Table Of Content1 Robust Controol Theory Volker Wagner
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A Course in Robust
Control Theory
a convex approach
Geir E. Dullerud Fernando G. Paganini
University of Illinois University of California
Urbana-Champaign Los Angeles
2 Robust Controol Theory Volker Wagner
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Contents
0 Introduction 1
0.1 System representations . . . . . . . . . . . . . . . . . . . 2
0.1.1 Block diagrams . . . . . . . . . . . . . . . . . . . 2
0.1.2 Nonlinear equations and linear decompositions . . 4
0.2 Robust control problems and uncertainty . . . . . . . . . 9
0.2.1 Stabilization . . . . . . . . . . . . . . . . . . . . . 9
0.2.2 Disturbances and commands . . . . . . . . . . . . 12
0.2.3 Unmodeled dynamics . . . . . . . . . . . . . . . . 15
1 Preliminaries in Finite Dimensional Space 18
1.1 Linear spaces and mappings . . . . . . . . . . . . . . . . 18
1.1.1 Vector spaces . . . . . . . . . . . . . . . . . . . . 19
1.1.2 Subspaces . . . . . . . . . . . . . . . . . . . . . . 21
1.1.3 Bases, spans, and linear independence . . . . . . 22
1.1.4 Mappings and matrix representations . . . . . . 24
1.1.5 Change of basis and invariance . . . . . . . . . . 28
1.2 Subsets and Convexity . . . . . . . . . . . . . . . . . . . 30
1.2.1 Some basic topology . . . . . . . . . . . . . . . . 31
1.2.2 Convex sets . . . . . . . . . . . . . . . . . . . . . 32
1.3 Matrix Theory . . . . . . . . . . . . . . . . . . . . . . . . 38
1.3.1 Eigenvalues and Jordan form . . . . . . . . . . . 39
1.3.2 Self-adjoint, unitary and positive de(cid:12)nite matrices 41
1.3.3 Singular value decomposition . . . . . . . . . . . 45
1.4 Linear Matrix Inequalities . . . . . . . . . . . . . . . . . 47
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ii Contents
1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2 State Space System Theory 57
2.1 The autonomous system . . . . . . . . . . . . . . . . . . 58
2.2 Controllability . . . . . . . . . . . . . . . . . . . . . . . . 61
2.2.1 Reachability . . . . . . . . . . . . . . . . . . . . . 61
2.2.2 Properties of controllability . . . . . . . . . . . . 66
2.2.3 Stabilizability and the PBH test . . . . . . . . . . 69
2.2.4 Controllability from a single input . . . . . . . . . 72
2.3 Eigenvalue assignment . . . . . . . . . . . . . . . . . . . 74
2.3.1 Single input case . . . . . . . . . . . . . . . . . . 74
2.3.2 Multi input case . . . . . . . . . . . . . . . . . . . 75
2.4 Observability . . . . . . . . . . . . . . . . . . . . . . . . 77
2.4.1 The unobservable subspace. . . . . . . . . . . . . 78
2.4.2 Observers . . . . . . . . . . . . . . . . . . . . . . 81
2.4.3 Observer-BasedControllers . . . . . . . . . . . . 83
2.5 Minimal realizations . . . . . . . . . . . . . . . . . . . . 84
2.6 Transfer functions and state space . . . . . . . . . . . . . 87
2.6.1 Real-rational matrices and state space realizations 89
2.6.2 Minimality . . . . . . . . . . . . . . . . . . . . . . 92
2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3 Linear Analysis 97
3.1 Normed and inner product spaces. . . . . . . . . . . . . 98
3.1.1 Complete spaces . . . . . . . . . . . . . . . . . . 101
3.2 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.2.1 Banach algebras . . . . . . . . . . . . . . . . . . . 107
3.2.2 Some elements of spectral theory . . . . . . . . . 110
3.3 Frequency domain spaces: signals . . . . . . . . . . . . . 113
3.3.1 The space L^2 and the Fourier transform . . . . . 113
3.3.2 The spaces H2 and H2? and the Laplace transform 115
3.3.3 Summarizing the big picture . . . . . . . . . . . . 119
3.4 Frequency domain spaces: operators . . . . . . . . . . . . 120
3.4.1 Time invariance and multiplication operators . . 121
3.4.2 Causality with time invariance . . . . . . . . . . . 122
3.4.3 Causality andH . . . . . . . . . . . . . . . . . 124
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3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4 Model realizations and reduction 131
4.1 Lyapunov equations and inequalities . . . . . . . . . . . 131
4.2 Observability operator and gramian . . . . . . . . . . . . 134
4.3 Controllability operator and gramian . . . . . . . . . . . 137
4.4 Balanced realizations . . . . . . . . . . . . . . . . . . . . 140
4.5 Hankel operators . . . . . . . . . . . . . . . . . . . . . . 143
4.6 Model reduction . . . . . . . . . . . . . . . . . . . . . . . 147
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Contents iii
4.6.1 Limitations . . . . . . . . . . . . . . . . . . . . . 148
4.6.2 Balanced truncation . . . . . . . . . . . . . . . . 151
4.6.3 Inner transfer functions . . . . . . . . . . . . . . . 154
4.6.4 Bound for the balanced truncation error . . . . . 155
4.7 Generalized gramians and truncations . . . . . . . . . . . 160
4.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
5 Stabilizing Controllers 167
5.1 System Stability . . . . . . . . . . . . . . . . . . . . . . . 169
5.2 Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . 172
5.2.1 Static state feedback stabilization via LMIs . . . 173
5.2.2 An LMI characterization of the stabilization prob-
lem . . . . . . . . . . . . . . . . . . . . . . . . . . 174
5.3 Parametrizationof stabilizing controllers . . . . . . . . . 175
5.3.1 Coprime factorization . . . . . . . . . . . . . . . . 176
5.3.2 Controller Parametrization . . . . . . . . . . . . . 179
5.3.3 Closed-loop maps for the general system . . . . . 183
5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
6 H2 Optimal Control 188
6.1 Motivation for H2 control . . . . . . . . . . . . . . . . . 190
6.2 Riccati equation and Hamiltonian matrix . . . . . . . . . 192
6.3 Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
6.4 State feedback H2 synthesis via LMIs . . . . . . . . . . . 202
6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
7 H Synthesis 208
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7.1 Two important matrix inequalities . . . . . . . . . . . . 209
7.1.1 The KYP Lemma . . . . . . . . . . . . . . . . . . 212
7.2 Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
7.3 Controller reconstruction . . . . . . . . . . . . . . . . . . 222
7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
8 Uncertain Systems 227
8.1 Uncertainty modeling and well-connectedness . . . . . . 229
8.2 Arbitrary block-structured uncertainty . . . . . . . . . . 234
8.2.1 A scaled small-gain test and its su(cid:14)ciency . . . . 236
8.2.2 Necessity of the scaled small-gain test. . . . . . . 239
8.3 The Structured Singular Value . . . . . . . . . . . . . . . 245
8.4 Time invariant uncertainty . . . . . . . . . . . . . . . . . 248
8.4.1 Analysis of time invariant uncertainty. . . . . . . 249
8.4.2 The matrix structured singular value and its upper
bound . . . . . . . . . . . . . . . . . . . . . . . . 257
8.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
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iv Contents
9 Feedback Control of Uncertain Systems 270
9.1 Stability of feedback loops . . . . . . . . . . . . . . . . . 273
9.1.1 L2-extended and stability guarantees . . . . . . . 274
9.1.2 Causality and maps on L2-extended . . . . . . . . 277
9.2 Robust stability and performance . . . . . . . . . . . . . 280
9.2.1 Robust stability under arbitrary structured uncer-
tainty. . . . . . . . . . . . . . . . . . . . . . . . . 281
9.2.2 Robust stability under LTI uncertainty . . . . . . 281
9.2.3 Robust Performance Analysis . . . . . . . . . . . 282
9.3 Robust Controller Synthesis . . . . . . . . . . . . . . . . 284
9.3.1 Robust synthesis against (cid:1)a;c . . . . . . . . . . 285
9.3.2 Robust synthesis against (cid:1)TI . . . . . . . . . . . 289
9.3.3 D-K iteration: a synthesis heuristic . . . . . . . . 293
9.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
10 Further Topics: Analysis 298
10.1 Analysis via Integral Quadratic Constraints . . . . . . . 298
10.1.1 Analysis results . . . . . . . . . . . . . . . . . . . 303
10.1.2 The search for an appropriateIQC . . . . . . . . 308
10.2 Robust H2 Performance Analysis . . . . . . . . . . . . . 310
10.2.1 Frequencydomainmethodsandtheirinterpretation 311
10.2.2 State-Space Bounds Involving Causality . . . . . 316
10.2.3 Comparisons . . . . . . . . . . . . . . . . . . . . 320
10.2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . 321
11 Further Topics: Synthesis 323
11.1 Linear parameter varying and multidimensional systems 324
11.1.1 LPV synthesis . . . . . . . . . . . . . . . . . . . . 327
11.1.2 Realization theory for multidimensional systems . 333
11.2 A Framework for Time Varying Systems: Synthesis and
Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
11.2.1 Block-diagonaloperators . . . . . . . . . . . . . 338
11.2.2 The system function . . . . . . . . . . . . . . . . 340
11.2.3 Evaluating the ‘2 induced norm . . . . . . . . . . 344
11.2.4 LTV synthesis . . . . . . . . . . . . . . . . . . . . 347
11.2.5 Periodic systems and (cid:12)nite dimensional conditions 349
A Some Basic Measure Theory 352
A.1 Sets of zero measure . . . . . . . . . . . . . . . . . . . . 352
A.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . 355
A.3 Comments on norms and Lp spaces . . . . . . . . . . . . 357
B Proofs of Strict Separation 359
C (cid:22)-Simple Structures 365
6 Robust Controol Theory Volker Wagner
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0
Introduction
In this course we will explore and study a mathematical approach aimed
directlyatdealingwith complexphysicalsystemsthat arecoupledinfeed-
back. The general methodology we study has analytical applications to
both human-engineered systems and systems that arise in nature, and the
context of our course will be its use for feedback control.
The direction we will take is based on two related observations about
models for complex physical systems. The (cid:12)rst is that analytical or com-
putational models which closely describe physical systems are di(cid:14)cult or
impossible to precisely characterize and simulate. The second is that a
model, no matter how detailed, is never a completely accurate represen-
tation of a real physical system. The (cid:12)rst observation means that we are
forced to use simpli(cid:12)ed system models for reasons of tractability; the lat-
ter simply states that models are innately inaccurate. In this course both
aspects will be termed system uncertainty, and our main objective is to
developsystematic techniquesand tools forthe designand analysis of sys-
tems which are uncertain. The predominant idea that is used to contend
with such uncertainty or unpredictability isfeedback compensation.
There are several ways in which systems can be uncertain, and in this
course we will target the main three:
The initial conditions of a system may not be accuratelyspeci(cid:12)ed or
(cid:15)
completely known.
Systemsexperiencedisturbancesfromtheirenvironment,andsystem
(cid:15)
commands are typically not known a priori.
7 Robust Controol Theory Volker Wagner
2 0. Introduction
Uncertainty in the accuracy of a system model itself is a central
(cid:15)
source. Any dynamical model of a system will neglect some physi-
cal phenomena, and this means that any analyticalcontrolapproach
based solely on this model will neglect some regimes of operation.
Inshort: the majorobjectiveoffeedbackcontrolistominimizethe e(cid:11)ects
of unknown initial conditions and external in(cid:13)uences on system behavior,
subjecttotheconstraintofnothavingacompleterepresentationofthesys-
tem.Thisisaformidablechallengeinthatpredictablebehaviorisexpected
from a controlled system, and yet the strategies used to achieve this must
do so using an inexact system model. The term robust in the title of this
course refers to the fact that the methods we pursue will be expected to
operatein anuncertainenvironmentwith respecttothe systemdynamics.
The mathematical tools and models we use will be primarily linear, moti-
vatedmainlybytherequirementofcomputabilityofourmethods;however
thetheorywedevelopisdirectlyaimedatthecontrolofcomplexnonlinear
systems.Inthisintroductorychapterwewilldevotesomespacetodiscuss,
at an informal level, the interplay between linear and nonlinear aspects in
this approach.
The purpose of this chapter is to provide some context and motivation
for the mathematical work and problems we will encounter in the course.
For this reason we do not provide many technical details here, however it
might be informative to refer back to this chapter periodically during the
course.
0.1 System representations
We will now introduce the diagrams and models used in this course.
0.1.1 Block diagrams
We will often view physical or mathematical systems a mappings. From
this perspective a system maps an input to an output; for dynamical sys-
tems these are regarded as functions of time. This is not the only or most
primitive way to view systems, although we will (cid:12)nd this viewpoint to be
veryattractivebothmathematicallyandforguidingandbuildingintuition.
In this section we introduce the notion of ablock diagramforrepresenting
systems, and most importantly for specifying their interconnections.
We use the symbol P to denote a system that maps an input function
u(t) to an output function y(t). This relationship is denoted by
y=P(u):
Figure 1 illustrates this relationship. The direction of the arrows indicate
whether a function is an input or an output of the system P. The details
8 Robust Controol Theory Volker Wagner
0.1. Systemrepresentations 3
y u
P
Figure 1. Basic blockdiagram
of how P constructs y from the input u is not depicted in the diagram,
insteadthebene(cid:12)tofusingsuchblockdiagramsisthatinterconnectionsof
systems can be readily visualized.
Consider the so-called cascade interconnection of the two subsystems.
This interconnection represents the equations
v
y u
P2 P1
v =P1(u)
y =P2(v):
We see that this interconnection takes the two subsystems P1 and P2 to
form a system P de(cid:12)ned by P(u)=P2(P1(u)). Thus this diagram simply
depicts a composition of maps. Notice that the input to P2 is the output
of P1.
z w
P
u
y
Q
Another type of interconnection involves feedback. In the (cid:12)gure above
wehavesuchanarrangement.HereP hasinputsgivenby theorderedpair
(w; u) and the outputs (z; y). The system Q has input y and output u.
This block diagram therefore pictorially represents the equations
(z; y)=P (w; u)
y=Q(y):
Since part of the output of P is an input to Q, and conversely the output
of Q is an input to P, these systems are coupled in feedback.
9 Robust Controol Theory Volker Wagner
4 0. Introduction
We will now move on to discussing the basic modeling concept of this
course and in doing so will immediately make use of block diagrams.
0.1.2 Nonlinear equations and linear decompositions
We have just introduced the idea of representing a system as an input-
output mapping, and did not concern ourselves with how such a mapping
might be de(cid:12)ned. We will now outline the main idea behind the modeling
framework used in this course, which is to represent a complex system as
a combination of a perturbation and a simpler system. We will illustrate
this by studying two important cases.
Isolating nonlinearities
The(cid:12)rstcaseconsideredisthedecompositionofasystemintoalinearpart
and a static nonlinearity. The motivation for this is so that later we can
replace the nonlinearity using objects more amenable to analysis.
To start consider the nonlinear system described by the equations
x_ =f(x;u) (1)
y=h(x;u);
with the initial condition x(0). Here x(t), y(t) and u(t) are vector valued
functions, and f and h are smooth vector valued functions. The (cid:12)rst of
theseequationsisadi(cid:11)erentialequationandthesecondispurelyalgebraic.
Givenaninitialconditionandsomeadditionaltechnicalassumptions,these
equationsde(cid:12)neamappingfromutoy.Ourgoalisnowtodecomposethis
system into a linear part and a nonlinear part around a speci(cid:12)ed point; to
reduce clutter in the notation we assume this point is zero.
De(cid:12)ne the following equivalent system
x_ =Ax+Bu+g(x; u) (2)
y=Cx+Du+r(x; u);
whereA,B,C andD providealinearapproximationtothedynamics,and
g(x; u)=f (x; u) Ax Bu
(cid:0) (cid:0)
r(x; u)=h(x; u) h(0; 0) Cx Du:
(cid:0) (cid:0) (cid:0)
For instance one could take the Jacobian linearization
A=d1f(0; 0); B =d2f(0; 0);
C =d1h(0; 0); and D=d2h(0; 0);
whered1 andd2 denotevectordi(cid:11)erentiationbythe(cid:12)rstandsecondvector
variables respectively. The following discussion, however, does not require
this assumption. The system in (2) consists of linear functions and the
possibly nonlinear functions g and r. It is clear that the solutions to this
10 Robust Controol Theory Volker Wagner
0.1. Systemrepresentations 5
systemhaveaone-to-onecorrespondencewiththesolutionsof(1),sincewe
have simply rewritten the functions. Further let us write these equations
in the equivalent form
x_ =Ax+Bu+w1 (3)
y =Cx+Du+w2 (4)
(w1; w2)=(g(x; u); r(x; u)): (5)
Now let G be the mapping described by (3) and (4) which satis(cid:12)es
G:(w1; w2; u) (x; u; y);
7!
given an initial condition x(0). Further let Q be the mapping which takes
(x; u) (w1; w2)asdescribedby(5).Thusthesystemofequationsde(cid:12)ned
7!
by (3 - 5) has the block diagram below. The system G is totally described
Q
G
y u
Figure 2. Systemdecomposition
by linear di(cid:11)erential equations, and Q is a static nonlinear mapping. By
static we mean that the output of Q at any point in time depends only on
the input at that particular time, or equivalently that Q has no memory.
Thus all of the nonlinear behavior of the initial system (1) is captured in
Q and the feedback interconnection.
We will almost exclusively work with the case where the point (0; 0),
around which this decomposition is taken, is an equilibrium point of (1).
Namely
f(0; 0)=0:
In this case the functions g and r satisfy g(0; 0) =0 and r(0; 0)= 0, and
therefore Q(0; 0)=0. Also the linear system described by
x_ =Ax+Bu
y =Cx+Du
is the linearization of (1) around the equilibrium point. The linear system
G is thus an augmented version of the linearization.
