Module 8 p4 - ECE147C Lecture Notes on Nonlinear control...

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Unformatted text preview: ECE147C Lecture Notes on Nonlinear control Jo˜ ao P. Hespanha May 21, 2004 1 1 Revisions from May 1, 2004 version: Fixed typo in equations (3.20), Exercises 7 and 8, and several equations in Section 4.1. Contents 1 Introduction 2 2 Feedback linearization 3 2.1 Application to mechanical systems . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Lyapunov stability 8 3.1 Lyapunov Stability Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 LaSalle’s Invariance Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 Li´ enard equation and generalizations . . . . . . . . . . . . . . . . . . . . . . . 13 4 Lyapunov-based designs 15 4.1 Application to mechanical systems . . . . . . . . . . . . . . . . . . . . . . . . 17 5 To probe further 19 5.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.2 Sidebars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.3 MATLAB hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1 Chapter 1 Introduction In this section we consider the control of nonlinear systems such as the one shown in Fig- ure 1.1. 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 mea- sured. 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 1.1: Nonlinear process with m in- puts Figure 1.2: Closed-loop nonlinear system with state-feedback Pre-requisites 1. Basic knowledge of Laplace z-transform, discrete-time transfer functions, and im- pulse/step response. 2. Basic knowledge of Laplace transform, continuous-time transfer functions, and im- pulse/step response. 3. Familiarity with basic vector and matrix operations. 4. Knowledge of MATLAB/Simulink. 2 Chapter 2 Feedback linearization In feedback linearization control design, we decompose the control signal u into two com- ponents with distinct functions: u = u nl + v, where 1. u nl is used to “cancel” the process’ nonlinearities, and 2. v is used to control the resulting linear system. u y F drag g From Newton’s law: m ¨ y = F drag- mg + u =- 1 2 cρA ˙ y | ˙ y | - mg + u, where m is the vehicle’s mass, g grav- ity’s acceleration, F drag =- 1 2 cρA ˙ y | ˙ y | the drag force, and u an applied force. Figure 2.1: Dynamics of a vehicle moving vertically in the atmosphere To understand how this is done, consider the vehicle shown in Figure 2.1 moving vertically in the atmosphere. By choosing Sidebar 2: For large objects moving through air, the air resistance is approximately proportional to the square of the velocity, with a drag force given by F drag =- 1 2 cρA ˙ y | ˙ y | where ρ is the air density,...
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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 8 p4 - ECE147C Lecture Notes on Nonlinear control...

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