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# Lect_11 - Nonlinear Systems and Control Lecture 11...

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Nonlinear Systems and Control Lecture # 11 Exponential Stability & Region of Attraction – p. 1/1

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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 ; 0 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
Proof: Choose c > 0 small enough that { k 1 bardbl x bardbl a c } ⊂ D V ( x ) c k 1 bardbl x bardbl a c Ω c = { V ( x ) c } ⊂ { k 1 bardbl x bardbl a c } ⊂ D Ω c is compact and positively invariant; x (0) Ω 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

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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 k 2 bardbl x (0) bardbl a e ( k 3 /k 2 ) t k 1 bracketrightBigg 1 /a = parenleftbigg k 2 k 1
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Lect_11 - Nonlinear Systems and Control Lecture 11...

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