Lecture Notes in Control and Information Sciences

Transcription

1 Lecture Notes in Control and Information Sciences Edited by M.Thoma and A.Wyner 138 D. C. McFarlane, K. Glover Robust Controller Design Using Normalized Coprime Factor Plant Descriptions Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong

2 Series Editors M. Thoma A. Wyner Advisory Board L D. Davisson A. G. J. MacFarlane H. Kwakernaak J. L Massey Ya Z. Tsypkin A. J. Viterbi Authors Duncan C. McFarlane BHP Melbourne Research Laboratories P.O. Box 264 Clayton 3168 Victoria Australia Keith Glover Dept. of Engineering University of Cambridge Trumpington Street Cambridge CB2 1PZ, UK ISBN Springer-Verlag Berlin Heidelberg NewYork ISBN Springer-Verlag NewYork Berlin Heidelberg Library of Congress Cataloging-in-Publication Data Glover, K. (Keith), Robust controller design using normalized coprime factor plant design / K. Glover, D. C. McFarlane. (Lecture notes in control and information sciences ; 138) Includes bibliographical references. ISBN (U.S. : alk. paper) 1. Automatic control. 2. Control theory. I. McFarlane, D. C. (Duncan C.). II. Title. III. Series. TJ213.G dc This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9,1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. Springer-Vedag Bedin, Heidelberg 1990 Printed in Germany The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printing: Mercedes-Druck, Berlin Binding: B. Helm, Berlin 2161/ Printed on acid-free paper.

3 Acknowledgements Duncan McFarlane would like to thank the B.H.P. Co. Ltd., Australia, for their support of this work, which was performed while he was a Ph.D. s~udent at the University of Cmnbridge. The authors are pleased to acknowedge the following: M. Vidyasagar for valuable discussions in the early stages of this work; B. Frmlcis, N.J. Young, R.J.~ Ober, J.R. Partington, D. Mustafa and P.I. Iglesias for helpful commeats on the manuscript; J.M. Maciejowski and M. Noton for their cooperation with the design exazalples; aald E. Donaldson for her assistance in preparing the manuscript.

4 TABLE OF CONTENTS 1, 2, INTRODUCTION Background and Problem Description Organization... PRELIMINARIES Introduction State-Space Systems State-Space Forms Controllability and Observability Inner and Co-Inner Systems... 2,3 Frequency Domain Spaces Linear Fractional Transformations Coprime Factorizations Definitions State-Space Properties of Normalized Coprime Factors Stability Results WeU-Posedness and Internal Stability Closed-Loop Stability Results... ROBUST STABILIZATION OF UNCERTAIN SYSTEMS Introduction Modelling of Uncertainty Background Frequency Domain Error Modelling Robust Stability Analysis Introduction Frequency Domain Uncertainty Analysis Robust Controller Design Using H~ Optimization The Robust Stabilization Problem Nominal Performance Problems H,,~ Problem Specification Solution to the H ~ Optimization Problem Summary... ROBUST STABILIZATION OF NORMALIZED COPRIME FACTOR. PLANT DESCRIPTIONS Introduction and Problem Statement Characterizing all Solutions Solution Via a Nehari Extension Approach Parametrizing AU Controllers A State Space Representation of Sub-Optimal Controllers Characterizing All Controllers The Central Controller Solution Using the Procedure of Glover and Doyle, (1988) Related Issues H~ Problems with Exact Solutions

5 Robust Stability of Passive Systems... Review of Results... V REDUCED ORDER CONTROLLER DESIGN Introduction Review Model Reduction Techniques Coprime Factor Model Reduction Reduced Order Controller Design Problem Statement Reduced Order ControUers by Plant Model Reduction Reduced Order Controllers by Controller Model Reduction 5.4 Summary A LOOP SHAPING DESIGN PROCEDURE Introduction Loop Shaping Methods The Design Procedure Using e as a Design Indicator Guarantees on the Achieved Loop Shape Assessing Stabilizing Controllers using e Shaping Functions and Closed-Loop Behaviour Behaviour of Standard Closed-Loop Objectives Bounds on the Normalized Coprime Factors Discussions The Role of the Shaping Function in Design Model Reduction Summary DESIGN EXAMPLES Introduction Example 1: Attitude Control of a Flexible Spacecraft Problem Description Design Objectives The Loop Shaping Design Procedure Results AnMysis of Results Example 2: Attitude Control of a Flexible Space Platform The Dynamic Model Design Objectives The Loop Shaping Design Procedure Results Analysis of Results Spillover Mode Analysis Design on a Model with Reduced Number of Inputs Smnmary Example 3: Vertical Plalm Dynamics Control of an Aircraft Problem Description Design Objectives

6 7.5 Vi Tlle Loop Shaping Design Procedure Results Analysis of Results Comparison with Maciejowski's Results... Concluding Remarks... IG Appendix A The Algebraic tziccatl Equation Appendix B Sub-Optimal Nehari Extensions Appendix C Proof of Miscellaneous Results Appendix D State-Space Systems for Chapter REFERENCES

7 LIST OF FIGURES Figure 2.1 Figure 2.2 Figure 2.3 Figure 2.4 Figure 3.1 Figure 3.2 Figure 3.3 Figure 3.4 Figurc 4.1 Figure 4.2 Figure 4.3 Figure 6.1 Figure 6.2 Figure 6.3 Figure 6.4 Figure 6.5 Figure 6.6 Figure 7.1 Figure 7.2 Figure 7.3 Figure 7.4 Figure 7.5 Figure 7.6 Figure 7.7 Figure 7.8 Figure 7.9 Figure 7.10 Figure 7.11 Figure 7.12 Lower Linear Fractional Transformation (LLFT)... Upper Linear Fractional Transformation (ULFT)... Generalized Feedback Framework with Additional Inputs... Closed-Loop System... Unstructured Uncertainty Models... Generalized Uncertainty Model (with Feedback Controller)... Hoo Objective Function... Example 3.13: Feedback Arrangements... Coprime Factor Robust Stabilization Problem... Construction of K from Coprime Factors... Augmented Problem... Closed-Loop System with Loop Break Points... Open-Loop Systcm... Open-Loop Singular Value Shaping... The Loop Shaping Design Procedure... Specified and Achieved Loop Shapes for k ,1,5,20... Specified and Achieved Loop Shapes... Bode Plots for Nominal, Shaped, and Stabilized Systems - Design (1)... Bode Plots for Nominal, Shaped, and Stabilized Systems - Design (2)... Bode Plot for K~ - Design (2)... Output Response to Input Step - Designs (1), (2)... Input Response to Input Step - Designs (1),. (2)... Frequency Domain Indicators - Designs (1), (2)... Frequency Domain Indicators - Design (2) and Bounds... Elementary Model of a Spacecraft... Open-Loop Singular Values - Nominal Plant... Open-Loop Singular Values - Perturbed Plant... Open-Loop Singular Values - Shaped Plant... Open-Loop Singular Values - Stabilized Plant

8 Figurc 7.13 Figure 7.14 Figurc 7.15 Figure 7.16 Figure 7.17 Figure 7.18 Figure 7.19 Figurc 7.20 Figurc 7.21 Figure 7.22 Figure 7.23 Figure 7.24 Figurc 7.25 Figurc 7.26 Figure 7.27 Figurc 7.28 Figure 7.29 Figurc 7.30 Figurc 7.31 Figurc 7.32 Figure 7.33 Figure 7.34 Figure 7.35 Figure 7.36 Figure 7.37 Figure 7.38 Figurc 7.39 VIII Output Responses to 0.1Nm Input Steps (Nominal and Perturbed Systems)... Input Responses to 0.1Nm Input Steps (Nominal and Perturbed Systems)... ~'((I - GK) -1 (7) V's Frequency... 1/(-~(GK(I - GK) -1)) V's Frequency... - GK) -1)) and v's Frequency Output Responses to 0.1Nm Input Steps (NominM azld Perturbed Spillover Systems)... Input Responses to 0.1Nm Input Steps (Nominal and Perturbed Spillover Systems)... Responses for Reduced Inputs Design (1)... Rcsponses for Rcduced Inputs Design (2)... Open-Loop Singular Values-Nominal Plant... Open-Loop Singular Values-Shaped Plmlt (1)... Open-Loop Singular Values-Stabilized Plant (1)... Frequency Domain Indlcators-Design (1)... Open-Loop Singular Values-Shapcd Plant (2)... Opcn-Loop Singular Values-Stabilized Plant (2)... Frequency Domain Indicators-Design (2)... Output Responses to Input Steps - Designs (1),~ (2)... Input Responses to Input Steps - Designs (1) &: (2)... Effective Weight for LTR ProbIem... Weights for Mixed Sensitivity Problem... "~((I - GK) -1 ) V's Frequency... 1/(-5(GK(I - GK) -1)) V's Frequency... Output Responses to Input Steps - LTR and S K: T Designs.. 1/('5(1(7(1 - GK) -1 )) V's Frequency... Input Responses to Input Steps - LTR and S & T Designs... Open-Loop Singular Values-Stabilized Plant, S & T Design... Open-Loop Singular Values-Stabilized Plant, LTR Design

9 NOTATION Symbols Note: Symbols in boldface are functions of the Laplace Transform variable 'd. All transfer functions matrices are taken to be matrices with real-rational elements only, and constant matrices are taken to be matrices with real elements only. C j~ R RL2 RH2 RLoo RH o ;~i(') p(.),n(.) ~(.) z(,) c(.) t~i( ) ai l[ 112 II lifo I 1 II IIn A± A>0 A>B A>O A* G* Re(,) Rio( ) E(.) field of complex numbers field of real numbers the imaginary axis of the complex plane see Definition 2.9 see Definition 2.10 see Definition 2.10 see Definition 2.11 see Definition 2.11 the i th eigenvalue =X,,,az(*) - spectral radius the i th singular value the maximum singular value the minimum singular value = ~(,)]a_(*) - the condition number the i a* characteristic value - see Theorem 4.26 the i th Hankel singular value - see Definition 2.2 L2/H2 norm - see Definition 2.10 Loo/ttoo norm- see Definition 2.11 modulus Hankel Norm - see Definition 2.2 orthogonal complement A positive definite A - B is a positive definite matrix A positive semi-definite complex conjugate transpose n n identity matrix = G(-$)* - complex conjugate transpose real part Riccati operator - see Section expectation

10 ,(.).T'L(P, K) J:v(P,a) TI(G,K) D :2:),, ~s,, :Du, (A,B,C,D) ~_, <.%, ~ Acronyms SISO MIMO LTI RHP, LHP RCF LCF LFT LLFT ULFT ARE GCARE GFAI%E LTR LQR LQG LSDP BTA HNA S&T number of closed right half-plane poles X Lower Linear Fractional Transformation - see Definition 2.13 Upper Linear Fractional Transformation - see Definition 2.13 Alternative form for a Lower Linear Fractional Transformation - see Lemma 2.14 closed-loop transfer matrix - Section 5.2 class of uncertainty classes of bounded uncertainty - see Definition 3.1 state-space realization - Section 2.2 degrees asymptotically equal to, lcss than or cqual to, greater than or equal to - Section Single-Input, Single-Output Multi-Input, Multi-Output Linear, Time-Inv~riant Right Half-Plane, Left Half-Plane Right Coprime Factorization Left Coprime Factorization Linear Fractional Transformation Lower Linear Fractional Transformation Upper Linear Fractional Transformation Algebraic Riccati Equation Generalized Control Algebraic Riceati Equation Generalized Filter Algebraic Riccati Equation Loop Transfer Recovery Linear Quadratic Regulator Lineaz Quadratic Gaussian Loop Shaping Design Procedure Balanced Truncation Approximation Hankel Norm Approximation Sensitivity and Complementary Sensitivity