An Approach of Robust Iterative Learning Control for Uncertain Systems

Transcription

1 ,,, :, Lyapunov( ),,.,,,.,,. :,,, An Approach of Robust Iterative Learning Control for Uncertain Systems Mingxuan Sun, Chaonan Jiang, Yanwei Li College of Information Engineering, Zhejiang University of Technology, Hangzhou, Abstract: By applying Lyapunov s second method, the traditional robust control approach to controller design explicitly deals with uncertainty. The controller designs can be conducted through modifying the discontinuous control synthesis, e.g., the unit vector continuous robust control method. The robust control guarantees all the variables in the closed-loop system are bounded and make the tracking error converge to a neighborhood of the origin. With minimal modification to the existing robust control design, in this paper, a robust iterative learning control approach is proposed by installing the iterative learning mechanism. It is shown that the tracking error converges to zero, as iteration increases, and the tracking performance is improved dramatically. In addition, the fully saturated learning is shown useful in order to make the estimated parameters are enforced to stay in the pre-specified range. Key Words: Robust control, iterative learning control, convergence, uncertain systems (Introduction),,..,, [, 2, 3].,., [4].,,.,., : 67434, 63743, ,,,,.,,., Lyapunov [5, 6, 7, 8]..,,. [9],.[],.[] Lyapunov,, Lipschitz. [2, 3, 4] /6/$3. c 26 IEEE 6853

2 ,..,,,,.,,,. 2 (Problem Formulation) [,T] ẋ(t) = Ax + B[η(x, t)+ u(t)] () x(t) R n, u(t) R, η(x, t) R, A R n n B R n A =....,B = {A, B}. e = x x d. (), ė = Ae + B[η(x, t)+u ẋ dn ] (2), η(x, t) η(x, t) ρ(x, t) (3), ρ(x, t).,, η(x, t) η(x, t) η(x d,t) α(x, x d,t) e (4), α(x, x d,t). (),, u = Kx + u (5), K, u,, {A BK}. A K = A BK., Q R n n, P R n n Lyapunov A T KP + PA K = Q Qλ Q min. (), x d. η(x, t),. (5), (2) ė = A K e + B[Ke + η(x, t)+u ẋ dn ] (6), : i) (),, ; ii) ( ),. 3 (Robust Control) (),, u = u r Ke +ẋ dn (7) μ u r = ρ (8) μ + ε, ε>, μ = B T Peρ. (8). (3)(), (7) (8),. Lyapunov V (e, t) = 2 et Pe (9) 2 λ P min e 2 V (e, t) 2 λ P max e 2 () λ P min, λ P max P. (9)Lyapunov, (6) (7) (8) V = 2 (ėt Pe+ e T P ė) = 2 et Qe + e T PB(Ke + η + u ẋ dn ) 2 λ Q min e 2 μ 2 μ + ε + e T PB ρ 2 λ Q min e 2 + ε () 2ε, e, V., e λ Q min. x d, x. (3)ρ. (7) (8),u,u r th Chinese Control and Decision Conference (CCDC)

3 ,,., ε μ u r = ρ μ + ε ρsgnμ, ρsgnμ., ε,. 4 (Robust Iterative Learning Control),,,..,,.,. η(x d,t)=θ, η(x d,t) θ max. u k = u r,k ˆθ k Ke k +ẋ dn (2) ˆθ k = ˆθ k + γb T Pe k (3) ˆθ k = sat(ˆθ k) (4) μ k u r,k = (α k e k +2θ max ) (5) μ k + ε μ k = B T Pe k (α k e k +2θ max ). k, ε>. γ>, ˆθ k, u r,k, sat(ˆθ k )ˆθ k, θ max, { ˆθ sat(ˆθ k)= k, ˆθ k θ max sgnˆθ k θ max, ˆθ k >θ (6) max, [4]. a, b, b max b min b, b min <a<b max, (a sat(b))(b sat(b)) (7) 2 (4)(), (2) (5), k, e k [,T], lim k e k(t) =, t [,T]. ). (9)Lyapunov, (6) V k = 2 (ėt k Pe k + e T k P ė k ) = 2 et k Qe k + e T k PB(Ke k + η k η d +η d + u k ẋ dn ) 2 λ Q min e k 2 μ k 2 μ k + ε + e T k PB (α k e k +2θ max ) 2 λ Q min e k 2 + ε (8) 2ε, e k, V k. λ Q min, e k. x d, x k. (2) (5) sat(ˆθ k ), ˆθ k, u r,k, u k. 2). Lyapunov-like L k (t) = 2 e λt e T k Pe k + θ k (t) =θ(t) ˆθ k (t). L k (t) e λτ θ2 k (τ)dτ (9) ΔL k = 2 e λt e T k Pe k 2 e λt e T k Pe k + e λτ [ θ 2 k(τ) θ 2 k (τ)]dτ (2) (9) 2 e λt e T k Pe k = 2 λ e λτ e T k (τ)pe k (τ)dτ + e λτ [ė T k (τ)pe k (τ)+e T k (τ)p ė k (τ)]dτ et k ()Pe k () (2) (2) (5) 2 (ėt k Pe k + e T k P ė k ) = 2 et k Qe k + e T k PB(Ke k + η k η d +η d + u k ẋ dn ) 2 λ Q min e k 2 + e T k PB(θ ˆθ k ) + e T k PB α k e k μ k 2 μ k + ε 2 λ Q min e k 2 + e T k PB α k e k +e T k PB θ k (22) 2 e λt e T k Pe k 2 λ Q min + e λτ e k (τ) 2 dτ e λτ ( λ 2 et k (τ)pe k (τ) e T k PB α k e k )dτ e λτ e T k PB θ k dτ + 2 et k ()Pe k () (23) 26 28th Chinese Control and Decision Conference (CCDC) 6855

4 (2), (3) e λτ [ θ k(τ) 2 θ k (τ)]dτ 2 = e λτ (ˆθ k ˆθ k ) [2(θ ˆθ k )+(ˆθ k ˆθ k )]dτ γ e λτ (ˆθ k ˆθ k ) θ k dτ (24) (23)(24)e k () = ΔL k 2 λ Q min e λτ [e T k PB θ k + γ (ˆθ k ˆθ k ) θ k ]dτ e λτ [ λ 2 et k (τ)pe k (τ) e T k PB α k e k ]dτ e λτ e k (τ) 2 dτ (25) e T k PB θ k + γ (ˆθ k ˆθ k ) θ k = γ (ˆθ k ˆθ k ) θ k + γ (ˆθ k ˆθ k ) θ k = γ (ˆθ k ˆθ k ) θ k = γ (ˆθ k sat(ˆθ k))(θ sat(ˆθ k)) (26) ΔL k 2 λ Q min λ, e λτ e k (τ) 2 dτ e λτ [ λ 2 et k (τ)pe k (τ) e T k PB α k e k ]dτ λ 2 et k (t)pe k (t) e T k PB α k e k () λ 2 et k (t)pe k (t) λ 2 λ P min e k 2 (27) e T k PB α k e k e T k P B α k e k λ P max α k e k 2 λ, λ 2 λ P min e k 2 λ P max α k e k 2 λ 2λ P maxα k λ P min (27) ΔL k 2 λ Q min e λτ e k (τ) 2 dτ (28) t [,T], k k =,,m, m m ΔL k (t) = L m (t) L (t) k= 2 λ Q mine λt m (9), L m (t). 2 λ Q mine λt k =, m k= L (t) = 2 e λt e T Pe + k= e k 2 dτ(29) e 2 k dτ L (t) L m (t) L (t) (3) e λτ θ2 (τ)dτ (3) L (t) [,T], L (t) [,T]. m, 2 λ Q mine λt lim m m k= lim k e k 2 dτ < (32) e k 2 dτ = (33), ė k, Barbalat [4]. lim e k = (34) k 5 (Numerical Simulation) ẋ = x 2 ẋ 2 = ax 2 + bu + η(x, t) (35) y = x, a =, b = 24, x (deg), x 2 (deg/s), u (V), η(x, t) =.x sin x 2 sin t. (3) ρ(x, t) =. x, α(x, x d,t)=.+. x. 5(t 3 5t 4 +6t 5 ), t< 5, t<.5 y d (t) = 5((2.5 t) 3 5(2.5 t) 4 +6(2.5 t) 5 ),.5 t< th Chinese Control and Decision Conference (CCDC)

5 T =2.5, x()=[ ] T. (7) (8) (2) (5). A K,K,P,Q P = ( A K = 2 8 ( ),K = ( 2 8 ) ) (,Q= θ max =7. J i,k = max t [,T ] e i(t),i=, 2, k.,2, ε =.6.,, [.5,.2]. e(deg) ) 3 6, ε =.6, γ =5. 5 ˆθ., 2 4., ε,,. e2(deg/s) e(deg) 3: k = e e 2((2)) e2(deg/s) : e e 2((7)) J,k J2,k k 4: J,k J 2,k ((2)) k u(v) 2: u((7)) 6 (Conclusion).,,,,.,,. [] Corless M J, Leitmann G. Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic 26 28th Chinese Control and Decision Conference (CCDC) 6857

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