Download Passivity Based Control-Non Linear Control Systems-Lecture Slides and more Slides Nonlinear Control Systems in PDF only on Docsity! Nonlinear Systems and Control Lecture # 29 Stabilization Passivity-Based Control – p. 1/?? Docsity.com ẋ = f(x, u), y = h(x) f(0, 0) = 0 uT y ≥ V̇ = ∂V ∂x f(x, u) Theorem 14.4: If the system is (1) passive with a radially unbounded positive definite storage function and (2) zero-state observable, then the origin can be globally stabilized by u = −φ(y), φ(0) = 0, yT φ(y) > 0 ∀ y 6= 0 – p. 2/?? Docsity.com Example ẋ1 = x2, ẋ2 = −x 3 1 + u V (x) = 1 4 x4 1 + 1 2 x2 2 With u = 0 V̇ = x3 1 x2 − x2x 3 1 = 0 Take y = ∂V ∂x G = ∂V ∂x2 = x2 Is it zero-state observable? with u = 0, y(t) ≡ 0 ⇒ x(t) ≡ 0 u = −kx2 or u = −(2k/π) tan−1(x2) (k > 0) – p. 5/?? Docsity.com Feedback Passivation Definition: The system ẋ = f(x) + G(x)u, y = h(x) is equivalent to a passive system if there is u = α(x) + β(x)v such that ẋ = f(x) + G(x)α(x) + G(x)β(x)v, y = h(x) is passive – p. 6/?? Docsity.com Theorem [31]: The system ẋ = f(x) + G(x)u, y = h(x) is locally equivalent to a passive system (with a positive definite storage function) if it has relative degree one at x = 0 and the zero dynamics have a stable equilibrium point at the origin with a positive definite Lyapunov function Example: m-link Robot Manipulator M(q)q̈ + C(q, q̇)q̇ + Dq̇ + g(q) = u M = MT > 0, (Ṁ − 2C)T = −(Ṁ − 2C), D = DT ≥ 0 – p. 7/?? Docsity.com How does passivity-based control compare with feedback linearization? Example 13.20 ẋ1 = x2, ẋ2 = −h(x1) + u h(0) = 0, x1h(x1) > 0, ∀ x1 6= 0 Feedback linearization: u = h(x1) − (k1x1 + k2x2) ẋ = [ 0 1 −k1 −k2 ] x – p. 10/?? Docsity.com Passivity-based control: V = ∫ x1 0 h(z) dz + 1 2 x2 2 V̇ = x2h(x1) − x2h(x1) + x2u = x2u Take y = x2 With u = 0, y(t) ≡ 0 ⇒ h(x1(t)) ≡ 0 ⇒ x1(t) ≡ 0 u = −σ(x2), [σ(0) = 0, yσ(y) > 0 ∀ y 6= 0] ẋ1 = x2, ẋ2 = −h(x1) − σ(x2) – p. 11/?? Docsity.com Linearization: [ 0 1 −h′(0) −k ] , k = σ′(0) s2 + ks + h′(0) = 0 Sketch the root locus as k varies from zero to infinity One of the two roots cannot be moved to the left of Re[s] = − √ h′(0) – p. 12/?? Docsity.com
