Nonlinear Systems and Control: Tracking, Feedback Linearization, and Sliding Mode Control, Slides of Nonlinear Control Systems

Download Nonlinear Systems and Control: Tracking, Feedback Linearization, and Sliding Mode Control and more Slides Nonlinear Control Systems in PDF only on Docsity! Nonlinear Systems and Control Lecture # 35 Tracking Feedback Linearization & Sliding Mode Control – p. 1/11 Docsity.com SISO relative-degree ρ system: ẋ = f(x) + g(x)u, y = h(x) f(0) = 0, h(0) = 0 LgL i−1 f h(x) = 0, for 1 ≤ i ≤ ρ − 1, LgL ρ−1 f h(x) 6= 0 Normal form: η̇ = f0(η, ξ) ξ̇i = ξi+1, 1 ≤ i ≤ ρ − 1 ξ̇ρ = L ρ fh(x) + LgL ρ−1 f h(x)u y = ξ1 f0(0, 0) = 0 – p. 2/11 Docsity.com v = −Ke ⇒ ė = (Ac − BcK) ︸ ︷︷ ︸ Hurwitz e lim t→∞ e(t) = 0 ⇒ lim t→∞ [y(t) − r(t)] = 0 e(t) is bounded ⇒ ξ(t) = e(t) + R(t) is bounded What about η(t)? η̇ = f0(η, ξ) Local Tracking (small ‖η(0)‖, ‖e(0)‖, ‖R(t)‖): Minimum Phase ⇒ The origin of η̇ = f0(η, 0) is asymptotically stable ⇒ η is bounded for sufficiently small ‖η(0)‖, ‖e(0)‖, and ‖R(t)‖ – p. 5/11 Docsity.com Global Tracking (large ‖η(0)‖, ‖e(0)‖, ‖R(t)‖): What condition on η̇ = f0(η, ξ) is needed? Example 13.21 ẋ1 = x2, ẋ2 = −a sin x1 − bx2 + cu, y = x1 e1 = x1 − r, e2 = x2 − ṙ ė1 = e2, ė2 = −a sin x1 − bx2 + cu − r̈ u = 1 c [a sin x1 + bx2 + r̈ − k1e1 − k2e2] ė1 = e2, ė2 = −k1e1 − k2e2 See simulation in the textbook – p. 6/11 Docsity.com Sliding Mode Control ẋ = f(x) + g(x)[u + δ(t, x, u)], y = h(x) Lgh(x) = · · · = LgL ρ−2 f h(x) = 0, LgL ρ−1 f h(x) ≥ a > 0 η̇ = f0(η, ξ) ξ̇1 = ξ2 ... ... ξ̇ρ−1 = ξρ ξ̇ρ = L ρ fh(x) + LgL ρ−1 f h(x)[u + δ(t, x, u)] y = ξ1 e = ξ − R – p. 7/11 Docsity.com s = (k1e1 + · · · + kρ−1eρ−1) + eρ = ρ−1 ∑ i=1 kiei + eρ ṡ = ρ−1 ∑ i=1 kiei+1+L ρ fh(x)+LgL ρ−1 f h(x)[u+δ(t, x, u)]−r (ρ)(t) u = − 1 LgL ρ−1 f h(x) [ ρ−1 ∑ i=1 kiei+1 + L ρ fh(x) − r (ρ)(t) ] + v ṡ = LgL ρ−1 f h(x)v + ∆(t, x, v) ∣ ∣ ∣ ∣ ∣ ∆(t, x, v) LgL ρ−1 f h(x) ∣ ∣ ∣ ∣ ∣ ≤ ̺(x) + κ0|v|, 0 ≤ κ0 < 1 – p. 10/11 Docsity.com v = −β(x) sat ( s ε ) , ε > 0 β(x) ≥ ̺(x) (1 − κ0) + β0, β0 > What properties can we prove for this control? – p. 11/11 Docsity.com