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Unformatted text preview: Nonlinear Systems and Control Lecture # 8 Lyapunov Stability p.1/11 Let V ( x ) be a continuously differentiable function defined in a domain D R n ; D . The derivative of V along the trajectories of x = f ( x ) is V ( x ) = n summationdisplay i =1 V x i x i = n summationdisplay i =1 V x i f i ( x ) = bracketleftBig V x 1 , V x 2 , . .. , V x n bracketrightBig f 1 ( x ) f 2 ( x ) . . . f n ( x ) = V x f ( x ) =: L f V ( x ) It is the Lie Derivative of V with respect to f or along f p.2/11 If ( t ; x ) is the solution of x = f ( x ) that starts at initial state x at time t = 0 , then V ( x ) = d dt V ( ( t ; x )) vextendsingle vextendsingle vextendsingle vextendsingle t =0 If V ( x ) is negative, V will decrease along the solution of x = f ( x ) If V ( x ) is positive, V will increase along the solution of x = f ( x ) p.3/11 Lyapunovs Theorem: If there is V ( x ) such that V (0) = 0 and V ( x ) > , x D \ { } V ( x ) ,...
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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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