Module 4 p3 - ECE 514 Nonlinear & Adaptive Control...

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ECE 514 onlinear & Adaptive Control Nonlinear & Adaptive Control Module 4-Part 3 LYAPUNOV STABILITY 1
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Topics 1. Defintions of Lyapunov Stability for Time-varying or non-autonomous systems. niform and exponential stability 2. Uniform and exponential stability 3. Comparison (class K) functions yapunov theorem for non- tonomous systems 4. Lyapunov theorem for non autonomous systems. 5. Linear time-varying systems. aSalle- ke theorems. 6. LaSalle like theorems. Material covered in Chapter 4 of Khalil. 2
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Beyond Lyapunov Stability Lyapunov stability theory for ( ) is generalized to xf x = ± (1) ( , ) ) ( ) ( ) t x f t x g t x = + ± ± (2) ( , )( , ) (3) Stability if , then stability versus If stable, then (Converse Theorem) V V =+ "" (4) Other types of stability? u y bounded bounded ? System 3
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tability of Time- arying Systems Stability of Time varying Systems . where : ) , ( n n R D R D R f x t f x × = + ± (1) f is piecewise continuous in t and Lipschitz in x . Why time-varying systems? (i) parameters change in time. (ii) investigation of stability of trajectories of time invariant system, e.g. solution a is ) ( where ) ( * t x x f x = ± )) ( ( ) ( ) ( ) ( * * * * * * t x z f t x z t x z x t x x z + = + + = = ± ± ± ± z z F z t A z z 0 ) ( ioin Linearizat = = = ± 4 ) , ( ) ( )) ( ( z t F t x t x z f z = + =
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Stability for Non-Autonomous Systems Definition : The origin is an equilibrium of (1) if ( ,0) 0, 0 f tt = ∀≥ 0 0 Definition: The eq. point 0 of (1) is stable if 0 and 0, ( , ) such that t t ε δε ∀> ∃> 00 0 0 0 (, , ) , w h e n ( ,) x xtt t t x t εδ <∀ < efinition: The eq point 0 of (1) s uniformly stable if 0 > Definition: The eq. point 0 of (1) is uniformly stable if ( ) such that 0 0 0 , ) , , ( t t t t x x < 0 when ( ) x < 5
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Stability for Non-autonomous Systems ε () δε 6
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Example 1 ) 2 sin 6 ( ) 2 sin 6 ( dt t t t x dx x t t t x = = ± 2 0 0 0 0 2 0 cos 6 sin 6 cos 6 sin 6 ln ln ) 2 sin 6 ( ln 0 0 t t t t t t t t x x dt t t t x t t x x + + = = Then ] cos 6 sin 6 cos 6 sin 6 [ 0 2 0 0 0 0 2 ) ( ) ( t t t t t t t t e t x t x + + = Hence ] cos 6 sin 6 cos 6 sin 6 [ 0 2 0 0 0 0 2 0 sup ) ( let t t t t t t t t t t e t c + + = Then 0 0 0 ), ( ) ( ) ( t t t c t x t x < stable. is origin that the shows ) ( choice the , 0 any For 0 t c ε δ = > 7 case. each in later seconds evaluated is ) ( suppose and , 2 , 1 , 0 , 2 values successive on takes Suppose 0 0 π t x n n t t " = =
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Example 1(cont.) hen Then ( ) p{ } ) 2 ( ) 2 cos( ) 2 ( 6 ) 2 sin( 6 } ) 1 2 ( ) 1 2 cos( ) 1 2 ( 6 ) 1 2 sin( 6 exp{ ) ( ) ( 2 2 2 2 0 0 π + + + + + + = + n n n n n n n n t x t x ( ) } 6 4 24 exp{ ) ( } 4 12 ) 1 4 4 ( ) 6 12 ( exp{ ) ( } ) 2 ( ) 2 ( 6 ) 1 2 ( ) 1 2 ( 6 exp{ ) ( 2 2 0 2 2 2 2 0 0 + = + + + + + = + + + + = n n t x n n n n n t x n n n n t x )} 6 )( 1 4 ( exp{ ) ( )} 6 4 24 ( exp{ ) ( 0 0 + = + = n t x n n t x r plies his + n t x t x t x as ) ( ) ( , 0 ) ( for implies, This 0 0 0 that f dependen o there iven hus 8 . in uniformly t requiremen he satisfy t would of t independen no is , 0 given Thus 0 0 t t δ ε>
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Asymptotic & Exponential Stability efinition: The eq point 0 of (1) s asymptotically stable if it is table and 10 00 0 Definition: The eq. point 0 of (1) is asymptotically stable if it is stable and ( ) such that ( , , ) 0 as if ( ) ( ) t xx t t t xt t δ →→ < 1 Definition: The eq. point 0 of (1) is uniformly asymptotically stable if it is uniformly stable and such that () 0 a s i f ( ) x t t t < 0 1 ( ,, )0 as if 1 Definition: The eq. point 0 of (1) is
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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 p3 - ECE 514 Nonlinear &amp; Adaptive Control...

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