Module 4 p1 - CE 514 ECE 514 Nonlinear & Adaptive...

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Unformatted text preview: CE 514 ECE 514 Nonlinear & Adaptive Control Module 4-Part 1 LYAPUNOV STABILITY 1 Topics 1. Definitions of Lyapunov Stability yapunov 2 nd ethod Theorems 2. Lyapunov 2 Method Theorems 3. Instability Theorems Material covered in Chapter 4 of Khalil. 2 Translating Equilibrium points In this Module we assume that the origin is the equilibrium point. If the equilibrium is then one can translate the dynamics to the origin as follows: e x ≠ ( ); ( ) 0; ) e x f x f x x x z f z x = = − ⇒ = + i i So that the origin is an equilibrium point in the z coordinates. ( ) e e z x x z f z x ⇒ + 3 Why Lyapunov Stability? n Assume that is such that the solution exists for all and is an equilibrium point ( ) : n n x f x f R R = → ¡ t < ∞ ) x f ⇒ 0 is an equilibrium point. What happens if we start close to 0? ” (0) f = ⇒ Continuous dependence on the initial conditions 0, ε δ ∀ > ∃ 1 || ( , , ) 0 || , [ , ] if || 0 || x x t t t t t x ε δ − < ∈ − < whenever is bounded. But what happens on infinite time intervals? || || 1 t 4 Lyapunov Stability Definitions is ) ( of point m equilibriu The x f x = ¡ δ < ≥ ∀ ε < > ε δ = δ ∃ > ε ∀ stable not if unstable- ) ( if , ) ( such that ) ( , if stable- x t t x ∞ = δ δ < = δ ∃ ∞ → 1 1 1 if and stable ally asymptotic is it if stable ally asymptotic globally - ) ( if ) ( lim such that and stable is it if stable ally asymptotic- x t x t 1 x 2 x x 2 x 1 x 2 x 1 asymptotically able stable unstable 5 stable Stability : continuous dependency on initial conditions at infinity. Lyapunov Stability in Pictures 2 x ) t (0) x 2 x ( ) x t (0) x 1 x ( ) x t R e g i o n o f A t t r a c t i o n 1 x 2 x ( ) x t (0) x Lyapunov Stability Asymptotic Stability 1 x 6 Global Asymptotic Stability Lyapunov Stability Concepts Stable Unstable Asymptotically Stable 7 Methods of Nonlinear Analysis 1. Qualitative method : finding properties of solution without actually finding the solution. 2. Quantitative method : concerned with explicitly finding Q p y g closed forms of approximate or exact solution. 3. Computer (numerical) method : centered around e eloping n merical techniq e for sol tion on comp ter developing numerical technique for solution on computer. Lyapunov stability ⇒ most comprehensive idea efinition tuitive- 10 10 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − 10 10 10 for stable not is it 10 for stable is it definition intuitive 8 10 10 Stability Types 1. Lyapunov stability 2. BIBO Stability 3. Input-State Stability 4. Input-output stability 5. Orbital stability 6. Stability under persistent perturbation 7. Lagrange stability 8. Finite-Time Stability 9 Positive Definite Functions n dissipatio energy - Idea Basic ) , , , ( : 2 1 n x x x V R D V " → Definition: is locally positive definite if , {0} such that ( ) 0 {0}, (0) 0....
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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 4 p1 - CE 514 ECE 514 Nonlinear & Adaptive...

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