1-2-HighlighSOThiSB(Q 7 Chapt3trats thCoptimal control olin tim variant syst with quadratic pTormancerit Tia8i 8H.probl ns1 WConsid Pa dynamical syst (md srib dby an LFT with AB。B. G(s)= C.0 D.. C D.. 0 De 71 T 40 B h.=6c2c.64266C2D. D2 C.B2+ 420。1C2 J.= 6B.B26A66B.D2 D.2C+ X.=R%6(H.)-0=￥.=R%(J.)-0 F:=6(B2x.+D2C)=C.=6(KC2+BD2)x Th品thGi.optimal Controll8i①th Controll尼that minimiz忌kT:wk8 is giv by K(因：=A+BF+C.C6Ce F ，0 Chapt,4 first cousid rs a simplifiH control probl m with theli plant G(s)as giv nin Chapt星，3 I WChow that thrCasts an admissibl controll尼 such that kTk iff thCollowing thre conditions hold: (d）7te·do(ig)and:=Rb(H）-0whGe 。A1B.B26B.B2' H=6C2C. 6A2 (）Je·dom(Ric)and Y=Rb(J)-0whGe 142,1c2C6c2c.': =6B.B2 64 ()p(XY)⊥y1
Highlights of This Book Chapter treats the optimal control of linear timeinvariant systems with quadratic performance criteria ie H problems We consider a dynamical system described by an LFT with Gs A B B C D C D G K z y w u Dene H A C C A B C D D C B J A BB A C BD DB C X RicH Y RicJ F B X D C L YC BD Then the H optimal controller ie the controller that minimizes kTzwk is given by Kopts A BF LC L F Chapter rst considers a simplied H control problem with the generalized plant Gs as given in Chapter We show that there exists an admissible controller such that kTzwk i the following three conditions hold i H domRic and X RicH where H A BB BB C C A ii J domRic and Y RicJ where J A C C C C BB A iii XY
8 IN TRODU TION Mor an admissibl ontroll such that kTk is givby Ksub= A6ZC▣ whre Aoo :A+y+B2B24oo+BoFoo +Zoo CooC Joo=6 Boo=Loo=6 Yoo C=Zoo=(16 y+Yoco)+2e wh品consid rth尼thTalH Control probl品1WEndicat Chow various assumptions can bxdto accommodat Cothmor Ccomplicatd probl s such as singular control probl sl Welso consid PthGual coutrol in thel andH thy and show how the al H solution can bGs d to solv thlo filt Ting probl品l Chapt omithCm fordoofon rfction!Sp al att ion is pad to th conrollruction mthods that pre thClos d loop stabiltpand prormanc Mthods arCprs nt d that givCsufficint Conditions in t ris o.f.cqu ncy wfight d mod Lr ductionl Chapt,firssoly星a spal minin吨ation prob1品lLeP=i+2vbC normaliz coprimChctorizationl ThwChow that v.o 2 ]川a This implithat thGs a robustly stabilizing controll P4=(M+△m)+2w+△N with End ony f IANAM]‖上e efi V,6] Using this stabilization rGult8a loop shapingd gn t niqu s proposl Th pro postdtmiquCus on th basic concpt olloop aping mthods and th a robust stabilization controll Por thChormalizd coprimClctor p Turb syst is usd to Construct th Chinal Controll .l Chapt,7 introducCth Capgnicic and the-gap mtlicl TherGucy domain int rpr tation and applications olthG-gap mitic anCdiscussd in sometaill The controll ordr duction in th ap or i-gap mttic Famork is also considdl
INTRODUCTION Moreover an admissible controller such that kTzwk is given by Ksub A ZL F where A A BB X BF ZLC F B X L YC Z I YX 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 H and H theory and show how the general H solution can be used to solve the H ltering problem Chapter considers the design of reduced order controllers by means of controller reduction Special attention is paid to the controller reduction methods that preserve the closedloop stability and performance Methods are presented that give su cient conditions in terms of frequency weighted model reduction Chapter rst solves a special H minimization problem Let P M N be a normalized left coprime factorization Then we show that inf K stabilizing K I I P K I P inf K stabilizing K I I P KM q N M H This implies that there is a robustly stabilizing controller for P M M N N with N M if and only if q N M H Using this stabilization result a loop shaping design technique is proposed The pro posed technique uses only the basic concept of loop shaping methods and then a robust stabilization controller for the normalized coprime factor perturbed system is used to construct the nal controller Chapter introduces the gap metric and the gap metric The frequency domain interpretation and applications of the gap metric are discussed in some detail The controller order reduction in the gap or gap metric framework is also considered
1.3.Notes and References 9 Chapter 18 considers first the characterization of the largest set of linear,time- invariant plants that can,a priori be guaranteed to be stabilized by any controller which stabilizes a nominal plant model and satisfies a specified Ho norm bound on some closed-loop transfer function.Then it also introduces briefly the problems of model validation and the mixed real and complex u analy sis and sy nthesis. We have incl uded in this book some brief explanation of MATLABTM,SIMULINKTM Control System Toolbox and u Analysis and Synthesis Toolbox'commands.In par- ticular,this book is written consistently with the u Analysis and Synthesis Toolbox. (Robust Control Toolbox may equally be used with this book.Hence it is very helpful for readers to be familar with this toolbox.It is suggested at this point to try the following demo programs from this toolbox. >msdemol >msdemo2 It is assumed that the readers have some basic working knowledge about the basic MATLABTM functions(for example,how to input vectors and matrics,etc.) 1.3 Notes and References 1MATLA BTM and SIMULINKTM are registered trademarks of The MathWorks,Inc.,A-Analysis and Synthesis is a trademark of The MathWorks,Inc.and MUSYN Inc.,Control System Toolbox and Robust Control Toolbox are trademarks of The Math Works,Inc
Notes and References Chapter considers rst the characterization of the largest set of linear time invariant plants that can a priori be guaranteed to be stabilized by any controller which stabilizes a nominal plant model and satises a specied H norm bound on some closedloop transfer function Then it also introduces brie!y the problems of model validation and the mixed real and complex analysis and synthesis We have included in this book some brief explanation of MATLABTM SIMULINKTM Control System Toolbox and Analysis and Synthesis Toolbox commands In par ticular this book is written consistently with the Analysis and Synthesis Toolbox Robust Control Toolbox may equally be used with this book Hence it is very helpful for readers to be familar with this toolbox It is suggested at this point to try the following demo programs from this toolbox msdemo msdemo It is assumed that the readers have some basic working knowledge about the basic MATLABTM functions for example how to input vectors and matrics etc Notes and References MATLABTM and SIMULINKTM are registered trademarks of The MathWorks Inc Analysis and Synthesis is a trademark of The MathWorks Inc and MUSYN Inc Control System Toolbox and Robust Control Toolbox are trademarks of The MathWorks Inc
10 INTRODUCTION
INTRODUCTION
Chapter 2 Lin er Alge bra Some basiclinear algebra facts will be reviewed in this chapter.The detailed treatment of this topic can be found in the referenceslisted at the end of the chapter.Hence we shall omit most proofs and provide proofs only for those results that either cannot be easily found in the standard linear algebra textbooks or are insightful to the understanding of some related problems. 2.1 Linear Subsp aces Let R denote the real scalar field and C the complex scalar field.For the interest of this chapter,let F be either Ror C and let F be the vector space over F,i.e.,Fm is either R or C".Now let iE.Then an element of the form a+ak with ai E F is a linear combination over F of 13k.The set of all linear combinations of 123kEF is a subspace called the span of 12ock,denoted by spanfi3o31113ck}:={x=a11+111+axtk aiE F)1 A set of vectors 1Fn are said to be linearly dependent over F if there exists aF not all zero such thata=0;otherwise they are said to be linearly indep endent. Let S be a subspace of Fn,then a set of vectors frES is called a basis for S if 123111ck are linearly independent and S=Spanti3r31ck}.However, such a basis for a subspace S is not unique but all bases for S have the same number of elements.This number is called the dimension of S,denoted by dim(S). A set of vectors f23ck}in F are mutually ortho gonal if irj =0 for all ij and orthonormalifxzi=ij,where the superscript denotes complex conjugate transpose and 6ij is the Kronecker delta function with 6ij =1 for i=j and 6ij=0 for ij.More generally,a collection of subspaces 5135311135 of F are mutually orthogonal if x*y=0 whenever x∈Si and y∈Si for i≠j. 11
Chapter Linear Algebra Some basic linear algebra facts will be reviewed in this chapter The detailed treatment of this topic can be found in the references listed at the end of the chapter Hence we shall omit most proofs and provide proofs only for those results that either cannot be easily found in the standard linear algebra textbooks or are insightful to the understanding of some related problems Linear Subspaces Let R denote the real scalar eld and C the complex scalar eld For the interest of this chapter let F be either R or C and let Fn be the vector space over F ie Fn is either Rn or C n Now let x xxk Fn Then an element of the form x kxk with i F is a linear combination over F of xxk The set of all linear combinations of x xxk Fn is a subspace called the span of x xxk denoted by spanfx xxkg fx x kxk i Fg A set of vectors x xxk Fn are said to be linearly dependent over F if there exists k F not all zero such that x kxk otherwise they are said to be linearly independent Let S be a subspace of Fn then a set of vectors fx xxkg S is called a basis for S if x xxk are linearly independent and S spanfx xxkg However such a basis for a subspace S is not unique but all bases for S have the same number of elements This number is called the dimension of S denoted by dimS A set of vectors fx xxkg in Fn are mutually orthogonal if x i xj for all i j and orthonormal if x i xj ij where the superscript denotes complex conjugate transpose and ij is the Kronecker delta function with ij for i j and ij for i j More generally a collection of subspaces S SSk of Fn are mutually orthogonal if xy whenever x Si and y Sj for i j