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.....··· 251PREFACE 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 ￾ ￾ ￾ ￾ ￾ ￾ ￾ ￾ ￾ ￾ ￾ ￾ ￾ ￾ ￾ ￾ ￾ ￾ ￾ ￾ ￾ ￾ ￾ ￾ ￾ ￾ ￾ ￾ ￾ ￾ ￾ ￾ ￾ ￾ ￾