Lect_18 - Nonlinear Systems and Control Lecture # 18...

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Unformatted text preview: Nonlinear Systems and Control Lecture # 18 Boundedness & Ultimate Boundedness p. 1/1 8 Definition: The solutions of x = f ( t,x ) are uniformly bounded if c > and for every < a < c, = ( a ) > such that bardbl x ( t ) bardbl a bardbl x ( t ) bardbl , t t uniformly ultimately bounded with ultimate bound b if b and c and for every < a < c, T = T ( a,b ) such that bardbl x ( t ) bardbl a bardbl x ( t ) bardbl b, t t + T Globally if a can be arbitrarily large Drop uniformly if x = f ( x ) p. 2/1 8 Lyapunov Analysis: Let V ( x ) be a cont. diff. positive definite function and suppose that the sets c = { V ( x ) c } , = { V ( x ) } , = { V ( x ) c } are compact for some c > > c p. 3/1 8 Suppose V ( t,x ) = V x f ( t,x ) W 3 ( x ) , x , t W 3 ( x ) is continuous and positive definite c and are positively invariant...
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Lect_18 - Nonlinear Systems and Control Lecture # 18...

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