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

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Nonlinear Systems and Control Lecture # 7 Stability of Equilibrium Points Basic Concepts & Linearization – p. 1/1
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˙ 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
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Definition: The equilibrium point x = 0 of ˙ x = f ( x ) is stable if for each ε > 0 there is δ > 0 (dependent on ε ) such that bardbl x (0) bardbl < δ ⇒ bardbl x ( t ) bardbl < ε, t 0 unstable if it is not stable asymptotically stable if it is stable and δ can be chosen such that bardbl x (0) bardbl < δ lim t →∞ x ( t ) = 0 – p. 3/1
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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 ε δ requirement for stability is violated if xf ( x ) > 0 on either side of the origin f(x) x f(x) x f(x) x Unstable Unstable Unstable – p. 4/1
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The origin is stable if and only if xf ( x ) 0 in some neighborhood of the origin f(x) x f(x) x f(x) x Stable Stable Stable – p. 5/1
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The origin is asymptotically stable if and only if xf ( x ) < 0 in some neighborhood of the origin f(x) x -a b f(x) x (a) (b) Asymptotically Stable Globally Asymptotically Stable – p. 6/1
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