nonlinearnotes

nonlinearnotes - Undergraduate Lecture Notes on Nonlinear...

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Unformatted text preview: Undergraduate Lecture Notes on Nonlinear Control Jo ao P. Hespanha April 1, 2007 Disclaimer: This is a draft and probably contains several typos. Comments and information about typos are welcome. Please contact the author at ( hespanha@ ece.ucsb.edu ). c Copyright to Jo ao Hespanha. Please do not distribute this document without the authors consent. Contents 1 Feedback linearization controllers 3 1.1 Feedback linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Generalized model for mechanical systems . . . . . . . . . . . . . . . . . . . . 4 1.3 Feedback linearization of mechanical systems . . . . . . . . . . . . . . . . . . 7 1.4 To probe further . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Lyapunov stability 9 2.1 Lyapunov stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Lyapunov Stability Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 LaSalles Invariance Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.5 Li enard equation and generalizations . . . . . . . . . . . . . . . . . . . . . . . 14 2.6 To probe further . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3 Lyapunov-based designs 19 3.1 Lyapunov-based controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.3 Application to mechanical systems . . . . . . . . . . . . . . . . . . . . . . . . 22 3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Attention! When a margin sidebar finishes with . . . , more information about the topic can be found at the end of the lecture in the To probe further section. i ii Jo ao P. Hespanha Introduction to nonlinear control In this section we consider the control of nonlinear systems such as the one shown in Figure 1. Our goal is to construct a state-feedback control law of the form Sidebar 1. This control-law implicitly assumes that the whole state x can be measured. u = k ( x ) that results in adequate performance for the closed-loop system x = f ( x, k ( x ) ) . Typically, at least we want x to be bounded and converge to some desired reference value r . u ( t ) R m x ( t ) R n x = f ( x, u ) u ( t ) R m x ( t ) R n u = k ( x ) x = f ( x, u ) Figure 1. Nonlinear process with m inputs Figure 2. Closed-loop nonlinear system with state-feedback Pre-requisites 1. Basic knowledge of nonlinear ordinary differential equations....
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This note was uploaded on 04/06/2010 for the course ECE 145 taught by Professor Rodwell during the Spring '07 term at UCSB.

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nonlinearnotes - Undergraduate Lecture Notes on Nonlinear...

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