System Identification and Robust Control

1. Introduction.- 1.1 The Organization of the Book.- 1.1.1 Part I, Robust Control — Theory and Design.- 1.1.2 Part II, System Identification and Estimation of Model Error Bounds.- 1.1.3 Part III, A Synergistic Control Systems Design Philosophy.- 1.1.4 Part VI, Conclusions.- I. Robust Control — Theory and Design.- 2. Introduction to Robust Control.- 2.1 ? Theory.- 2.1.1 ? Synthesis.- 2.2 An Overview.- 3. Spaces and Norms in Robust Control Theory.- 3.1 Normed Spaces.- 3.2 Vector and Matrix Norms.- 3.2.1 Singular Values.- 3.3 Operator Norms.- 3.3.1 Scalar Systems.- 3.3.2 Multivariable Systems.- 3.4 Banach and Hilbert Spaces.- 3.4.1 Convergence and Completeness.- 3.5 Lebesgue and Hardy Spaces.- 3.5.1 Time Domain Spaces.- 3.5.2 Frequency Domain Spaces.- 3.6 Summary.- 4. Robust Control Design using Singular Values.- 4.1 Nominal Stability.- 4.2 Nominal Performance.- 4.3 Robust Stability.- 4.3.1 The Small Gain Theorem.- 4.4 Robust Performance.- 4.5 Computing the H? Optimal Controller.- 4.5.1 Remarks on the H? Solution.- 4.6 Discrete-Time Results.- 4.7 Summary.- 5. Robust Control Design using Structured Singular Values.- 5.1 ? Analysis.- 5.1.1 Robust Stability.- 5.1.2 Robust Performance.- 5.1.3 Computation of ?.- 5.2 ? Synthesis.- 5.2.1 Complex ? Synthesis — D-K Iteration.- 5.2.2 Mixed ? Synthesis — D,G-K Iteration.- 5.2.3 Mixed ? Synthesis — ?-K Iteration.- 5.3 Summary.- 6. Mixed ? Control of a Compact Disc Servo Drive.- 6.1 Complex ? Design.- 6.2 Mixed ? Design.- 6.3 Summary.- 7. ? Control of an Ill-Conditioned Aircraft.- 7.1 The Aircraft Model.- 7.1.1 Plant Scaling.- 7.1.2 Dynamics of the Scaled Aircraft Model.- 7.2 Control Objectives.- 7.2.1 Robustness.- 7.2.2 Performance.- 7.3 Formulation of Control Problem.- 7.4 Evaluation of Classical Control Design.- 7.5 Controller Design using ?.- 7.6 Summary.- II. System Identification and Estimation of Model Error Bounds.- 8. Introduction to Estimation Theory.- 8.1 Soft versus Hard Uncertainty Bounds.- 8.2 An Overview.- 8.3 Remarks.- 9. Classical System Identification.- 9.1 The Cramér-Rao Inequality for any Unbiased Estimator.- 9.2 Time Domain Asymptotic Variance Expressions.- 9.3 Frequency Domain Asymptotic Variance Expressions.- 9.4 Confidence Intervals for ??N.- 9.5 Frequency Domain Uncertainty Bounds.- 9.6 A Numerical Example.- 9.6.1 Choosing the Model Structure.- 9.6.2 Estimation and Model Validation.- 9.6.3 Results.- 9.7 Summary.- 10. Orthonormal Filters in System Identification.- 10.1 ARX Models.- 10.1.1 Variance of ARX Parameter Estimate.- 10.2 Output Error Models.- 10.2.1 Variance of OE Parameter Estimate.- 10.3 Fixed Denominator Models.- 10.3.1 Variance of Fixed Denominator Parameter Estimate.- 10.3.2 FIR Models.- 10.3.3 Laguerre Models.- 10.3.4 Kautz Models.- 10.3.5 Combined Laguerre and Kautz Structures.- 10.4 Summary.- 11. The Stochastic Embedding Approach.- 11.1 The Methodology.- 11.1.1 Necessary Assumptions.- 11.1.2 Model Formulation.- 11.1.3 Computing the Parameter Estimate.- 11.1.4 Variance of Parameter Estimate.- 11.1.5 Estimating the Model Error.- 11.1.6 Recapitulation.- 11.2 Estimating the Parameterizations of f? and fv.- 11.2.1 Estimation Techniques.- 11.2.2 Choosing the Probability Distributions.- 11.2.3 Maximum Likelihood Estimation of ?.- 11.3 Parameterizing the Covariances.- 11.3.1 Parameterizing the Noise Covariance Cv.- 11.3.2 Parameterizing the Undermodeling Covariance C?.- 11.3.3 Combined Covariance Structures.- 11.4 Summary.- 11.4.1 Remarks.- 12. Estimating Uncertainty using Stochastic Embedding.- 12.1 The True System.- 12.2 Error Bounds with a Classical Approach.- 12.3 Error Bounds with Stochastic Embedding Approach.- 12.3.1 Case 1, A Constant Undermodeling Impulse Response.- 12.3.2 Case 2: An Exponentially Decaying Undermodeling Impulse Response.- 12.3.3 Case 3: A First Order Decaying Undermodeling Impulse Response.- 12.4 Summary.- III. A Synergistic Control Systems Design Philosophy.- 13. Combining System Identification and Robust Control.- 13.1 System Identification for Robust Contro l.- 13.1.1 Bias and Variance Errors.- 13.1.2 What Can We Do with Classical Techniques.- 13.1.3 The Stochastic Embedding Approach.- 13.1.4 Proposed Approach.- 13.2 Robust Control from System Identification.- 13.2.1 The H? Approach.- 13.2.2 The Complex ? Approach.- 13.2.3 The Mixed ? Approach.- 13.3 A Synergistic Approach to Identification Based Control.- 13.4 Summary.- 14. Control of a Water Pump.- 14.1 Identification Procedure.- 14.1.1 Estimation of Model Uncertainty.- 14.1.2 Constructing A Norm Bounded Perturbation.- 14.2 Robust Control Design.- 14.2.1 Performance Specification.- 14.2.2 H? Design.- 14.2.3 Mixed ? Design.- 14.3 Summary.- IV. Conclusions.- 15. Conclusions.- 15.1 Part I, Robust Control — Theory and Design.- 15.2 Part II, System Identification and Estimation of Model Error Bounds.- 15.3 Part III, A Synergistic Control Systems Design Methodology.- 15.4 Future Research.- V. Appendices.- A. The Generalized Nyquist Criterion.- D. Rigid Body Model of ASTOVL Aircraft.- G.1 Transforming the Residuals.- I. Partial Derivatives of the Noise Covariance.- J. Partial Derivatives of the Undermodeling Covariance.- K. ARMA(1) Noise Covariance Matrix.- L. Extracting Principal Axis from Form Matrix.- M. Determining Open Loop Uncertainty Ellipses.