INTRODUCTION Chapter 5 introduces the feedback structure and discusses its stability. We define that the above closed-loop system is internally stable if and only if (I-KP)- K(I-Pk)- P(-6P)1(-P)-1∈Rx Alternative characterizations of internal stability using coprime factorizations are also Chapter 6 considers the feedback system properties and design limitations. The formulations of optimal H2 and Hoo control problems and the selection of weighting functions are also considered in this chapter hapter 7 considers the problem of reducing the order of a linear multivariable dynamical system using the balanced truncation method. Suppose A1141 B ∈R is a balanced realization with controllability and observability Gramians P=Q=2 diag(∑1,∑2) ∑1=diag(a1ls1,02l2,…,rIs) diag(r+1Is+1,0r+2lsn+2,……,NIsN) Then the truncated system Gr(s)=_AnBL 1D Is stable and satisfies an additive error bound G(s)-G(s川≤2∑a i=r+ Frequency-weighted balanced truncation method is also discussed Chapter 8 derives robust stability tests for systems under various modeling assump- tions through the use of the small gain theorem. In particular, we show that a system shown at the top of the following page, with an unstructured uncertainty A E RH
4 INTRODUCTION Chapter 5 introduces the feedback structure and discusses its stability. e e + + + e + 2 e1 w2 w1 ✛ ✛ ✲ ✻ ❄ ✲ Kˆ P We define that the above closed-loop system is internally stable if and only if I −Kˆ −P I −1 = (I − KPˆ )−1 Kˆ (I − PKˆ )−1 P(I − KPˆ )−1 (I − PKˆ )−1 ∈ RH∞. Alternative characterizations of internal stability using coprime factorizations are also presented. Chapter 6 considers the feedback system properties and design limitations. The formulations of optimal H2 and H∞ control problems and the selection of weighting functions are also considered in this chapter. Chapter 7 considers the problem of reducing the order of a linear multivariable dynamical system using the balanced truncation method. Suppose G(s) = A11 A12 A21 A22 B1 B2 C1 C2 D ∈ RH∞ is a balanced realization with controllability and observability Gramians P = Q =Σ= diag(Σ1, Σ2) Σ1 = diag(σ1Is1 ,σ2Is2 ,...,σrIsr ) Σ2 = diag(σr+1Isr+1 ,σr+2Isr+2 ,...,σN IsN ). Then the truncated system Gr(s) = A11 B1 C1 D is stable and satisfies an additive error bound: kG(s) − Gr(s)k∞ ≤ 2 X N i=r+1 σi. Frequency-weighted balanced truncation method is also discussed. 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 at the top of the following page, with an unstructured uncertainty ∆ ∈ RH∞
1.2. Highlights of This Book with Alloo I is robustly stable if and only if T2w l s 1, where Tzw is the matrix transfer function from w to nominal system Chapter 9 introduces the LFT in detail. We show that many control problems can be formulated and treated in the lF T framework. In particular, we show that every analysis problem can be put in an LFT form with some structured A(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 designe y M 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 Hoo performance criterion for all△i∈ RHoo with l△l‖l。< I if and only if the structured singular value (u) of the corresponding interconnection model is no nominal system
1.2. Highlights of This Book 5 with k∆k∞ < 1 is robustly stable if and only if kTzwk∞ ≤ 1, where Tzw is the matrix transfer function from w to z. ∆ z w nominal system Chapter 9 introduces the 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 ∆(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. z w ✛ ✛ ✛ ✲ M ∆ y u z w ✛ ✛ ✛ ✲ G K 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 ∆i ∈ RH∞ with k∆ik∞ < 1 if and only if the structured singular value (µ) of the corresponding interconnection model is no greater than 1. nominal system ∆ ∆ ∆ ∆ 1 2 3 4
INTRODUCTION Chapter 1l characterizes in state-space all controllers that stabilize a given d namical system G(s). For a given generalized plant G G(s we show that all stabilizing controllers can be parameterized as the transfer matrix from y to u below where F and L are such that A+ LC2 and A+ B2 F are stable and where Q is any stable proper transfer matrix G Chapter 12 studies the stabilizing solution to an algebraic Riccati equation(are) A solution to the following ARE F X+XA+XRX+Q=0 is said to be a stabilizing solution if A+ RX is stable. Now let A R and let a-(H) be the stable H invariant subspace and where X1, X2E Cxn. If X1 is nonsingular, then X: X2X1 is uniquely determined by H, denoted by X= Ric(H). A key result of this chapter is the so-called bounded
6 INTRODUCTION Chapter 11 characterizes in state-space all controllers that stabilize a given dynamical system G(s). For a given generalized plant G(s) = G11(s) G12(s) G21(s) G22(s) = A B1 B2 C1 D11 D12 C2 D21 D22 we show that all stabilizing controllers can be parameterized as the transfer matrix from y to u below where F and L are such that A + LC2 and A + B2F are stable and where Q is any stable proper transfer matrix. cc c c c ✻ ✻ ✛ ✛ ✛ − u1 y1 z w y u ✲ ✛ ✛ Q ❄ G ✛ ❄ ✛ ✲ ✻ ✲ ✲ ✛ ✛ ✛ ✛ ✛ D22 −L F A B2 R C2 Chapter 12 studies the stabilizing solution to an algebraic Riccati equation (ARE). A solution to the following ARE A∗X + XA + XRX + Q = 0 is said to be a stabilizing solution if A + RX is stable. Now let H := A R −Q −A∗ and let X−(H) be the stable H invariant subspace and X−(H) = Im X1 X2 , where X1,X2 ∈ Cn×n. If X1 is nonsingular, then X := X2X−1 1 is uniquely determined by H, denoted by X = Ric(H). A key result of this chapter is the so-called bounded
1.2. Highlights of This Book eal lemma, which states that a stable transfer matrix G(s) satisfies G(s)ll y if d only if there exists an X such that A+BBX/ is stable and XA+A'X+XBBX/2+C"C=0 The Hoo control theory in Chapter 14 will be derived based on this lemma Chapter 13 treats the optimal control of linear time-invariant systems with quadratic performance criteria (i. e, H2 problems). We consider a dynamical system described b an LFt with A B B D 21 R1=D12D12>0,R2=D21D21>0 A-B2RT DiCI B2ri B2 -C(I-D2R1D12)C1-(-B2R11D12C1) (A- BiDE, R2 C2 CR2 C2 B1(1-D21R21D21)Br-(A-B1Dt12C2) (H2)≥0,Y:=Ric(2)≥ F2:=-R1(B2X2+D12C1),L2:=-(Y2C2+B1D21)21 Then the H2 optimal controller (i.e, the controller that minimizes T2wll2) is given by A+B2F2+L2C2-L2 s Chapter 14 first considers an Ho control problem with the generalized plant G(s) given in Chapter 13 but with some additional simplifications: R 1,B2=I, Di2C1=0, and B1D:1=0. We show that there exists an admissible controller such that Tzu loo <y if and only if the following three conditions hold: (i) Hoo E dom(Ric)and Xoo: =RicHoo)>0, where A B1B1-B2 B2
1.2. Highlights of This Book 7 real lemma, which states that a stable transfer matrix G(s) satisfies kG(s)k∞ < γ if and only if there exists an X such that A + BB∗X/γ2 is stable and XA + A∗X + XBB∗X/γ2 + C∗C = 0. The H∞ control theory in Chapter 14 will be derived based on this lemma. Chapter 13 treats the optimal control of linear time-invariant systems with quadratic performance criteria (i.e., H2 problems). We consider a dynamical system described by an LFT with G(s) = A B1 B2 C1 0 D12 C2 D21 0 . G K z y w u ✛ ✛ ✛ ✲ Define R1 = D∗ 12D12 > 0, R2 = D21D∗ 21 > 0 H2 := A − B2R−1 1 D∗ 12C1 −B2R−1 1 B∗ 2 −C∗ 1 (I − D12R−1 1 D∗ 12)C1 −(A − B2R−1 1 D∗ 12C1)∗ J2 := (A − B1D∗ 21R−1 2 C2)∗ −C∗ 2R−1 2 C2 −B1(I − D∗ 21R−1 2 D21)B∗ 1 −(A − B1D∗ 21R−1 2 C2) X2 := Ric(H2) ≥ 0, Y2 := Ric(J2) ≥ 0 F2 := −R−1 1 (B∗ 2X2 + D∗ 12C1), L2 := −(Y2C∗ 2 + B1D∗ 21)R−1 2 . Then the H2 optimal controller (i.e., the controller that minimizes kTzwk2) is given by Kopt(s) := A + B2F2 + L2C2 −L2 F2 0 . Chapter 14 first considers an H∞ control problem with the generalized plant G(s) as given in Chapter 13 but with some additional simplifications: R1 = I, R2 = I, D∗ 12C1 = 0, and B1D∗ 21 = 0. We show that there exists an admissible controller such that kTzwk∞ < γ if and only if the following three conditions hold: (i) H∞ ∈ dom(Ric) and X∞ := Ric(H∞) > 0, where H∞ := A γ−2B1B∗ 1 − B2B∗ 2 −C∗ 1C1 −A∗ ;
INTRODUCTION (ii)Joo E dom(Ric)and Yoo: =Ric(Joo)>0, where B1B Moreover, an admissible controller such that IT2ulloo y is given by K-[÷ Aoo: =A+2-2B1 B*Xoo+B2 Foo+ Zoo LooC2 F:=-BX,L:=-YC,Z:=(I-12YxX)-1. We then consider further the general Hoo control problem. We indicate how various ssumptions can be relaxed to accommodate other more complicated problems, such as singular control problems. We also consider the integral control in the H2 and Hoo theory and show how the general Hoo solution can be used to solve the Hoo filtering Chapter 15 considers the design of reduced-order controllers by means of controller reduction. Special attention is paid to the controller reduction methods that preserve the closed-loop stability and performance. Methods are presented that give sufficient conditions in terms of frequency-weighted model reduction Chapter 16 first solves a special Hoo minimization problem. Let P=M-N be a norn malized left coprime factorization. Then we show that K stabilizing I(I+Pk)-[I P 期=4]u+P21=(=Hx1Im) This implies that there is a robustly stabilizing controller for =(M+△M)-1(N+△N) with I[△N△M]‖ if and only if ≤y1-|[NM
8 INTRODUCTION (ii) J∞ ∈ dom(Ric) and Y∞ := Ric(J∞) > 0, where J∞ := A∗ γ−2C∗ 1C1 − C∗ 2C2 −B1B∗ 1 −A ; (iii) ρ(X∞Y∞) < γ2 . Moreover, an admissible controller such that kTzwk∞ < γ is given by Ksub = Aˆ∞ −Z∞L∞ F∞ 0 where Aˆ∞ := A + γ−2B1B∗ 1X∞ + B2F∞ + Z∞L∞C2 F∞ := −B∗ 2X∞, L∞ := −Y∞C∗ 2 , Z∞ := (I − γ−2Y∞X∞) −1. We then consider further 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 H2 and H∞ theory and show how the general H∞ solution can be used to solve the H∞ filtering problem. Chapter 15 considers the design of reduced-order controllers by means of controller reduction. Special attention is paid to the controller reduction methods that preserve the closed-loop stability and performance. Methods are presented that give sufficient conditions in terms of frequency-weighted model reduction. Chapter 16 first solves a special H∞ minimization problem. Let P = M˜ −1N˜ be a normalized left coprime factorization. Then we show that inf K stabilizing K I (I + PK) −1 I P ∞ = inf K stabilizing K I (I + PK) −1M˜ −1 ∞ = q 1 − N˜ M˜ 2 H −1 . This implies that there is a robustly stabilizing controller for P∆ = (M˜ + ∆˜ M) −1(N˜ + ∆˜ N ) with ∆˜ N ∆˜ M ∞ < if and only if ≤ q 1 − N˜ M˜ 2 H.