. SYNTHESIS
106
Similar to Proposition 5.2, a [B^c (^)]ij satisfying Eq. 5.87 is the saddle point ([B^c (^)]ij ):
2
4 0T
Dyw
3 2
3
2
3
Dyw 5 4 B^c (^)T 5
Cy (^) 5
4
=
;
^)T X^ (^) :
;I ij
B
(
w
ij
ij
(5.88)
The condition (1; 2) = 0 immediately implies
h^ ^i
h
i
Ac () ij = ;A(^)T ; I ; 1=L2 (^)L1 (^)T ij
h
i
= ;A(^)T ; I ; 1=(X^ (^)Bw (^) + B^c(^)Dyw )Bw (^)T ij : (5.89)
After some simple matrix manipulations, we can obtain Eqs. 5.85 { 5.86.
5.3.3
2
H2-Norm
The problem we consider here is to design an LPV controller (Eq. 5.3) such that minimizes
an upper bound of the H2 -norm of the closed-loop system (Eq. 5.4). In contrast to the
L2-gain, it will be assumed that [B^c(^)]ij is ane on ^ij , i.e.,
2
h^ ^ i ^
X
Bc() ij = Bcij0 + ^ijk B^cijk :
k=1
(5.90)
The synthesis follows the previous standard approach (Refer to section 5.3.1 for details).
Lemma 5.7 Suppose there exist Q and dierentiable functions [P^cl (^)]ij 's such that for all
ij ,
Lcl (^; ^_ ) + Ccl (^)T Ccl (^) < 0;
(5.91)
2
4
ij
3
Q
Bcl (^)T P^cl (^) 5
>0
P^cl (^)Bcl(^)
P^cl (^)
ij
(5.92)
for all (^ij ; ^_ ij ) 2 P^ . Then, the QPALPV system (Eq. 5.4) is uniformly asymptotically
p
p
stable about x = 0 for all 2 F 2 and its H2 -norm is less than TrQ, i.e., kk2 < TrQ.
Proof: The proof is similar to that of Lemma 5.1. An upper bound of the H2-norm
can be derived using Proposition 3.4. Both the QPALPV system and the QPAL are used
for Proposition 3.4. First, consider condition (II) of Proposition 3.4. As in the proof of
CHAPTER 5. SYNTHESIS
107
Lemma 5.1, condition (II) over each Iij becomes
D+V (xcl ; )(x_ cl ; 0) + D+ V (xcl ; )(0; _) + z T z < 0:
(5.93)
Since V (xcl ; ) is dierentiable,
2 T^ ^ 3
@ xTcl P^cl (^)xcl ^_ T
@ xcl Pcl()xcl
x_ cl +
Eq. 5.93 = 4
+ z z5
@xcl
@ ^
ij
_
= xTcl Lcl (^; ^)xcl + xTcl Ccl (^)T Ccl (^)xcl
< 0:
(5.94)
< 0; 8(^ij ; ^_ ij ) 2 P^ :
(5.95)
ij
Therefore, condition (II) is satised if for all ij ,
Lcl (^; ^_ ) + Ccl (^)T Ccl (^)
ij
As in Proposition 4.4, an upper bound of the H2 -norm is given:
h
i
kk22 < sup sup Tr Bcl(^)T P^cl (^)Bcl(^) ij :
ij ^ij 2P^
(5.96)
Eq. 5.96 immediately implies kk22 < TrQ such that
2
3
T P^cl (^)
^
Q
B
(
)
cl
5 >0
Q ; Bcl (^)T P^cl (^)Bcl () ij > 0 () 4 ^ ^ ^
Pcl ()Bcl ()
P^cl (^)
ij
h
i
^
(5.97)
for all ^ij 2 P^ and all ij . Condition (I) can be replaced by [P^ (^)]ij > 0 for all ^ij 2 P^ and
all ij . However, it is redundant because Eq. 5.97 implies [P^ (^)]ij > 0.
2
Lemma 5.7 is equivalently restated using the congruence transformation.
Lemma 5.8 Suppose there exist Q, [X^ (^)]ij , [Y^ (^)]ij , [A^c(^)]ij , [B^c(^)]ij and [C^c(^)]ij 's
such that for all ij ,
2
0
13
T+
^
^
^
Y
(
)
C
(
)
z
_
66 LY I (^; ^) A(^) + A^c(^)T @
A 77
T
77 < 0
66
C^c(^)T Dzu
77
66 _
LXI (^; ^)
Cz (^)T
5
4
;I
ij
(5.98)
CHAPTER 5. SYNTHESIS
and
108
2
0
13
T X^ (^)+
^
B
(
)
w
66 Q Bw (^)T @ T
A 77
66
77 > 0
Dyw B^c (^)T
66 Y^ (^)
77
I
4
5
X^ (^)
ij
(5.99)
for all (^ij ; ^_ ij ) 2 P^ Then, the QPALPV system (Eq. 5.4) is uniformly asymptotically
p
p
stable about x = 0 for all 2 F 2 and its H2 -norm is less than TrQ, i.e., kk2 < TrQ.
Proof: Perform the congruence transformation on Eqs. 5.91 { 5.92.
2
3T 2
3 2
3
_
T
^
^
^
^
^
S
(
)
0
S
(
)
0
L
(
;
)
C
(
)
cl
5 4 cl
5 4 1
5 < 0;
Eq. 5.91 < 0 , 4 1
^
0 I ij Ccl ()
;I ij 0 I ij
(5.100)
where [S1 (^)]ij is dened at Eq. 5.17. After some matrix manipulations, we can obtain
0
13
2
^ (^)Cz (^)T +
Y
_
_
66 T1(^; ^) T2(^; ^) @
A 77
T
66
77 < 0;
N^ (^)Cc (^)T Dzu
66 T3(^; ^_ )
77
Cz (^)T
4
5
;I
ij
(5.101)
where [T1 (^; ^_ )]ij , [T2 (^; ^_ )]ij and [T3 (^; ^_ )]ij are dened at Eqs. 5.26 { 5.28. With the
intermediate controller variables at Eqs. 5.29 - 5.31, Eq. 5.101 is immediately equal to
Eq. 5.98. Similarly,
2
I
Eq. 5.92 < 0 , 4
3T
2
3
0 5
4 I 0 5 < 0:
(Eq.
5.92)
0 S1 (^) ij
0 S1 (^) ij
After simple matrix manipulations, we can obtain Eq. 5.99.
(5.102)
2
Some of intermediate controller variables of Lemma 5.8 will be even further eliminated
applying the elimination lemma. However, [Bc (^)]ij remains to simplify the construction of
the controller dynamics.
CHAPTER 5. SYNTHESIS
109
Lemma 5.9 Suppose there exist Q, [X^ (^)]ij , [Y^ (^)]ij and [B^c(^)]ij 's such that for all ij ,
_
LXI (^; ^) + Cz (^)T Cz (^) < 0;
(5.103)
ij
2
3
_
T
T
^
^
^
^
^
^
^
^
L
(
A;
;
)
;
B
(
)
B
(
)
Y
(
)
C
(
)
u
u
z1
4 Y
5 < 0;
^
^
^
Cz1 ()Y ()
;I
ij
2
3
T
T
T
T
66 Q Bw (^) Bw (^) X^ (^) + Dyw B^c(^) 77
64 Y^ (^)
75 > 0
I
X^ (^)
ij
(5.104)
(5.105)
for all (^ij ; ^_ ij ) 2 P^ , where [A^(^)]ij = [A(^) ; Bu (^)Cz2 ]ij . Then, the QPALPV system
(Eq. 5.4) is uniformly asymptotically stable about x = 0 for all 2 F 2 , and its H2 -norm is
p
p
less than TrQ, i.e., kk2 < TrQ.
Proof: The result is directly obtained by applying Lemma 2.7 (Elimination Lemma) to
Eq. 5.98. We dene some matrices:
2
LY (A; ^; ^_ ) A(^) Y^ (^)Cz (^)T
6
G(^; ^_ ) = 664 A(^)T LXI (^; ^_ ) Cz (^)T
ij
Cz (^)Y^ (^) Cz (^)
;I
2
3
2 3
66 0 Bu(^) 77 h i 66 I 77
h ^i
U () ij = 64 I 0 75 ; V (^) ij = 64 0 75 ;
0 Dzu
0 ij
ij
3
77
75
ij
2
3
^c (^)
A
K () ij = 4 ^ ^ 5 :
h^ ^i
A simple matrix manipulation shows that
Eq. 5.98 = G(^; ^_ ) + U (^)K^ (^)V (^)T + V (^)K^ (^)T U (^)T
Cc ( )
ij
< 0:
ij
(5.106)
According to Lemma 2.7, the above inequality is equivalent to a pair of LMIs such that
U?(^)T G(^; ^_ )U?(^)
ij
< 0 and V?(^)T G(^; ^_ )V? (^)
ij
< 0:
(5.107)
CHAPTER 5. SYNTHESIS
110
Here, [U?(^)]ij and [V?(^)]ij are orthogonal complements of [U (^)]ij and [V (^)]ij , respectively:
2
3
2
3
0
I
66
77
0
0
77
h ^ i 66 2 0 3 2 0 3 77 h ^ i 66
U?() ij = 6
;
V
(
)
=
6
75 :
I
0
? ij 4
64 4 I 5 4 0 5 775
0 I ij
0
;Bu(^)T ij
After calculating Eq. 5.107 with these [U?(^)]ij and [V? (^)]ij , we can eliminate unknown
intermediate controller dynamics [A^c (^)]ij and [C^c (^)]ij . Then, we can derive Eqs. 5.103 5.104.
2
Each of Eqs. 5.103 - 5.105 leads to an innite number of LMIs. Therefore, we need the
\convexifying" technique to derive a nite number of LMIs.
Proposition 5.5 Suppose there exist Q, Lijk ; Mijk ; Nijk (> 0), B^cij0, B^cijk , (Xa1 ; Ya1 )
(a = 1; ; m1 + 1) and (X1b ; Y1b ) (b = 1; ; m2 + 1) such that for all ij ,
h
2
i X
T
LXI (w; ) + Cz (w) Cz (w) ij + wk2 Nijk < 0;
k=1
2
66
66
66
4
(5.108)
2
3 2
^ w; ) ; Bu (w)Bu (w)T Y^ (w)Cz1 (w)T
L
(
A;
Y
4
5 + X wk2 Mijk < 0;
(5.109)
Cz1 (w)Y^ (w)
;I
k
=1
ij
3
X^ (w)
7
2
3
2
T
^
Bw1(w) X (w)
Q1
L
777 + X
0
wk2 4 ijk 5 > 0 (5.110)
7
T
T
T
^
^
Bw2X (w) + Bc(w)
Q2
Q3
75 k=1
0 0
Bw1(w) Bw2 Y^ (w)
I
for all (w; ) 2 and
ij
T B^ T + N 0;
ATijk Xijk + Xijk Aijk + B^cijkCyijk + Cyijk
ijk
cijk
2
3
^ijk Yijk + Yijk A^Tijk Yijk CzT1ijk
A
4
5 + Mijk 0;
Cz1ijk Yijk
0
2
3
0
X
B
ijk
w
1
ijk
4 T
5 + Lijk 0
Bw1ijk Xijk
0
(5.111)
(5.112)
(5.113)
CHAPTER 5. SYNTHESIS
111
2
3
Q
Q
1
2
5. Then, the QPALPV system (Eq. 5.4) is uniformly
for all k. Here, Q = 4 T
Q2 Q3
p
asymptotically stable about x = 0 for all 2 F 2 and its H2 -norm is less than TrQ, i.e.,
kk2 < pTrQ.
Proof: The proof of Eqs. 5.103 - 5.104 is similar to that of Proposition 5.1. Consider
Eq. 5.105. By the elementary transformation, Eq. 5.105 is equivalent to
2
X^ (^)
66
h ^ i 66 Bw1(^)T X^ (^)
G() ij = 6 T ^ ^ ^ ^ T
64 Bw2X () + Bc()
I
Q1
QT2
Q3
^
Bw1 () Bw2 Y^ (^)
3
77
77 > 0
77
5
(5.114)
ij
for all ^ij 2 P^ . According Lemma 2.10, this equation is always true on all ^ij 2 P^ if
2
3
2
X
L
0
5>0
[G(w)]ij + wk2 4 ijk
k=1
for all w 2 and
for all k.
2
4
0
Cz1ijk Yijk
0
0
3
Yijk CzT1ijk 5
+ Lijk 0
0
p
(5.115)
(5.116)
2
As minimizing TrQ, we compute the smallest upper bound of the H2 -norm of the
QPALPV system (Eq. 5.4). This smallest value is often used as the H2 -norm. This minimization problem is a standard convex optimization problem. Furthermore, we can construct X () and Y () with solutions of Proposition 5.5.
A central controller formula is constructed in terms of the PALPV system dynamics
(Eq. 4.3) and solutions of LMIs of Proposition 5.5.
Proposition 5.6 Suppose [B^c(^)]ij , [X^ (^)]ij and [Y^ (^)]ij are solutions of Proposition 5.5.
Then, a central controller of Eq. 5.3, which makes the H2 -norm of the QPALPV system
p
(Eq. 5.4) less than TrQ, is given as follows: for (t) 2 Pij , Eqs. 5.45 { 5.47 and
h^ ^i
h
i
Ac () ij = ;A(^)T ; Cz1(^)T Cz1 (^)Y^ (^) + CzT2Bu (^)T ij
(5.117)
CHAPTER 5. SYNTHESIS
112
h^ ^ i
h
i
Cc() ij = ; Bu(^)T + Cz2 Y^ (^) ij :
(5.118)
Proof: We derive Eqs. 5.117 { 5.118. The derivations are similar to those of Proposition 5.2.
We dene some matrices
h ^ i h^ ^ ^ T ^ ^ T T i
h i h
i
L1 () ij = Y ()Cz () + Cc() Dzu ij and L2 (^) ij = Cz (^)T ij :
Eq. 5.98 can be formulated
2
3 2
3 2
3
^; ^_ ) A(^) + A^c(^)T
^)
^) T
L
(
L
(
L
(
Y
I
1
1
5 +4
5 4
5
Eq. 5.98 = 4
L2 (^) ij L2 (^) ij
LXI (^; ^_ ) ij
2
3
_
T
T
T
^
^
^
^
^
^
^
^
^
L
(
;
)
+
L
(
)
L
(
)
A
(
)
+
A
(
)
+
L
(
)
L
(
)
1
1
c
1
2
5 < 0:
= 4 YI
_
^
^
^
^
LXI (; ) + L2 ()L2 ()T
ij
The idea is to nd [A^c (^)]ij and [C^c (^)]ij such that make (1; 1) < 0 and (1; 2) = 0, respectively. With a x and any ,
2 3T
2 3
2 3T 2
3 2 3
^; ^_ ) L1 (^)
x
x
x
L
(
Y
I
4 5 (1; 1) 4 5 = 4 5 4
5 4 x 5<0
T
x
x
x
L1(^)
;I ij x
2
3 2
3 2
3
T
^c (^)x T
^c (^)x
C
0
D
C
zu
5 4
5 4
5
= 4
x ij Dzu ;I ij
x ij
3 2
3
2
^)T x T C^c(^)x
B
(
u
5 4
5
+ 24
Cz (^)Y^ (^)x ij
x ij
_
+ xT LY (A; ^; ^) x < 0:
ij
A [C^c (^)]ij satisfying Eq. 5.119 is the saddle point ([C^c (^)]ij ):
2
4 0
Dzu
3 2
3
2
3
T
T
Dzu
5 4 C^c (^) 5 = ; 4 Bu(^) 5 :
;I ij Cz (^)Y^ (^) ij
ij
(5.119)
(5.120)
CHAPTER 5. SYNTHESIS
113
The condition (1; 2) = 0 immediately implies
h^ ^i
h
i
Ac() ij = ;A(^)T ; L2 (^)L1 (^)T ij
h
i
= ;A(^)T ; Cz (^)T Cz (^)Y^ (^) + Dzu C^c (^) ij :
After some simple matrix manipulations, we can obtain Eqs. 5.117 { 5.118.
(5.121)
2
5.3.4 Passivity
The problem we consider here is to design an LPV controller (Eq. 5.3) such that the the
closed-loop system (Eq. 5.4) is strictly passive. It is assumed that the PALPV system
(Eq. 5.186) has nonzero Dzw term:
h
i
T
Dzw + Dzw
>0
ij
2
3
D
D
z
1
w
1
z
1
w
2
5 :
[Dzw ]ij = 4
with
Dz2w1 Dz2w2
ij
The corresponding QPALPV system (Eq. 5.4) is the same as in Eq. 5.4 except for [Dcl ]ij =
[Dzw ]ij .
The synthesis follows the previous standard approach (Refer to section 5.3.1 for details).
Lemma 5.10 Suppose there exist dierentiable functions [P^cl (^)]ij 's such that for all ij ,
h^ ^ i
Pcl () ij > 0
and
2
4
3
Lcl (^; ^_ )
P^cl (^)Bcl (^) ; CclT (^) 5 < 0
BclT (^)P^cl (^) ; Ccl (^)
;(Dcl + DclT )
ij
(5.122)
(5.123)
for all (^ij ; ^_ ij ) 2 P^ .
Then, the QPALPV system (Eq. 5.4) is uniformly asymptotically stable about x = 0 for
all 2 F 2 and also strictly passive.
Proof: The proof is similar to that of Lemma 5.1. Sucient conditions of the strict passivity
can be derived using Proposition 3.5. Both the QPALPV system and QPAL are used for
CHAPTER 5. SYNTHESIS
114
Proposition 3.5. First, we consider condition (II) of Proposition 3.5. As in the proof of
Lemma 5.1, condition (II) over each Iij becomes
D+V (xcl; )(x_ cl ; 0) + D+V (xcl ; )(0; _) ; 2wT z < 0:
(5.124)
Since V (xcl ; ) is dierentiable,
2 T^ ^ 3
@ xTcl P^cl (^)xcl ^_
@ xclPcl ()xcl
Eq. 5.124 = 4
x_ cl +
; 2wT z5
^
@xcl
@
ij
_
= xTcl Lcl (^; ^)xcl + xTcl P^cl (^)Bcl (^)w + wT Bcl (^)T P^cl (^)xcl
i
;xT Ccl (^)T w ; wT Ccl(^)x ; wT (Dcl + DclT )w ij < 0: (5.125)
Therefore, condition (II) is satised if all ij ,
2
4
3
Lcl (^; ^_ )
P^cl (^)Bcl (^) ; CclT (^) 5 < 0
BclT (^)P^cl (^) ; Ccl (^)
;(Dcl + DclT )
ij
(5.126)
for all (^ij ; ^_ ij ) 2 P^ .
Condition (I) of Proposition 3.5 is satised if [P^cl (^)]ij > 0 for all ^ij 2 P^ and all ij . 2
Lemma 5.10 is equivalently restated, using the congruence transformation.
Lemma 5.11 Suppose there exist [X^ (^)]ij , [Y^ (^)]ij , [A^c(^)]ij , [B^c(^)]ij and [C^c(^)]ij 's such
that for all ij ,
2
3
^ (^) I
X
4
5 > 0;
(5.127)
I Y^ (^) ij
2
66 LY I (^; ^_ ) A + A^c(^)T
66
66
66 LXI (^; ^_ )
64
for all (^ij ; ^_ ij ) 2 P^ .
0
T
@ Bw (^) ; Y^ (T^)CTz (^)
^
0 ;C^c() Dzu T
@ X^ (^)Bw (^) ; Cz (^)
+B^c (^)Dyw
T )
;(Dzw + Dzw
13
A 77
1 777
A 777 < 0
75
ij
(5.128)
CHAPTER 5. SYNTHESIS
115
Then, the QPALPV system (Eq. 5.4) is uniformly asymptotically stable about x = 0 for
all 2 F 2 and also strictly passive.
Proof: Perform the congruence transformation on Eqs. 5.122 { 5.123. Since the derivation
of Eq. 5.122 is obvious, we consider Eq. 5.123.
2
3T
2
3
^
^
S ( ) 0 5
S () 0 5
Eq. 5.123 < 0 , 4 1
(Eq. 5.123) 4 1
< 0;
0
I
0
ij
I
ij
(5.129)
where [S1 (^)]ij is dened at Eq. 5.17. After some matrix manipulations, we can obtain
2
66 T1 (^; ^_ ) T2 (^; ^_ )
66
66
66 T3 (^; ^_ )
64
0
T
@ Bw (^) ; Y^ (^)CT z (T^)
^ ^ ^
0 ;N ()Cc() Dzu T
@ X^ (^)Bw (^) ; Cz (^)
+M^ (^)Bc (^)Dyw
T )
;(Dzw + Dzw
13
A 77
1 777
A 777 < 0;
75
(5.130)
ij
where [T1 (^; ^_ )]ij , [T2 (^; ^_ )]ij and [T3 (^; ^_ )]ij are dened at Eqs. 5.26 { 5.28. With the
intermediate controller variables at Eqs. 5.29 - 5.31, Eq. 5.130 is immediately equal to
Eq. 5.128.
2
The intermediate controller variables of Lemma 5.11 will be even further eliminated
applying the elimination lemma. For convenience, we dene the following matrices:
D11 = Dz1w1 + DzT1w1; D21 = Dz2w1 + DzT1w2 ; D22 = Dz2w2 + DzT2w2
Lemma 5.12 Suppose there exist [X^ (^)]ij and [Y^ (^)]ij 's such that for all ij ,
2
3
^ (^) I
X
4
5 > 0;
I
Y^ (^)
ij
(5.131)
20
1 0
13
_ ) + Bu (^)B T
^ (^)Cz1 (^)T + Bw1 (^)
^
^
^
;
Y
L
(
A;
;
Y
w
2
66 @
A @
A 77
64 +Bw2Bu(^)T ; Bu(^)D22 Bu(^)T
75 < 0 (5.132)
;Bu(^)D21
;D11
ij
CHAPTER 5. SYNTHESIS
116
20
_ ) + C T Cy (^) + Cy (^)T Cz2 1 0 X^ (^)Bw1 (^) ; Cz1 (^)T 1 3
^
^
~
L
(
A;
;
X
z2
66 @
A @
A 77
^
64
75 < 0
+Cy (^)T D21
;Cy ()T D22 Cy (^)
;D11
ij
(5.133)
_
^
^
^
^
^
^
^
^
~
for all (ij ; ij ) 2 P , where [A()]ij = [A() ; Bu ()Cz2 ]ij and [A()]ij = [A(^) ;
Bw2Cy (^)]ij .
Then, the QPALPV system (Eq. 5.4) is uniformly asymptotically stable about x = 0 for
all 2 F 2 and also strictly passive.
Proof: The result is directly obtained by applying Lemma 2.7 (Elimination Lemma) to
Eq. 5.128. For convenience, dene some matrices such as
2
LY (A; ^; ^_ )
A(^)
Bw (^) ; Y^ (^)Cz (^)T
6
G(^; ^_ ) = 664
LX (A; ^; ^_ ) X^ (^)Bw (^) ; Cz (^)T
ij
T )
;(Dzw + Dzw
2
3
2
3
^)
0
B
(
I
0
u
6
7
7
h ^i
h ^ i 66
U () ij = 664 I 0 775 ;
V () ij = 64 0 Cy (^)T 775
T
0 ;Dzu ij
0 Dyw
ij
2
3
h^ ^i
A^ (^) B^c(^) 5
K ( ) = 4 c
:
C^c (^)
ij
0
3
77
75
ij
ij
A simple matrix manipulation shows that
Eq. 5.128 = G(^; ^_ ) + U (^)K^ (^)V (^)T + V (^)K^ (^)T U (^)T
ij
< 0:
(5.134)
According to Lemma 2.7, the above inequality is equivalent to a pair of LMIs such that
U?(^)T G(^; ^_ )U?(^) < 0 and V?(^)T G(^; ^_ )V?(^) < 0:
ij
ij
(5.135)
CHAPTER 5. SYNTHESIS
117
Here, [U?(^)]ij and [V?(^)]ij are orthogonal complements of [U (^)]ij and [V (^)]ij , respectively:
2
66 0
h ^ i 66 2 0 3 2
U?() ij = 6
64 4 I 5 4
0
3
2
3
0
0
66
77
I
77
6
77
0
I
0 3 77 h ^ i 66
77 :
; V?() ij = 66 2 3 2
7
3
77
0 5 75
66 I
0
5 75
44 5 4
Bu(^)T
ij
0
;Cy (^) ij
After calculating Eq. 5.135 with these [U? (^)]ij and [V? (^)]ij , we can eliminate unknown
intermediate controller dynamics [A^c (^]ij , [B^c (^)]ij and [C^c (^)]ij and then derive Eqs. 5.132 5.133.
2
Each of Eqs. 5.132 - 5.133 leads to an innite number of LMIs. Therefore, we need the
\convexifying" technique to derive a nite number of LMIs.
Proposition 5.7 Suppose there exist Mijk ; Nijk (> 0), (Xa1 ; Ya1 ) (a = 1; ; m1 +1), and
(X1b ; Y1b ) (b = 1; ; m2 + 1) such that for all ij ,
2
3
^
4 X (w) I 5 > 0;
(5.136)
I
Y^ (w)
ij
20
1 0
13
^ w; ) + Bu (w)BwT 2 +
^ (w)Cz1 (w)T +
L
(
A;
;
Y
Y
2
66 @
A @
A7 X
64 Bw2Bu(w)T ; Bu(w)D22 Bu(w)T
Bw1 (w) ; Bu (w)D21 775 + wk2 Mijk < 0;
k=1
;D11
ij
(5.137)
20
1 0
13
T
~
^
2
66 @ LX (TA; w; ) + Cz2TCy (w)+ A @ X (wT )Bw1 (w);T A 77 X
64 Cy (w) Cz2 ; Cy (w) D22 Cy (w)
Cz1(w) + Cy (w) D21 75 + wk2 Nijk < 0
k=1
;D11
ij
for all (w; ) 2 and
2
3
T
T
^
^
Y
A
+
A
Y
;
Y
C
4 ijk ijk ijk ijk ijk z1ijk 5 + Mijk 0;
;Cz1ijk Yijk
0
(5.138)
(5.139)
CHAPTER 5. SYNTHESIS
118
2 T
3
~ijk Xijk + Xijk A~ijk Xijk Bw1ijk
A
4
5 + Nijk 0
T
Bw1ijk Xijk
0
(5.140)
for all k. Then, the QPALPV system (Eq. 5.4) is uniformly asymptotically stable about
x = 0 for all 2 F 2 and also strictly passive.
Proof: The result can be easily obtained by Lemma 2.10 (Refer the proof of Proposition 4.1
2
for details).
We can construct X () and Y () with solutions of Proposition 5.7. A central controller
formula is constructed in terms of the PALPV system dynamics (Eq. 5.1) and solutions of
Proposition 5.7.
Proposition 5.8 Suppose [X^ (^)]ij and [Y^ (^)]ij are solutions of Proposition 5.7. Then, a
central controller of Eq. 5.3, which makes the QPALPV
(Eq. 5.4)
strictly passive,
is
h ^ ^ system
i
h
i
given as follows: for (t) 2 Pij , Eqs. 5.45 { 5.47, M ()N^ (^)T ij = I ; X^ (^)Y^ (^) ij and
h^ ^i
h
T );1
Ac() ij = ;A(^)T ; X^ (^)Bw (^) + B^c (^)Dyw ; Cz (^)T (Dzw + Dzw
^T
i
Bw () ; Cz (^)Y^ (^) ; DzuC^c(^) ij ;
(5.141)
2
32
3
2
3
T
^
^
^
0
D
B
(
)
C
(
)
yw
y
4 T
54 c
5 = ;4
5
T
T
^
^) ; Cz (^) ; (5.142)
^
Dyw ;(Dzw + Dzw )
B
(
)
X
(
w
ij
ij
2
3
2
3
2
3
T
;Dzu
C^c (^) 5
Bu(^)T
5
4
4
5 : (5.143)
4 0
=
;
T )
;Dzu ;(Dzw + Dzw
ij
Bw (^)T ; Cz (^)Y^ (^) ij
Proof: We derive Eqs. 5.141 { 5.143. The derivations are similar to those of Proposition 5.2.
We dene some matrices
h ^i
h
i
T
L1 () ij = Bw (^) ; Y^ (^)Cz (^)T ; C^c(^)T Dzu
h ^i
h^ ^ ^ ^ ^
i ij
T
^
L2 () ij = X ()Bw () + Bc ()Dyw ; Cz () ij
h
i
T
ij = Dzw + Dzw ij :
CHAPTER 5. SYNTHESIS
119
Eq. 5.128 can be formulated
2
3 2
3
2
3T
_
T
^
^
^
^
^
^
^
L
(
;
)
A
(
)
+
A
(
)
L
(
)
L
(
)
c
5 + 4 1 5 ;ij1 4 1 5
Eq. 5.128 = 4 Y I
_
^
^
L2(^) ij
L2 (^) ij
LXI (; ) ij
2
3
^; ^_ ) + L1 (^);1 L1 (^)T A(^) + A^c (^)T + L1 (^);1 L2 (^)T
(
L
Y
I
5 < 0:
= 4
_
;
1
T
^
^
^
^
LXI (; ) + L2 () L2 ()
ij
The idea is to nd [A^c (^)]ij , [B^c (^)]ij and [C^c (^)]ij such that make (1; 2) = 0, (2; 2) < 0 and
(1; 1) < 0, respectively. First, consider the condition (2; 2) < 0. With a x and ,
2 3T
2 3
2 3T 2
3 2 3
_
^
^
^
x
x
x
4 5 (2; 2) 4 5 = 4 5 4 LXI (;T) L2() 5 4 x 5 < 0
x
x
x
L2 (^)
; ij x
2
3
2
3 2
3
T
^
^c (^)T x T
^
B
0
D
B
(
)
x
yw
5 4 T
5 4 c
5
= 4
T
x
D
;
(
D
+
D
)
x
zw
yw
zw ij
ij
ij
2
3T 2
3
Cy (^)x
B^c (^)T x 5
5
4
+ 24
Bw (^)T X^ (^)x ; Cz (^)x ij
x
ij
_
+ xT LX (A; ^; ^) x < 0:
(5.144)
ij
A [B^c (^)]ij satisfying Eq. 5.144 is the saddle point ([B^c (^)]ij ):
2
4 0T
Dyw
3 2
3
2
3
Dyw
B^c(^)T 5
Cy (^)
5
4
4
5
=
;
T )
^)T X^ (^) ; Cz (^) :
;(Dzw + Dzw
B
(
w
ij
ij
ij
Similarly, we can derive the following equation:
(5.145)
2 3T
2 3
2 3T 2
3 2 3
^; ^_ ) L1 (^)
x
x
x
L
(
Y
I
4 5 (1; 1) 4 5 = 4 5 4
5 4 x 5<0
T
^
x
x
x
L1 ()
; ij x
2
3 2
3 2
3
T
^c (^)x T
^c(^)x
C
0
;
D
C
zu
5 4
5 4
5
= 4
T )
x ij ;Dzu ;(Dzw + Dzw
x
ij
ij
2
3
2
3
T
Bu(^)T x
C^c (^)x 5
5
4
+ 24
Bw (^)T x ; Cz (^)Y^ (^)x ij
x ij
CHAPTER 5. SYNTHESIS
120
+ xT LY (A; ^; ^_ ) x < 0:
ij
(5.146)
Similarly, a [C^c (^)]ij satisfying Eq. 5.146 is the saddle point ([C^c (^)]ij ):
2
3 2 3
2
3
T
T
^
^
^
0
;
D
C
(
)
B
(
)
u
zu
4
5 4 c 5 = ;4
5
T
T
^
^)Y^ (^) : (5.147)
;Dzu ;(Dzw + Dzw ) ij B
(
)
;
C
(
w
z
ij
ij
The condition (1; 2) = 0 immediately implies
h^ ^i
h
i
Ac() ij = ;A(^)T ; L2 (^);1 L1(^)T ij
T ;1
= ;A(^)T ; X^ (^)Bw (^) + B^c (^)Dyw ; Cz (^)T Dzw + Dzw
^T
i
Bw () ; Cz (^)Y^ (^) ; DzuC^c (^) ij :
(5.148)
This result is also true for all ij .
2
Note that Eqs. 5.142 - 5.143 can be at least a least-squares solution with full rank
matrices [Dyw ]ij and [Dzu ]ij [Gah96].
5.3.5 Robust L2-Gain
The problem we consider here is to design an LPV controller (Eq. 5.3) that minimizes an
upper bound of the robust L2 -gain (^2 ) of the closed-loop system, the open-loop system
of which is subject to structured time-varying parametric uncertainties ((t)) depicted in
Fig. 3.2 (Refer to section 3.5 for details). These types of uncertainties stem from several
sources such as modeling error, estimation error on parameters, and parameters themselves
that are highly noisy.
We dene a PALPV system subject to structured time-varying parametric uncertainties
((t)). The input (w) and output (z ) channels are partitioned into wT = [w0T pT ] and
z T = [z0T qT ], respectively. Both the channel w0 and z0 are even further partitioned into
w0T = [w1T w2T ] and z0T = [z1T z2T ], respectively. Then, we dene a PALPV system (^ ) subject
CHAPTER 5. SYNTHESIS
121
to structured time-varying parametric uncertainties: for (x; ; w; u) 2 D F 2 W U ,
2
66
66
66
66
64
x_
z1
z2
q
y
3
2
77
66
77 X
66
m2
77 = m1 X
6
77 i=1 j=1 ij () 666
75
64
where
3
77
77
77
77
75
A(^) Bw1(^) Bw2 Bp(^) Bu (^)
Cz1 (^)
0
0
0
0
0
0
0
Inu
Cz 2
^
Cq ()
0
0
0
0
Cy (^)
0
Iny
0
0
2
^
66 A(^) T
66 Bw1()
66 Bp(^)T
66
66 Bu(^)T
66 Cz1(^)
66
4 Cq (^)
Cy (^)
3 2
77 66 ATij0
77 66 Bw1ij0
T
77 66 Bpij
77 = 66 T 0
77 66 Buij0
77 66 Cz1ij0
77 66
5 4 Cqij0
ij
Cyij0
2
66
66
66
66
64
ij
3
2
77
66 ATijk
77
66 Bw1ijk
T
77 2 66 Bpijk
77 + X ^ 66 T
ijk 6 Buijk
77
66
k
=1
77
66 Cz1ijk
77
64 Cqijk
5
x
w1
w2
p
u
3
77
77
77 ; (5.149)
77
75
3
77
77
77
77 :
77
77
77
5
Cyijk
(5.150)
With pT = [pT1 pTr ] and qT = [q1T qrT ], it is assumed that pi = i (t)qi , where i (t) is
a time-varying parametric uncertainty with unit magnitude, i.e., i (t) = i (t)I 2 Si =
fi I j i (t) 2 [;1; 1] 8t 2 Ig. Without loss of generality, we assume for notational convenience that every i -block is square. We also some conventional matrices:
h
i
h
i h
i h
i
2
3
2 3
2 3
0
0
0
5 ; [Dz0u]ij = 4 5 ; [Dyw0]Tij = 4 0 5 ;
[Dz0w0 ]ij = 4
T
Bw0 (^) ij = Bw1(^) Bw2 ij ; Cz0 (^) ij = Cz1 (^)T CzT2 ij ;
0 0
Inu
= fdiag[1 r ] j i 2 Si 8i = 1; ; rg ;
S = fdiag[S1 Sr ]g :
Iny
Now, we close the PALPV system (Eq. 5.149) with an LPV controller (Eq. 5.3). With
Dcl R2n, the QPALPV is then dened such that for (xcl ; ; w) 2
xTcl = [xT xTc ] and
Dcl F 2 W ,
2
3 2 3
2 3 m1 m2
_
^
^
^
X
X
x
_
A
(
;
)
B
(
)
cl
5 4 xcl 5
4 cl 5 =
ij () 4 cl
z
i=1 j =1
Ccl (^)
0
ij
w
(5.151)
CHAPTER 5. SYNTHESIS
where
122
2
3
^
^
^
A
(
)
B
(
)
C
(
)
Acl (^; ^_ ) = 4 ^ ^ u ^ c^_ 5
ij
Bc()Cy () Ac(; ) ij
3
2
h ^i
^) Bw2 Bp (^)
B
(
w
1
5
Bcl() ij = 4
0
Bc(^) 0 ij
2
3
^
C
(
)
0
6 z1
7
h ^i
Ccl() ij = 664 Cz2 Cc(^) 775 :
Cq (^)
0
(5.152)
(5.153)
(5.154)
ij
For computational simplicity, we consider special types of scaling matrices ^ cl (), cl ()
and ;cl () such that
2 2
3
I
0
5 ;
^ cl () =
ij ()^ clij =
ij () 4
0 ij
i=1 j =1
i=1 j =1
2
3
m1 X
m2
m1 X
m2
X
X
I
0
5 ;
cl () =
ij ()clij =
ij () 4
0
i=1 j =1
i=1 j =1
2
3 ij
m
m
m
m
1
2
1
2
XX
XX
;cl () =
ij ();clij =
ij () 4 0 0 5 ;
m1 X
m2
X
i=1 j =1
m1 X
m2
X
i=1 j =1
0 ; ij
(5.155)
(5.156)
(5.157)
where ij = Tij = diag[1 ; ; r ]ij (> 0) and ;ij = ;;Tij = diag[;1 ; ; ;r ]ij . The
synthesis follows the previous standard approach.
Lemma 5.13 Suppose there exist ^ clij , clij , ;clij and dierentiable functions [P^cl (^)]ij 's
such that for all ij ,
h^ ^ i
Pcl () ij > 0
(5.158)
and
2
_
T
T
66 Lcl(^; ^) P^cl (^)Bcl (^) + Ccl(^) ;cl Ccl(^)
64 ;^ cl
0
;;cl1
3
77
75 < 0
ij
(5.159)
CHAPTER 5. SYNTHESIS
123
^ . Then, the QPALPV system (Eq. 5.151) is uniformly asymptotically
for all (^ij ; ^_ ij ) 2 P
stable about x = 0 for all 2 F 2 and (t) 2 S , and the robust L2 -gain from w0 to z0 is
less than , i.e., ^2 < .
Proof: The proof is similar to that of Lemma 5.1. An upper bound of the robust L2-gain
can be derived using Proposition 3.8. Both the QPALPV system and the QPAL are used
for Proposition 3.8. First, we consider condition (II) of Proposition 3.8. As in the proof of
Lemma 5.1, condition (II) over each Iij becomes
2 3T 2
3 2 3
^
D+V (xcl ; )(x_ cl ; 0) + D+ V (xcl ; )(0; _) ; 4 w 5 4 Tcl ;cl 5 4 w 5 < 0: (5.160)
z
;cl ;cl ij z
Note that [^ cl ]ij , [cl ]ij and [;cl ]ij are assumed to be dened over the smallest open set
that includes the sub-domain Pij . Since V (x; ) is dierentiable,
3 2 3T 2
2 T^ ^ 3 2 3
^ cl ;cl
@ xclP ()xcl
@ xTcl P^ (^)xcl ^_
w
w5
5
4
5
4
5
4
x
_
+
Eq. 5.160 = 4
;
cl
T
@xcl
@ ^
z
;cl ;cl ij z
ij
= xTcl Lcl (^; ^_ ) + Ccl (^)T cl Ccl (^) xcl + xTcl P^cl (^)Bcl (^) ; Ccl (^)T ;Tcl w
i
+wT Bcl (^)T P^cl (^) ; ;cl Ccl (^) xcl ; wT ^ cl w ij < 0:
(5.161)
Therefore, condition (II) is satised if for all ij ,
2
Lcl(^; ^_ )
P^cl (^)Bcl(^) + Ccl(^)T ;cl Ccl (^)T
66
64 Bcl(^)T P^cl (^) + ;Tcl Ccl(^)
;^ cl
0
Ccl(^)
0
;;cl1
3
77
75 < 0
(5.162)
ij
for 8(^ij ; ^_ ij ) 2 P^ . Condition (I) of Proposition 3.8 is satised if [P^cl (^)]ij > 0 for all
^ij 2 P^ and all ij .
2
Lemma 5.13 is equivalently restated using the congruence transformation.
CHAPTER 5. SYNTHESIS
124
Lemma 5.14 Suppose there exist ij , ;ij , [X^ (^)]ij , [Y^ (^)]ij , [A^c(^)]ij , [B^c(^)]ij and [C^c(^)]ij
such that for all ij ,
2
3
^ (^) I
X
4
5 >0
(5.163)
I
and
Y^ (^)
ij
2
3
^; ^_ ) A(^) + A^c (^)T H1 (^)
^)
L
(
H
(
Y
I
2
66
7
66 LXI (^; ^_ ) H3 (^) [Cz (^)T Cq (^)T ] 777 < 0
66 77
;^ cl
0
4
5
;;cl1
ij
(5.164)
for all (^ij ; ^_ ij ) 2 P^ , where
h ^i
h
i
H1() ij = Bw0 (^) Bp(^) + Y^ (^)Cq (^)T ; ij ;
h ^i
h
i
H2() ij = Y^ (^)Cz0(^)T + C^c (^)T DzT0u Y^ (^)Cq (^)T ij ;
h ^i
h
i
H3() ij = X^ (^)Bw0 (^) + B^c(^)Dyw0 X^ (^)Bp(^) + Cq (^)T ; ij :
Then, the QPALPV system (Eq. 5.151) is uniformly asymptotically stable about x = 0 for
all 2 F 2 and 2 S , and the robust L2 -gain from w0 to z0 is less than , i.e., ^2 < .
Proof: Perform the congruence transformation on Eqs. 5.158 { 5.159. Since the derivation
of Eq. 5.158 is obvious, we consider Eq. 5.159.
2
3
2
3
^) 0 0
^) 0 0 T
S
(
S
(
1
1
6
7
6
7
Eq. 5.159 < 0 , 664 0 I 0 775 (Eq. 5.159) 664 0 I 0 775 < 0;
0
0 I ij
0
0 I ij
where [S1 (^)]ij is dened at Eq. 5.17. After some matrix manipulations, we can obtain
2
3
^; ^_ ) T2 (^; ^_ ) H1 (^)
^)
T
(
H
(
1
2
66
7
66 T3 (^; ^_ ) H3(^) [Cz (^)T Cq (^)T ] 777 < 0;
66 77
;^ cl
0
4
5
;;cl1
ij
(5.165)
CHAPTER 5. SYNTHESIS
125
where [T1 (^; ^_ )]ij , [T2 (^; ^_ )]ij and [T3 (^; ^_ )]ij are dened at Eqs. 5.26 { 5.28. With the
intermediate controller variables at Eqs. 5.29 - 5.31, Eq. 5.165 is equal to Eq. 5.164.
2
The intermediate controller variables of Lemma 5.14 will be even further eliminated
applying the elimination lemma.
Lemma 5.15 Suppose there exist ij , ;ij , [X^ (^)]ij and [Y^ (^)]ij 's such that for all ij ,
2
4 X^ (^)
I
3
I 5 > 0;
Y^ (^) ij
(5.166)
2
3
^ ^; ^_ ) ; Bu (^)Bu (^)T
L
(
A;
Y
66
77
2
3
2
3
^
66
77
C
(
)
;
I
0
4 z1 5 Y^ (^)
4
5
66
77 < 0;
Cq (^)
0 ;;1 2
66 2
77
3
3
2I 0
64 4
75
Bw0(^)T
;
5
4
5
0
Bp(^)T + ;T Cq (^)Y^ (^)
0 ;
ij
2
3
_
2
T
66 L2X (A;~ ^; ^) ; Cy (^) Cy (^3) 2
77
3
2I 0
66 4
77
Bw1 (^)T X^ (^)
;
66 Bp(^)T X^ (^) + ;T Cq (^) 5 4 0 ; 5
7
66
2
3
2
3 777 < 0
64
4 Cz0(^) 5
4 ;I 0;1 5 75
0
Cq (^)
0 ;
ij
for all (^ij ; ^_ ij ) 2 P^ , where
h^ ^i
h
A() ij = A(^) ; Bu (^)Cz2
i
and
ij
h~ ^i
h
(5.167)
(5.168)
i
A() ij = A(^) ; Bw2 Cy (^) ij :
Then, the QPALPV system (Eq. 5.151) is uniformly asymptotically stable about x = 0 for
all 2 F 2 and 2 S , and the robust L2 -gain from w0 to z0 is less than , i.e., ^2 < .
CHAPTER 5. SYNTHESIS
126
Proof: The result is directly obtained by applying Lemma 2.7 (Elimination Lemma) to
Eq. 5.164. We dene some matrices
2
LY (A; ^; ^_ )
66
66
A(^)T
LX (A; ^; ^_ )
66
Bw (^)T 1 0 Bw0 (^)T X^ (^) 1
_
6
0
G(^; ^) = 666 @ Bp(^)T + A @ Bp (^)T X^ (^) A
ij
66 ;T C (^)Y^ (^)
+;T Cq (^)
q
66
64
Cz0 (^)Y^ (^)
Cz0 (^)
Cq (^)T Y^ (^)
Cq (^)
2
3
2
0
66 0 Bu(^) 77
66 I
66 I 0 77
66 0 Cy (^)T
66
77 h i 66
T
h ^i
U () ij = 66 0 0 77 ; V (^) ij = 66 0 Dyw0
66 0 0 77
66 0 0
66 0 Dz0u 77
66 0 0
4
5
4
0
0
0
0
ij 3
2
h^ ^i
^ ^ ^ ^
K () = 4 Ac() Bc() 5 :
ij
C^c(^)
0
;^ cl
3
77
77
77
77 ;
77
77
5
3
77
77
77
7
777 ;
77
77
7
;;cl1 5
0
ij
ij
ij
A simple matrix manipulation shows that
Eq. 5.164 = G(^; ^_ ) + U (^)K^ (^)V (^)T + V (^)K^ (^)T U (^)T
ij
< 0:
According to Lemma 2.7, the above inequality is equivalent to a pair of LMIs such that
U?(^)T G(^; ^_ )U?(^) < 0 and V?(^)T G(^; ^_ )V?(^) < 0:
ij
ij
(5.169)
CHAPTER 5. SYNTHESIS
127
Here, [U?(^)]ij and [V?(^)]ij are an orthogonal complement of [U (^)]ij and [V (^)]ij , respectively:
2
I
0
66 0
66 0
0
0
66 0
0
I
66
h ^i
U?() ij = 66 2 0 3 2 0 3 0
66 I
66 4 5 4 0 T 5 0
64 0
;Bu(^)
3
77
77
77
77
77
77
0 0 77
75
0
0
0
I
0
0
0
0
0 0 I ij
2
3
0
0
0
0
0
66
7
66 2 0 3 2 I 3 0 0 0 777
66 I
7
66 4 5 4 0 5 0 0 0 777
h ^i
V?() ij = 66 0
77 :
;Cy (^)
66 0
0
0 0 I 777
66
64 0
0
I 0 0 775
0
0
0 I 0 ij
0
0
After calculating Eq. 5.169 with these [U?(^)]ij and [V?(^)]ij , we can eliminate unknown intermediate controller dynamics [A^c (^)]ij , [B^c (^)]ij and [C^c (^)]ij and then derive Eqs. 5.167 5.168.
2
Each of Eqs. 5.166 - 5.168 leads to an innite number of LMIs. Therefore, we need the
\convexifying" technique to derive a nite number of LMIs.
Proposition 5.9 Suppose there exist Mijk ; Nijk (> 0), ij , ;ij , (Xa1 ; Ya1 ) (a = 1; ; m1 +
1) and (X1b ; Y1b ) (b = 1; ; m2 + 1) such that for all ij ,
2
3
^ (w) I
X
4
5 > 0;
(5.170)
I
Y^ (w)
ij
CHAPTER 5. SYNTHESIS
128
20
3
1
^ w; ) ; Bu (w)Bu (w)T
L
(
A;
Y
66 @
A
77
66
77 X
+12= 2 Bw0 (w)3Bw0 (w)T
2
2
3
66
7
2
+
7
66
4 Cz1(w) 5 Y^ (w)
4 ;I 0;1 5 77 k=1 wk Mijk < 0;
64
75
Cq (w)
0 ;
Bp(w)T + ;T Cq (w)Y^ (w)
0
; ij
(5.171)
20
1
3
2
T
~
66 @ LX (A;Tw; ) ; Cy (w) TCy (w) A
77 2
66 +C2z0(w) Cz0(w) + Cq (w) Cq3(w) 2
77 + X w2 N < 0
3
66
Bw1(w)T X^ (w)
; 2I 0 5 775 k=1 k ijk
4
4
5
4
Bp(w)T X^ (w) + ;T Cq (w)
0 ;
ij
(5.172)
for all (w; ) 2 and
2
3
T
^
^
Y
A
+
A
Y
66 ijk2 ijk 3ijk ijk
77
66 4 Cz1ijk 5 Y
7
0 77 + Mijk 0;
ijk
66
75
Cqijk
4
T
; Cqijk Yijk
(5.173)
0 0 ij
2 T
3
~ijk Xijk + Xijk A~ijk A
66 2 T 3
77
64 4 Bw1ijk 5 X
75 + Nijk 0
0
ijk
T
Bpijk
(5.174)
ij
for all k. Then, the QPALPV system (Eq. 5.151) is uniformly asymptotically stable about
x = 0 for all 2 F 2 and 2 S , and the robust L2-gain from w0 to z0 is less than , i.e.,
^2 < .
Proof: The proof is similar to that of Proposition 4.1. Consider Eq. 5.168 (Eq. 5.167 can
be similarly treated).
2
_
62
G(^; ^) = 664 4
ij
~ ^; ^_ )
LX (A;
3
2
3
2I 0
Bw1(^)T X^ (^)
;
5 4
5
Bp (^)T X^ (^) + ;T Cq (^)
0 ;
3
77
75
ij
CHAPTER 5. SYNTHESIS
129
20 2
1 3
^)T Cy (^)
;
C
(
y
66 BB
CC 77
T
^
^
6
B
C 07
+
C
(
)
C
(
)
+ 66 @ z0 ^ T z0 ^ A 77 < 0:
64 +Cq () Cq ()
75
0
(5.175)
0 ij
According to Lemma 2.10, Eq. 5.168 is satised on all ^ij 2 P^ if
[G(w; )]ij +
and
2
X
k=1
wk2 Nijk < 0; 8(w; ) 2 2 T
3
~ijk Xijk + Xijk A~ijk A
66 2 T 3
77
64 4 Bw1ijk 5 X
75 + Nijk 0; 8k:
0
ijk
T
Bpijk
(5.176)
(5.177)
ij
Note that the coupling terms of the second matrix of Eq. 5.175 can be eliminated by the
schur complement. Thus, the second matrix does not have the ^k2 terms.
2
Remark 5.5 As minimizing , we compute the smallest upper bound of the robust L2-gain
of the QPALPV system (Eq. 5.151). This smallest (opt ) is often used as the robust L2 -
gain. However, this minimization problem is not LMIs but BMIs because of coupling terms
such as [Y^ (w)Cq (w)T ;]ij . Nevertheless, these BMIs can be eciently and eectively solved
by an iterative method, which nds Pij for xed ij and ;ij , and converse. This iterative
method, while a heuristic method without guarantee of convergence, seems to converge the
local minima in many practical applications (Refer to [Ban97] for details). Note that these
BMIs can give tighter upper-bound than the typical D-K iteration approach in the LPV
control with constant D [AHN97,LeeL97].
A central controller formula is constructed in terms of the PALPV system dynamics
(Eq. 5.149) and solutions of Proposition 5.9.
Proposition 5.10 Suppose , ij , ;ij , [X^ (^)]ij and [Y^ (^)]ij are solutions of Proposition 5.9. Then, a central controller of Eq. 5.3, which makes the robust L2 -gain of the
CHAPTER 5. SYNTHESIS
130
QPALPV system (Eq. 5.151) less than , is given as follows: for (t) 2 Pij , Eqs. 5.45 {
5.47 and
2
3
h^ ^i
h
i
^
^
^
^
^
C
(
)
Y
(
)
+
D
C
(
)
z0u c
5
Ac() ij = ;A(^)T ; Cz0 (^)T Cq (^)T ij clij 4 z0
^
^
^
Cq ()Y ()
ij
20
1
3
T
T
2
3
T
66 @ Bw0T(^) X^ (^T) A 77
^
B
(
)
w0
1 77
6
+Dyw0 B^c (^) 1 77 ^ ;1 66 0
T+
6
0
^
;6
(5.178)
6
7
B
(
)
p
7
64 @ Bp(^)T X^ (^) A 75 clij 4 @ T ^ ^ ^ A 5
; Cq ()Y ()
+;T Cq (^)
h^ ^ i
h 2 ^ T ^ ^ i ij
Bc() ij = ; Cy () + X ()Bw2 ij
h^ ^ i
h
i
Cc() ij = ; Bu (^)T + Cz2 Y^ (^) ij :
ij
(5.179)
(5.180)
Proof: We derive Eqs. 5.178 { 5.180. The derivations are similar to those of Proposition 5.2.
We dene some matrices
h ^i
h
i
L1 () ij = Bw0(^) Bp (^) + Y^ (^)Cq (^)T ; ij
h ^i
h
i
L2 () ij = X^ (^)Bw0(^) + B^c(^)Dyw0 X^ (^)Bp (^) + Cq (^)T ; ij
h ^i
h
i
L3 () ij = Y^ (^)Cz0 (^)T + C^c(^)T DzT0u Y^ (^)Cq (^)T ij
h ^i
h
i
T
T
L4 ()
=
ij
Cz0(^)
Eq. 5.164 can be formulated
2
Eq. 5.164 = 4
Cq (^)
ij
:
3
2
3
2
3
T
LY I (^)
A(^) + A^c(^)T 5 + 4 L1 (^) 5 ^ ;1 4 L1 (^) 5
A(^)T + A^c(^)
LXI (^)
L2 (^) ij clij L2 (^) ij
ij
2
3
2
3
^)
^) T
L
(
L
(
3
3
+ 4 ^ 5 clij 4 ^ 5 < 0:
L4 () ij
L4 () ij
CHAPTER 5. SYNTHESIS
131
The idea is to nd [A^c (^)]ij , [B^c (^)]ij , and [C^c (^)]ij such that make (1; 2) = 0, (2; 2) < 0
and (1; 1) < 0, respectively. First, consider the condition (2; 2) < 0.
2 3T
2 3
2 3T 2
_
x
x
66 77
66 77
66 x 77 66 LXI (^; ^) L2(^) L4(^)
64 y 75 (2; 2) 64 y 75 = 64 y 75 64 L2(^)T ;^ cl 0
z
z
z
L4 (^)T
0 ;;cl1
2
2
3
T
T
6 0T Dyw 0
66 B^c(^) x 77 66 Dyw
= 64 y 75 66
;^ cl
64 0
z
ij
0
0
2
Cy (^)x
66
66
Bw (^)T X^ (^)x
+ 2 666 Bp(^)T X^ (^)x + ;T Cq (^)x
66
Cz (^)x
4
Cq (^)x
_
+ xT LX (A; ^; ^) x < 0:
3
77
75
2 3
66 x 77
64 y 75
z
ij
3
0 7 2 ^ ^T
77 6 Bc() x
0 77 664 y
75
z
;
1
;cl ij
3T
77 2
77 6 B^c(^)T x
77 66 y
77 4
75
z
ij
ij
(5.181)
A [B^c (^)]ij satisfying Eq. 5.181 is the saddle point ([B^c (^)]ij ):
0
2
Cy (^)
66
3
66 0 Bw (^)T X^ (^) 1
Dyw0 0 0 7 2 ^ ^ T 3
B
(
)
77 6 c
66 B (^)T X^ (^)
7
;^ cl
0 77 664 775 = ; 66 @ p T ^ A
75
66 +; Cq ()
66
;
1
ij
0
;cl ij
Cz (^)
4
Similarly, we can derive the following equation:
ij
3
77
75
ij
2
66 T0
66 Dyw0
66 0
4
3
77
75
Cq (^)
2 3T
2 3
2 3T 2
3 2 3
_
^
^
^
^
x
x
x
L
(
;
)
L
(
)
L
(
)
1
3
66 77
66 77
66 77 66 Y I
77 66 x 77
64 y 75 (1; 1) 64 y 75 = 64 y 75 64 L1(^)T ;^ cl 0 75 64 y 75
z
z
z
L3 (^)T
0 ;;cl1 ij z
3
77
77
77
77 : (5.182)
77
77
5
ij
CHAPTER 5. SYNTHESIS
132
2
h T
i3
3T 6
0
0
Dz0u 0 7 2 ^ ^
77 6 Cc()x
77 66
^ cl
0
;
0
77 66 y
3
75 66 2
75 4
64 4 Dz0u 5
;1
0
;
z
z
cl
ij
0
ij
3
2
T
Bu(^)T x
77 2
66
3
77 6 C^c(^)x 7
66
Bw (^)T x
+ 2 666 Bp(^)T x + ;T Cq (^)Y^ (^)x 777 664 y 775
77
66
Cz (^)Y^ (^)x
z
5
4
ij
Cq (^)Y^ (^)x
ij
_
2
66 C^c(^)x
= 64 y
+ xT LY (A; ^; ^) x < 0:
Similarly, a [C^c (^)]ij satisfying Eq. 5.183 is the saddle point ([C^c (^)]ij ):
0
3
77
77
77
5
ij
2 66 C^c (^)
64 ij
(5.183)
ij
2
h T
i
0
0
D
0
z 0u
66
66 2 0 3 ;^ cl
0
66 D
4 4 z0u 5 0
;;cl1
3
77
75
2
Bu (^)T
66
3
66 0 Bw (^)T 1
66
77
^T
75 = ; 66 @ TBp(^) ^+ ^ A
66 ; Cq ()Y ()
66 Cz (^)Y^ (^)
ij
4
Cq (^)Y^ (^)
The condition (1; 2) = 0 immediately implies
h^ ^i h ^T
i
Ac() ij = ;A() ; L2 (^)^ ;cl1 L1 (^)T ; L4(^)cl L3 (^)T ij :
After some simple matrix manipulations, we can obtain Eqs. 5.178 { 5.180.
3
77
77
77
77 :
77
77
5
ij
(5.184)
(5.185)
2
Note that other robust performance synthesis problems, such as robust L1-gain and
robust H2 -norm, can be similarly characterized.
CHAPTER 5. SYNTHESIS
133
5.4 Generalized Results
We extend the results for s = 2. The extension is conceptually straightforward except the
construction of the QPAL. We will show that the derived formulations are very similar to
counterparts of the simple case (s = 2).
4 i i . It is also dened that
As in section 4.4, we make some denitions for index. l =
1
s
4
4
1(s) = 1| {z 1} for s 1 with 1(0) = ;.
s
5.4.1 QPALPV system
We dene a PALPV system for the general case of s 3 (Refer to section 4.4.1 for details):
for (x; ; w; u) 2 D F s W U ,
2 3
2
32 3
^) Bw1 (^) Bw2 Bu (^)
x
_
A
(
66 77 m m
66
77 66 x 77
1
s
66 z1 77 = X X () 66 Cz1 (^) 0
0
0 77 66 w1 77
l 6
66 z2 77
77 66 w2 77 ;
6
C
0
0
I
z
2
n
i
=1
i
=1
u
s
4 5
4
54 5
Cy (^)
y
where
2
^
66 A(^) T
66 Bw1()
66 Bu(^)T
66
4 Cz1(^)
3 2
77 66 ATl0
77 66 Bw1l0
77 = 66 BulT 0
77 66
5 4 Cz1l0
0
Iny
0
l
3
2
77
66 ATlk
77 X
66 Bw1lk
s
T
77 + ^lk 66 Bulk
77 k=1 66
5
4 Cz1lk
(5.186)
u
3
77
77
77 :
77
5
(5.187)
Cy (^) l
Cyl0
Cylk
An LPV controller and the QPALPV system can, respectively, be dened over hyperrectangle P Rs : for (xc ; ; y) 2 D F s Y ,
2 3 m 1 ms
2
32 3
_
^
^
^
X
X
x
_
A
(
;
)
B
(
)
c
4 c 5 = l () 4 c
5 4 xc 5 ;
^
u
i=1
is =1
Cc ( )
0
y
l
(5.188)
for (xcl ; ; w) 2 Dcl F s W ,
2
32 3
2 3 m1 ms
^; ^_ ) Bcl (^)
X
X
x
_
A
(
cl
cl
5 4 xcl 5 ;
4 5 = l () 4
^
z
i=1
is =1
Ccl()
0
l
w
(5.189)
CHAPTER 5. SYNTHESIS
where
134
2
3
^
^
^
A
(
)
B
(
)
C
(
)
Acl (^; ^_ ) = 4 ^ ^ u ^ c^_ 5
l
B ()C () Ac(; ) l
2 c y
3
h ^i
^
B
(
)
B
w2 5
Bcl () l = 4 w1
0
Bc(^) l
2
3
h ^i
^
C
(
)
0
5:
Ccl () = 4 z1
Cz2
l
Cc(^)
5.4.2 QPAL with s 3
(5.190)
(5.191)
(5.192)
l
We extend the QPAL to the general case of s 3 (Refer to section 4.4.2 for details). The
QPAL, V (xcl ; ) = xTcl Pcl ()xcl , is dened such that
Pcl () =
m1
X
i=1
m1
X
ms
X
is =1
ms
X
h
i
l () P^cl (^)
l
2 ^^
X ()
l ( ) 4
T
3
(5.193)
M^ (^) 5
=
T X^ (^) ; Y^ ;1 (^) ;1 M
^
^
^
^
^
^
M
(
)
M
(
)
(
)
i=1 is =1
l
2
3
X ()
M ( )
5
= 4
;
M ()T M ()T X () ; Y ;1 () ;1 M ()
Continuous, piecewise-ane X (): over Pl ,
s
h^ ^i
X
X () l = Xl0 + ^lk Xlk
k=1
with
(5.194)
Xl0 = 21 X(i1 +1)1(s;1) + Xi1 1(s;1) + + X1(k;1) (ik +1)1(s;k) + X1(k;1) ik 1(s;k)
+ + X1(s;1) (is +1) + X1(s;1) is ; 2(s ; 1)X1(s)
and
Xlk =
X1(k;1) (ik +1)1(s;k) ; X1(k;1) ik 1(s;k)
:
k
CHAPTER 5. SYNTHESIS
135
Continuous, piecewise-ane Y (): over Pl ,
s
h^ ^ i
X
Y () l = Yl0 + ^lk Yiprk
(5.195)
k=1
with
Yl0 = 21 Y(i1 +1)1(s;1) + Yi1 1(s;1) + + Y1(k;1)(ik +1)1(s;k) + Y1(k;1) ik 1(s;k)
+ + Y1(s;1) (is +1) + Y1(s;1) is ; 2(s ; 1)Y1(s)
and
Y1(k;1)(ik +1)1(s;k) ; Y1(k;1) ik 1(s;k)
:
k
Continuous, piecewise-smooth M () such that over Pl ,
Ylk =
h
i
h
i
I ; X^ (^)Y^ (^) l = M^ (^)N^ (^)T l :
Note that constructing the [X^ (^)]l needs only Xa1(s;1) , , X1(s;1) b (a = 1; ; m1 + 1;
, b = 1; ; ms + 1). Furthermore, for w 2 ,
h^
i n
|
o
}
X (w) l 2 Xi1 is ; X(i1 +1)is ; ; X(i1 +1)is ; X(i1 +1)(is +1) ;
{zs
2
where
Xi1 is = Xi1 1(s;1) + + X1(k;1) ik 1(s;k) + + X1(s;1) is ;(s ; 1)X1(s) :
|
{z
s
}
(5.196)
These relations are true for [Y^ (^)]l when replacing Xi1 is with Yi1 is .
5.4.3 Synthesis formulations
We can derive various synthesis formulations for the generalized PALPV system using the
nonsmooth dissipative systems framework with the generalized QPAL. The derivations are
exactly same as those of the sample cases. Thus, we present only the results of the L2 -gain
synthesis.
CHAPTER 5. SYNTHESIS
136
Let i1 = 1; ; m1 , , is = 1; ; ms and k = 1; ; s. We dene
h^ ^i
h
h~ ^i
h
i
s
i
k=1
s
X
X
A() l = A(^) ; Bu (^)Cz2 l = A^l0 + ^lk A^lk
(5.197)
A() l = A(^) ; Bw2 Cy (^) l = A~l0 +
(5.198)
k=1
^lk A~lk :
Proposition 5.11 Suppose there exist Mlk ; Nlk (> 0) (Xa1(s;1) ; Ya1(s;1) ) (a = 1; ; m1 +
1), , (X1(k;1) b1(s;k) ; Y1(k;1) b1(s;k) ) (b = 1; ; mk + 1), and (X1(s;1) c ; Y1(s;1) c ) (c =
1; ; ms + 1) such that for all l,
2
3
^ (w) I
X
4
5 > 0;
(5.199)
I
Y^ (w)
l
2
3 s
^ w; ) ; Bu (w)Bu (w)T + 1=Bw (w)Bw (w)T Y^ (w)Cz1 (w)T
L
(
A;
Y
4
5 + X wk2 Mlk < 0;
Cz1 (w)Y^ (w)
;I
l k=1
(5.200)
2
3 s
T
T
4 LX (A;~ w; ) ; Cy (w) CyT(w) + 1=Cz (w) Cz (w) X^ (w)Bw1 (w) 5 + X wk2 Nlk < 0
Bw1(w) X^ (w)
;I
l k=1
(5.201)
for all (w; ) 2 and
2
3
^Tlk + A^lk Ylk Ylk CzT1lk
Y
A
lk
4
5 + Mlk 0;
(5.202)
2 T
3
~lk Xlk + Xlk A~lk Xlk Bw1lk
A
4
5 + Nlk 0
T
(5.203)
Cz1lk Ylk
Bw1lk Xlk
0
0
for all k. Then, the QPALPV system (Eq. 5.189) is uniformly asymptotically stable about
x = 0 for all 2 F s and its L2-gain (2 ) is less than , i.e., 2 < .
Proof: The proof is exactly same as that of Proposition 5.1.
2
A central controller formula is constructed in terms of the PALPV system dynamics and
solutions of Proposition 5.11.
CHAPTER 5. SYNTHESIS
137
Proposition 5.12 Suppose , [X^ (^)]l and [Y^ (^)]l are solutions of Proposition 5.1. Then,
a central controller, which makes the L2 -gain of the QPALPV system less than , is given
as follows: for (t) 2 Pl ,
_
;
1
^
^
^
^
Ac(; ) = M () A^c (^) + X^ (^)Y^_ (^) + M^ (^)N^_ (^)T ; X^ (^)A(^)Y^ (^)
l
i
;X^ (^)Bu(^)C^c(^) ; B^c(^)Cy (^)Y^ (^) N^ (^);T l
(5.204)
h ^i
h ^ ^ ;1 ^ ^ i
Bc() = M () Bc() l
(5.205)
h ^ il
h ^ ^ ^ ^ ;T i
h
Cc()
l
= Cc()N ()
i
h
l
(5.206)
i
where M^ (^)N^ (^)T l = I ; X^ (^)Y^ (^) l and
h^ ^i
h
Ac() l = ;A(^)T ; ;1 X^ (^)Bw1(^)Bw1 (^)T + Cy (^)T BwT 2
i
; ;1 Cz1(^)T Cz1(^)Y^ (^) + Cz2(^)T Bu(^)T l
h^ ^i
h
i
Bc () l = ; CyT (^) + X^ (^)Bw2 l
h^ ^ i
h
i
Cc () l = ; BuT (^) + Cz2 Y^ (^) l :
Proof: The proof is exactly same as that of Proposition 5.2.
(5.207)
(5.208)
(5.209)
2
Chapter 6
Numerical Studies
Throughout numerical examples, we will address the usefulness, reliability, and conservatism
of the developed techniques in the previous chapters. Stability margin, L2 -gain analysis and
synthesis problems are included to demonstrate that our design techniques can lead to less
conservative, guaranteed results than the published techniques because they use a general
class of PDLFs. With a missile autopilot design, we demonstrate the whole process from the
PALPV modeling of a nonlinear system to nonlinear numerical simulation. We also address
several important issues: the impact of the model used in the synthesis on the performance
and reliability of the designed controller; and the feasibility of solving and implementing
the LPV control problem.
6.1 Stability Margin and L2-Gain Problems
We will perform a comparative study to investigate several aspects of our approach. The
stability margin will be used as a measure of conservatism in various analysis methods.
Consider the benchmark mass-spring system with a time-varying uncertainty [GAC96]
2
x_ = 4
0
3
15
x
;1 ; (t) ;1
(6.1)
with the bound on time variation j_(t)j _max tr ; where tr = 1:8 sec is the rise time of the
nominal system.
138
CHAPTER 6. NUMERICAL STUDIES
139
We choose 100 points for the normalized bound _max between [0:01; 60]. For each _max,
we calculate the parameter margin max , which is the largest value of j(t)j over which an
LPV system (Eq. 6.1) can still be guaranteed to be uniformly asymptotically stable. This
problem is obviously a maximization problem.
Since the LPV system is ane in , we assume that the optimal class of PDLFs is an
ane parameter-dependent Lyapunov function (ALF). This assumption stems from the fact
that the Lure-Postnikov Lyapunov function has been widely used for the analysis of Lure
systems [BGFB94]. It is also a popular heuristic rule of the gridding technique that the
PDLF in the dissipative systems framework has the same parameter-dependency as the LPV
system [Beck96]. Therefore, we consider the published methods [FAG95,GAC96], called S procedure and multi-convexity method, that use the smooth dissipative systems framework
with ALF over the entire range of _max. Using dierent \convexifying" techniques, these
two methods provide guaranteed, (possibly conservative) lower bound results (Note that
this is a maximization problem so typical meanings of lower and upper bound are reversed).
We also consider the gridding technique [Wu95] with the same ALF over the entire range
of _max. Note that the gridding technique itself does not guarantee the analysis result,
but does provide a upper bound. Therefore, when using the same class of PDLFs, the
result from the gridding technique is useful in indicating how a technique is conservative.
Finally, our approach based on Proposition 4.1, called PAL, is also considered for a number
of dierent partitions in the parameter, N 2 f1; 2; 3; 5g. It is straightforward to derive a
PALPV system (Eq. 4.3) from Eq. 6.1 for given N . Note that in the case of N = 1 (i.e.,
no partition), our developed technique is equivalent to that in Ref. [GAC96].
All the methods are incorporated with a bisection algorithm to nd the maximum
stability margin. All LMI-related computations are then performed with the MATLAB
LMI Control Toolbox [GNLC95]. Ten grid points are used for the gridding method.
Figure 6.1 plots the resulting curves of the stability margin max vs. _max (normalized
bound on time variation of the parameter). This gure shows that as _max ! 0 (i.e. an
unknown-but-constant parameter), the results of all three methods are consistent with those
of the real -analysis and Popov analysis which predicted max = 1. Also, as _max ! 1,
all the techniques are consistent with the quadratic stability using a quadratic Lyapunov
function which predicted max = 0:8660. Most importantly, this gure shows the prediction
from each approach for intermediate values of _max. Each of the curves has the same basic
CHAPTER 6. NUMERICAL STUDIES
140
1
Gridding Tech
S−Procedure
0.98
PAL’s
0.96
max
0.94
0.92
N =1
0.9
2
3 5
0.88
0.86 −2
10
−1
10
0
10
_max
1
10
Fig. 6.1: max vs. _max from the gridding technique, multi-convexity, S -procedure, and our
PAL with a number of dierent partitions (N ). The multi-convexity approach is equivalent
to our PAL with N = 1.
shape which indicates that, as expected, the techniques can only guarantee asymptotic
stability for smaller values of the parameter as the rate of change in the parameter increases.
Figure 6.1 shows which of the techniques predicts the largest stability margin for intermediate values of _max, which should provide a direct measure of conservatism in various
approaches. As shown, the previously published methods using the S -procedure or multiconvexity requirement (N = 1) are more conservative than the heuristic gridding technique
for N 0:03. In particular, the S -procedure approach is very conservative (except in the
case with arbitrary slowly-varying parameters). In contrast, our PAL approach with N = 2
predicts a larger stability margin over the entire range of _max than the gridding technique.
Therefore, our PAL approach with a small number of partitions (N 2) leads to the largest
stability margin of the presented techniques, and thus is the least conservative. Furthermore, as N increases, our PAL approach predicts even larger stability margins. Note that
CHAPTER 6. NUMERICAL STUDIES
22
20
141
16
_max = 0
_max = 1
15
P11 ()
P11 ()
18
14
16
14
13
12
−0.5
12
−1
0.5
233
_max = 20
232.5
139
P11 ()
P11 ()
139.5
0
−0.5
0
0.5
1
_max = 1000
232
231.5
138.5
231
230.5
138
−0.5
0
0.5
230
−0.5
0
0.5
Fig. 6.2: P () from our PAL with N = 5 at dierent values of _max's.
in this example, these improvements in the analysis should make up for the polynomiallyincreased computational time due to the increased number of variables and LMIs.
The question arises why our approach predicts larger stability margins than the gridding
method with ALF. The answer can be found in Fig. 6.2, which plots P () from our PAL
approach with N = 5 vs. 2 [;max; max ]. For convenience, Fig. 6.2 plots four P11 ()'s,
(1; 1) elements of P ()'s, for several values of _max's. In each case, a dashed line connecting
the maximum and minimum value of P11 () has been included as a reference (Note that
an ALF looks like a line in this plot). The gure shows that P11 () is almost ane in for small _max and then changes to a nonlinear S -shaped curve for intermediate values of
_max. Finally, P11 () is constant at _max = 1000, indicating that the desired PDLF is essentially parameter-independent. This new result shows an important relationship between
the optimal class of PDLFs and time variation of the parameter (_max ): the optimal PDLF
does not have the same parameter-dependency as the LPV system but has very dierent
parameter-dependency at dierent values of _max. Therefore, the gridding technique using
CHAPTER 6. NUMERICAL STUDIES
142
ALF over the entire range of _max is more conservative than our PAL approach. Obviously,
this result can be also used to select the optimal class of PDLFs in the framework of the
gridding technique.
We continue a similar comparative study with the L2 -gain problem. We consider another
simple benchmark system
2
3 2 3
0
1
5x + 4 0 5w
x_ = 4
;1:25 ; (t) ;1
1
h
i
z =
1 0 x
(6.2)
(6.3)
and j(t)j 0:70; j_(t)j _max tr ; where tr = 1:61(sec) is the rising time of the nominal
system. We then calculate the L2 -gain of the LPV system (Eqs. 6.2 { 6.3) vs. _max using
various analysis techniques.
As in the previous case, we assume that the optimal class of PDLFs is ALF. We then
consider the published methods [GAC96, LimH97], called multi-convexity method and S procedure, using ALF over the entire range of _max. These two methods provide guaranteed,
(possibly conservative) upper bound results. We also consider the gridding technique [Wu95]
with the same ALF over the entire range of _max, which can provide a lower bound. Finally,
our approach based on Proposition 4.2, called PAL, is also used for a number of dierent
partitions in the parameter, N 2 f1; 2; 3; 5g. Note that in the case of N = 1 (i.e., no
partition), our PAL technique is equivalent to that in Ref. [GAC96].
All the methods lead to standard convex optimization problems because the L2 -gain is
dened as the minimum upper bound ( ) of Proposition 4.2. All LMI-related computations
are performed with the MATLAB LMI Control Toolbox [GNLC95], especially the mincx
subroutine. Ten grid points are used for the gridding method.
Fig. 6.3 plots the L2 -gain vs. _max of various analysis techniques. This gure demonstrates similar behavior of analysis techniques to the stability margin problem of section 6.1:
each of the curves has the same basic shape which indicates that the techniques can only
predict larger value of the L2 -gain as the rate of change in the parameter increases (In
this case, a smaller L2 -gain is better performance). However, the gure shows that for
_max 0:01, the multi-convexity approach (N = 1) yields a slightly conservative result
while the S -procedure leads to the same L2 -gain as the gridding approach. Conservatism
is mainly due to the \convexifying" technique itself. However, as N increases, our PAL
CHAPTER 6. NUMERICAL STUDIES
143
3.2
3
N=1
2 3
5
L2-Gain
2.8
2.6
2.4
Gridding Tech
S−Procedure
2.2
PAL’s
2
1.8 −2
10
−1
10
0
10
_max
1
10
Fig. 6.3: L2 -gain vs. _max from the gridding technique, multi-convexity, S -procedure
and our PAL with a number of dierent partitions (N ). The multi-convexity approach is
equivalent to our PAL with N = 1.
technique reduces the conservatism of the \convexifying" technique because it \convexies"
an innite-dimensional LMIs over each smaller partitioned parameter subspace rather than
the original parameter space.
6.2 L2-Gain Synthesis
We will perform a comparative study of conservatism in various synthesis methods. Consider
a benchmark problem (see Figure 6.4) with a time-varying parameter (t) such that
CHAPTER 6. NUMERICAL STUDIES
144
1/2 1/2
x2
x1
1/2 -1/2
w21
w11
u
M1 = 1
z1
M2 = 1
y
k = 1:25 + (t)
z2
w2
Fig. 6.4: Block diagram of a benchmark problem
2
66 x_
66 z1
66 z2
4
y
2
0
0
66
0
0
3 666
77 66 ;1:25 ; (t) 1:25 + (t)
77 = 66 1:25 + (t) ;1:25 ; (t)
77 66
0:5
;0:5
5 666
0:5
0:5
66
64
0
0
0
1
0
0
0
0
0
0
1 0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0
0
1
0
3
77
77 2
77
77 66
77 66
77 66
77 4
77
75
x
w1
w2
u
3
77
77
77
5
(6.4)
with j(t)j 0:7 and j_(t)j _max tr , where tr = 1:61 sec is the rise time of the nominal
system. It is assumed that (t) and _(t) are measurable in real time.
Similar to section 6.1, we choose 50 points for the normalized bound _max between
[0:01; 60]. For each _max, we nd the opt and the LPV controller that makes the L2 -gain
of the closed-loop system less than opt . Note that opt means the smallest upper bound of
the L2 -gain (2 ).
Since the LPV system (Eq. 6.4) is ane in , we assume that, as before, the optimal
class of PDLFs is a quasi-ane parameter-dependent Lyapunov function (QAL) such that
X () and Y () of Pcl () are ane in . We then consider the gridding technique [Wu95],
multi-convexity and S -procedure [LimH98] with QAL over the entire range of _max . Finally,
our QPAL approach based on Proposition 5.1 is used with a number of dierent partitions,
CHAPTER 6. NUMERICAL STUDIES
145
22
20
N=1
2 3
5
18
L2-Gain
16
14
PCS
12
Gridding Tech
S−Procedure
10
QPAL
8
6
−2
10
−1
10
0
10
_max
1
10
Fig. 6.5: L2 -gain vs. _max from the gridding technique, S -procedure, multi-convexity
approach and our QPAL. The multi-convexity approach is equivalent to our QPAL with
N = 1. PCS means the Popov controller for _max = 0. The dot-line indicates the minimum
L2-gain that can be obtained by the pointwise H1 control for _max = 0.
N 2 f1; 2; 3; 5g. Note that the multi-convexity approach is equivalent to our QPAL approach with N = 1. As mentioned, all other approaches except the gridding technique can
provide the guaranteed upper bound of the L2 -gain, while the gridding technique yields
a lower bound of the L2 -gain. To investigate the parametric robust control and the LPV
control, we have also designed a Popov controller from the Popov Control Synthesis (PCS)
technique [Ban97] for the special case _max = 0. As before, all the methods lead to standard
convex optimization problems. Ten gridding points are used for the gridding method.
Fig. 6.5 plots the L2 -gain's vs. _max from various synthesis techniques. In the gure,
the dot-line indicates the minimum achievable L2 -gain from the pointwise H1 control at
_max = 0. Fig. 6.5 shows the prediction of the L2 -gain's from each approach for a broad
range of _max. This gure shows similar behaviors to the previous results. Our QPAL
CHAPTER 6. NUMERICAL STUDIES
146
22
20
18
L2-Gain
16
14
12
Gridding Tech
S−Procedure
10
QPAL
8
6
−2
10
−1
10
0
10
_max
1
10
Fig. 6.6: L2-gain vs. _max from the same synthesis techniques as in Fig. 6.5 except that
Y () is constrained to be constant. All the labels are same as Fig. 6.5.
approach with even small N = 2 reduces conservatism due to \convexifying" and then
recovers the result of the gridding technique. Furthermore, our QPAL with N 3 can yield
even better results than the gridding technique because of the richness of PDLFs. Note
that the results approach to the achievable optimal value (dot-line) at _max ! 0.
Fig. 6.5 also shows that as _max ! 0, the results of all LPV controls are much better
than that of PCS [Ban97]. The PCS technique is based on the Lure-Postnikov Lyapunov
function of the form
2
3
X
(
)
M
5;
Pcl () = 4 T
M S
where X () is ane in and the others are constant. Thus, this Lyapunov function is much
more restrictive than QAL or QPAL, thereby yielding results which are too conservative.
This fact can also be conrmed in Fig. 6.6, which plots L2 -gain's vs. _max from the same
synthesis techniques as Fig. 6.5 except that Y () is constrained to be constant. For example,
CHAPTER 6. NUMERICAL STUDIES
the Lyapunov function is of the form
2
Pcl () = 4
147
3
X ()
;X () + Y ;1 5 ;
;X () + Y ;1 X () ; Y ;1
where X () is (piecewise-) ane in . Clearly, the restricted QAL or QPAL is more similar
to the Lure-Postnikov Lyapunov function than the original QAL or QPAL. Thus, it is
expected that the results from this restricted QAL or QPAL are more similar to that of
the PCS approach. Fig. 6.6 exactly demonstrates this expectation. At _max = 0:01, all
LPV controllers have almost the same performance as the Popov controller. These results
demonstrate the strong relationship between the richness of PDLFs and conservatism of the
synthesis.
It should be mentioned that the type of the controller determines the type of PDLF. For
example, consider the Popov controller. The Popov controller should treat a parametric
uncertainty which is not measurable in real time. Thus, the Popov controller should be
uncertainty-independent. Since M and S are related to the controller dynamics [Ban97], it
is obvious that M and S should be constant. Therefore, the Popov controller limits the class
of PDLFs for the synthesis. In contrast, an LPV controller can be parameter-dependent so
that the LPV controller allows us to use a general class of PDLFs for the synthesis.
To analyze further benets of an LPV controller, we perform a post-analysis to calculate
H1-norm's of the closed-loop systems from several design techniques. For this purpose, we
consider the static case (_max = 0). We select 50 frozen 's in [;0:7 0:7] and construct an
open-loop perturbed system for each value of 's. These open-loop systems are then closed
by various controllers. After closing, we calculate nominal H1 -norm's of these closed-loop
systems over selected 's. The controllers considered here are the Popov controller [Ban97]
and the LPV controller from our QPAL approach. We also include the results from the
pointwise-H1 control which is the best approach for the static case.
Figure 6.7 plots H1 -norm's vs. of the closed-loop systems from three design techniques.
In the gure, the smaller value of the H1-norm implies a controller of better performance.
The gure shows that the Popov controller yields almost the same actual performance as
the worst performance guaranteed by the synthesis over the entire parameter variation. In
contrast, LPV controllers yield better performance than the worst performance guaranteed
by the synthesis. The results shows a clear dierence between a parameter-independent and
a parameter-dependent controller: the Popov controller should be tuned to the worst case
CHAPTER 6. NUMERICAL STUDIES
148
13
12
11
10
Pointwise H1
PCS
9
H1
QPAL
8
7
6
N=1
N=3
5
N=2
4
3
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
Fig. 6.7: H1-norm vs. of the closed-loop system for _max = 0, the controller of which is
designed by pointwise-H1 , PCS, and QPAL approach with N = f1; 2; 3g, respectively.
scenarios, while the LPV controller is not necessarily. This is another benet of using LPV
controllers.
As N increases, our QPAL yields even better actual performance over the entire parameter variation. Furthermore, the actual performance is likely to converge to the best achievable performance curve of the pointwise-H1 controller. However, the large gap between the
actual performances still remains, while there is little gap between the worst performances.
Therefore, it could be desirable to use a parameter-dependent , as suggested in [LeeL97].
In this case, one of good candidates is a piecewise-constant .
CHAPTER 6. NUMERICAL STUDIES
149
6.3 Autopilot Design
Through the L2 -gain synthesis of a realistic LPV model, this section addresses the usefulness, reliability, and conservatism of the newly developed techniques. While the previous
design examples emphasize conservatism of the analysis and synthesis technique associated
with the class of PDLFs, this section demonstrates the whole process from the PALPV modeling of a nonlinear system to nonlinear numerical simulation. Furthermore, it addresses
several important issues: the impact of the model used in the synthesis on the performance
and reliability of the designed controller, and the feasibility of solving and implementing
the LPV control problem.
6.3.1 Missile Model and Performance Objective
The following pitch-axis missile dynamics have been widely used as a benchmark problem [NRR93,WPB95]. Therefore, we directly formulate the missile model as an LPV model
over P = [2 4] [;20 20] and = [;1:5 1:5] [;1 1],
2 3
2
32 3 2
_
f
(
M;
)
1
4 5 = 4 1
5 4 5 + 4 f3(M; )
q_
f (M; ) 0
q
f (M; )
2 3
2 2
32 3 2 4
4 5 = 4 f5(M; ) 0 5 4 5 + 4 f6(M; )
q
0
where
f1 (M; )
f2 (M; )
f3 (M; )
f4 (M; )
f5 (M; )
f6 (M; )
=
=
=
=
=
=
h
1
q
0
3
5 c;
3
5 c;
i
M 0:00012 ; 0:0112jj ; 0:2010(2 ; M=3) cos();
h
i
M 2 0:01522 ; 1:3765jj + 3:6001(;7 + 8M=3) ;
;0:0403M cos();
;14:542M 2 ;
h
i
M 2 0:00012 ; 0:0063jj ; 0:1130(2 ; M=3) ;
;0:0226M 2 :
(6.5)
(6.6)
CHAPTER 6. NUMERICAL STUDIES
150
The actuator dynamics describing the tail deection is
2 3 2
32 3 2
3
_
0
1
0
4 5=4
5 4 5 + 4 2 5 c :

;1502 ;210 _
150
(6.7)
The various variables in the plant model are dened as follows:
q
M
c
c
angle of attack (deg)
pitch rate (deg/sec)
Mach number
command tail deection angle (deg)
actual tail deection angle (deg)
command normal acceleration in g's
actual normal acceleration in g's:
As shown in Eqs. 6.5 - 6.6, variables and q are measured for the output feedback control.
M and are assumed to be scheduling parameters in the process of modeling the missile
dynamics as an LPV system. The prescribed bound of these parameters assumes that
2 M (t) 4 with ;1:5 M_ (t) 1:5 and ;20 (t) 20 with ;1 _ (t) 1.
The assumption on the rate of the Mach number variation enables us to use a PDLF in
the L2 -gain synthesis. Furthermore, it reects real situations in that the Mach number is
relatively slowly varying [Hel95]. In contrast, the angle-of-attack (t) is actually one of the
states in the missile model (In fact, Eqs. 6.5 - 6.6 are typically referred to a quasi linear
parameter-varying (QLPV) system [WPB95]). Therefore, the actual variation of (t) could
be larger than the prescribed value. An iterative design process might be necessary to
design an LPV controller for which the closed-loop system satises the prescribed bounds.
The performance goals for the closed-loop system are dened as follows:
(1) Robust Stability: maintain robust stability over the prescribed bounds of parameters, i.e., 2 M 4 with jM_ (t)j 1:5 and ;20o 20o with j_ (t)j ! 1.
(2) Robust Performance: Track step commands in c(t) with time constant no greater
than 0:35 sec, maximum overshoot no greater than 10%, and steady-state error no
greater than 1%.
CHAPTER 6. NUMERICAL STUDIES
151
Wref
ec
Wc
c
-
err
y
Controller
c
Actuator
Missile
model
q
-
e
er
We
Wd
d
Wn2
dn2
Wn1
dn1
Fig. 6.8: Weighted open-loop interconnection of the missile plant
(3) Bandwidth: Maximum tail deection rate for 1g step command in c(t) does not
exceed 25 deg/sec.
Similar to the H1 design procedure for LTI systems, we use rational weighting functions
to characterize the overall closed-loop performance objective, which are similar to those in
Ref. [WPB95].
;0:05s + 1) ; W (s) = 17:321 ; W (s) = s + 0:25 ;
Wref (s) = s2 +144(
2 0:8 12s + 144 e
s + 0:0577 c
25(0:005s + 1)
Wd (s) = 0:01; Wn1 (s) = Wn2 (s) = 0:001; Act(s) = 1:
Wref reects the designated step response, which exhibits no more than 0:35 sec time
constant and non-minimum phase characteristics of the missile plant [WPB95]. The weighting function We has a low frequency gain of 300, which corresponds to a tight (0:33%) tracking error. To regulate the overshoot of the response, We is desired to have nite DC-gain.
However, a strictly proper We is used for the augmented open-loop system dynamics to satisfy the assumption that Bw2 and Cz2 of Eq. 5.1 are parameter-independent. In particular,
Wd and Wc are included to yield a stable LPV controller [BB91].
CHAPTER 6. NUMERICAL STUDIES
152
The open-loop interconnection for the synthesis is shown in Fig. 6.8. In this gure,
the missile model tends to vary one synthesis approach to another because of dierent
assumptions about the open-loop dynamics. For example, the LFT- approach uses an
LFT model of the missile dynamics, while our proposed approach uses a PALPV model.
The actuator dynamics are deliberately neglected in the open-loop interconnection to keep
the order of open-loop system as small as possible [WPB95]. However, the actuator model
is included in a nonlinear simulation. Furthermore, the direct feedforward term from d to
is small enough and then neglected so that the augmented open-loop system dynamics
satises the assumption that Cz2 is constant. Note that a low-pass lter Wd (s) eliminates
the direct feedforward term, but increases the size of the augmented system dynamics.
6.3.2 PALPV Modeling
This section investigates a simple process to derive PALPV models for the synthesis from
the original missile dynamics (Eqs. 6.5 { 6.6) with a number of dierent partitions (N ). The
process used here is to determine the best t (in the least-squares sense) of the functions
(f1 { f6 ) in Eqs. 6.5 { 6.6 with a piecewise-linear function comprised of N linear functions.
Note that in the general case, an identication technique for a linear-fractionally parameterdependent LPV system [LeeL97] could be used to derive a PALPV model from experimental
data.
There are many methods within the LFT- framework to derive an LFT model from a
nonlinear system dynamics (See Ref. [CM96] and references therein). In the missile autopilot
design, the LFT model, however, corresponds to a PALPV model with N = 1. The gridding
technique for the LPV controller design in [WPB95] requires the missile dynamics (Eqs. 6.5 {
6.6) themselves.
The modeling process is outlined below. Functions f1 { f6 are symmetric in , so we
consider the half parameter space P = [2 4] [0 20]. To begin with, partition P into
N N parameter subspace Pij 's. Over each Pij , an approximate of each function fk is
assumed of the form,
h^ ^ i ^
fk (M; ^ ) ij = fkij 0 ; M^ ij f^kij 1 ; ^ ij f^kij2;
where M^ ij and ^ ij (jM^ ij j 1=N and j^ ij j 10=N ) are local values of M and measured
from the local coordinates with the origin at the center of Pij . The approximate functions
CHAPTER 6. NUMERICAL STUDIES
Table 6.1:
153
Relative RMS of Ek (M; ) in (%)
Partitions RMS(E1 ) RMS(E2 ) RMS(E3 ) RMS(E4 ) RMS(E5 ) RMS(E6 )
N =1
4.59
30.8
0.59
3.10
6.87
3.10
N =2
1.21
8.26
0.16
0.81
1.85
0.81
N =3
0.55
3.83
0.07
0.37
0.86
0.37
N =4
0.32
2.24
0.04
0.22
0.50
0.22
N =5
0.22
1.50
0.03
0.14
0.34
0.14
are then t to the actual functions over each Pij . This problem is formulated as follows:
Problem: Find ff^kij0; f^kij1; f^kij2g such that
h ^ ^ i min fk (M; ) ; fk (M; ^ ) ij 8(M; ) 2 P ij :
2
Note that this optimization problem is innite-dimensional, but in practice it is enough
to solve this problem over a dense nite grid (We use 50 50 grid).
The optimal solution for given N is easily found using MATLAB. A post-analysis was
performed to calculate the approximation error (Ek (M; )) from the t. To compare the
approximation errors, we normalize them as relative RMS errors, which is dened as the
ratio of RMS gain of Ek (M; ) to that of fk (M; ). Table 6.1 shows the relative RMS errors
of Ek (M; )'s vs. N . As expected, the LFT model (N = 1) yields large modeling errors,
especially on f2 (M; ). To design a reliable controller, the LFT model for the synthesis
should take into account these large approximation errors as extra uncertainties of large
size. In contrast, large N leads to a PALPV model that is a good approximation of the
original missile dynamics. In this case, all Ek (M; )'s are small enough that they can be
ignored.
Determining the size of the uncertainty depends upon nding the minimum and maximum value of Ek (M; ). It is assumed that in the modeling process, approximately 5%
relative RMS error in the t of the fk (M; ) is acceptable (This assumption will be veried
by a post-analysis discussion later). With this assumption, it is enough for one to treat only
E2 (M; ) as a uncertainty. The actual value and its actual error of f2 (M; ) for given N
are plotted in Fig. 6.9. Note that f2 (M; ) and E2 (M; ) are plotted along one-dimensional
slices of the two-dimensional parameter space. From this gure, we can nd the minimum
CHAPTER 6. NUMERICAL STUDIES
154
E2 (M; ) at N = 1
200
200
100
100
E2 (M; )
f2(M; )
f2(M; )
0
−100
0
−100
−200
0
500
−200
0
1000 1500 2000
data
500
E2 (M; ) at N = 5
200
200
100
100
E2 (M; )
E2 (M; )
E2(M; ) at N = 3
1000 1500 2000
data
0
−100
0
−100
−200
0
500
−200
0
1000 1500 2000
data
500
1000 1500 2000
data
Fig. 6.9: f2(M; ) and its approximation error (E2 (M; )) with a number of dierent
partitions (N = 1; 3; 5)
and maximum value of E2 (M; ). Then, we can derive a PALPV system with uncertainty
to represent the missile model:
2 3 N N
2
3 2 3
^ ^ ) B (M;
^ ^ )
X
X
x
_
A
(
M;
4 5=
5 4 x 5;
ij (M; ) 4
y
where
2
h ^ i
f^
A(M; ^ ) ij = 4 ^1ij0
f
2 2ij0
h ^ i
f^
B (M; ^ ) = 4 3ij0
ij
^ ^ ) D(M;
^ ^ )
C (M;
i=1 j =1
3
2
3
2
ij
(6.8)
u
3
2
3
1 5 ^ 4 f^1ij 1 0 5
f^ 0
0 05
+ Mij ^
+ ^ ij 4 ^1ij 2 5 + ij 4
;
0
f2ij 1 0
f2ij2 0
1 0
2
2
3
3
3
^3ij 1
^3ij 2
f
f
5 + M^ ij 4
5 + ^ij 4
5;
f^4ij0
f^4ij1
f^4ij 2
CHAPTER 6. NUMERICAL STUDIES
2
h ^ i
f^
C (M; ^ ) ij = 4 5ij0
0
2
h ^ i
f^
D(M; ^ ) = 4 6ij0
ij
3
2
155
3
2
3
0 5 ^ 4 f^5ij 1 0 5
f^ 0
+ Mij
+ ^ ij 4 5ij 2 5 ;
1
0 0
0 0
3
2
3
2
3
5 + M^ ij 4 f^6ij1 5 + ^ij 4 f^6ij2 5 :
0
0
0
where jM^ ij j 1=N and j^ ij j 10=N . The bound of ij (M; ) depends on N . For example,
;30:6 ij (M; ) 123:2 for N = 1 and ij (M; ) = 0 for N 3 because relative RMS
error is less than 5%. For the autopilot design in section 6.3.4, we will select a PALPV model
without uncertainty so we will pick up the PALPV model with N = 3. And the LFT model
is the PALPV model with N = 1. This PALPV model can be easily converted to an LFT
form with the minimum size of uncertainties using singular value decomposition [Hel95].
Note that Eq. 6.8 is an approximation of the missile model. Therefore, this approximation should be augmented by the introduced weighting matrices for the synthesis.
6.3.3 Comparison of LPV control techniques
Before we perform a comparison of dierent autopilot design techniques, we investigate
the conservatism and reliability of the LPV control techniques, such as gridding technique
(GRID) [WPB95] and our new approach (QPAL).
The GRID approach is based on LMIs of Theorem 5.3.1 in [WPB95] (Similar to Lemma 5.3).
As mentioned, the model for the synthesis is the augmented missile dynamics. To solve the
problem, GRID requires the selection of a PDLF and a nite grid set. The PDLFs used here
are the quadratic Lyapunov function [WPB95] and the quasi-ane parameter-dependent
Lyapunov function (QAL) such that X (M ) and Y (M ) are ane in M , Mach number (Note
that a PDLF should be -independent because j_ (t)j ! 1 [GAC96, LimH97, LimH98]).
The rst and second case are called GRID and GRID1, respectively. GRID1 is used to
explore the impact of a PDLF on the gridding technique for the LPV control. For convenience, the grid set for GRID and GRID1 is selected that the grid points are corner points
of the N N parameter subspaces that are partitioned for the PALPV modeling.
The QPAL approach is based on Proposition 5.1. The open-loop system is comprised
of the PALPV model (Eq. 6.8) and the weighting matrices. In this case, the model error
ij (M; ) is intentionally ignored for all N 's in order to investigate the impact of the modeling
error on the reliability of the synthesis technique. The PDLFs used here are two special
cases: QPAL such that X (M ) and Y (M ) is piecewise-ane in M . QPAL1 that X (M ) and
CHAPTER 6. NUMERICAL STUDIES
Table 6.2:
156
Features of LPV control techniques
Method
Model
Type of Lyapunov Function
GRID Missile dynamics
Constant X , Y
GRID1 Missile dynamics
Ane X (M ), Y (M )
QPAL1 PALPV models
Ane X (M ), Y (M )
QPAL PALPV models Piecewise-ane X (M ), Y (M )
Y (M ) are only ane in M . QPAL1 allows us to investigate the conservatism of our approach
to the GRID1 technique. The key dierences between these four cases are summarized in
Table 6.2.
We calculate L2 -gain's from these control techniques for a number of dierent partitions (N). While the results are guaranteed values for the missile model, they may not be
guaranteed values for the original nonlinear missile dynamics. Therefore, it is necessary
to perform a post-analysis: to check eigenvalues of LMIs of Theorem 5.3.1 in [WPB95]
using the solutions (opt , X (M ) and Y (M )) on a dense post-analysis grid (50 50). The
design and post-analysis results from four design techniques are plotted in Fig. 6.10. In this
gure, the L2 -gain's veried by the post-analysis are plotted by black 2; otherwise, they
are plotted by white . In Fig. 6.10, a smaller L2 -gain indicates better performance of the
controller for given N .
Fig. 6.10 shows that GRID with constant X and Y is reliable, but yields overly conservative results. This result is consistent with [WPB95]. This gure also shows that QPAL
with N 3 yields the best performance and also is reliable. As N increases, the performance is improved even further. As pointed out in previous case studies, the improvement
is primarily due to the richness of PDLFs used for the synthesis. As expected, the results
of QPAL with N = 1; 2 are unreliable because the PALPV model does not account for
the modeling errors. This result supports our assumption that 5% relative RMS error is
acceptable in modeling. Note that if ij (M; ) is included as a uncertainty, QPAL with
N = 1; 2 can yield a reliable result, but the performance of the controller degrades.
We continue to investigate the LPV control techniques. Consider GRID1 with ane
X (M ) and Y (M ). Fig. 6.10 shows that GRID1 produces unreliable results over quite dense
grids (up to N = 20). This result indicates that GRID1 is very sensitive to small modeling
errors. This sensitivity is likely related to the type of PDLFs used in the synthesis. A
CHAPTER 6. NUMERICAL STUDIES
157
3.5
3
GRID
GRID1
QPAL1
QPAL
L2-Gain
2.5
2
1.5
1
0
2
4
6
N
8
10
20
Fig. 6.10: L2-gain v.s. N (number of partitions) from GRID, GRID1, QPAL and QPAL1.
Black 2 indicates the result is veried by the post-analysis, while white means the result
is not.
Table 6.3:
Maximum eigenvalues from the synthesis and the post-analysis
QPAL1
QPAL
SYNTHESIS POSTANALYSIS SYNTHESIS POSTANALYSIS
N=3
-1.57E-6
-1.14E-6
-1.97E-6
-1.73E-5
N=4
-7.79E-7
-4.06E-6
-1.53E-6
-1.16E-5
N=5
-9.02E-7
5.91E-7
-1.47E-6
-7.98E-6
similar trend is also found in the QPAL1 approach. As N increases, QPAL1 improves the
performance of the controller. However, the result of QPAL1 with N = 5 is unreliable.
To analyze this reliability problem further, Table 6.3 shows the maximum eigenvalues
from both the synthesis and the post-analysis of the QPAL and QPAL1 technique. The
maximum eigenvalues from the synthesis are directly available from the control design.
CHAPTER 6. NUMERICAL STUDIES
158
Note that in the table, eigenvalues should be negative for a reliable result. The table shows
that the maximum absolute values of eigenvalues from QPAL1 are smaller than those from
QPAL for given N . Furthermore, the change of the maximum eigenvalues from QPAL1 is
as much as that from QPAL in magnitude, but it can be made in the positive direction (or
the worst direction), as shown in the result of N = 5. Therefore, QPAL1 is more sensitive
to the modeling error than QPAL with large N . Note that QPAL has the same parameterdependency as the system, while QPAL1 has the dierent parameter-dependency from the
system.
We recall that the GRID approach is based on the parameter-independent Lyapunov
function. In contrast, the GRID1 approach can improve the performance using a PDLF.
However, the reliability issues appear to stem from the fact that the parameter-dependency
of the PDLF diers from the parameter-dependency of the missile dynamics. A potential
remedy is to use a PDLF that emulates the parameter-dependency of the nonlinear missile
dynamics, which is consistent with the heuristic rule [Beck95]. In the gridding technique,
the desired PDLF should be selected by users. However, our new approach provides a
systematic approach for selecting the desired PDLF by gridding the parameter space prior
to the synthesis.
6.3.4 Autopilot Design and Simulations
Autopilot Design
We have discussed the LPV control techniques. Now, we design several pitch-axis missile
autopilots. For this purpose, we consider several design techniques: naive gain-scheduling
(NGS) [NRR93], complex- (C-), LPV control techniques such as our approach based on
Proposition 5.1 (QPAL) and the typical gridding technique (GRID) [WPB95]. Characteristics of these techniques are summarized in Table 6.4.
The NGS technique designs a controller by two design steps. First, it design 9 nominal H1 controllers. Each controller is based on an augmented open-loop model, which
consists of the missile dynamics frozen at one of nine dierent parameter values, i.e.,
(M; ) 2 [2 3 4] [0 10 20], and the weighting matrices. Second, the NGS technique
linearly interpolate the gain, zeros, and poles of these nine H1 controllers to design a
gain-scheduling controller. The C- technique uses the LFT model (or PALPV model with
CHAPTER 6. NUMERICAL STUDIES
159
Characteristics of design techniques and their results. Note that the
NGS technique does not provide any guaranteed L2 -gain.
Method
Missile Model
Synthesis Tool
L2-Gain
NGS set of 9 frozen missile dynamics H1 control and scheduling
()
QPAL
PALPV model with N = 3
X and piecewise-ane Y (M ) 1:44
GRID
missile dynamics
X and Y
3:10
C-
PALPV model with N = 1
scaled H1 control
37:7
Table 6.4:
N = 1) augmented with the weighting matrices to design a robust controller. Since parameters and uncertainties are time-varying, a constant scaling matrix is used to treat the
structured uncertainties.
The QPAL approach is based on the augmented PALPV model and a special class of
QPAL (Eq. 5.8) with N = 3 such that X is constant and Y (M ) is piecewise-ane in M .
This PDLF makes the LPV controller dynamics depend on only parameters (M; ) (See
remark 5.4). The GRID approach is based on the augmented missile dynamics and the
quadratic Lyapunov function with constant X and Y . The grid set for the GRID approach
is selected (M; ) 2 f2; 2:4; 2:8; 3:2; 3:6; 4g f0; 4; 8; 12; 16; 20g.
Several autopilots are designed with these dierent techniques. All LMI-related computations are performed with the MATLAB LMI Control Toolbox [GNLC95]. Except for
NGS, all other techniques can provide the guaranteed L2 -gain's. The results are shown in
Table 6.4. Table 6.4 shows that our QPAL approach yields the best guaranteed performance. Furthermore, the constraint of X being constant in QPAL does not degrade the
performance of the LPV controller because the L2 -gain is equal to that shown in Fig. 6.10.
As expected, the C- approach yields the worst guaranteed performance. The very poor
performance of the C- controller is primarily due to the large uncertainty for E2 (M; )
associated with the LFT model. This result is conrmed by repeating the synthesis with
the LFT model without the uncertainty for E2 (M; ). In this case, the performance (L2 gain) is improved to 6:11. Note that the performance is still limited by the fact that the
controller is parameter-independent. Conservatism in the LFT model can be reduced using
the PALPV model with an appropriate number of partitions.
CHAPTER 6. NUMERICAL STUDIES
160
Nonlinear Simulations
We perform nonlinear numerical simulations to verify the performances of the designed
controllers. Nonlinear simulations are used to nd the multi-step response of the missile
dynamics (Eqs. 6.5 { 6.6) closed by each controller designed in the previous section. The
reference command (c (t)) is assumed
8
>
>
< 10 0 t < 1
c(t) = > ;15 1 t < 2
> 0 2 t 3:
:
All parameter-dependent controllers need to measure the scheduling parameters (M; ), so
we rst address the method to measure these parameters. Since the Mach number is an
exogenous variable, the Mach number prole is assumed that
; [Case I] M (t) = ;0:05 sin 6 t + 2:1
; [Case II] M (t) = ;0:8 sin 3 t + v(t) + 3.
In Case II, the noise (v(t)) is a uniformly distributed random value between [;0:05; 0:05],
[;0:15; 0:15] and [;0:25; 0:25] (standard deviations are = 0:023; 0:070; 0:116). Case I is
used for both of the missile dynamics and the parameter-dependent controller. However,
Case II is used only for the controller, while the noise-free prole of Case II (v(t) = 0) is
used for the missile dynamics. Therefore, Case II is of special interest because it allows us
to investigate the sensitivity of an LPV controller with respect to the error in scheduling
parameters.
While the parameter ((t)) is not measurable, a simple nonlinear static estimator in
Ref. [WPB95] could be used to generate the estimated (~) in real-time. This static
estimator is a polynomial approximation of an inverse function of the output equation
(Eq. 6.6). Based on the result [WPB95],
~ = ;1:396 ; 0:33421Mn ; 3:7653n ; 0:91681n Mn
+n (;46:03 + 21:26Mn ; 8:8362Mn2 ; 0:33564n + 0:385n Mn + 0:32892n Mn2 )
+n3 (61:367 ; 69:756Mn + 30:44Mn2 + 3:9589n ; 15:668n Mn + 11:498n Mn2 )
+n5 (;54:655 + 94:381Mn ; 48:212Mn2 ; 4:7973n + 18:807n Mn ; 13:871n Mn2 )
CHAPTER 6. NUMERICAL STUDIES
161
where the normalized variables are n = =60, n = ( ; 10)=25 and Mn = M ; 3. This
approximation results in some approximation errors (the mean error is approximately 8%).
In implementing our QPAL controller, a numerical integration may be dicult around
the boundary of partitioned parameter subspaces Pij 's because Y () is piecewise-dierentiable.
However, there is a simple and practical way to smooth Y () in the small region () around
the boundary of Pij 's. For this purpose, a 4th polynomial function (f () = 4 + a3 + b2 +
c + d) is used and the coecients are chosen to match the values of Y () and @Y ()=@
at = .
Case I
The simulation results are shown in Fig. 6.12. The controller from NGS yields the worst
performance. This result is due to the fact that as the H1 controllers have been optimized at
local nominal models, the modern approach to gain-scheduling can be sensitive to coupling
and other nonlinear eects that are not included in the control design models [NRR93].
As shown in [NRR93], an improved method is to use a sophisticated scheduling scheme
that takes into account the ignored system dynamics. However, this approach still does not
guarantee the synthesis result. Furthermore, developing such a complicated scheme can be
a very time-consuming task for many MIMO systems.
The simulation results of the other three cases are consistent with the synthesis results. As expected from Table 6.4, the worst case among the three cases (C- and LPV
controllers) is the C- controller. The conservatism of the C- controller (opt = 37:7) is
captured in terms of a relatively large steady-state error during 1 { 2 (sec). In contrast,
our QPAL controller yields the best performance and satises design objectives for various
input commands. Note that in this specic example, all LPV controllers can tolerate some
estimation error on (t).
Case II
We repeat the same nonlinear simulations for Case II. First, we consider the case that
v(t) = 0. Fig. 6.14 shows that the time responses of the C-, GRID and QPAL controller
are similar to those of Case I. This result implies that the guaranteed performances from
the design techniques are captured in the simulation for any prole of reference commands.
Next, we consider cases with nonzero v(t) to investigate the sensitivity of the LPV controller
to the error on the Mach number. The simulation results for our QPAL approach are plotted
CHAPTER 6. NUMERICAL STUDIES
162
in Figs. 6.16 - 6.18. Fig. 6.16 shows that the LPV controller from QPAL is insensitive to
error on the Mach number up to v(t) 2 [;0:25; 0:25]. Figs. 6.17 - 6.18 also demonstrate
that the LPV controller can tolerate errors on the Mach number and still achieve the
design objectives up to v(t) 2 [;0:15; 0:15]. However, with larger v(t) 2 [;0:25; 0:25],
the LPV controller yields unrealistic tail deection rate responses. As a result, the LPV
controller from our QPAL approach can tolerate a 4%- 5:5% error on the Mach number and
approximately an 8% error on the angle-of-attack (See g. 6.19).
Through an autopilot design of a realistic missile pitch-axis dynamics, we observe that
our QPAL technique can yield a more reliable and better performance LPV controller than
the published techniques. Furthermore, the LPV controller can tolerate some error on
parameters.
CHAPTER 6. NUMERICAL STUDIES
163
Case I
4
3.8
3.6
Mach number
3.4
3.2
M (t) = ;0:05 sin( 6 t) + 2:1
3
2.8
2.6
2.4
2.2
2
0
0.5
1
1.5
time (sec)
2
2.5
3
Fig. 6.11: Mach number prole for Case I
15
NGS
C−µ
GRID
QPAL
10
5
η(t) in g
0
−5
−10
−15
−20
−25
0
0.5
1
1.5
time (sec)
2
2.5
3
Fig. 6.12: Normal acceleration (t) from NGS, C-, GRID, and QPAL for Case I
CHAPTER 6. NUMERICAL STUDIES
164
Case II with v(t) = 0
4
3.8
3.6
M (t) = ;0:8 sin( 3 t) + 3
Mach number
3.4
3.2
3
2.8
2.6
2.4
2.2
2
0
0.5
1
1.5
time (sec)
2
2.5
3
Fig. 6.13: Mach number prole for Case II without noise
15
C−µ
GRID
QPAL
10
5
η(t) in g
0
−5
−10
−15
−20
0
0.5
1
1.5
time (sec)
2
2.5
3
Fig. 6.14: Normal acceleration (t) from C-, GRID, and QPAL for Case II without noise
CHAPTER 6. NUMERICAL STUDIES
165
4
σ=0
σ = 0.023
σ = 0.070
σ = 0.116
3.8
3.6
Mach number
3.4
3.2
3
2.8
2.6
2.4
2.2
2
0
0.5
1
1.5
time (sec)
2
2.5
3
Fig. 6.15: Mach number prole for Case II with and without noise
15
σ=0
σ = 0.023
σ = 0.070
σ = 0.116
10
5
η(t) in g
0
−5
−10
−15
−20
0
0.5
1
1.5
time (sec)
2
2.5
3
Fig. 6.16: Normal acceleration (t) from the QPAL approach for Case II with and without
noise
CHAPTER 6. NUMERICAL STUDIES
166
Tail deflection rate(deg/sec)
Tail Deflection rate at σ = 0
400
200
0
−200
−400
−600
0
0.5
1
1.5
time (sec)
2
2.5
3
2
2.5
3
Tail deflection rate(deg/sec)
Tail Deflection rate at σ = 0.023
400
200
0
−200
−400
−600
0
0.5
1
1.5
time (sec)
Fig. 6.17: Response of tail deection rate (_ (t)) from the QPAL approach for Case II with
v(t) = 0 and v(t) 2 [;0:05; 0:05]
Tail deflection rate(deg/sec)
Tail Deflection Rate atσ = 0.070
600
400
200
0
−200
−400
−600
0
0.5
1
1.5
time (sec)
2
2.5
3
2
2.5
3
Tail deflection rate(deg/sec)
Tail Deflection Rate atσ = 0.116
600
400
200
0
−200
−400
−600
0
0.5
1
1.5
time (sec)
Fig. 6.18: Response of tail deection rate (_ (t)) from the QPAL approach for Case II with
v(t) 2 [;0:15; 0:15] and v(t) 2 [;0:25; 0:25]
CHAPTER 6. NUMERICAL STUDIES
167
Responses of Angle−Of−Attack at σ=0 and σ=0.116
25
Exact α
Estimated α
20
15
α (deg)
10
5
0
−5
−10
−15
0
0.5
1
1.5
time (sec)
2
2.5
3
Fig. 6.19: Angle-of-attack () and its estimated value (~) from the QPAL approach for
Case II with v(t) = 0 and v(t) 2 [;0:25; 0:25]
Chapter 7
Conclusions
7.1 Summary
The area of the analysis and control of LPV systems has received much recent attention
because of its role in developing systematic techniques for gain-scheduling. Associated with
the analysis and synthesis technique of LPV systems are the scaled small-gain approach and
the dissipative systems framework. Since the dissipative systems framework can directly
treat real time-varying parameters and yield an LPV-type controller, it is preferred for
the LPV control. However, the current dissipative systems framework has two potential
challenges: (1) to select an optimal PDLF in order to reduce conservatism of the dissipative
systems approach; (2) to exactly solve convex optimization problems involving an innite
number of LMIs.
This thesis has developed new analysis and synthesis tools to avoid these two potential
issues of the dissipative systems framework. The new techniques are based on a nonsmooth dissipative systems framework comprised of the continuous, (quasi-) piecewise-ane
parameter-dependent Lyapunov function (PAL) and the proper supply rate. This nonsmooth dissipative systems framework is used for the analysis and synthesis of a piecewiseane parameter-dependent linear parameter-varying (PALPV) system which is a new class
of LPV systems. As anely approximating the local behavior of an LPV system and a
PDLF, our approach enjoys the nice property of \ane parameter-dependency," which
leads to a nite-dimensional convex optimization problem. Furthermore, the global behavior of an LPV system and a PDLF is eventually approximated by the piecewise-ane term.
168
CHAPTER 7. CONCLUSIONS
169
Thus, our new approach can reduce conservatism of the dissipative systems framework
associated with \ane parameter-dependency."
Reducing conservatism is achieved by three factors: accurate modeling, the use of general class of PDLFs and the reduction of conservatism in the \convexifying" technique.
Typically, the system dynamics of an LPV system are nonlinear functions of parameters.
However, they should be simplied as an LFT model or an ane LPV model in order to
recover \convexity." To design a reliable controller, the modeling error created by simplication should be considered as uncertainties which degrade the performance of the designed
controller (or increase conservatism of the design technique). However, the PALPV modeling can reduce the modeling error and then conservatism of the design technique, as shown
in a missile autopilot design (section 6.3).
A PDLF in the dissipative systems framework should be optimized for the synthesis
technique to yield a controller of high performance. For computational tractability, the
PDLF is typically assumed to be a weighted sum of some known basis functions, which are
heuristically dened to have the same parameter-dependency as the LPV system. However,
this thesis shows in section 6.1 that the parameter-dependency of the optimal PDLF is not
relevant to that of the LPV system when parameters are time-varying. In contrast, our
approach can approximate the optimal PDLF itself in the form of \piecewise-ane" terms.
The approximation is automatically used in the analysis and synthesis problem. Thus, our
technique can reduce conservatism of the dissipative systems framework because of the use
of optimal PDLFs.
The no free lunch theorem exists in every engineering problem including the LPV control.
For example, using a general PDLF for the reduction of conservatism increases computational size and time. Therefore, it is a desirable feature of the developed technique to
allow a trade-o between these two players in a zero-sum game. Tuning the number of
piecewise-ane terms or partitions, our approach can tradeo between conservatism and
computational size and time. In other words, it can reproduce the existing result with no
partition and improve the analysis and synthesis result by increasing the number of partitions. Several benchmark problems show that the improved result can be achieved with
even a small number of partitions.
Results from numerical studies, such as section 6.1, shows that the existing \convexifying" techniques create some conservatism in the developed techniques. The degradation
by a \convexifying" technique is inevitable. However, our approach reduces conservatism
CHAPTER 7. CONCLUSIONS
170
by applying the \convexifying" technique to smaller problems over partitioned parameter
subspaces.
In chapters 4 - 5, this thesis fully characterizes several interesting analysis and synthesis
problems{such as stability, L2 -gain, L1 -gain, H2 -norm, passivity and robust counterparts{
for LPV systems within a nonsmooth dissipative systems framework. The result shows that
the derived formulations are similar to each other and counterparts of LTI systems. The
rst similarity stems from the fact that the analysis and synthesis problems studied here are
related to similar types of supply rates, i.e., quadratic form. The second similarity stems
from the fact that as many interesting analysis problems of LTI systems can be formulated
with the quadratic Lyapunov function and the proper supply rate [BGFB94,SGC97], so can
counterparts of LPV systems or nonlinear systems with smooth or nonsmooth Lyapunov
functions and the same supply rates [JR96,SGC97]. However, using nonsmooth Lyapunov
functions can lead to diculty in deriving \dierential" versions of analysis and synthesis
formulations and numerically solving the derived formulations (See [AC84, Clar83]). By
focusing on a Lipschitz Lyapunov function, the developed nonsmooth dissipative systems
framework enables us to derive the \dierential" versions of analysis and synthesis formulations as easily as in the smooth dissipative systems framework.
Finally, several LPV controllers were used in the designed synthesis formulations. Several benchmark problems, including a missile autopilot design, were considered. As mentioned earlier, a couple of benchmark problems demonstrate that the new technique yields a
less conservative result than the published techniques using a general class of PDLFs for the
analysis and synthesis problem. Furthermore, this improvement is even increased using an
accurate modeling. The improved performance of the designed LPV controllers is veried
by nonlinear simulations.
7.2 Conclusions and Contributions
This thesis has developed new analysis and synthesis tools based on a nonsmooth
dissipative systems framework comprised of a continuous, (quasi-) piecewise-ane
parameter-dependent Lyapuov function and the proper supply rate. Associated with
this nonsmooth dissipative systems framework is a piecewise ane parameter-dependent
CHAPTER 7. CONCLUSIONS
171
linear parameter-varying system which is a new class of linear parameter-varying systems. The new approach can reduce the conservatism of the dissipative systems framework associated with \ane parameter-dependency," which is a typical approach in
the literature. The improvement is made using an accurate modeling and a general
class of parameter-dependent Lyapunov functions. Furthermore, this new approach
can tradeo between conservatism and computational size and time by tuning the
number of piecewise-ane terms or partitions.
This thesis fully characterizes several interesting analysis and synthesis problems{
such as stability, L2 -gain, L1 , H2 -norm, passivity and robust counterparts{for LPV
systems within a nonsmooth dissipative systems framework. The derived formulations
are similar to counterparts of LTI systems. These results show the feasibility of
the developed nonsmooth dissipative systems framework. Furthermore, these results
demonstrate that the dissipative systems framework can deal with uncertainties in
the same manner as the scaled small-gain approach.
7.3 Recommendations
The new approach developed in this thesis can yield convex optimization problems
involving (possibly) many LMIs. Furthermore, reducing conservatism may require a
greater number of partitions or piecewise-ane terms of system dynamics and PDLFs.
It may then largely increase computational size and time of the analysis and synthesis
formulation. Thus, applications of the new approach may be practically limited to
relatively simple systems. However, this limitation could be eliminated by improving
current LMI solvers. These improvements would be made by exploiting the spatial
variation of the PDLF in the parameter space because the increased computation effort mainly stems from the use of a PDLF, or spatially-varying Lyapunov function in
the parameter space. As mentioned in section 2.6.2, the overall computational eort of
interior-point methods is dominated by the least-squares problem that must be solved
by each iteration. The least-squares problem involving a spatially-varying PDLF is
very similar to some numerical methods for the boundary-value problem, such as computational uid dynamics (CFD). Thus, the improvements would be made by using
the parallel processing [EPST95] and multi-partitioning technique [PTVF92] which
CHAPTER 7. CONCLUSIONS
172
are popular techniques in the CFD eld. The rst approach is to separate the original least-squares problem into several small problems and solve them using dierent
process simultaneously. The second approach is to solve the least-squares problem at
a reduced number of partitions, interpolate, and then rene the solution onto that of
the original least-squares problem at the original number of partition [PTVF92]. This
second approach can also be applied to speed up iterations.
This thesis fully characterizes several interesting analysis and synthesis problems using
a nonsmooth dissipative systems framework. Each result can serve as a basic result for
the multi-objective control which is practically more attractive. Similar to [SGC97],
the present results can be easily combined for the multi-objective control using QPAL.
However, it will be more desirable to explore methods that use a QPAL for each
performance synthesis problem rather than a single QPAL for all dierent performance
synthesis problems.
This thesis addresses full-order LPV controller design problems. In the case of high
order physical models, simple controllers are often preferred over complicated controllers for several reasons, such as implementation. Therefore, a reduced-order control design problem should be investigated. The problem in designing a minimum
order controller in LTI systems falls into the nonconvex optimization problem that
is equivalent to the minimum rank problem [GG93, IS93]. Recently, this nonconvex
problem has been approximately solved by the lower and upper convex optimization problem [Mes98,MP97]. These results could be extended for the development of
reduced-order LPV controllers. Note that reduced-order LPV controllers were mainly
designed for the singular case of the L2 -gain synthesis problem [LeeL97].
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