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Extensions of Input-Output Stability Theory and the Control of Aerospace Systems

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Extensions of Input-Output Stability Theory and the Control of Aerospace Systems Extensions of Input-Output Stability Theory and the Control of Aerospace Systems by James Richard Forbes A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Aerospace Science and Engineering University of Toronto Copyright c 2011 by James Richard Forbes Abstract Extensions of Input-Output Stability Theory and the Control of Aerospace Systems James Richard Forbes Doctor of Philosophy Graduate Department of Aerospace Science and Engineering University of Toronto 2011 This thesis is concerned with input-output stability theory. Within this framework, it is of interest how inputs map to outputs through an operator that represents a system to be controlled or the controller itself. The Small Gain, Passivity, and Conic Sector Stability Theorems can be used to assess the stability of a negative feedback interconnection involving two systems that each have specific input-output properties. Our first contribution concerns characterization of the input-output properties of linear time- varying (LTV) systems. We present various theorems that ensure that a LTV system has finite gain, is passive, or is conic. We also consider the stability of various negative feedback intercon- nections. Motivated by the robust nature of passivity-based control, we consider how to overcome passivity violations. This investigation leads to the hybrid conic systems framework whereby systems are described in terms of multiple conic bounds over different operating ranges. A special case relevant to systems that experience a passivity violation is the hybrid passive/finite gain framework. Sufficient conditions are derived that ensure the negative feedback interconnection of two hybrid conic systems is stable. The input-output properties of gain-scheduled systems are also investigated. We show that a gain-scheduled system composed of conic subsystems has conic bounds as well. Using the conic bounds of the subsystems along with the scheduling signal properties, the overall conic bounds of the gain-scheduled system can be calculated. We also show that when hybrid very strictly passive/finite gain (VSP/finite gain) subsystems are gain-scheduled, the overall map is also hybrid VSP/finite gain. Being concerned with the control of aerospace systems, we use the theory developed in this thesis to control two interesting plants. We consider passivity-based control of a spacecraft endowed with magnetic torque rods and reaction wheels. In particular, we synthesize a LTV input strictly passive controller. Using hybrid theory we control single- and two-link flexible manipulators. We present two controller synthesis schemes, each of which employs numerical optimization techniques and attempts to have the hybrid VSP/finite gain controllers mimic a controller. One of our synthesis methods uses the Generalized Kalman-Yakubovich-Popov H2 Lemma, thus realizing a convex optimization problem. ii Acknowledgements I would like to thank my Mom and Dad for their never ending love, encouragement, and support. My Dad taught me to be curious, while my Mom taught me how to work hard. I value both these lessons immensely. Thanks, Mom and Dad. I would like to thank my Supervisor, Prof. Christopher Damaren, for his guidance, support, and friendship. Prof. Damaren has, without a doubt, influenced my academic growth more than any other individual. Thanks for everything, Chris. I couldn’t have done it without you. During my doctoral work, Prof. Timothy Barfoot and Prof. Hugh Liu served on my Doctoral Examination Committee, for which I am very grateful. Finally, I would like to thank my friends at the University of Toronto Institute for Aerospace Studies (UTIAS). UTIAS was a fun, interesting, and exciting place because of all of you. Thanks. James Richard Forbes April 2011 iii Contents Abstract ii Acknowledgements iii List of Figures viii List of Tables x Acronyms xi 1 Introduction 1 1.1 SomeHistory ..................................... 1 1.1.1 ControlanditsChallanges . .... 1 1.1.2 Lyapunov Stability and Input-Output Stability: A Historical Account . 2 1.2 Thesis Motivation, Objectives, and Outline . ............ 4 1.2.1 MotivationandObjectives. ..... 4 1.2.2 SummaryofObjectives . 6 1.2.3 Outline ...................................... 6 2 Mathematical Preliminaries 10 2.1 Function Spaces, Norms, and Inner Products . .......... 10 2.1.1 The L2 Space and Extended L2 Space ..................... 10 2.1.2 The L Space .................................. 11 ∞ 2.1.3 Useful Inequalities and Equalities Related to Norms . ............ 12 2.2 OptimizationProblems. ...... 12 2.2.1 General Nonlinear Optimization Problems . ......... 12 2.2.2 Convex Optimization and Linear Matrix Inequalities . ............ 13 2.3 Summary ......................................... 16 3 Classic Input-Output Stability Theory 17 3.1 System Operators and Input-Output Stability . ............ 17 3.1.1 Operators,Causality,andAdjoints . ........ 17 3.1.2 L2 Stability.................................... 18 3.2 State-SpaceRealizations . ........ 18 3.2.1 NonlinearSystems .............................. 19 3.2.2 LinearTime-VaryingSystems . ..... 19 3.2.3 LinearTime-InvariantSystems . ...... 19 3.3 FiniteGainSystems ............................... .... 20 3.3.1 Linear Time-Invariant Finite Gain Systems . ......... 20 3.4 PassiveSystems.................................. .... 22 iv 3.4.1 Passive Linear Time-Invariant Systems. ......... 22 3.5 ConicSystems .................................... 25 3.5.1 LinearTime-InvariantConicSystems . ....... 27 3.5.2 Conic Representation of Finite Gain and Passive Systems .......... 28 3.5.3 VariableConicSectors. .... 31 3.6 DissipativeSystems .............................. ..... 31 3.6.1 Finite Gain Systems in Terms of the Dissipation Inequality ......... 32 3.6.2 Passive Systems in Terms of the Dissipation Inequality............ 33 3.6.3 Conic Systems in Terms of the Dissipation Inequality . ........... 33 3.7 Stability of Interconnected Dissipative Systems . ............... 34 3.7.1 TheSmallGainTheorem . 35 3.7.2 ThePassivityTheorem . 35 3.7.3 TheConicSectorTheorem . 36 3.7.4 A Geometric Interpretation of the Conic Sector Theorem .......... 38 3.8 Summary ......................................... 42 I Extensions of Input-Output Stability Theory 43 4 State-Space Realizations of Linear Time-Varying Systems 44 4.1 LinearTime-VaryingSystems . ....... 44 4.2 State-Space Realization of Linear Time-Varying Finite GainSystems . 45 4.2.1 Boundary Condition and Solution of P( ) ................... 46 · 4.3 State-Space Realizations of Passive Linear Time-VaryingSystems. 47 4.3.1 SystemsWithaFeedthroughMatrix . ..... 48 4.3.2 SystemsWithoutaFeedthroughMatrix . ...... 51 4.4 LinearTime-VaryingConicSystems . ........ 52 4.5 Closed-Loop Stability of Passive Linear Time-Varying Systems ........... 54 4.6 Control of a Nonlinear, Time-Varying System . ........... 57 4.6.1 The System Differential Equation and its Output Strictly Passive Nature . 57 4.6.2 Controller Design Inspired by Optimal Control . .......... 58 4.6.3 MemorylessNonlinearControl . ..... 60 4.7 Summary ......................................... 61 5 Hybrid Input-Output Systems 62 5.1 AMotivatingExample.............................. .... 62 5.2 HybridDissipativeSystems . ....... 66 5.3 The Causal, Bounded, Linear Time-Invariant Operators Ai ............. 67 5.4 HybridConicSystems .............................. .... 69 5.4.1 Linear Time-Invariant Hybrid Conic Systems . ......... 70 5.4.2 Variable Conic Sectors and Their Relation to Hybrid ConicSystems . 73 5.4.3 Approximating Variable Cones with Hybrid Cones . ......... 74 5.5 HybridPassiveandFiniteGainSystems . ......... 76 5.5.1 Approximating α and β.............................. 78 5.5.2 Linear Time-Invariant Hybrid Passive/Finite Gain Systems ......... 81 5.6 Closed-Loop Stability of Hybrid Input-Output Systems . .............. 84 5.6.1 The Structure and Positive Definite Nature of Q ............... 88 5.7 Summary ......................................... 92 v 6 Scheduling of Conic and Hybrid VSP/Finite Gain Systems 93 6.1 SchedulingofPassiveSystems. ........ 95 6.2 SchedulingofConicSystems . ...... 98 6.3 Scheduling of Hybrid Passive/Finite Gain Systems . ............. 103 6.4 Summary ......................................... 107 II Applications to Aerospace Systems 108 7 Linear Time-Varying Passivity-Based Attitude Control Using Magnetic and Mechanical Actuation 109 7.1 Review of Passive, Input Strictly Passive, and Linear Time-Varying Systems . 110 7.2 Spacecraft Kinematics, Dynamics, Disturbances, and Actuation. 111 7.3 ControlFormulation .............................. ..... 112 7.3.1 Passivity Properties of a Spacecraft Compensated by Proportional Control 113 7.3.2 InputStrictlyPassiveRateControl. ........ 113 7.3.3 Distribution of Control Between Wheel Torques and Magnetic Torques . 114 7.4 ControllerDesignandSynthesis. ......... 114 7.5 NumericalExample................................ .... 116 7.6 Practical Considerations and Additional Comments . .............. 121 7.7 Summary ......................................... 122 8 Single-Link Flexible Manipulator Control Accommodating Passivity Viola- tions: Theory and Experiments 123 8.1 Single-Link Flexible Manipulator Dynamics . ............ 124 8.1.1 Input-OutputModel .. .. .. .. .. .. .. 124 8.1.2 ViolationofPassivity . .... 125 8.2 Review of Hybrid Passive/Finite Gain Systems Theory in a Linear Time-Invariant, Single-InputSingle-OutputContext . ....... 126 8.3 ControllerDesignProblem. ......
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