Download Integral Control-Control of Non Linear Systems-Lecture Slides and more Slides Nonlinear Control Systems in PDF only on Docsity! Nonlinear Systems and Control Lecture # 41 Integral Control – p. 1/17 Docsity.com ẋ = f(x, u, w) y = h(x, w) ym = hm(x, w) x ∈ Rn state, u ∈ Rp control input y ∈ Rp controlled output, ym ∈ Rm measured output w ∈ Rl unknown constant parameters and disturbances Goal: y(t) → r as t → ∞ r ∈ Rp constant reference, v = (r, w) e(t) = y(t) − r – p. 2/17 Docsity.com Integral Control via Linearization State Feedback: u = −K1x − K2σ − K3e Closed-loop system: ẋ = f(x, −K1x − K2σ − K3(h(x, w) − r), w) σ̇ = h(x, w) − r Equilibrium points: 0 = f(x̄, ū, w) 0 = h(x̄, w) − r ū = −K1x̄ − K2σ̄ Unique equilibrium point at x = xss, σ = σss, u = uss – p. 5/17 Docsity.com Linearization about (xss, σss): ξδ = [ x − xss σ − σss ] ξ̇δ = (A − BK)ξδ A = [ A 0 C 0 ] , A = ∂f ∂x (x, u, w) ∣ ∣ ∣ ∣ eq , C = ∂h ∂x (x, w) ∣ ∣ ∣ ∣ eq B = [ B 0 ] , B = ∂f ∂u (x, u, w) ∣ ∣ ∣ ∣ eq K = [ K1 + K3C K2 ] – p. 6/17 Docsity.com (A, B) is controllable if and only if (A, B) is controllable and rank [ A B C 0 ] = n + p Task: Design K, independent of v, such that (A − BK) is Hurwitz for all v (xss, σss) is an exponentially stable equilibrium point of the closed-loop system. All solutions starting in its region of attraction approach it as t tends to infinity e(t) → 0 as t → ∞ – p. 7/17 Docsity.com Output Feedback: We only measure e and ym σ̇ = e = y − r ż = Fz + G1σ + G2ym u = Lz + M1σ + M2ym + M3e Task: Design F , G1, G2, L, M1, M2, and M3, independent of v, such that Ac is Hurwitz for all v Ac = A + BM2Cm + BM3C BM1 BL C 0 0 G2Cm G1 F Cm = ∂hm ∂x (x, w) ∣ ∣ ∣ ∣ eq – p. 10/17 Docsity.com Integral Control via Sliding Mode Design η̇ = f0(η, ξ, w) ξ̇1 = ξ2 ... ... ξ̇ρ−1 = ξρ ξ̇ρ = b(η, ξ, u, w) + a(η, ξ, w)u y = ξ1 a(η, ξ, w) ≥ a0 > 0 Goal: y(t) → r as t → ∞ ξss = [r, 0, . . . , 0] T – p. 11/17 Docsity.com Steady-state condition: There is a unique pair (ηss, uss) that satisfies the equations 0 = f0(ηss, ξss, w) 0 = b(ηss, ξss, uss, w) + a(ηss, ξss, w)uss ė0 = y − r z = η − ηss, e = e1 e2 ... eρ = ξ1 − r ξ2 ... ξρ – p. 12/17 Docsity.com ż = f̃0(z, e, w, r) ζ̇ = Aζ + Bs (A is Hurwitz) ṡ = −a(·)β(e) sat ( s µ ) + ∆(·) ζ = [e0, . . . , eρ−1] T α̃1(‖z‖) ≤ V1(z, w, r) ≤ α̃2(‖z‖) ∂V1 ∂z f̃0(z, e, w, r) ≤ −α̃3(‖z‖), ∀ ‖z‖ ≥ γ̃(‖e‖) V2(ζ) = ζ T Pζ, PA + AT P = −I – p. 15/17 Docsity.com Ω = {|s| ≤ c} ∩ {V2 ≤ c 2ρ1} ∩ {V1 ≤ c0} Ωµ = {|s| ≤ µ} ∩ {V2 ≤ µ 2ρ1} ∩ {V1 ≤ α̃2(γ̃(µρ2))} All trajectories starting in Ω enter Ωµ in finite time and stay in thereafter Inside Ωµ there is a unique equilibrium point at (z = 0, e = 0, e0 = ē0), s̄ = k0ē0, uss = −β(0) s̄ µ Under additional conditions (the origin of ż = f̃0(z, 0, w, r) is exponentially stable), local analysis inside Ωµ shows that for sufficiently small µ all trajectories converge to the equilibrium point as time tends to infinity – p. 16/17 Docsity.com Output Feedback: Only e1 is measured High-gain Observer: ė0 = e1 u = −β sat ( k0e0 + k1e1 + k2ê2 + · · · + êρ µ ) ˙̂ei = êi+1 + ( αi εi ) (e1 − ê1), 1 ≤ i ≤ ρ − 1 ˙̂eρ = ( αρ ερ ) (e1 − ê1) β = β(e1, ê2, . . . , êρ) – p. 17/17 Docsity.com
