System Identification and Robust Control : A Case Study Approach

The series Advances in Industrial Control aims to report and encourage technology transfer in control engineering. The rapid development of control technology impacts all areas of the control discipline. New theory, new controllers, actuators, sensors, new industrial processes, computer methods, new applications, new philosophies, ... , new challenges. Much of this development work resides in industrial reports, feasibility study papers and the reports of advanced collaborative projects. The series offers an opportunity for researchers to present an extended exposition of such new work in all aspects of industrial control for wider and rapid dissemination. The present text Steen T0ffner-Clausen deals with both system identification and robust control. It provides a very comprehensive tutorial introduction to some of the most difficult topics in robust control theory before considering applications problems. Traditional Hoo robust control design concepts for multivariable systems are first considered and the problems of robust stability and performance are discussed. The following chapter introduces the idea of the structured singular value and applies this to both analysis and synthesis problems. The author manages to provide a very straightforward introduction to this subject and also introduces some new ideas.

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.