Module 4 p2a - ECE 514 Nonlinear Adaptive Control Module...

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1 ECE 514 Nonlinear & Adaptive Control Module 4-Part 2 LYAPUNOV STABILITY
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2 Topics 1. Lyapunov Function Candidates 2. Domain of Attraction Material covered in Chapter 4 of Khalil.
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3 Lyapunov Function Candidates Quadratic form: Works for linear system. Quadratic form plus integral of nonlinearity: Px x V P Q Q Q PA P A T T T = > = = + 0 , () ( ) ) ( ) ( ) ( ) ( ) ( 2 0 , 0 ) ( ) ( ) ( 2 2 2 1 1 2 1 1 2 2 2 1 0 1 1 2 2 1 1 2 2 2 2 1 1 x g x x g x x g x g x V d g x V x x g x x g x g x x x x i i i i = + = + = > = = ± ± ± σ
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4 Krasovkii’s Method x f x A R x x f f R R f x f x n n n = = = ) ( able differenti ly continuous is ) ( 0 ) 0 ( : ) ( ± stable. asym. globally is it then unbounded, radially is ) ( ) ( and n.d ) ( ˆ If l.n.d is ) ( ) ( ) ( ˆ if stable asym. is ) ( of 0 point eq. Then the point eq. isolated for l.p.d ) ( ) ( ) ( Choose x f x f x A x A x A x A x f x x f x f x V T T T + = = = ± Idea :
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5 Proof: [] analogous. is proof the of rest the Thus l.n.d 0 ) ( ) ( if 0 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( < + < + = + = + = + = = x A x A f x A x A f f x A f x f f x A t x x f x f x f t x x f x f x f x f x f x V x f x f x V T T T T T T T T T T ± ± ±
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6 Example 9: () 11 3 2122 22 2 2 2 23 2 2 0 is an eq. point 10 1 0 1 1 ˆ 3 1( 1 3 ) 0( 1 3 ) 21 ˆ 0 asym. stable 12 6 and as T xx x x Ax A x A x ff x x x x x =− =−− −− ⎡⎤ == + ⎢⎥ −+ ⎣⎦ −= > =+ ± ± the equilibrium is globally asymtotically stable.
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7 Variable Gradient Method j i j i i j j i i j i j j i T T x x V x V x x g x x V x x V x V x x g x g x V x g x f x g x V x g x g x V x f x x f x V V x V x V x V = = = = = < = = = = > 2 2 2 by d constructe is ) ( that Note . 0 ) ( that so ) ( Choose ) ( ) ( ) ( ctor. column ve a is ) ( where ) ( ) ( ) ( 0 ) ( that so parameters free choose and ) ( determine free, parameters some leave ), (
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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 p2a - ECE 514 Nonlinear Adaptive Control Module...

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