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Unformatted text preview: Nonlinear Systems and Control Lecture # 11 Exponential Stability & Region of Attraction p. 1/1 8 Exponential Stability: The origin of x = f ( x ) is exponentially stable if and only if the linearization of f ( x ) at the origin is Hurwitz Theorem: Let f ( x ) be a locally Lipschitz function defined over a domain D R n ; D . Let V ( x ) be a continuously differentiable function such that k 1 bardbl x bardbl a V ( x ) k 2 bardbl x bardbl a V ( x ) k 3 bardbl x bardbl a for all x D , where k 1 , k 2 , k 3 , and a are positive constants. Then, the origin is an exponentially stable equilibrium point of x = f ( x ) . If the assumptions hold globally, the origin will be globally exponentially stable p. 2/1 8 Proof: Choose c > small enough that { k 1 bardbl x bardbl a c } D V ( x ) c k 1 bardbl x bardbl a c c = { V ( x ) c } { k 1 bardbl x bardbl a c } D c is compact and positively invariant; x (0) c V k 3 bardbl x bardbl a k 3 k 2 V dV V k 3 k 2 dt V ( x ( t )) V ( x (0)) e ( k 3 /k 2 ) t p. 3/1 8 bardbl x ( t ) bardbl bracketleftbigg V ( x ( t )) k 1 bracketrightbigg 1 /a bracketleftBigg V ( x (0)) e ( k 3 /k 2 ) t k 1 bracketrightBigg 1 /a bracketleftBigg...
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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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