Lect_30 - Nonlinear Systems and Control Lecture # 30...

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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 = ( x ) such that the origin of x = f ( x ) + g ( x ) ( x ) is asymptotically stable By the converse Lyapunov theorem, there is V ( x ) such that V x [ f ( x ) + g ( x ) ( x )] < , x D, x negationslash = 0 If u = ( x ) is globally stabilizing, then D = R n and V ( x ) is radially unbounded p. 2/1 2 V x [ f ( x ) + g ( x ) ( x )] < , x D, x negationslash = 0 V x g ( x ) = 0 for x D, x negationslash = 0 V x f ( x ) < Since ( x ) is continuous and (0) = 0 , given any > , > such that if x negationslash = 0 and bardbl x bardbl < , there is u with bardbl u bardbl < 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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Lect_30 - Nonlinear Systems and Control Lecture # 30...

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