Download Robust Stabilization of Nonlinear Systems: Sliding Mode Control and more Slides Nonlinear Control Systems in PDF only on Docsity! Nonlinear Systems and Control Lecture # 33 Robust Stabilization Sliding Mode Control – p. 1/15 Docsity.com Regular Form: η̇ = fa(η, ξ) ξ̇ = fb(η, ξ) + g(η, ξ)u + δ(t, η, ξ, u) η ∈ Rn−1, ξ ∈ R, u ∈ R fa(0, 0) = 0, fb(0, 0) = 0, g(η, ξ) ≥ g0 > 0 Sliding Manifold: s = ξ − φ(η) = 0, φ(0) = 0 s(t) ≡ 0 ⇒ η̇ = fa(η, φ(η)) Design φ s.t. the origin of η̇ = fa(η, φ(η)) is asymp. stable – p. 2/15 Docsity.com sṡ ≤ −g(η, ξ)(1 − κ0)β0|s| ≤ −g0β0(1 − κ0)|s| v = −β(x) sat ( s ε ) , ε > 0 sṡ ≤ −g0β0(1 − κ0)|s|, for |s| ≥ ε The trajectory reaches the boundary layer {|s| ≤ ε} in finite time and remains inside thereafter Study the behavior of η η̇ = fa(η, φ(η) + s) What do we know about this system and what do we need? – p. 5/15 Docsity.com α1(‖η‖) ≤ V (η) ≤ α2(‖η‖) ∂V ∂η fa(η, φ(η) + s) ≤ −α3(‖η‖), ∀ ‖η‖ ≥ γ(|s|) |s| ≤ c ⇒ V̇ ≤ −α3(‖η‖), for ‖η‖ ≥ γ(c) α(r) = α2(γ(r)) V (η) ≥ α(c) ⇔ V (η) ≥ α2(γ(c)) ⇒ α2(‖η‖) ≥ α2(γ(c)) ⇒ ‖η‖ ≥ γ(c) ⇒ V̇ ≤ −α3(‖η‖) ≤ −α3(γ(c)) The set {V (η) ≤ c0} with c0 ≥ α(c) is positively invariant Ω = {V (η) ≤ c0} × {|s| ≤ c}, with c0 ≥ α(c) – p. 6/15 Docsity.com α(.) α(ε) α(c) c 0 ε c V |s| Ω = {V (η) ≤ c0} × {|s| ≤ c}, with c0 ≥ α(c) is positively invariant and all trajectories starting in Ω reach Ωε = {V (η) ≤ α(ε)} × {|s| ≤ ε} in finite time – p. 7/15 Docsity.com ∆(x) = θ2x 2 2 + kθ1x1 sin x2 |∆(x)| ≤ ak|x1| + bx 2 2 β(x) = ak|x1| + bx 2 2 + β0, β0 > 0 u = −x1 − kx2 − β(x) sgn(s) Will u = −x1 − kx2 − β(x) sat ( s ε ) stabilize the origin? – p. 10/15 Docsity.com Example: 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 View ξρ as input to the system η̇ = f0(η, ξ1, · · · , ξρ−1, ξρ) ξ̇i = ξi+1, 1 ≤ i ≤ ρ − 2 ξ̇ρ−1 = ξρ Design ξρ = φ(η, ξ1, · · · , ξρ−1) to stabilize the origin – p. 11/15 Docsity.com s = ξρ − φ(η, ξ1, · · · , ξρ−1) Minimum Phase Systems: The origin of η̇ = f0(η, 0) is asymptotically stable s = ξρ + k1ξ1 + · · · + kρ−1ξρ−1 η̇ = f0(η, ξ1, · · · , ξρ−1, −k1ξ1 − · · · − kρ−1ξρ−1) ξ̇1 ... ξ̇ρ−1 = 1 . . . 1 −k1 −kρ−1 ξ1 ... ξρ−1 – p. 12/15 Docsity.com
