P FEEACE xi bliography 1?? Jdex 151
PREFACE xi Bibliograph y Index
N。tati0 aa a d Sy m b。1s R and and mb fi fa8fith TRor opaund closhae and op and closright-hallplane R imaginary axis 1 bG品mgto 2 subs 3 union 7 intis ction Gaofroof ◇ of品ark = dintd as 之and≤ asyuptotically grand I than 4 and 5 much great rand Ies than a lal absolut CatCr Re(a) ra part oln 1∪ In n 6 n id nity matrix [ais a matrix with a as its i-th row and j-th column diag(a1,...,an) an n6n dagonal marix with a as its i-th diegonal i AT and A* transpos Cand compl conjugat Ceranspos ColA AI and A+ in TsOindpsGdo in TsCoL4 A* shorthand or l 1)* dA) d t.Tmpant o4 Trac tracCol4 xiii
Notation and Symbols R and C elds of real and complex numbers F eld either R or C C and C open and closed lefthalf plane C and C open and closed righthalf plane jR imaginary axis belong to subset union intersection end of proof end of remark dened as and asymptotically greater and less than and much greater and less than complex conjugate of C jj absolute value of C Re real part of C In n n identity matrix aij a matrix with aij as its ith row and jth column element diagaan an n n diagonal matrix with ai as its ith diagonal element AT and A transpose and complex conjugate transpose of A A and A inverse and pseudo inverse of A A shorthand for A detA determinant of A TraceA trace of A xiii
xiv NCTAIKNAN SAMEOS 7(A) 3(A) 3RA) A)ald(A) Cang etalaTCauetSular valu(CA 9(A) i-h cular valuCfA 4(A) cOd(hb金fA IA‖ PEI血①A‖天(A） Im A)2R(A) inag ArhalcfpacCA KA)2N(A) EFI OHPacCPA 1(A) SGbIGEriaIbpacCA Ric) tetbEiISRTCOTARE g*1 cCIEIG alal ame (9 inGCet X⊥y D3 cooaeo3! DiSIitry 2 S3 GiCCcpIGEKEXbSace261273 C2(-∞9o) eQairSuarereeaS C2+:=C2I09o) Tb pacC(fC2(-009o)wil fulc(CRt50 C21:=C2(-∞90 bipacCfc2(-0o9o wilh fulcCt 6 0 C2(j®) 3匹ereabiG1dqp1a 7L(j®) bBacCfc2(iR)wil fulct(nytcie)60 H(jR biTacCf(i wilh fulci(Slalylic i5 0 C(jR lciasmlacd (e-oiludlato H(jR) eOC (iR fulcE(SHyteiI)60 H以GR) Get(e(iR)fulctOSIy ticilke)50 closed uibal2正1B prifik R 2省6x-.aln%,2t Rp(s) ra fOprOeTaITCma Tic(S G7(s) SrtialarGT(-s) a AB C D SGtaldrsEpaceAiaf(sI-A)1B+D F.(M9Q) IQPLFT FuM9Q) M8N 裙t
xiv NOTATION AND SYMBOLS A eigenvalue of A A spectral radius of A RA real spectrum radius of A A and A the largest and the smallest singular values of A iA ith singular value of A A condition number of A kAk spectral norm of A kAk A ImA RA image or range space of A KerA NA kernel or null space of A XA stable invariant subspace of A RicH the stabilizing solution of an ARE g f convolution of g and f angle h i inner product x y orthogonal hx yi D orthogonal complement of D ie D D or D D is unitary S orthogonal complement of subspace S eg H L time domain square integrable functions L L subspace of L with functions zero for t L L subspace of L with functions zero for t LjR square integrable functions on C including at HjR subspace of LjR with functions analytic in Res H jR subspace of LjR with functions analytic in Res LjR functions bounded on Res including at HjR the set of LjR functions analytic in Res HjR the set of LjR functions analytic in Res prex B closed unit ball eg B prex Bo open unit ball prex R real rational eg RH and RH etc Rps rational proper transfer matrices G s shorthand for GT s A B C D shorthand for state space realization CsI AB D FMQ lower LFT FuMQ upper LFT M N star product
L ist Of A cr ony m S ARE algebraic Riccati equation FDLTI finite dimensional linear time in ariant 证 if and only if lcf left coprie factorization LFT linear fractional trarsformation Ihp cr LHP left-half plane Res)9 0 LQG linear quadratic Gaussian LTI linear time in ariant MIMO miti input milti-output ncf ncrmalized left coprime factoriation NP nommal performalke Icf narmalized righ coprime factarization NS mommal stability rcf right coprime factariation rhp cr RHP right-half plane Res)8 0 RP rcbust performanke RS rcbust stability SISO sigle iput sigle output SSV stritured singular value (1) SVD sigular value decomposit ion xV
List of Acronyms ARE algebraic Riccati equation FDLTI nite dimensional linear time invariant i if and only if lcf left coprime factorization LFT linear fractional transformation lhp or LHP lefthalf plane Res LQG linear quadratic Gaussian LTI linear time invariant MIMO multiinput multioutput nlcf normalized left coprime factorization NP nominal performance nrcf normalized right coprime factorization NS nominal stability rcf right coprime factorization rhp or RHP righthalf plane Res RP robust performance RS robust stability SISO singleinput singleoutput SSV structured singular value SVD singular value decomposition xv
Ch a p t e r. I n t r o d u c t io n This chapter gives a brief description of the problems considered in this book and the key results presented in each chapterl 212 WhatThiSB CQ ISAbat This book is about some basic robust and H control theory1 We consider a control system with possibly multiple sources of uncertainties2noises2and disturbances as shown in Figure 3131 uncertainty disturbance other controlled signals uncertainty System Interconnection uncertainty tracking errors noise controller reference signals Figure 313:General System Interconnection 3
Chapter Introduction This chapter gives a brief description of the problems considered in this book and the key results presented in each chapter What This Book Is About This book is about some basic robust and H control theory We consider a control system with possibly multiple sources of uncertainties noises and disturbances as shown in Figure controller reference signals tracking errors noise uncertainty uncertainty other controlled signals uncertainty disturbance System Interconnection Figure General System Interconnection