ESSENTIALS OF ROBUST CONTROL Kemin Zhou March 18,1997
ESSENTIALS OF ROBUST CONTROL Kemin Zhou March
Contents Ffe Notation and Symbols xiii List o.Acronyms 1 Introductipp 1 1.1 What his 1 1.2 Highlights his ook 3 1.3 Lotes and 9 8 Lin Alg ra 11 2.1 11 2.2 Ligenvalues and e nvectors 12 2.3 Mat rix version Brmuas· 13 2.4 ←bspaces ，。。。。。。。。。。。 14 2.5 ector ms and Matrix 6 ngular Value Decomposition. 18 2 emidefinits Matrices 3 . Lotes and leferences. 2.9 Problems 24 Linr Syst(品s 89 3.1 scriptions of Dy namical 。。 ntrollability 2 rvers and rver- ontrollers stems.·· izations for ansfer Matrices 3 6 stem Poles and Zeros Lotes and eferences·· 。。。。。。。。 4 3.8 Problems 42 vii
Contents Preface v Notation and Symbols xiii List of Acronyms xv Introduction What This Book Is About Highlights of This Book Notes and References Linear Algebra Linear Subspaces Eigenvalues and Eigenvectors Matrix Inversion Formulas Invariant Subspaces Vector Norms and Matrix Norms Singular Value Decomposition Semidenite Matrices Notes and References Problems Linear Systems Descriptions of Linear Dynamical Systems Controllability and Observability Observers and ObserverBased Controllers Operations on Systems State Space Realizations for Transfer Matrices Multivariable System Poles and Zeros Notes and References Problems vii
vi进i PREFA 4 H.aldH SpacS 45 41 Hilb rt Space 47 H.andH ac 。。。 47 43 Computing and H. Nos·· 44 Computing and只o Norms 415 Cand R 1539 46 ob1品s 9 5 IIeR stoint 61 51 e 57 Ptt whi lostdns olFf dback Loop 60 53 Int Thal Stability 54 Coprim Fact ization ov E 55 56 食Re 73 6 PROnaldet faSlLimitfas 75 61 Ffdback rop rtie 67 W ight dHe and Ho rmnce 63 sacction olw igh色g Functions 6 Bods Gain and haseR lation 57988 65 Bode s nsitivity Int gral... 63 Analyticity Constraints 67 paa金€ 990 618 7 Balalded md①REucio 99 71 Lyapunoy Equations 。。。。。。 00 77 Balancd R alizations 73 Mod R duction by Balancd Truncation ,0 74 nCcy-wehte Balanced Mod r eduction.... 75 Cand R 0 76 fobl品s ,70 8 vIetiI alR⑥stress 123 81 Mod LUnc rtainty ,73 87 Small Gain Th品.· Stability undUrstructurUncfainti 3, 83 , 831 AdditivCUnc rtainty.······· ,35 837 Multiplicativ Unc rtainty.. 36 833 Coprim CFactor Unc rtainty.. 7 834 Uuctur d Robust Stability Ts ,38 84 Robust【ormanc已 ,4, 85 Sk ed Sp (cfications ,44
viii PREFACE H and H Spaces Hilbert Spaces H and H Spaces Computing L and H Norms Computing L and H Norms Notes and References Problems Internal Stability Feedback Structure WellPosedness of Feedback Loop Internal Stability Coprime Factorization over RH Notes and References Problems Performance Speci cations and Limitations Feedback Properties Weighted H and H Performance Selection of Weighting Functions Bodes Gain and Phase Relation Bodes Sensitivity Integral Analyticity Constraints Notes and References Problems Balanced Model Reduction Lyapunov Equations Balanced Realizations Model Reduction by Balanced Truncation FrequencyWeighted Balanced Model Reduction Notes and References Problems Uncertainty and Robustness Model Uncertainty Small Gain Theorem Stability under Unstructured Uncertainties Additive Uncertainty Multiplicative Uncertainty Coprime Factor Uncertainty Unstructured Robust Stability Tests Robust Performance Skewed Specications
REFACE 8.6 Classical Control far MIMO Systems ............ 148 8.7 Nctes and References.··,··················· 152 8.8 Prcblems 152 5 Lin Fractional Transormation 135 9.1 Linear Fractional Transfcrmations 159 9.2 Basic Principle..... 166 9.3 Redheffer Star-Products ...... 171 9.4 Nctes and Refererces....... 173 9.5 Prcblems··············· 173 18.and.7Synth s 193 10.1 General Framewark for Sy stem Rcbustness..·...····. 176 l0.2 Strictured Singular Valve.···..···.················ 179 10.2.1 Definitions cf.,·:························· 179 10.2.2B0ms................................ 184 10.2.3 Well Posedness and Perfarmancefar Constant LFTs .... 188 l0.3 Structured Robust Stability and Perfarmarke..,,·.··,···· 192 10.3.1 Rcbust Stability 。。。。···。···。··。········ 192 10.3.2 Rcbust Pefcrmarce.·········::··········· 194 l0.3.3 TwoBlock.:Rcbust Perfarmance Revisited.····· 198 10.3.4 Appro imation cf Multiple Full Block.·.········.· 204 l0.4 Overview an.Sy nthesis..·.·.················ 205 l0.5 Nctes and Refererces.........·········.··· 208 10.6 Prcblems...·..········· 209 11 Controleamzation 81, 11.1 Existence of Stabilizing Controllers.············· 214 11.2 Pararreterization of All Stabiliing Controllers ...... 216 11.3 Coprime Factarization Approach 220 l1.4 Nctes and References.····················· 223 11.5 Prcblems 223 18 Alg braic Riccati Equations 883 l2.1 Stabiliing Solution and Riccati Operatcr...,.,..·.·.·.... 226 12.2 Imer Furktions..······················· 237 12.3 Nctes and References......·..···.·····.········ 239 239 1,H2 Optimal Control 873 13.1 Standard7H2 Prcblem....................,·. 245 l3.2 Stability Margins of H2 Controllers,··············· 249 13.3 Nctes and References. 250 13.4 Prcblems.....··· 251
PREFACE ix Classical Control for MIMO Systems Notes and References Problems Linear Fractional Transformation Linear Fractional Transformations Basic Principle Redheer StarProducts Notes and References Problems and Synthesis General Framework for System Robustness Structured Singular Value Denitions of Bounds Well Posedness and Performance for Constant LFTs Structured Robust Stability and Performance Robust Stability Robust Performance Two Block Robust Performance Revisited Approximation of Multiple Full Block Overview on Synthesis Notes and References Problems Controller Parameterization Existence of Stabilizing Controllers Parameterization of All Stabilizing Controllers Coprime Factorization Approach Notes and References Problems Algebraic Riccati Equations Stabilizing Solution and Riccati Operator Inner Functions Notes and References Problems H Optimal Control Standard H Problem Stability Margins of H Controllers Notes and References Problems
PREFACE 14H.Control 253 14.1 PrcblemFarmilat ion.··························· 253 14.2 A Simplified7H.Control Prcblem,··········· 254 14.3 Optimality and Limiting Behav icr 263 14.4 MinimmEntr0 py Controller....,.···.·.·..···· 267 14.5 An Optimal Controller..,··················· 268 14.6 General7H.Solutions··。·················· 269 14.7 Relaxing Assumpt ions.···················· 273 14.8L.ad7H.Integral Control.·..,.,····...·:· 276 1497H.Ftag·························· 279 l4.10 Nctes and References.····················· 281 14.11 Prcblems........·. 281 15 Controller Reduction 285 15.1L.Cortroller Reductions.························· 286 l51.1 Additive Redution.························· 287 l5.l.2 Coprime Factor Reduction.·············· 289 15.2 Nctes and References...................... 。 292 15.3 Prcblems 293 16 H.Loop Shaping 295 l6.1 Robust Stabiliation cf Coprime Factars..·...·.··. 295 16.2L0 cp Shaping Design...................... 。 305 16.3 Justification far Ht.Locp Shapirg ............. 308 16.4 Further Guidelines far Locp Shaping 314 l6.5 Nctes and References,.··················· 321 16.6 Prcblems..... 322 17 Gap Metric and 8Gap Metric 327 17.1 Gap Metric·....······.·············· 328 17.28 Gap Metric......,:·················· 335 l7.3 Geometric Interpretat ion of8 Gap Metric.,.,,····· 346 17.4 Extended Locp Shaping Design................. 350 17.5 Controller Order Reduction....·...·.··,······· 354 17.6 Nctes and References.....··..·.············· 354 354 18 Miscellaneous Topics 357 18.1 Max imally Stabilized Set of Sy stems.··.·....··.· 357 l8.2 Model Validation..................,..·.· 363 18.3 Mixed1 Analysis and Sy Ithesis..·············· 367 18.4 Nctes and Refererces..................... 375 18.5 Prcblems...... 375
x PREFACE H Control Problem Formulation A Simplied H Control Problem Optimality and Limiting Behavior Minimum Entropy Controller An Optimal Controller General H Solutions Relaxing Assumptions H and H Integral Control H Filtering Notes and References Problems Controller Reduction H Controller Reductions Additive Reduction Coprime Factor Reduction Notes and References Problems H Loop Shaping Robust Stabilization of Coprime Factors Loop Shaping Design Justication for H Loop Shaping Further Guidelines for Loop Shaping Notes and References Problems Gap Metric and Gap Metric Gap Metric Gap Metric Geometric Interpretation of Gap Metric Extended Loop Shaping Design Controller Order Reduction Notes and References Problems Miscellaneous Topics Maximally Stabilized Set of Systems Model Validation Mixed Analysis and Synthesis Notes and References Problems
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