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# Lect_8 - Nonlinear Systems and Control Lecture 8 Lyapunov...

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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 φ ( 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 ( φ ( 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 Lyapunov’s Theorem: If there is V ( x ) such that V (0) = 0 and V ( x ) > , ∀ x ∈ D \ { } ˙ V ( x ) ≤ ,...
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Lect_8 - Nonlinear Systems and Control Lecture 8 Lyapunov...

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