Module 10 p1 - ECE 517: Nonlinear and Adaptive Control Fall...

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ECE 517: Nonlinear and Adaptive Control Fall ’07 Lecture Notes Daniel Liberzon November 14, 2007
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Disclaimer These lecture notes are posted for class use only. This is a very rough draft which contains many errors. I don’t always give proper references to sources from which results are taken. A lack of reference does not mean that the result is original. In fact, all results presented in these notes (with possible exception of some simple examples) were borrowed from the literature and are not mine. 2
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Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1 Motivating example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 1.2 Course logistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Weak Lyapunov functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1 LaSalle and Barbalat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Connection with observability . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Back to the adaptive control example . . . . . . . . . . . . . . . . . . . . . . 15 3 Lyapunov-based design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.1 Control Lyapunov functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Back to the adaptive control example . . . . . . . . . . . . . . . . . . . . . . 24 4 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.1 Gradient method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.2 Parameter estimation: stable case . . . . . . . . . . . . . . . . . . . . . . . . 29 4.3 General case: adaptive laws with normalization . . . . . . . . . . . . . . . . . 36 4.3.1 Linear plant parameterizations (parametric models) . . . . . . . . . 39 4.3.2 Gradient method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.3.3 Least squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.3.4 Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.4 SuFciently rich signals and parameter identi±cation . . . . . . . . . . . . . . 48 4.5 Case study: model reference adaptive control . . . . . . . . . . . . . . . . . . 53 4.5.1 Direct MRAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.5.2 Indirect MRAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5 Stability of slowly time-varying systems . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3
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5.2 Application to adaptive stabilization . . . . . . . . . . . . . . . . . . . . . . . 65 5.3 Detectability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6 Minimum-phase systems and universal regulators . . . . . . . . . . . . . . . . . . . . 71 6.1 Universal regulators for scalar plants . . . . . . . . . . . . . . . . . . . . . . . 73 6.1.1 The case b> 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 6.1.2 General case: non-existence results . . . . . . . . . . . . . . . . . . . 74 6.1.3 Nussbaum gains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 6.2 Relative degree and minimum phase . . . . . . . . . . . . . . . . . . . . . . . 78 6.3 Universal regulators for higher-dimensional plants . . . . . . . . . . . . . . . 82 7 Backstepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7.1 Integrator backstepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 7.2 Adaptive integrator backstepping . . . . . . . . . . . . . . . . . . . . . . . . . 89 8 Input-to-state stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 8.1 Weakness of certainty equivalence . . . . . . . . . . . . . . . . . . . . . . . . 92 8.2 Input-to-state stability and stabilization . . . . . . . . . . . . . . . . . . . . . 94 8.2.1 ISS backstepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 8.3 Adaptive ISS controller design . . . . . . . . . . . . . . . . . . . . . . . . . . 103 8.3.1 Adaptive ISS backstepping . . . . . . . . . . . . . . . . . . . . . . . 104 8.3.2 Modular design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 9 Switching adaptive control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 9.1 The supervisor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 9.1.1 Multi-estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 9.1.2 Monitoring signal generator . . . . . . . . . . . . . . . . . . . . . . . 112 9.1.3 Switching logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 9.2 Example: linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 9.3 Modular design objectives and analysis steps . . . . . . . . . . . . . . . . . . 118 9.3.1 Achieving detectability . . . . . . . . . . . . . . . . . . . . . . . . . 120 9.3.2 Achieving bounded error gain and non-destabilization . . . . . . . . 123 10 Singular perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 10.1 Unmodeled dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 10.2 Singular perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 10.3 Direct MRAC with unmodeled dynamics . . . . . . . . . . . . . . . . . . . . 130 11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 4
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Here is the interdependency chart among the chapters: 1 2 3 45 6 7 8 9 10 5
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Lecture 1 1 Introduction The meaning of “nonlinear” should be clear, even if you only studied linear systems so far (by exclusion). The meaning of “adaptive” is less clear and takes longer to explain. Sometimes one gives the circular deFnition that a control law is adaptive if it involves adaptation. .. Perhaps it’s easier to Frst explain the class of problems it studies: modeling uncertainty. This includes (but is not limited to) the presence of unknown parameters in the model of the plant.
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This note was uploaded on 05/06/2010 for the course ECE 514 taught by Professor Chaoukit.abdallah during the Spring '09 term at University of New Brunswick.

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Module 10 p1 - ECE 517: Nonlinear and Adaptive Control Fall...

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