7_stability

7_stability - Control of Nonlinear Dynamic Systems: Theory...

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Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard © 2005 84 7 Stability of Nonlinear Systems We will consider the case of unforced, autonomous systems, as represented by the equation: ) ( x f x = & Key points Stability, asymptotic stability, uniform stability, global stability The Second Method of Lyapunov allows one to determine (global, asymptotic, uniform) stability of an equilibrium point without explicitly solving for system solutions. The hardest condition to meet in Lyapunov’s conditions is finding a Lyapunov function candidate such that 0 < V & . Oftentimes 0 V & . LaSalle’s invariance principle can be used if this is the case for an autonomous or periodic system. Barbalat’s lemma and a Lyapunov-like lemma from Slotine and Li can be used if 0 V & for non-autonomous systems (such as adaptive systems). Domains of attraction can be calculated or approximated for some equilibrium points. The Aizerman and Kalman conjectures deal with global asymptotic stability of a nonlinear system for a whole class of nonlinearities. Both are false, but two different criteria, the circle criterion and the Popov criterion, are applicable. References: Khalil and Slotine and Li
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Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard © 2005 85 We will not consider disturbances, and we will restrict the analysis to systems that do not have an explicit time dependence (in a first time). So far, we have started by looking for the equilibrium points : 0 ) ( = e x f We have then considered perturbations about the equilibrium points : x x x e δ + = HOT x x f x e + = . & e x f J = and if 0 ) Re( i λ , then local stability can be determined from the eigenvalues of J . If 0 ) Re( = i , one can use the center manifold theorem to determine local stability. What about global stability ? Definition: Stability in the sense of Lyapunov Assume 0 = e x . Stable: The equilibrium x = 0 is stable iff 0 2 2 0 0 , ) ( ) ( 0 , 0 , 0 t t t x t x t < < > > ε . x 1 δ ε
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Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard © 2005 86 “That is, if I start within δ , I stay within ε . In general, I give you an ε , you give me the corresponding δ . Things remain bounded.” Asymptotically stable: The equilibrium x = 0 is asymptotically stable iff: (i) x = 0 is a stable equilibrium (ii) 0 ) ( lim ) ( ) ( , 0 2 0 0 0 = < +∞ t x t x t t t δ Uniformly stable: The equilibrium x = 0 is uniformly stable iff: (i) x = 0 is a stable equilibrium (ii) ) ( ) , ( 0 ε = t These conditions refer to stability in the sense of Lyapunov . The Second Method of Lyapunov - Originally proposed by Lyapunov (around 1890) to investigate stability in the small (local stability) - Later extended to cover global stability - Stability can be determined without explicitly solving for the system solutions.
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7_stability - Control of Nonlinear Dynamic Systems: Theory...

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