2 IN TRODU TION We consider mainly twoty pes cf prcblems Analy sis Prcblens Given a controller.determine if the comtrolled signals (in cluding tracking errors.control signals etc)satisfy the desired properties far all admissible noises disturbarces and model ukertairties. Synthesis Prcblems:Design a controller sothat the cortrolled signals satisfy the desired properties for all admissible noises disturbarces and model ucertainties. Most of our aralysis and synthesis will be done ona uified linear fractional trarsfar mation framrewark.To that end we shall show that the systemshown in Figue 1.1 canbe put in ageneral diagraminFigure1.2 whereP is the intercommection matrix.K is the cortroller.A is the set of all possible ucertaintys w is a vectar signal including noises disturbances and referere signals.z is a vector signal including all controlled signals and tracking erars.u is the comtrol signal and y is the measurerrert. Figure1.2:General Framewark The block diagramin Figure1.2 represerts the follow ing equations: 93 93 7 79=P7D9 y 7=△v u Ky. Lt thetraner marix from toz bederted byand asuetht all adnissible ucertainty A satisfies 9(A)5 1fi-.Thenour analy sis prcblemis to anwer if the closed locp systemis stablefar all adnissible A andhll.5 fcrse prespecified-6 0 wherelo is the Ht.normdefined as ll.sup 9 (jw)).The sy nthesis problemis to design a controller K so that the abovercbust stability and performance conditions are satisfied Inthe simplest farm we have either A=0 cr w =0.The farmer becomes the well knownH.control problemand the later becomes the rcbut stability problem The
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 unied linear fractional transfor mation framework To that end we shall show that the system shown in Figure can be put in a general diagram in Figure 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 y u K P Figure General Framework The block diagram in Figure represents the following equations v z y P w u v u Ky Let the transfer matrix from w to z be denoted by Tzw and assume that all admissible uncertainty satises Then our analysis problem is to answer if the closed loop system is stable for all admissible and kTzwk for some prespecied where kTzwk is the H norm dened as kTzwk sup Tzwj The synthesis problem is to design a controller K so that the above robust stability and performance conditions are satised In the simplest form we have either or w The former becomes the well known H control problem and the later becomes the robust stability problem The
--Highlights of hisook 3 two probl s aruiva whA is a singl olock unstructur uncrainty through th Cpplication ola small gain thormt is probably th Ccasethat this robust stability cons qu nc was thGmain motpvation or thCa opm oft.mithods1 ThCnalysis and synth s or syst s with multipI block A can bductdin most cas to an quival n 7t.probl m with suitabl scalings1 Thus a solution to thet. Control probl盒is thekto all robustne忌Probl ms Cons率d in this bookl In the已 nts ction8w Cshall give chap Tby chapt Tsummary olthGmain r ults prd in this bookl WGdstot book Rebust and Optimal Coutrol by this author&1C1Dae and KIGlov(,996)r brif historical r ot.and robust control and r somCa(tailtdtr tm ofsomedvanctd topics1 Highlights of This Book ThrGts in G chapt arigligh1NgChat somGfd Gtate ms in this sction ar Cot prs⑧thar CruCindCcrtain assun呗tigos thatar已 not plicitly stat dl Rdrs should consult theorronding chaptrsor thCeact stat m nts and conditionsl Chapt somasic lin ag a fictsl Chapt3rs someyst thcal concts coutrollabilityS obs Tvability8 stabilizability8d t.ctability8pol plac m nt8 obs rVTth aly8 syst m pol Cand z ros8 and stat pacizations1 Chapt urocducCthespacCnd spac SonCCacGChodsof raional transPmatrix norms arersnedl For ampl8 1R. th品 kGk=tracQB)=tracPC") and kGk.=maxfy:H has an fig nvaluCon themaginary ais whre and Q arCth ontrollability and obs rvability Gramians and Chapt inroduc th back structured discust stabilityl
Highlights of This Book two problems are equivalent when is a single block unstructured uncertainty through the application of a small gain theorem It is probably the case that this robust stability consequence was 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 giveachaper by chapter summary of the main results presented in this book We refer readers to the book Robust and Optimal Control by this author J C Doyle and K Glover for a brief historical review of H and robust control and for some detailed treatment of some advanced topics Highlights of This Book The key results in each chapter are highlighted below Note that some of the state ments in this section are not precise they are true under certain assumptions that are not explicitly stated Readers should consult the corresponding chapters for the exact statements and conditions Chapter reviews some basic linear algebra facts Chapter reviews some system theoretical concepts controllability observability stabilizability detectability pole placement observer theory system poles and zeros and state space realizations Chapter introduces the H spaces and the H spaces Some state space methods of computing real rational H and H transfer matrix norms are presented For example let Gs A B C RH then kGk traceBQB traceCPC and kGk maxf H has an eigenvalue on the imaginary axisg where P and Q are the controllability and observability Gramians and H A BB CC A Chapter introduces the feedback structure and discusses its stability
IN TRODU TION W We define that the above closed-loop sy stem is internally stable if and only if I8K「8)1 K(I8PK)-1 8P I P(I8P)-1I8P-1 1 RHoo. erative characterizations of interal stabilityusing opime factorizations arels presented. hapterthe feedbackproperties and degnimitation formulations of optimal Hoand Hoo control problems and the selection of weighting functions are also considered in this chapter. Chapterde the obem ofedungtheofa linear mutivaiable dy namical sy stem using the balanced truncation method.Kuppose A11 B G(s)=1 A1A∞ 1 RHoo D is a balanced realization with controllability and observability Gramians P =Q=E= diag(,∑ ∑=diag(9Is1,9ls2,,9,Is,） ∑o=diag(9n◇Isr+1,9n◇ls,+2,9NIsw） hen the truncated system G(s) AB is stable and satisfies an additive CD error bound: N kG(s)8G,(s)k≤2∑9. i=r◇ Chapterevesobustability teonde various modingsm tions through the use of a small gain theorem.particular,we show that an uncertain system described below with an unstructured uncertainty A 1 RHoo with kAk 5 1 is robustly stable if and only if the transfer function from w to z has Hoo norm no greater than 1
INTRODUCTION e e e e w w K P We dene that the above closedloop system is internally stable if and only if I K P I I KP K I P K P I KP I P K RH Alternative characterizations of internal stability using coprime factorizations are also presented Chapter considers the feedback system properties and design limitations The formulations of optimal H and H control problems and the selection of weighting functions are also considered in this chapter Chapter considers the problem of reducing the order of a linear multivariable dynamical system using the balanced truncation method Suppose Gs A A A A B B C C D RH is a balanced realization with controllability and observability Gramians P Q diag diagIs Is rIsr diagrIsr rIsr N IsN Then the truncated system Gr s A B C D is stable and satises an additive error bound kGs Gr sk X N ir i Chapter derives robust stability tests for systems under various modeling assump tions through the use of a small gain theorem In particular we show that an uncertain system described below with an unstructured uncertainty RH with kk is robustly stable if and only if the transfer function from w to z has H norm no greater than
1-8 Highlights o.This Book △ W nominal system Chapter 9 introduces the linear fractional transformation(LFT).Weshow that many control problems can be formilated and treated in the LFT framewark.Inparticular. we show that evey analysis problemcanbeput in anLFT farmwith some strutured A(s)and somre intercommection matrix M(s)and every synthesis prcblemcanbeput in anLFT formwith a generalized plant G(s)and a controller K(s)tobe desig ned. Chapter 10 considers rcbust stability and pefarmarce for systens with miltiple sources of ucertainties We show that an uertain system is rcbutly stable and satisfies someH.pefarnancecriterionfar all A).I.withkA)k 9 1 if and only if the strutured singular value (1)of the carreponding intercommection model is no greater than1. nominal system △3 Chapter 11 characterizes all controllers that stabilieagivendy namical systemG(s) using the state space approach For a givengereralized plant G(s)= G1(s)G12(s) TA B1 B2 G21(s)G22(s) C D11 D12 C2D21D22
Highlights of This Book Δ z w nominal system Chapter introduces the linear fractional transformation LFT 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 Ms and every synthesis problem can be put in an LFT form with a generalized plant Gs and a controller Ks to be designed z w M y u z w G K Chapter considers robust stability and performance for systems with multiple sources of uncertainties We show that an uncertain system is robustly stable and satises some H performance criterion for all i RH with kik if and only if the structured singular value of the corresponding interconnection model is no greater than nominal system Δ Δ Δ Δ 1 2 3 4 Chapter characterizes all controllers that stabilize a given dynamical system Gs using the state space approach For a given generalized plant Gs Gs Gs Gs Gs A B B C D D C D D
6 INTRODU CTION we show that all stabiliing controllers canbeparapreterized as the transfer matrix from y to u below where F and L are suchthat A+Lnd A+B.F are stable Z W G D -L 12 2 Chapter 12 studies the stabilizing solutionto an Algebraic Riccati Equation (ARE). A solutiontothe following ARE AX+XA+XRX+Q=0 is said to be a stabilizing solutionif A+RX is stable Now let H:= A R -Q-A* and let _(H)bethe stable inv ariant subspace and [X2 X(H)=Im where Cnxm.If X is norsingular.then=is uniquely deternined by H:dencted by X=Rid).A key result of this chapter is the so-called the Bounded Real Lema.which states that a stable transfer matrix G(S hav ing G(9-if and only if there exists an X suchthat A+BB*Xfi-is stable and XA+AX+XBB*Xfi-+ CCo The Ho control theary in Chapter 14 will be derived based onthis lemma
INTRODUCTION 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 LC and A BF are stable c c c c c u y z w y u Q G D L F A B R C Chapter studies the stabilizing solution to an Algebraic Riccati Equation ARE A solution to the following ARE AX XA XRX Q is said to be a stabilizing solution if A RX is stable Now let H A R Q A and let XH be the stable H invariant subspace and XH Im X X where X X C nn If X is nonsingular then X XX is uniquely determined by H denoted by X RicH A key result of this chapter is the socalled the Bounded Real Lemma which states that a stable transfer matrix Gs having kGsk if and only if there exists an X such that A BBX is stable and XA AX XBBX CC The H control theory in Chapter will be derived based on this lemma