Module 4 p2b - ECE 514 Nonlinear & Adaptive Control...

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1 ECE 514 Nonlinear & Adaptive Control Module 4-Part 2 LYAPUNOV STABILITY
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2 Topics 1. Invariance Theorems (LaSalle) 2. Lyapunov 1 st or Indirect Method 3. Lyapunov Function Candidates 4. Domain of Attraction Material covered in Chapter 4 of Khalil.
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3 The Need for Invariance Theorems What if for a Lyapunov function candidate , the derivative is only negative semi- definite? Asymptotic stability may still be guaranteed if one can show that the Lyapunov function decreases everywhere except at the origin. More generally, we can show that the trajectory is eventually confined to an INVARIANT SET. () Vx i
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4 Limit Sets . as and ) ( y with analogousl defined is set limit Negative ). ( of , set, limite (positive) the called is ) ( of points limit all of set The . as ) ( and as such that in ) ( sequence a if ) ( of point limit (positive) a is point A . ), ( of y) (trajector solution a be ) ( Let 1 1 2 2 −∞ = = + = n t t t x L t x n z t x n t R t t x R z R x x f x t x n n n n n n n ±
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5 Positive & Negative Limit Sets 1 x 2 x positive limit set 1 x 2 x )) 0 ( , ( x t x )) 0 ( , ( x t x positive limit set 1 x 2 x Negative limit set exists. Positive limit set does not.
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6 Invariant Sets A set is said to be an invariant set of if A set is said to be a positively invariant set if M () x f x = i (0) ( ) , xM x t M t ⇒∈ \ ( ) , 0 x t M t M
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7 Approaching a Set Asymptotically We say that approaches a set as if for each there is a such that () xt M t →∞ 0 ε> 0 T > (( ) , ) , ( , ) inf || || xM dist x t M t T where dist p M p x ε < ∀> =
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8 A Trajectory Approaching M 1 ε () xt (0) x 1 T 2 T 2
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9
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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 p2b - ECE 514 Nonlinear &amp; Adaptive Control...

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