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 By
 Kemin Zhou
 December 2000

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 Robust and Optimal Control provides much more detailed and advanced
treatment of many topics in robust and optimal control. It is a must for
every researcher and graduate student.

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 We give an overview of robust and optimal control:
 What is this book about?
 Highlights of this book.

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 This book is about basic robust and H_{¥}_{ } control theory. We consider a control
system with possibly multiple sources of uncertainties, noises, and
disturbances as shown in Figure 1.1.

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 Analysis problems: Given a controller, determine if the controlled
signals (including tracking errors, control signals, etc.) satisfy the
desired properties for all admissible noises, disturbances, and model
uncertainties.
 Synthesis problems: Design a controller so that the controlled signals
satisfy the desired properties for all admissible noises, disturbances,
and model uncertainties.

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 Almost every problem can be put in a linear fractional transformation
(LFT) form:
 P: interconnection matrix, K: controller, D: the set of all possible uncertainty
 w: noises, disturbances, and reference signals; z: all controlled
signals and tracking errors; u: control signal; y: measurement.

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 Chapter 2 reviews some basic linear algebra and matrix facts.
 Chapter 3 reviews system theoretical concepts: controllability,
observability, stabilizability, detectability, pole placement, observer
theory, system poles and zeros, and state space realizations.
 Chapter 4 introduces the H_{2 }and H_{¥}_{ }spaces. State space methods for
computing real rational H_{2 }and H_{¥} transfer matrix norms are presented.

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 For example, let
 Then
 G(s)_{2}^{2}=trace(B*QB)=trace(CPC*)
 and
 G(s)_{¥}=max{g: H has an eigenvalue on jwaxis}
 where P and Q are the controllability and observability Gramians and

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 Chapter 5 introduces the feedback structure and discusses its stability.
 We define that the above closedloop system is internally stable if and
only if

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 Chapter 6 considers the feedback system properties and design
limitations. The formulations of optimal H_{2} and H_{¥}_{ }control
problems and the selection of weighting functions are also considered in
this chapter.
 Chapter 7 considers the problem of model reduction using balanced
truncation method. Suppose
 is a balanced realization with controllability and observability
Gramians P=Q=S=diag(S_{1}, S_{2})

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 S_{1}=diag(s_{1}I_{s1},
s_{2}I_{s2},…,
s_{r}I_{sr}),
 S_{1}=diag(s_{r+}_{1 }I_{sr+1},
s_{r+2}I_{sr+1},…,
s_{N}I_{sN})
 Then the truncated system
 is stable and satisfies an additive error bound:
 G(s)G_{r}(s)_{¥} £2(s_{r+}_{1}+s_{r+2}+…+s_{N})
 Frequency weighted balanced truncation method is also presented.

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 Chapter 8 derives robust stability tests for systems under various
modeling assumptions through the use of the small gain theorem. In
particular, we show that a system shown below, with an unstructured
uncertainty D ÎRH_{¥} with D_{¥} <1, is
robustly stable if and only if T_{zw}_{¥} £1 , where T_{zw}
is the matrix transfer function from w to z.

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 Chapter 9 introduces LFT in detail. We show that many control problems
can be formulated and treated in the LFT framework. In particular, we
show that every analysis problem can be put in an LFT form with some
structured D(s) and some interconnection matrix M(s) and
every synthesis problem can be put in an LFT form with a generalized
plant G(s) and a controller K(s) to be designed.

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 Chapter 10 considers robust stability and performance for systems with
multiple sources of uncertainties. We show that an uncertain system is
robustly stable and satisfies some H_{¥} performance criterion for all D_{i} ÎRH_{¥} with D_{i}_{¥} <1 if and only if the structured singular
value of the corresponding interconnection model is no greater than 1.

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 Chapter 11 characterizes all stabilizing controllers for a given
dynamical system G(s).
 All stabilizing controllers can
be parameterized as the transfer matrix from y to u where F and L are
such that A+B_{2}F and A+LC_{2} are stable and where Q
is any stable proper transfer matrix.

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 Chapter 12 studies the stabilizing solution to an Algebraic Riccati
Equation (ARE):
 A*X+XA+XRX+Q=0
 I.e., A+RX is stable.
 Suppose H has no jwaxis
eigenvalues and X_{}(H) is the stable invariant subspace of H and
suppose X_{1 }is nonsingular, then X=Ric(H):=X_{2 }X_{1}^{1
}is the stabilizing solution. A key result of this chapter is the
socalled Bounded Real Lemma: G(s)ÎRH_{¥} with G(s)_{¥} <1 if and only if there exists an X such that
A+BB*X/g^{2
}is stable and
 XA+A*X+XBBX *X/g^{2
}+C*C=0.

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 Chapter 13 treats the H_{2 }optimal control.

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 Chapter 14 treats H_{¥} control with some simplified assumptions: R_{1}=I,
R_{2}=I, D_{12}*C_{1}=0, and B_{1}D_{21}*=0.
We show that there exists an admissible controller such that T_{zw}_{¥} <g if and only if the
following three conditions hold:
 (i) H_{¥}
Îdom(Ric) and
X_{¥}
:=Ric(H_{¥})>0, where
 (ii) J_{¥}
Îdom(Ric) and
Y_{¥}
:=Ric(J_{¥})>0, where
 (iii) r(X_{¥}Y_{¥})< g^{2}.

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 Moreover, an admissible controller such that T_{zw}_{¥} <g is given by
 We then consider the general H_{¥} control problem. We indicate how various
assumptions can be relaxed to accommodate other more complicated
problems, such as singular control problems. We also consider the
integral control in the H_{2} and H_{¥} theory and show how the general H_{¥} solution
can be used to solve the H_{¥} filtering problem.

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 Chapter 15 considers the design of reducedorder controllers by means of
controller order reduction. Special attention is paid to the controller
reduction methods that preserve the closedloop stability and
performance. Methods are presented that give sufficient conditions in
terms of frequencyweighted model reduction.

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 Chapter 16 first solves a special H_{¥} minimization problem. Let be a normalized left
coprime factorization. Then we show that
 This implies that there is a robustly stabilizing controller for
with
 if and only if
 Using this stabilization result, a loopshaping design technique is
proposed. This technique uses only the basis concept of loopshaping
methods, and then a robust stabilization controller for the normalized
coprime factor perturbed system is used to construct the final
controller.

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 Chapter 17 introduces the gap metric and the ngap metric. The frequency domain interpretation
and applications of the ngap metric are discussed. The controller order
reduction in the gap or ngap metric framework is also considered.
 Chapter 18 considers briefly the problems of model validation and the
mixed real and complex m analysis and synthesis.
 All computations in the book are done using MATLAB and the m Analysis and Synthesis
Toolbox. Many other toolboxes may be used in the place of mtoolbox.
