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 22–2 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. • Lyapunov’s 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 22–3 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 sufficient 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 ε ≥ , there is δ = δ ( ε ) > such that x (0) < δ ⇒ x ( t ) ≤ ε, ∀ t ≥ 0; • Asymptotically stable if stable, and there exists δ > s.t. x (0) < δ lim x ( t ) = 0 ⇒ t → + ∞ • Exponentially stable if there exist δ,α,β > s.t. x (0) < δ ⇒ x ( t ) < βe − αt , ∀ t ≥ 0; Unstable if not stable. • x e x (0) δ ISL or Marginally Stable Unstable November 27, 2010 Fall 2010 16.30/31 22–4 • 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 Lyapunov’s 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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