Experiment Design with Applications in Identification for Control Henrik Jansson TRITA-S3-REG-0404 ISSN 1404–2150 ISBN 91-7283-905-8 Automatic Control Department of Signals, Sensors and Systems Royal Institute of Technology (KTH) Stockholm, Sweden, 2004 Submitted to the School of Electrical Engineering, Royal Institute of Technology, in partial fulfillment of the requirements for the degree of Doctor of Philosophy. Copyright c 2004 by Henrik Jansson Experiment Design with Applications in Identification for Control Automatic Control Department of Signals, Sensors and Systems Royal Institute of Technology (KTH) SE-100 44 Stockholm, Sweden Abstract The main part of this thesis focuses on optimal experiment design for system identification within the prediction error framework. A rather flexible framework for translating optimal experiment de- sign into tractable convex programs is presented. The design variables are the spectral properties of the external excitations. The framework allows for any linear and finite-dimensional parametrization of the design spectrum or a partial expansion thereof. This includes both continuous and discrete spectra. Constraints on these spectra can be included in the design formulation, either in terms of power bounds or as frequency wise constraints. As quality constraints, general linear functions of the asymptotic covariance matrix of the estimated parameters can be in- cluded. Here, different types of frequency-by-frequency constraints on the frequency function estimate are expected to be an important contri- bution to the area of identification and control. For a certain class of linearly parameterized frequency functions it is possible to derive variance expressions that are exact for finite sample sizes. Based on these variance expressions it is shown that the optimiza- tion over the square of the Discrete Fourier Transform (DFT) coefficients of the input leads to convex optimization problems. The optimal input design are compared to the use of standard iden- tification input signals for two benchmark problems. The results show significant benefits of appropriate input designs. Knowledge of the location of non-minimum phase zeros is very useful when designing controllers. Both analytical and numerical results on input design for accurate identification of non-minimum phase zeros are presented. A method is presented for the computation of an upper bound on the maximum over the frequencies of a worst case quality measure, e.g. the worst case performance achieved by a controller in an ellipsoidal uncer- tainty region. This problem has until now been solved by using a fre- quency gridding and, here, this is avoided by using the Kalman-Yakubovich- iv Popov-lemma. The last chapter studies experiment design from the perspective of controller tuning based on experimental data. Iterative Feedback Tuning (IFT) is an algorithm that utilizes sensitivity information from closed- loop experiments for controller tuning. This method is experimentally costly when multivariable systems are considered. Several methods are proposed to reduce the experimental time by approximating the gradient of the cost function. One of these methods uses the same technique of shifting the order of operators as is used in IFT for scalar systems. This method is further analyzed and sufficient conditions for local convergence are derived. Acknowledgments The time as a PhD student has been a great experience. I would like to express my sincere gratitude to my supervisors during these years; Pro- fessor Bo Wahlberg, Professor H˚akan Hjalmarsson, and Docent Anders Hansson. You have all been very helpful. A special thanks goes to H˚akan. It has been really inspiring working with you. You are really creative and I have learned a lot. I really appreciate the time you spend on me. If I had to pay ”OB-till¨agg” I would be a poor guy. Many thanks goes to my collaborators and co-authors Kristian Lindqvist, Jonas M˚artensson and M¨arta Barenthin. It has been a great fun working with you. Without you, there would be no SYSID Lab. I would also like to express my gratitude to Karin Karlsson-Eklund for helping me with different problems and being supportive. It has been a pleasure to be at the department. For this, I would like to thank all colleagues, former and present. I have really enjoyed your company. Finally, I would like to thank my family for their support. Contents 1 Introduction 1 1.1 System Identification - A Starter .............. 2 1.2 Optimal Experiment Design - A Background ....... 5 1.2.1 Design for Full Order Models ............ 6 1.2.2 Design for High Order Models ........... 7 1.2.3 The Return of Full Order Modeling ........ 9 1.2.4 Other Contributions ................. 13 1.3 Iterative Feedback Tuning . ................. 14 1.4 Contributions and Outline . ................. 15 2 Parameter Estimation and Connections to Input Design 19 2.1 Parameter Estimation . ................. 19 2.2 Uncertainty in the Parameter Estimates .......... 21 2.2.1 Parameter Covariance ................ 21 2.2.2 Confidence Bounds for Estimated Parameters . 23 2.3 Uncertainty of Frequency Function Estimates ....... 24 2.3.1 Variance of Frequency Function Estimates ..... 25 2.3.2 Uncertainty Descriptions Based on Parametric Con- fidence Regions . ................. 27 2.4 Summary ........................... 29 3 Fundamentals of Experiment Design 31 3.1 Introduction .......................... 31 3.1.1 Quality Constraints ................. 33 3.1.2 Signal Constraints . ................. 42 3.1.3 Objective Functions ................. 42 3.1.4 An Introductory Example .............. 42 3.2 Parametrization and Realization of the Input Spectrum . 45 viii Contents 3.2.1 Introduction ..................... 45 3.2.2 Finite Dimensional Spectrum Parametrization . 47 3.2.3 Partial Correlation Parametrization ........ 50 3.2.4 Summary . ...................... 51 3.3 Parametrizations of the Covariance Matrix ......... 52 3.3.1 Complete Parametrizations of the Covariance Matrix 53 3.3.2 A Parametrization Based on a Finite Dimensional Spectrum....................... 57 3.3.3 Summary . ...................... 57 3.4 Parametrization of Signal Constraints ........... 58 3.4.1 Parametrization of Power Constraints ....... 58 3.4.2 Parametrization of Point-wise Constraints ..... 60 3.5 Experiment Design in Closed-loop ............. 60 3.5.1 Spectrum Representations .............. 62 3.5.2 Experiment Design in Closed-loop with a Fix Con- troller . ...................... 62 3.5.3 Experiment Design in Closed-loop with a Free Con- troller . ...................... 63 3.6 Quality Constraints ..................... 70 3.6.1 Convex Representation of Quality Constraints . 70 3.6.2 Application of the KYP-lemma to Quality Con- straints........................ 71 3.7 Quality Constraints in Ellipsoidal Regions ......... 72 3.7.1 Reformulation as a Convex Problem ........ 74 3.7.2 A Finite Dimensional Formulation ......... 77 3.8 Biased Noise Dynamics ................... 79 3.8.1 Weighted Variance Constraints ........... 80 3.8.2 Parametric Confidence Ellipsoids .......... 81 3.9 Computational Aspects ................... 82 3.10 Robustness Aspects ..................... 83 3.10.1 Input Spectrum Parametrization .......... 83 3.10.2 Working with Sets of a Priori Models ....... 84 3.10.3 Adaptation ...................... 84 3.10.4 Low and High Order Models and Optimal Input Design......................... 85 3.11 Framework Review and Numerical Illustration ....... 86 Contents ix 4 Finite Sample Input Design for Linearly Parametrized Models 95 4.1 Introduction .......................... 96 4.2 Discrete Fourier Transform Representation of Signals . 97 4.3 Least-squares Estimation . ................. 98 4.4 Input Design ......................... 102 4.4.1 Geometric Programming Solution .......... 103 4.4.2 LMI Solution ..................... 105 4.4.3 Numerical Illustration ................ 106 4.4.4 Closed-form Solution ................. 109 4.4.5 Input Design Based on Over-parametrized Models 111 4.5Conclusions.......................... 112 5 Applications 115 5.1 Introduction .......................... 115 5.2 A Process Control Application ............... 118 5.2.1 Optimal Design Compared to White Input Signals 119 5.2.2 Optimal Input Design in Practice .......... 120 5.3 A Mechanical System Application ............. 122 5.3.1 Optimal Design Compared to White Input Signals 123 5.3.2 Input Design in a Practical Situation ........ 124 5.3.3 Sensitivity of the Optimal Design .......... 125 5.4Conclusions.......................... 127 6 Input Design for Identification of Zeros 131 6.1 Introduction .......................... 132 6.2 Estimation of Parameters and Zeros ............ 133 6.2.1 Parameter Estimation ................ 133 6.2.2 Estimation of Zeros ................. 135 6.3 Input Design - Analytical Results .............. 136 6.3.1 Input Design for Finite Model Orders ....... 136 6.3.2 Input Design for High-order Systems ........ 142 6.3.3 Realization of Optimal Inputs ............ 143 6.4 Input Design - A Numerical Solution ............ 144 6.4.1 Numerical Example ................. 146 6.5 Sensitivity and Benefits . ................. 147 6.5.1 Numerical Example ................. 149 6.6 Using Restricted Complexity Models for Identification of Zeros.............................. 150 6.7Conclusions.......................... 151 x Contents 7 Convex Computation of Worst Case Criteria 153 7.1 Motivation of Quality Measure ............... 155 7.1.1 Parametric Ellipsoidal Constraints ......... 155 7.1.2 Worst Case