Lect_7 - Nonlinear Systems and Control Lecture # 7...

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Unformatted text preview: Nonlinear Systems and Control Lecture # 7 Stability of Equilibrium Points Basic Concepts & Linearization p. 1/1 9 x = f ( x ) f is locally Lipschitz over a domain D R n Suppose x D is an equilibrium point; that is, f ( x ) = 0 Characterize and study the stability of x For convenience, we state all definitions and theorems for the case when the equilibrium point is at the origin of R n ; that is, x = 0 . No loss of generality y = x x y = x = f ( x ) = f ( y + x ) def = g ( y ) , where g (0) = 0 p. 2/1 9 Definition: The equilibrium point x = 0 of x = f ( x ) is stable if for each > there is > (dependent on ) such that bardbl x (0) bardbl < bardbl x ( t ) bardbl < , t unstable if it is not stable asymptotically stable if it is stable and can be chosen such that bardbl x (0) bardbl < lim t x ( t ) = 0 p. 3/1 9 First-Order Systems ( n = 1 ) The behavior of x ( t ) in the neighborhood of the origin can be determined by examining the sign of f ( x ) The requirement for stability is violated if xf ( x ) > on either side of the origin f(x) x f(x) x f(x) x Unstable Unstable Unstable p. 4/1 9 The origin is stable if and only if xf ( x ) in some neighborhood of the origin f(x) x f(x) x f(x) x Stable Stable Stable p. 5/1 9 The origin is asymptotically stable if and only if xf ( x ) < in some neighborhood of the origin f(x) x-a b f(x) x (a) (b) Asymptotically Stable Globally Asymptotically Stable p. 6/1 9 Definition:...
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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.

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Lect_7 - Nonlinear Systems and Control Lecture # 7...

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