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Unformatted text preview: Nonlinear Systems and Control Lecture # 12 Converse Lyapunov Functions & Time Varying Systems – p.1/20 Converse Lyapunov Theorem–Exponential Stability Let x = 0 be an exponentially stable equilibrium point for the system ˙ x = f ( x ) , where f is continuously differentiable on D = {bardbl x bardbl < r } . Let k , λ , and r be positive constants with r < r/k such that bardbl x ( t ) bardbl ≤ k bardbl x (0) bardbl e − λt , ∀ x (0) ∈ D , ∀ t ≥ where D = {bardbl x bardbl < r } . Then, there is a continuously differentiable function V ( x ) that satisfies the inequalities – p.2/20 c 1 bardbl x bardbl 2 ≤ V ( x ) ≤ c 2 bardbl x bardbl 2 ∂V ∂x f ( x ) ≤ − c 3 bardbl x bardbl 2 vextenddouble vextenddouble vextenddouble vextenddouble ∂V ∂x vextenddouble vextenddouble vextenddouble vextenddouble ≤ c 4 bardbl x bardbl for all x ∈ D , with positive constants c 1 , c 2 , c 3 , and c 4 Moreover, if f is continuously differentiable for all x , globally Lipschitz, and the origin is globally exponentially stable, then V ( x ) is defined and satisfies the aforementioned inequalities for all x ∈ R n – p.3/20 Idea of the proof: Let ψ ( t ; x ) be the solution of ˙ y = f ( y ) , y (0) = x Take V ( x ) = integraldisplay δ ψ T ( t ; x ) ψ ( t ; x ) dt, δ > – p.4/20 Application: Consider the system ˙ x = f ( x ) where f is continuously differentiable in the neighborhood of the origin and f (0) = 0 . Show that the origin is exponentially stable only if A = [ ∂f/∂x ](0) is Hurwitz f ( x ) = Ax + G ( x ) x, G ( x ) → as x → Given any L > , there is r 1 > such that bardbl G ( x ) bardbl ≤ L, ∀ bardbl x bardbl < r 1 Because the origin of ˙ x = f ( x ) is exponentially stable, let V ( x ) be the function provided by the converse Lyapunov theorem over the domain {bardbl x bardbl < r } . Use V ( x ) as a Lyapunov function candidate for ˙ x = Ax – p.5/20 ∂V ∂x Ax = ∂V ∂x f ( x ) − ∂V ∂x G ( x ) x ≤ − c 3 bardbl x bardbl 2 + c 4 L bardbl x bardbl 2 = − ( c 3 − c 4 L...
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This note was uploaded on 07/25/2008 for the course ME 859 taught by Professor Choi during the Spring '08 term at Michigan State University.
 Spring '08
 CHOI

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