Converse Lyapunov Functions And Time Varying Systems-Non Linear Systems Control and Analysis-Lecture Slides, Slides of Nonlinear Control Systems

Download Converse Lyapunov Functions And Time Varying Systems-Non Linear Systems Control and Analysis-Lecture Slides and more Slides Nonlinear Control Systems in PDF only on Docsity! Nonlinear Systems and Control Lecture # 12 Converse Lyapunov Functions & Time Varying Systems – p. 1/18 Docsity.com Converse Lyapunov Theorem–Exponential Stability Let x = 0 be an exponentially stable equilibrium point for the system ẋ = f(x), where f is continuously differentiable on D = {‖x‖ < r}. Let k, λ, and r0 be positive constants with r0 < r/k such that ‖x(t)‖ ≤ k‖x(0)‖e−λt, ∀ x(0) ∈ D0, ∀ t ≥ 0 where D0 = {‖x‖ < r0}. Then, there is a continuously differentiable function V (x) that satisfies the inequalities – p. 2/18 Docsity.com Example: Consider the system ẋ = f(x) where f is continuously differentiable in the neighborhood of the origin and f(0) = 0. Show that the origin is exponentially stable only if A = [∂f/∂x](0) is Hurwitz f(x) = Ax+G(x)x, G(x) → 0 as x → 0 Given any L > 0, there is r1 > 0 such that ‖G(x)‖ ≤ L, ∀ ‖x‖ < r1 Because the origin of ẋ = f(x) is exponentially stable, let V (x) be the function provided by the converse Lyapunov theorem over the domain {‖x‖ < r0}. Use V (x) as a Lyapunov function candidate for ẋ = Ax – p. 5/18 Docsity.com ∂V ∂x Ax = ∂V ∂x f(x) − ∂V ∂x G(x)x ≤ −c3‖x‖ 2 + c4L‖x‖ 2 = −(c3 − c4L)‖x‖ 2 Take L < c3/c4, γ def = (c3 − c4L) > 0 ⇒ ∂V ∂x Ax ≤ −γ‖x‖2, ∀ ‖x‖ < min{r0, r1} The origin of ẋ = Ax is exponentially stable – p. 6/18 Docsity.com Converse Lyapunov Theorem–Asymptotic Stability Let x = 0 be an asymptotically stable equilibrium point for ẋ = f(x), where f is locally Lipschitz on a domain D ⊂ Rn that contains the origin. LetRA ⊂ D be the region of attraction of x = 0. Then, there is a smooth, positive definite function V (x) and a continuous, positive definite function W (x), both defined for all x ∈ RA, such that V (x) → ∞ as x → ∂RA ∂V ∂x f(x) ≤ −W (x), ∀ x ∈ RA and for any c > 0, {V (x) ≤ c} is a compact subset of RA When RA = Rn, V (x) is radially unbounded – p. 7/18 Docsity.com Example α(r) = tan−1(r) is strictly increasing since α′(r) = 1/(1 + r2) > 0. It belongs to class K, but not to class K∞ since limr→∞ α(r) = π/2 < ∞ α(r) = rc, for any positive real number c, is strictly increasing since α′(r) = crc−1 > 0. Moreover, limr→∞ α(r) = ∞; thus, it belongs to class K∞ α(r) = min{r, r2} is continuous, strictly increasing, and limr→∞ α(r) = ∞. Hence, it belongs to class K∞ – p. 10/18 Docsity.com β(r, s) = r/(ksr + 1), for any positive real number k, is strictly increasing in r since ∂β ∂r = 1 (ksr + 1)2 > 0 and strictly decreasing in s since ∂β ∂s = −kr2 (ksr + 1)2 < 0 Moreover, β(r, s) → 0 as s → ∞. Therefore, it belongs to class KL β(r, s) = rce−s, for any positive real number c, belongs to class KL – p. 11/18 Docsity.com Definition: The equilibrium point x = 0 of ẋ = f(t, x) is uniformly stable if there exist a class K function α and a positive constant c, independent of t0, such that ‖x(t)‖ ≤ α(‖x(t0)‖), ∀ t ≥ t0 ≥ 0, ∀ ‖x(t0)‖ < c uniformly asymptotically stable if there exist a class KL function β and a positive constant c, independent of t0, such that ‖x(t)‖ ≤ β(‖x(t0)‖, t−t0), ∀ t ≥ t0 ≥ 0, ∀ ‖x(t0)‖ < c globally uniformly asymptotically stable if the foregoing inequality is satisfied for any initial state x(t0) – p. 12/18 Docsity.com Theorem: Suppose the assumptions of the previous theorem are satisfied with ∂V ∂t + ∂V ∂x f(t, x) ≤ −W3(x) for all t ≥ 0 and x ∈ D, where W3(x) is a continuous positive definite function on D. Then, the origin is uniformly asymptotically stable. Moreover, if r and c are chosen such that Br = {‖x‖ ≤ r} ⊂ D and c < min‖x‖=rW1(x), then every trajectory starting in {x ∈ Br | W2(x) ≤ c} satisfies ‖x(t)‖ ≤ β(‖x(t0)‖, t− t0), ∀ t ≥ t0 ≥ 0 for some class KL function β. Finally, if D = Rn and W1(x) is radially unbounded, then the origin is globally uniformly asymptotically stable – p. 15/18 Docsity.com Theorem: Suppose the assumptions of the previous theorem are satisfied with k1‖x‖ a ≤ V (t, x) ≤ k2‖x‖ a ∂V ∂t + ∂V ∂x f(t, x) ≤ −k3‖x‖ a for all t ≥ 0 and x ∈ D, where k1, k2, k3, and a are positive constants. Then, the origin is exponentially stable. If the assumptions hold globally, the origin will be globally exponentially stable. – p. 16/18 Docsity.com Example: ẋ = −[1 + g(t)]x3, g(t) ≥ 0, ∀ t ≥ 0 V (x) = 12x 2 V̇ (t, x) = −[1 + g(t)]x4 ≤ −x4, ∀ x ∈ R, ∀ t ≥ 0 The origin is globally uniformly asymptotically stable Example: ẋ1 = −x1 − g(t)x2 ẋ2 = x1 − x2 0 ≤ g(t) ≤ k and ġ(t) ≤ g(t), ∀ t ≥ 0 – p. 17/18 Docsity.com