DECOUPLING AND REDUCED ORDER MODELING OF TWO-TIME-SCALE CONTROL SYSTEMS Item Type text; Dissertation-Reproduction (electronic) Authors Anderson, Leonard Robert Publisher The University of Arizona. Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 01/10/2021 20:26:31 Link to Item http://hdl.handle.net/10150/298494 INFORMATION TO USERS This was produced from a copy of a document sent to us for microfilming. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the material submitted. The following explanation of techniques is provided to help you understand markings or notations which may appear on this reproduction. 1. 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Zeeb Road, Ann Arbor, MI 48106 DECOUPLING AND REDUCED ORDER MODELING OF TWO-TIME-SCALE CONTROL SYSTEMS by Leonard Robert Anderson A Dissertation Submitted to the Faculty of the AEROSPACE AND MECHANICAL ENGINEERING DEPARTMENT In Partial Fulfillment of the Requirements For the Degree of DOCTOR OF PHILOSOPHY • WITH A MAJOR IN MECHANICAL ENGINEERING In the Graduate College THE UNIVERSITY OF ARIZONA 19 7 9 STATEMENT BY AUTHOR This dissertation has been submitted in partial ful­ fillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library. Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledg­ ment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or Dean of the Graduate College when in his judgment the pro­ posed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author. SIGNED ACKNOWLEDGMENTS I wish to express my sincere gratitude to my dis­ sertation director, Dr. Robert E. O'Malley, Jr., for his guidance in this work, for financial support provided at several stages, and for the opportunity to visit with Dr. Petar V. Kokotovic of the Coordinated Science Laboratory, University of Illinois, Urbana, whose research and discus­ sions on singularly perturbed engineering systems strongly motivated this work. I would also like to thank the fol­ lowing persons: Dr. T. L. Vincent and Dr. M. R. Bottaccini for their careful review of this dissertation and many helpful comments; Dr. W. R. Sears for his assistance in obtaining fellowship support for this work; Dr. L. B. Scott, head of the Aerospace and Mechanical Engineering Department, for a teaching associateship and other financial assistance; Dr. B. S. Goh and Dr. J. M. Cushing who served on my examining committee; Mrs. Ginger Bierman and Mrs. Sarah Oordt for their excellent typing; and special thanks to my wife, Laurie, for her continuing encouragement and understanding. iii TABLE OF CONTENTS Page LIST OF TABLES vi LIST OF ILLUSTRATIONS vii ABSTRACT ix CHAPTER 1. INTRODUCTION 1 2. MODAL DECOUPLING AND REDUCED ORDER MODELING 9 2.1. A Review of Modal Variables 9 2.2. The LK Decoupling Transformation ... 11 2.3. The Reduced Order Models of Davison and Marshall 18 2.4. A New Reduced Order Model from Singular Perturbations 28 2.5. A Second-Order Example ......... 33 2.6. Feedback Matrix Design via the LK Transformation 40 3. EXISTENCE AND UNIQUENESS OF THE L AND K TIME-SCALE-DECOUPLING MATRICES 46 3.1. A Review of Matrix Eigenstructure ... 47 3.2. The General Solution to the Non- symmetric Algebraic Riccati Equation . 51 3.3. Conditions for Matrices L and K to be Real 58 3.4. Uniqueness of L and K Matrices ... 61 4. COMPUTING THE LK TRANSFORMATION 64 4.1. Computing the L Matrix 64 4.2. Computing the K Matrix 71 4.3. Ordering and Partitioning of the State Variables 72 4.4. The Relationship Between Ordering and Coupling 74 4.5. The Computational Method Summarized . 77 iv V TABLE OF CONTENTS—Continued . Page 5. NUMERICAL EXAMPLES 78 5.1. A Strongly Two-Time-Scale Example ... 78 5.2. A Weakly Two-Time-Scale Example .... 85 6. EXTENSIONS TO TIME-VARYING CONTROL SYSTEMS 94 6.1. The Time Varying LK Transformation 94 6.2. A Second-Order Example 108 APPENDIX A: DATA FOR EXAMPLES 5.1 AND 5.2 116 REFERENCES 121 LIST OF TABLES Table Page 5.1. Execution Times Corresponding to Figures 5.5 and 5.6 90 5.2. Data for Cases I, II and III 92 A.l. The A Matrix for the Turbofan Engine Model 119 A.2. The B Matrix for the Turbofan Engine Model 120 vi LIST OF ILLUSTRATIONS Figure Page 2.1. Response of reduced-order models to initial conditions for Example 2.5, 35 2.2. Response of reduced-order models to a step input for Example 2.5 37 2.3. Response of reduced-order models to a ramp input for Example 2.5 38 2.4. Response of reduced-order models to a sinusoidal input for Example 2.5 39 5.1. Convergence of Algorithms 4.1 and 4.2 for Example 5.1 81 5.2. Response of a slow variable, velocity variation, of Example 5.1 to initial conditions v(0) = 100 ft/sec, <5(0) = .5 rad/sec, a(0)=0, y(0) = 0 82 5.3. Response of a fast variable, angle of attack, of Example 5.1 to initial conditions v(0) = 100 ft/sec, <5(0) = .05 rad/sec, a(0) = 0, y(0) =0 83 5.4. Response of Example 5.1 to a .1 (rad) elevator deflection applied for one second 84 5.5. Convergence of Algorithms 4.1 and 4.2 for Example 5.2 with n^=3, y = .383 . 88 5.6. Convergence of Algorithms 4.1 and 4.2 for Example 5.2 with n^ = 5, y = .371 .... 89 5.7. Convergence of Algorithm 4.1 for Example 5.2 with n^ = 5 for alternative ordering of the state and alternative initial Lo 91 vii viii LIST OF ILLUSTRATIONS—Continued Figure Page 6.1. Eigenvalues of original and block- decoupled system for Example 6.2 110 6.2. L(t) and K(t) time histories for Example 6.2 Ill 6.3. Response of state variables to initial conditions for Example 6.2 113 6.4. Response of state variables to control inputs for Example 6.2 114 ABSTRACT This study presents new analytical findings and a new computational method for time-scale decoupling and order reduction of linear control systems. A transformation of variables is considered which reduces the system to block- diagonal form with slow and fast modes decoupled. Once the decoupled form of the system is obtained, a small parameter can be identified which provides a measure of the system's time scale separation, and the methods of singular perturba­ tions can be applied to yield reduced-order approximations of the original system. A particular reduced-order model so obtained overcomes certain difficulties encountered with other well known order-reduction methodologies. The type of control system considered here is a very general, widely used form. The ability to identify a specific small parameter in all such systems provides a useful link to singular perturbations theory which requires a small parameter to be identified. Although the examples of two-time-scale systems presented here are relatively low order, due to recent advances in computational methods for matrices these decoupling techniques can be directly applied to large scale systems with orders of 100 or more. ix X In Chapter 1, a two-time-scale property is defined in terms of system eigenvalues, and a general class of linear transformations which yield time scale decoupling is also defined.