CONTENTS 4 H2 and Hoo Spaces 4.1 Hilbert Spaces 45 4.3 Computing L2 and H2 Norms 4.4 Computing Loo and Ho Norm 4.5 Notes and references 4. 6 Problems 5 Internal Stability 5.1 Feedback Struct 5.2 Well-Posedness of Feedback Loop 5.3 Internal Stability 5.4 Coprime Factorization over RH 5.5 Notes and references 5.6 Problems 6 Performance Specifications and Limitations 6.1 Feedback Properties 6.2 Weighted H2 and Hoo Performance 6.3 Selection of Weighting Functions 6.4 Bode s Gain and phase relation 6.5 Bode's Sensitivity Integral 6.6 Analyticity Constraints 6.7 Notes and References 6.8 Problems 102 7 Balanced model reduction 1 7.1 Lyapunov Equations 7.2 Balanced realizations 7.3 Model Reduction by Balanced Truncation 117 7.4 Frequency-Weighted Balanced Model Reduction 7.5 Notes and references 126 7. 6 Problems 8 Uncertainty and robustness 129 8.1 Model Uncertainty 8.2 Small Gain Theorem 8.3 Stability under Unstructured Uncertainties 8. 4 Robust Performance 147 8.5 Skewed Specifications 150 8.6 Classical Control for MIMO Systems 8. 7 Notes and references 157 8.8 Problemsxii CONTENTS 4 H2 and H∞ Spaces 45 4.1 Hilbert Spaces ................................ 45 4.2 H2 and H∞ Spaces .............................. 47 4.3 Computing L2 and H2 Norms ........................ 53 4.4 Computing L∞ and H∞ Norms ....................... 55 4.5 Notes and References ............................. 61 4.6 Problems ................................... 62 5 Internal Stability 65 5.1 Feedback Structure .............................. 65 5.2 Well-Posedness of Feedback Loop ...................... 66 5.3 Internal Stability ............................... 68 5.4 Coprime Factorization over RH∞ ...................... 71 5.5 Notes and References ............................. 77 5.6 Problems ................................... 77 6 Performance Specifications and Limitations 81 6.1 Feedback Properties ............................. 81 6.2 Weighted H2 and H∞ Performance ..................... 85 6.3 Selection of Weighting Functions ...................... 89 6.4 Bode’s Gain and Phase Relation ...................... 94 6.5 Bode’s Sensitivity Integral .......................... 98 6.6 Analyticity Constraints ........................... 100 6.7 Notes and References ............................. 102 6.8 Problems ................................... 102 7 Balanced Model Reduction 105 7.1 Lyapunov Equations ............................. 106 7.2 Balanced Realizations ............................ 107 7.3 Model Reduction by Balanced Truncation ................. 117 7.4 Frequency-Weighted Balanced Model Reduction .............. 124 7.5 Notes and References ............................. 126 7.6 Problems ................................... 127 8 Uncertainty and Robustness 129 8.1 Model Uncertainty .............................. 129 8.2 Small Gain Theorem ............................. 137 8.3 Stability under Unstructured Uncertainties ................ 141 8.4 Robust Performance ............................. 147 8.5 Skewed Specifications ............................ 150 8.6 Classical Control for MIMO Systems .................... 154 8.7 Notes and References ............................. 157 8.8 Problems ................................... 158