NOTATION AND SYMBOLS A(a) eigenvalue of A P(A) spectral radius of A PR(A) real spectrum radius of A o(A)and g(a) the largest and the smallest singular values of A ith singular value of A (A) condition number of a ofA:‖A Im(A),R(A) image(or range) space of A Ker(A), N(A) kernel (or null) space of A x(4) stable invariant subspace of A the stabilizing solution of an arE g*f convolution of g and f Inner p roduct orthogonal, ( a, 9=0 D rthogonal complement of D orthogonal complement of subspace S, e.g., 72 C2(-∞,∞) time domain square integrable functions C2+:=C2[0,∞) subspace of L2(-∞,∞) with functions zero for t<0 C2-:=C2(-∞,0] subspace of L2(-∞,∞) with functions zero for t>0 C2 gR) square integrable functions on Co including at oo H subspace of C2GR)with functions analytic in Re(s>0 subspace of C2GR)with functions analytic in Re(s)<o C∞(jR) functions bounded on Re(s)=0 including at oo the set of Coo gR) functions analytic in Re(s)>0 the set of Coo gR) functions analytic in Re(s)<o refix B and Bo closed and open unit ball,e.g.B△andB°△ prefix R real rational, e.g., RHoo and RH2, etc. Rp(s) rational proper transfer matrices shorthand for G(s) shorthand for state space realization C(sI-A)B+D n(G(s) number of right-half plane poles 7(G(s) number of imaginary axis poles winding number Fe(m, Q) lower lft Fu(M, Q) upper LFT
xvi NOTATION AND SYMBOLS λ(A) eigenvalue of A ρ(A) spectral radius of A ρR(A) real spectrum radius of A σ(A) and σ(A) the largest and the smallest singular values of A σi(A) ith singular value of A κ(A) condition number of A kAk spectral norm of A: kAk = σ(A) Im(A), R(A) image (or range) space of A Ker(A), N(A) kernel (or null) space of A X−(A) stable invariant subspace of A Ric(H) the stabilizing solution of an ARE g ∗ f convolution of g and f ∠ angle h,i inner product x ⊥ y orthogonal, hx,yi = 0 D⊥ orthogonal complement of D S⊥ orthogonal complement of subspace S, e.g., H⊥ 2 L2(−∞,∞) time domain square integrable functions L2+ := L2[0,∞) subspace of L2(−∞,∞) with functions zero for t < 0 L2− := L2(−∞, 0] subspace of L2(−∞,∞) with functions zero for t > 0 L2(jR) square integrable functions on C0 including at ∞ H2 subspace of L2(jR) with functions analytic in Re(s) > 0 H⊥ 2 subspace of L2(jR) with functions analytic in Re(s) < 0 L∞(jR) functions bounded on Re(s) = 0 including at ∞ H∞ the set of L∞(jR) functions analytic in Re(s) > 0 H− ∞ the set of L∞(jR) functions analytic in Re(s) < 0 prefix B and Bo closed and open unit ball, e.g. B∆ and Bo∆ prefix R real rational, e.g., RH∞ and RH2, etc. Rp(s) rational proper transfer matrices G∼(s) shorthand for GT (−s) A B C D shorthand for state space realization C(sI − A)−1B + D η(G(s)) number of right-half plane poles η0(G(s)) number of imaginary axis poles wno(G) winding number F`(M,Q) lower LFT Fu(M,Q) upper LFT M?N star product
List of acronyms ARE algebraic Riccati equation FDLTI finite dimensional linear time invariant iff if and only if left coprime factorization LFT linear fractional transformation Thp or LHP left-half plane Re(s)<o LOG linear quadratic Gaussian LTI linear time invariant MIMO aput multioutput nlcf normalized left coprime factorization N nominal performance ormalized right coprime factorization nominal stability right coprime factorization rhp or RHP ight-half plane Re(s)>0 RP robust performance robust stability SISO single-input single-output SSV structured singular value(u) SVD singular value decomposition XVIl
List of Acronyms ARE algebraic Riccati equation FDLTI finite dimensional linear time invariant iff if and only if lcf left coprime factorization LFT linear fractional transformation lhp or LHP left-half plane Re(s) < 0 LQG linear quadratic Gaussian LTI linear time invariant MIMO multi-input multioutput nlcf normalized left coprime factorization NP nominal performance nrcf normalized right coprime factorization NS nominal stability rcf right coprime factorization rhp or RHP right-half plane Re(s) > 0 RP robust performance RS robust stability SISO single-input single-output SSV structured singular value (µ) SVD singular value decomposition xvii
Chapter 1 Introduction This chapter gives a brief description of the problems considered in this book and the key results presented in each chapter. 1.1 What is This book about? This book is about basic robust and Hoo control theory. We consider a control system with possibly multiple sources of uncertainties, noises, and disturbances as shown in disturbanc other controlled signals System Interconnection controlle reference signals Figure 1.1: General system interconnection
Chapter 1 Introduction This chapter gives a brief description of the problems considered in this book and the key results presented in each chapter. 1.1 What Is This Book About? 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. controller reference signals tracking errors noise uncertainty uncertainty other controlled signals uncertainty disturbance System Interconnection Figure 1.1: General system interconnection 1
INTRODUCTION We consider mainly two types of problems Analysis problems: Given a controller, determine if the controlled signals (in- cluding 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 Most of our analysis and synthesis will be done on a unified linear fractional transforma tion(LFT)framework. To that end, we shall show that the system shown in Figure 1 can be put in the general diagram in Figure 1.2, where P is the interconnection matrix, K is the controller, A is the set of all possible uncertainty, w is a vector signal including noises, disturbances, and reference signals, z is a vector signal including all controlled signals and tracking errors, u is the control signal, and y is the measurement Figure 1.2: General LFT framework The block diagram in Figure 1.2 represents the following equations y Let the transfe ansfer matrix from w to z be denoted by Tzw and assume that the ac missible uncertainty△ satisfies‖A‖。<1/ u for some y>0. Then our anal sis problem is to answer if the closed-loop system is stable for all admissible A and ITawlloo <tp for some prespecified p >0, where ITzulloo is the Hoo norm defined as IT2ulloo= sup o(T2u Gw)). The synthesis problem is to design a controller K so that the aforementioned robust stability and performance conditions are satisfied In the simplest form, we have either 4=0 or w=0. The former becomes the well known Hoo control problem and the later becomes the robust stability problem. The two
2 INTRODUCTION We consider mainly two types of problems: • 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. Most of our analysis and synthesis will be done on a unified linear fractional transformation (LFT) framework. To that end, we shall show that the system shown in Figure 1.1 can be put in the general diagram in Figure 1.2, where P is the interconnection matrix, K is the controller, ∆ is the set of all possible uncertainty, w is a vector signal including noises, disturbances, and reference signals, z is a vector signal including all controlled signals and tracking errors, u is the control signal, and y is the measurement. z w v η y u ✲ ✛ ✲ ✛ ✛ ✛ K ∆ P Figure 1.2: General LFT framework The block diagram in Figure 1.2 represents the following equations: v z y = P η w u η = ∆v u = Ky. Let the transfer matrix from w to z be denoted by Tzw and assume that the admissible uncertainty ∆ satisfies k∆k∞ < 1/γu for some γu > 0. Then our analysis problem is to answer if the closed-loop system is stable for all admissible ∆ and kTzwk∞ ≤ γp for some prespecified γp > 0, where kTzwk∞ is the H∞ norm defined as kTzwk∞ = supω σ¯ (Tzw(jω)). The synthesis problem is to design a controller K so that the aforementioned robust stability and performance conditions are satisfied. In the simplest form, we have either ∆ = 0 or w = 0. The former becomes the wellknown H∞ control problem and the later becomes the robust stability problem. The two
1.2. Highlights of This Book problems are equivalent when A is a single-block unstructured uncertainty through the pplication of the small gain theorem (see Chapter 8). This robust stability consequence was probably the main motivation for the development of Hoo methods. The analysis and synthesis for systems with multiple-block A can be reduced in most cases to an equivalent Hoo problem with suitable scalings. Thus a solution to the Hoo control problem is the key to all robustness problems considered in this book. In the next section, we shall give a chapter-by-chapter summary of the main results presented in this book We refer readers to the book Robust and Optimal Control by K. Zhou, J. C. Doyle, and K. Glover [1996 for a brief historical review of Hoo and robust control and for some detailed treatment of some advanced topics 1.2 Highlights of This Book The key results in each chapter are highlighted in this section. Readers should consult the corresponding chapters for the exact statements and condition Chapter 2 reviews some basic linear algebra facts Chapter 3 reviews system theoretical concepts: controllability, observability, sta bilizability, detectability, pole placement, observer theory, system poles and zeros, and tate-space realizations. Chapter 4 introduces the H2 spaces and the Hoo spaces. State-space methods of computing real rational H2 and Hoo transfer matrix norms are presented. For example let Co IG 2=trace(BQB)=trace(CPC") IG= maxy: H has an eigenvalue on the imagina where P and Q are the controllability and observability Gramians and A BB*/
1.2. Highlights of This Book 3 problems are equivalent when ∆ is a single-block unstructured uncertainty through the application of the small gain theorem (see Chapter 8). This robust stability consequence was probably the main motivation for the development of H∞ methods. The analysis and synthesis for systems with multiple-block ∆ can be reduced in most cases to an equivalent H∞ problem with suitable scalings. Thus a solution to the H∞ control problem is the key to all robustness problems considered in this book. In the next section, we shall give a chapter-by-chapter summary of the main results presented in this book. We refer readers to the book Robust and Optimal Control by K. Zhou, J. C. Doyle, and K. Glover [1996] for a brief historical review of H∞ and robust control and for some detailed treatment of some advanced topics. 1.2 Highlights of This Book The key results in each chapter are highlighted in this section. Readers should consult the corresponding chapters for the exact statements and conditions. Chapter 2 reviews some basic linear algebra 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 H2 spaces and the H∞ spaces. State-space methods of computing real rational H2 and H∞ transfer matrix norms are presented. For example, let G(s) = A B C 0 ∈ RH∞. Then kGk2 2 = trace(B∗QB) = trace(CPC∗) and kGk∞ = max{γ : H has an eigenvalue on the imaginary axis}, where P and Q are the controllability and observability Gramians and H = A BB∗/γ2 −C∗C −A∗