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Asymmetric Order-Doubling and Robust State Estimation for Uncertain Nonminimum Phase Systems By

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Asymmetric Order-Doubling and Robust State Estimation for Uncertain Nonminimum Phase Systems By Asymmetric Order-Doubling and Robust State Estimation for Uncertain Nonminimum Phase Systems by MURALIDHAR RAVURI B.Tech., Indian Institute of Technology, Madras (1993) M. S. in Mech. Eng., University of Houston, Houston, Texas (1995) Submitted to the Department of Mechanical Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mechanical Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 1999 @ Massachusetts Institute of Technology 1999. All rights reserved. A u th or ........................................ .................... Department of Mechanical Engineering May 15, 1999 I / ~'-~1 Certified by ................. Cy... b . .. .. ....... ...... Haruhiko Asada Professor _ wjis Supervisor A ccepted by .......................... .. .. ......... .... Ain Sonin Chairman, Department Committee on Graduate Students ENG Asymmetric Order-Doubling and Robust State Estimation for Uncertain Nonminimum Phase Systems by MURALIDHAR RAVURI Submitted to the Department of Mechanical Engineering on May 15, 1999, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mechanical Engineering Abstract In this thesis, a new control design method is developed to achieve high performance for a class of uncertain nonminimum phase systems. First, a new method called the asymmetric order-doubling method is developed for the nominal system that gives high speed of response. The presence of uncertainty is then shown to deteriorate the estimation of states significantly. A robust filter is designed to improve the state estimation, thereby improving the overall performance for both the nominal and perturbed systems. The proposed method is implemented on an industrial coordinate measuring machine (CMM) to demonstrate the effectiveness of the method. The new asymmetric order-doubling method generalizes symmetric root locus method of linear quadratic regulators to allow for asymmetric placement of additional poles and zeros. Extensive simulations and experiments on an industrial coordinate measuring machine revealed that the asymmetric order-doubling method has good nominal performance for nonminimum phase systems. Other applications for which this method could be used include numerically controlled machines, machine tools, and robots. Presence of parametric uncertainties and mild nonlinearities degrade per- formance of the system. Careful analysis indicated that large state estimation errors is the prominent reason for poor performance due to uncertainty in these applications. A robust filter was designed to improve the state estimation. The robust filtering theory proposed, estimates the states of an uncertain, nonlinear system whenever the uncertainties and nonlinearities are described in terms of integral quadratic con- straints (IQC's). The robust filter derived using the S-procedure losslessness theorem for shift-invariant spaces, was shown to be represented as a convex problem in terms of linear matrix inequalities (LMI's). The robust filter was implemented on CMM and was shown to improve both state estimation and performance for nominal and perturbed systems. Thesis Supervisor: Haruhiko Asada Title: Professor Acknowledgements I am extremely grateful to my thesis advisor, Professor Haruhiko Asada, for his guidance and encouragement throughout the course of my study. I have benefited a lot from his physical insights and his emphasis on clarity in presentations. His supervision has also helped me appreciate the importance of the interaction of experimental justification with theoretical developments. I am also immensely grateful to my thesis committee member, Professor Alexandre Megretski, for all those numerous research discussions we had. Discussions with Sasha helped me understand the power of abstraction in mathematics, importance of creating counter examples, necessity to search for an intuitive and elegant solution to a rigorous proof, among others. I am grateful to Professor Kamal Youcef-Toumi for serving on my thesis committee and for his comments and many useful suggestions on my thesis. I also want to thank many of my colleagues at MIT for their assistance: Joe Spano for helping me on any hardware and design related questions I had and for his goodnaturedness and friendly support all along my four years stay at MIT, Sokwoo Rhee for debugging and helping me understand his software required for the real time windows NT implementation of my control algorithm, and Youngjae and Sokwoo for helping me develop the necessary electronic circuits for the computer control of the coordinate measuring machine. I also want to thank Masayoshi Wada, Ming Zhou, Shrikant Savant, Brandon Gordon, Rolland Doubleday, and other members of the d'Arbeloff Laboratory for Information Systems and Technology for their assorted words of wisdom. I wouldn't have been able to do anything without the support of my parents, wife, brother and sister. A very special thanks goes to my wife Sridevi for her love and affection and also to her everlasting patience, and to my brother Vidyasagar and Sridevi for taking the immense extra responsibility to create a nice comfortable atmosphere allowing me to work only on research for the last six months. I owe everything to them. 3 I dedicate my work to my parents Subbarao and Indiradevi, my wife Sridevi, my brother Vidyasagar and my sister Udayalaksmi. 4 Contents Contents 3 List of Figures 6 1 Introduction 12 1.1 Organization of the Thesis .... .................... 20 2 Asymmetric Order-Doubling Control Design 22 2.1 Introduction ................................ 22 2.2 Asymmetric Order Doubling Method .................. 24 2.3 Theoretical Results ............................ 29 2.4 Application ................................ 36 2.4.1 The System ............................ 36 2.4.2 Identification ........................... 38 2.5 Implementation and Experiments .................... 39 2.5.1 Control Tuning ............. ............. 39 2.5.2 Experimental Results ....................... 42 2.6 Robustness Margins ....................... ..... 47 2.6.1 Stability Margins ......................... 47 2.6.2 Sensitivities ............................ 47 2.6.3 Robustness ............... ............. 50 2.7 Summary .. ............. ............ ...... 52 5 3 Uncertainty and Robustness 54 3.1 Introduction . .. .. .. .. .. .. .. .. ....... .... 54 3.2 Sources of Uncertainty . .. .. .. .. .. .. ........ .. 55 3.3 Representation of Uncertainty .. .. .. .. .... ... ... 57 3.3.1 Linear Fractional Transformations (LFT's) . .... ... ... 58 3.3.2 Parametric Uncertainty .. .. .. .. .. ... .. ... .. 60 3.4 Modeling, Simulation and Experiments .. .. .. .. ... .. ... 62 3.4.1 Model of y-axis motion of CMM . ... .. .. ... .... .. 63 3.4.2 Simulation Results .. .. ... .. .. .. ... .. ... .. 65 3.4.3 Experimental Results .. .. .. .. ... .. .. ... .. .. 70 3.5 Sum m ary ... .. ... .. ... .. ... .. .. .. .... .... 73 4 Robust State Estimation 74 4.1 Introduction . .. ... .. ... .. ... .. .. .... .... ... 74 4.2 Standard Filtering Problem .. ... .. .. ... .. ... .. ... 76 4.2.1 Kalman-Yakubovich-Popov Lemma . .. .... ... ... 80 4.2.2 Necessary Conditions .. .. ... .. ... .. ... .... .. 83 4.2.3 Sufficient Conditions .. .. .. .. ... .. .... .... 85 4.3 Robust Filtering Problem . .. .. .. .. .. ... .... ... 89 4.3.1 Problem Formulation ... .. .. ... .. .... ....... 91 4.3.2 Assumptions for the Robust Filtering Problem .. ....... 92 4.4 S-Procedure Losslessness Theorem ... .. .. .. .. .... .... 94 4.5 Robust Filtering .. .. .. ... .. .. .. .. .. .. ... .... .. 99 4.6 Convexity of Robust Filtering Problem .. .. .. .. .... .... 100 4.7 LMI Formulation . ... .. ... .. .. ... .. .. .. ... .. .. 102 4.7.1 Useful Results . ... .. ... .. ... .. .. .... .... 105 4.7.2 Solvability Conditions .. ... ... ... .. ... .... .. 110 4.7.3 Filter Reconstruction .. .. ... .. ... ... .. ... .. 116 4.7.4 Special Case ... .... .... .... .. ...... ..... 117 4.8 Application to Coordinate Measuring Machine .. 122 6 4.8.1 Formulation . .. .. .. .. ... .. .. .. .. .. .. .. .. 123 4.8.2 Simulation Results . ... .. .. ... .. .. ... .. .. .. 127 4.8.3 Experimental Results. ... .. .. ... .. .. ... .. .. 132 5 Conclusions 142 References 145 7 List of Figures 1.1 Applications (Courtesy of SNK Co., LTD). ..... ......... 14 1.2 Applications (Courtesy of SNK Co., LTD). ....... ........ 15 1.3 Applications (Courtesy of SNK Co., LTD). ..... ......... 16 1.4 Overview of the coordinate measuring machine (Courtesy of SNK Co., Ltd) ........ ... .................................... 17 1.5 Symmetric Root locus of the LQR design . ......... 18 1.6 Desired asymmetric root locus for the coordinate measuring machine that gives good stability properties, high speed of response and other performance/robustness properties. .... ........ ....... 19 1.7 Setup for robust estimation problem. .... ..... ..... .... 20 2.1 Example of nonminimum phase system with zeros close to the imagi- nary axis (Pole-zero plot of Coordinate Measuring Machine described in Sections 2.4 and 2.5). ... ...... ..... ...... ..... 26 2.2 Symmetric root locus for the example system. .. ...... ..... 27 2.3 Asymmetric root-squared locus. .. ..... ..... ..... .... 28 2.4 Overview of the coordinate measuring machine (Courtesy of SNK Co., Ltd) ...... .. ... .................................... 37 2.5 Schematic diagram of the x-axis drive system. ..... ..... ... 38 2.6 Effect of change in location of mass m on the poles and zeros ..... 39 2.7 Bode plot fit for
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