MIT16_30F10_lec22

MIT16_30F10_lec22 - Topic #22 16.30/31 Feedback Control...

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Unformatted text preview: Topic #22 16.30/31 Feedback Control Systems Analysis of Nonlinear Systems Lyapunov Stability Analysis Fall 2010 16.30/31 222 Lyapunov Stability Analysis Very general method to prove (or disprove) stability of nonlinear sys- tems. Formalizes idea that all systems will tend to a minimum-energy state. Lyapunovs stability theory is the single most powerful method in stability analysis of nonlinear systems . Consider a nonlinear system x = f ( x ) A point x is an equilibrium point if f ( x ) = 0 Can always assume x = 0 In general, an equilibrium point is said to be Stable in the sense of Lyapunov if (arbitrarily) small devia- tions from the equilibrium result in trajectories that stay (arbitrar- ily) close to the equilibrium for all time. Asymptotically stable if small deviations from the equilibrium are eventually forgotten, and the system returns asymptotically to the equilibrium point. Exponentially stable if it is asymptotically stable, and the con- vergence to the equilibrium point is fast. November 27, 2010 Fall 2010 16.30/31 223 Stability Let x = 0 D be an equilibrium point of the system x = f ( x ) , where f : D R n is locally Lipschitz in D R f ( x ) is locally Lipschitz in D if x D I ( x ) such that | f ( y ) f ( z ) | L | y z | for all y , z I ( x ) . Smoothness condition for functions which is stronger than regular continuity intuitively, a Lipschitz continuous function is limited in how fast it can change. ( see here ) A sucient condition for a function to be Lipschitz is that the Jacobian f/ x is uniformly bounded for all x . The equilibrium point is Stable in the sense of Lyapunov (ISL) if, for each , there is = ( ) > such that x (0) < x ( t ) , t 0; Asymptotically stable if stable, and there exists > s.t. x (0) < lim x ( t ) = 0 t + Exponentially stable if there exist ,, > s.t. x (0) < x ( t ) < e t , t 0; Unstable if not stable. x e x (0) ISL or Marginally Stable Unstable November 27, 2010 Fall 2010 16.30/31 224 How do we analyze the stability of an equilibrium point? Already talked about how to linearize the dynamics about the equilib- rium point and use the conclusion from the linear analysis to develop a local conclusion Often called Lyapunovs first method How about a more global conclusion?...
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MIT16_30F10_lec22 - Topic #22 16.30/31 Feedback Control...

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