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Unformatted text preview: Nonlinear Systems and Control Lecture # 30 Stabilization Control Lyapunov Functions – p. 1/1 2 ˙ x = f ( x ) + g ( x ) u, f (0) = 0 , x ∈ R n , u ∈ R Suppose there is a continuous stabilizing state feedback control u = ψ ( x ) such that the origin of ˙ x = f ( x ) + g ( x ) ψ ( x ) is asymptotically stable By the converse Lyapunov theorem, there is V ( x ) such that ∂V ∂x [ f ( x ) + g ( x ) ψ ( x )] < , ∀ x ∈ D, x negationslash = 0 If u = ψ ( x ) is globally stabilizing, then D = R n and V ( x ) is radially unbounded – p. 2/1 2 ∂V ∂x [ f ( x ) + g ( x ) ψ ( x )] < , ∀ x ∈ D, x negationslash = 0 ∂V ∂x g ( x ) = 0 for x ∈ D, x negationslash = 0 ⇒ ∂V ∂x f ( x ) < Since ψ ( x ) is continuous and ψ (0) = 0 , given any ε > , ∃ δ > such that if x negationslash = 0 and bardbl x bardbl < δ , there is u with bardbl u bardbl < ε such that ∂V ∂x [ f ( x ) + g ( x ) u ] < Small Control Property – p. 3/1 2 Definition: A continuously differentiable positive definite function V ( x ) is a Control Lyapunov Function (CLF)...
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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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