CONTENTS 9 Linear Fractional Transformation 165 9.1 Linear fractional Transformations 9.2 Basic Principle 9.3 Redheffer Star Products 9. 4 Notes and references 10 u and u Synthesis 10.1 General Framework for System Robustness 10.2 Structured Singular value 10.3 Structured Robust Stability and Performance 10.4 Overview of u Synthesis 213 10.5 Notes and references 216 10.6 Problems 11 Controller parameterization 221 11.1 Existence of Stabilizing Controllers 11.2 Parameterization of All Stabilizing Controllers 11.4 Notes and references 11.5 Proble 12 Algebraic Riccati Equations 233 12.1 Stabilizing Solution and Riccati Operator 234 245 12.3 Notes and references 246 12.4 Problems 13 H2 Optimal Control 13.1 Introduction to Regulator Problem 253 13.2 Standard LQR Problem 13.3 Extended LQR Problem 13.4 Guaranteed Stability Margins of LQR 13.5 Standard H2 Problem 13.6 Stability Margins of H2 Controllers 265 13.7 Notes and References 13.8 Problems 14H。 Control 269 14.1 Problem formulation 14.2 A Simplified Hoo Control Problem 270 14.3 Optimality and Limiting Behavior 14.4 Minimum Entropy Controller 14.5 An Optimal ControllerCONTENTS xiii 9 Linear Fractional Transformation 165 9.1 Linear Fractional Transformations ..................... 165 9.2 Basic Principle ................................ 173 9.3 Redheffer Star Products ........................... 178 9.4 Notes and References ............................. 180 9.5 Problems ................................... 181 10 µ and µ Synthesis 183 10.1 General Framework for System Robustness ................ 184 10.2 Structured Singular Value .......................... 187 10.3 Structured Robust Stability and Performance ............... 200 10.4 Overview of µ Synthesis ........................... 213 10.5 Notes and References ............................. 216 10.6 Problems ................................... 217 11 Controller Parameterization 221 11.1 Existence of Stabilizing Controllers ..................... 222 11.2 Parameterization of All Stabilizing Controllers .............. 224 11.3 Coprime Factorization Approach ...................... 228 11.4 Notes and References ............................. 231 11.5 Problems ................................... 231 12 Algebraic Riccati Equations 233 12.1 Stabilizing Solution and Riccati Operator ................. 234 12.2 Inner Functions ................................ 245 12.3 Notes and References ............................. 246 12.4 Problems ................................... 246 13 H2 Optimal Control 253 13.1 Introduction to Regulator Problem ..................... 253 13.2 Standard LQR Problem ........................... 255 13.3 Extended LQR Problem ........................... 258 13.4 Guaranteed Stability Margins of LQR ................... 259 13.5 Standard H2 Problem ............................ 261 13.6 Stability Margins of H2 Controllers ..................... 265 13.7 Notes and References ............................. 267 13.8 Problems ................................... 267 14 H∞ Control 269 14.1 Problem Formulation ............................. 269 14.2 A Simplified H∞ Control Problem ..................... 270 14.3 Optimality and Limiting Behavior ..................... 282 14.4 Minimum Entropy Controller ........................ 286 14.5 An Optimal Controller ............................ 286