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  • Robustness Analysis of Simultaneous Stabilization and its Applications in Flight Control

    by

    Yasaman Saeedi

    A thesis submitted in conformity with the requirements for the degree of Master of Applied Science

    Graduate Department of Aerospace Science and Engineering University of Toronto

    Copyright c© 2011 by Yasaman Saeedi

  • Abstract

    Robustness Analysis of Simultaneous Stabilization and its Applications in Flight

    Control

    Yasaman Saeedi

    Master of Applied Science

    Graduate Department of Aerospace Science and Engineering

    University of Toronto

    2011

    Simultaneous stabilization is an important problem in the design of robust controllers.

    It is the problem of designing a single feedback controller which will simultaneously

    stabilize every member of a finite collection of liner time-invariant systems. This provides

    simplicity and reliability which is desirable in aerospace applications. It can be used as

    a back up control system in sophisticated airplanes, or an inexpensive primary one for

    small aircraft. In this work the robustness of the simultaneous stabilization problem,

    known as the Robust Simultaneous Stabilization (RSS) problem, is addressed. First, an

    optimization methodology for finding a solution to the Simultaneous Stabilization (SS)

    problem is proposed. Next, in order to provide simultaneous stability while maximizing

    the stability robustness bounds, a multiple-robustness optimization design methodology

    for the RSS problem is presented. The two proposed design methodologies are then

    compared in terms of robustness of the designed controller.

    ii

  • Acknowledgements

    I would like to express my extreme gratitude to my supervisor, Dr. Hugh H.T. Liu,

    for providing me with the opportunity to pursue my interests in this research field. His

    insight and perspective on the topic was of great value to me and none of this would

    be possible without his extreme support and guidance throughout the past few years. I

    would also like to thank him for his kind understanding and support while I was going

    through a difficult time in the past few months.

    I would also like to thank my co-supervisor, Dr. Ruben Perez, for his much valued

    support and insight. His knowledge on the topic of simultaneous stabilization was of

    great value to me, he always made time to answer my questions and provide guidance

    and suggestions, and I definitely owe much of what I know to him.

    I would like to thank all my friends in the FSC lab, all the past and present members.

    They made my experience at UTIAS enthusiating and inspiring and their appreciated

    friendship and support is what I will take away from this.

    Last, but not least, I would like to thank my family for their never ending love and

    support, and for always being there by my side. I also truly want to thank my circle of

    friends for their spiritual support and friendship whenever I needed it.

    iii

  • Contents

    1 Introduction 1

    1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

    1.2 Motivation & Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . 4

    1.3 Thesis Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

    2 Simultaneous Stabilization 6

    2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

    2.2 Necessary and Sufficient Condition . . . . . . . . . . . . . . . . . . . . . 7

    2.3 Simultaneous Stabilization by Linear State Feedback Control . . . . . . . 8

    2.4 Optimal Stabilization via Linear State Feedback Control . . . . . . . . . 12

    2.5 Bi-Level Decomposition-Based Strategy . . . . . . . . . . . . . . . . . . . 15

    2.5.1 Decomposition formulation . . . . . . . . . . . . . . . . . . . . . . 15

    2.5.2 Decomposed equivalent of necessary and sufficient conditions . . . 17

    2.6 Parameter Optimization Approach . . . . . . . . . . . . . . . . . . . . . 17

    2.6.1 Cost Function Definitions . . . . . . . . . . . . . . . . . . . . . . 18

    2.6.2 Multiple Objective Design . . . . . . . . . . . . . . . . . . . . . . 19

    2.7 Proposed SS Optimization Methodology . . . . . . . . . . . . . . . . . . 20

    3 Robustness Analysis 22

    3.1 Kharitonov’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

    3.2 Extreme Point Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

    3.2.1 Stability as a Nonsingularity Problem via the ‘Kronecker Lyapunov

    Matrix’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

    iv

  • 3.2.2 Necessary and Sufficient Vertex Solution for Robust Stability . . . 26

    3.3 Stability Robustness Bounds . . . . . . . . . . . . . . . . . . . . . . . . . 26

    3.4 Robustness Analysis of a Numerical Example . . . . . . . . . . . . . . . 29

    3.4.1 Extreme Point Solution Application . . . . . . . . . . . . . . . . . 30

    3.4.2 Kharitonov’s Theorem Application . . . . . . . . . . . . . . . . . 34

    3.4.3 Stability Robustness Bound Application . . . . . . . . . . . . . . 35

    4 Robust Simultaneous Stabilization Problem 38

    4.1 An Extended Decomposition-Based Strategy for the RSS Problem . . . . 39

    4.2 Multi-Objective Optimization . . . . . . . . . . . . . . . . . . . . . . . . 42

    4.2.1 Formulation & the Concept of Pareto Optimality . . . . . . . . . 42

    4.2.2 Weighted-Sum Method . . . . . . . . . . . . . . . . . . . . . . . . 43

    4.2.3 Multiple Robustness Optimization . . . . . . . . . . . . . . . . . . 46

    5 Linear Simulation: An F4-C Flight Control Case Study 49

    5.1 Introduction of the Test Case . . . . . . . . . . . . . . . . . . . . . . . . 49

    5.2 Robustness Investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

    5.2.1 Perturbations Due to CLα uncertainties . . . . . . . . . . . . . . . 52

    5.2.2 Robustness Optimization . . . . . . . . . . . . . . . . . . . . . . . 59

    6 Non-Linear Simulation: A CRJ-200 Flight Control Case Study 67

    6.1 Modelling of the CRJ-200 . . . . . . . . . . . . . . . . . . . . . . . . . . 68

    6.1.1 Nonlinear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

    6.1.2 Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

    6.2 Introduction of the Test Case . . . . . . . . . . . . . . . . . . . . . . . . 72

    6.3 Results: Ordinary and Gust-Encountered Flight . . . . . . . . . . . . . . 75

    6.4 Robustness Investigation & Optimization . . . . . . . . . . . . . . . . . . 77

    7 Conclusion and Future Developments 86

    7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

    7.2 Future Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

    v

  • Bibliography 90

    vi

  • List of Tables

    3.1 F4-E flight operating conditions . . . . . . . . . . . . . . . . . . . . . . . 30

    3.2 Allowed perturbation resulting in a stable matrix family . . . . . . . . . 33

    3.3 Allowed perturbation resulting in an unstable matrix family . . . . . . . 33

    3.4 Kharitonov’s polynomials coefficients . . . . . . . . . . . . . . . . . . . . 35

    3.5 Stability robustness bounds . . . . . . . . . . . . . . . . . . . . . . . . . 37

    5.1 F4-C flight operating conditions . . . . . . . . . . . . . . . . . . . . . . . 50

    5.2 Simultaneous Stabilization solution . . . . . . . . . . . . . . . . . . . . . 51

    5.3 Closed-loop system eigenvalues . . . . . . . . . . . . . . . . . . . . . . . 51

    5.4 F4-C characteristics at different flight conditions . . . . . . . . . . . . . . 55

    5.5 Maximum allowable deviation in ∆CLα . . . . . . . . . . . . . . . . . . . 56

    5.6 Kharitonov’s polynomials coefficients . . . . . . . . . . . . . . . . . . . . 57

    5.7 Solutions to the SS and RSS problems . . . . . . . . . . . . . . . . . . . 60

    5.8 Effect of relaxing the robustness on |∆CLα|max . . . . . . . . . . . . . . . 61

    5.9 Results from different RSS optimization methodologies and objective func-

    tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

    5.10 A comparison of the closed-loop eigenvalues for different optimization

    methodologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

    6.1 CRJ-200 flight operating conditions . . . . . . . . . . . . . . . . . . . . . 74

    6.2 Open-loop system eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . 75

    6.3 Closed-loop system eigenvalues . . . . . . . . . . . . . . . . . . . . . . . 76

    6.4 Maximum allowable deviation in ai,j . . . . . . . . . . . . . . . . . . . . 81

    6.5 Effect of relaxing the robustness on |∆ai,j|max . . . . . . . . . . . . . . . 83

    vii

  • 6.6 A comparison of the closed loop eigenvalues for different optimization

    methodologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

    viii

  • List of Figures

    2.1 Multidisciplinary optimization and simultaneous stabilization problem [27] 16

    3.1 response to initial condition of the perturbed nominal plant . . . . . . . . 34

    3.2 response to initial condition of the perturbed nominal plant . . . . . . . . 36

    4.1 Robust S