5Du.pdf INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL Special Issue on Fuzzy Sets and Applications (Celebration of the 50th Anniversary of Fuzzy Sets) ISSN 1841-9836, 10(6):812-824, December, 2015. Fuzzy Robust Tracking Control for Uncertain Nonlinear Time-Delay System Z.-B. Du, T.-C. Lin, T.-B. Zhao Zhenbin Du Yantai University Yantai, Shandong, 264005, China zhenbindu@126.com Tsung-Chih Lin Feng-Chia University Taichung, 40724, Taiwan tclin@fcu.edu.tw Tiebiao Zhao University of California Merced, CA, 95343, USA tzhao3@ucmerced.edu Abstract: The problem of fuzzy robust tracking control is investigated for uncer- tain nonlinear time-delay systems. The nonlinear time-delay system is modeled as fuzzy Takagi-Sugeno (T-S) system, and fuzzy logic systems are used to eliminate the uncertainties of the system. A sufficient condition for the existence of fuzzy controller is given in terms of linear matrix inequalities (LMIs) and adaptive law. Based on Lyapunov stability theorem, the fuzzy control scheme guarantees the desired tracking performance in sense that all the closed-loop signals are uniformly ultimately bounded (UUB). Simulation results of 2-link manipulator demonstrate the effectiveness of the developed control scheme. Keywords: fuzzy T-S model; fuzzy logic systems; nonlinear system; time-delay; tracking control. 1 Introduction Fuzzy control approach offers a powerful and systematical control methodology to handle nonlinear system. Owing to the superior approximation and reasoning abilities of the fuzzy controller, fuzzy control approach has been applied in different applications. With the extensive efforts of the researchers working on the fuzzy control discipline, fruitful stability analysis results have been obtained to aid the design of stable fuzzy controllers. In [1], a fuzzy T-S model was employed to represent the system dynamics of the nonlinear system. The fuzzy T-S model represents the nonlinear system as a weighted sum of some linear subsystems. This particular structure offers a general framework to represent the nonlinear system which is favorable for system analysis. Fuzzy controllers [2-4] were proposed to handle the nonlinear system represented by the fuzzy T-S model. To avoid the effect of the uncertainties, a matching condition is assumed in [5–7], and an upper bound on uncertainties is introduced in [8–10]. The matching condition and the upper bound in dealing with the uncertainties are effective and feasible. However, there exists certain conservatism. The matching condition is a very conservative assumption and the upper bound may be too big or too small, which adds some difficulties to the controller design. On the other hand, it is well known that fuzzy logic systems can uniformly approximate nonlinear continuous functions to arbitrary accuracy. Thus, fuzzy logic systems are used to model uncertain nonlinear systems in [11–13]. Copyright © 2006-2015 by CCC Publications Fuzzy Robust Tracking Control for Uncertain Nonlinear Time-Delay System 813 Time delays are frequently encountered in engineering systems. The existence of time delays usually becomes the source of instability and degrading performance of systems. Therefore, stability analysis and controller synthesis for nonlinear time-delay systems are important both in theory and in practice. By using fuzzy T-S model and fuzzy logic systems, we propose a novel robust tracking con- trol scheme for a class of uncertain nonlinear time-delay system. Fuzzy T-S model is used to approximate the nonlinear system, and a fuzzy state feedback controller is designed to guarantee the stability of the fuzzy system. A compensator based fuzzy logic systems is introduced to eliminate the uncertainties of the system. The fuzzy control scheme ensures the desired tracking performance in sense that all the closed-loop signals are uniformly ultimately bounded (UUB). The rest of the paper is organized as follows. Section 2 provides the problem formulation. Section 3 develops a procedure of the controller design. Section 4 gives the main result. Section 5 presents simulation examples to illustrate the effectiveness of the proposed method. These are followed by conclusions in Section 6. 2 Problem formulation Consider the following uncertain nonlinear time-delay system: ẋ1 = x2, · · · ẋ(β1−1) = xβ1, ẋβ1 = f1(x, x(t − τ1), · · · , x(t − τr), u) + f̃1(x, x(t − τ1), · · · , x(t − τr), u) + d1, ẋ(β1+1) = x(β1+2), · · · ẋn = fm(x, x(t − τ1), · · · , x(t − τr), u) + f̃m(x, x(t − τ1), · · · , x(t − τr), u) + dm, (1) where x = [x1, · · · , x (β1−1) 1 , · · · , x(n−βm+1), · · · , x (βm−1) (n−βm+1) ]T ∈ Rn with β1+β2+· · ·+βm = n and u ∈ Rm are the system state and control input, respectively. fi (i = 1, · · · , m) are known smooth nonlinear functions, f̃i (i = 1, · · · , m) are unknown nonlinear uncertainties, τi(i = 1, · · · , r) are time delays, and di (i = 1, · · · , m) are external bounded disturbances. The control objective of this paper is to find a fuzzy tracking controller such that, while maintaining all the closed-loop signals UUB, the system states of nonlinear system (1) follow those of the given stable reference model. 3 Fuzzy model, reference model and fuzzy controller A fuzzy-model-based control system, formed by a fuzzy model, a reference model, and fuzzy controller connected in a closed-loop, is introduced. 3.1 Fuzzy model A fuzzy dynamic model has been proposed by Takagi and Sugeno to represent a nonlinear system. The fuzzy dynamic model is described by the following fuzzy IF-THEN rules and will be employed here to deal with the control design problem for the nonlinear system in (1). 814 Z.-B. Du, T.-C. Lin, T.-B. Zhao Plant Rule i: IF z1(t) is F i 1 and, · · · , and zs(t) is F i s,THEN ẋ(t) = Aix(t) + r ∑ l=1 Ailx(t − τl) + Biu(t) + d, i = 1, · · · , L (2) where z1(t), · · · , zs(t) are the premise variables, F i j (j = 1, · · · , s) are the fuzzy sets, L is the number of IF-THEN rules, Ai,Bi and Ail are some constant matrices with compatible dimensions, Bi=[0, · · · , b T i1, · · · , 0, · · · , b T im] T ∈ Rn×m with bi1 ∈ R m, · · · , bim ∈ R m , and d = [0, · · · , d1, · · · , 0, · · · , dm] T . Then, the final output of the fuzzy system is inferred as follows: ẋ(t) = L ∑ i=1 µi[Aix(t) + r ∑ l=1 Ailx(t − τl)] + L ∑ i=1 µiBiu(t) + d, (3) where µi = vi(z(t)) / L ∑ i=1 vi(z(t)), vi(z(t)) = s ∏ j=1 F ij (zj(t)) (4) for all t ≥ 0, and F ij (zj(t)) is the grade of membership of zj(t) in F i j . It can be seen that L ∑ i=1 vi(z(t)) > 0, and vi ≥ 0(i = 1, · · · , r) for all t ≥ 0. We have µi ≥ 0(i = 1, · · · , r), L ∑ i=1 µi = 1. Hence, the nonlinear system (1) can be rearranged as the following equivalent system : ẋ(t) = L ∑ i=1 µi[Aix(t) + r ∑ l=1 Ailx(t − τl)] + L ∑ i=1 µiBiu(t) + B∆(x, x(t − τ)) + d, (5) where B∆(x, x(t−τ)) = B∆(x, x(t−τ1), · · · , x(t−τr)) denotes the uncertainties between the non- linear system (1) and the fuzzy model (3), and B = diag[B1, · · · , Bm] with Bi = [0, · · · , 0, 1]T ∈ Rβi . 3.2 Reference model The system states of nonlinear systems (1) are driven to follow those of the following stable reference model ẋr(t) = Arxr(t) + r(t), (6) where xr(t) is a reference state, r(t) is a bounded reference input, and Ar is an asymptotically stable matrix. 3.3 Fuzzy controller A fuzzy controller is chosen as u(t) = ul(t) − uf(t), (7) where ul(t) denotes the fuzzy state feedback control based on T-S model, and uf(t) is the adaptive compensator based on fuzzy logic systems. The former is used to stabilize the linear part of system (11), and the latter is used to compensate the uncertainties. ul(t) and uf(t) are designed as (8) and (10), respectively. For the fuzzy model represented by (2) or (3), fuzzy state feedback control ul(t) shares the same IF parts with the following structure. Fuzzy Robust Tracking Control for Uncertain Nonlinear Time-Delay System 815 Control Rule i: IF z1(t) is F i 1 and, · · · , and zs(t) is F i s , THEN ul(t) = Ki(x(t) − xr(t)), i = 1, · · · , L. Hence, the overall state feedback controller ul(t) is given by ul(t) = L ∑ i=1 µiKi(x(t) − xr(t)), (8) where Ki(i = 1, 2, · · · , L) are matrices with proper dimensions and satisfy the following inequal- ities ĀTijP + PĀij + r ∑ l=1 α −1 l PĀilĀ T ilP + r ∑ l=1 αlI + 1 ρ2 PP + Q̄ < 0, i, j = 1, · · · , L, (9) where Āij = [ Ai + BiKj −BiKj 0 Ar ] , Āil = [ Ail 0 0 0 ] , Q̄ = diag{2Q, 2Q}, P and Q are some symmetric and positive definite matrices, and αl(l = 1, · · · , r) are positive constants. The adaptive compensator based on fuzzy logic systems uf(t) are as follows: uf(t) = { E−1û(x, x(t − τ)|Θ), if E is nonsigular ET (I + EET )−1û(x, x(t − τ)|Θ), if E is sigular (10) where Ei = [b T i1, · · · , b T im] T ∈ Rm×m,E = L ∑ i=1 µiEi, and û(x, x(t − τ)|Θ) is constructed by fuzzy logic systems. The weight Θ is an adaptive parameter, which is adapted by Θ̇ = η1Ψ T (x, x(t − τ))B̄T Px̃, (11) where η1 is a positive constant, Ψ(x, x(t−τ)) is a fuzzy basis-function matrix, and x̃ = [x T , xTr ] T . In the following, we explain the solution of the inequalities (9) and the construction of fuzzy logic systems û(x, u|Θ). 1) By Schur complements, the inequalities (9) are transformed into the LMIs. For the conve- nience of design, P is chosen as the formP = diag{P1, P2}, where P1, P2 are some symmetric and positive definite matrices. The inequalities (9) are equivalent to the following matrix inequalities    S11 −P1BiKj 0 −(BiKj) T P1 S22 P2 0 P2 −ρ 2I    < 0, i, j = 1, 2, · · · , L, (12) Where S11 = P1(Ai + BiKj) + (Ai + BiKj) T P1 + r ∑ l=1 α −1 l P1AilA T il P1 + r ∑ l=1 αlI + 1 ρ2 P1P1 + 2Q, S22 = P2Ar + A T r P2 + r ∑ l=1 αlI + 2Q. The matrix inequalities (12) imply S11 < 0. Let W = P −1 1 and Yj = KjW . S11 < 0 is equivalent to the LMIs with prescribed Q and αl(l = 1, · · · , r),   S W W −( r ∑ l=1 αlI + 2Q) −1   < 0, i, j = 1, 2, · · · , L (13) 816 Z.-B. Du, T.-C. Lin, T.-B. Zhao where S = AiW + WA T i + BiYj + (BiYj) T + r ∑ l=1 α −1 l AilA T il + (ρ2)−1I. By solving the LMIs (13), P1 and Kj(j = 1, 2, · · · , L) could be obtained. And then, by substituting P1 and Kj(j = 1, 2, · · · , L) into (12), (12) becomes standard LMIs. We can easily solve P2 from (12). Therefore, the common solution P and Kj(j = 1, 2, · · · , L) could be found. Remark 1: Either the matching condition or the upper bound is related to a large number of matrix operations. Without the matching condition and the upper bound, the dimension of the LMIs of this paper is reduced. 2) Fuzzy adaptive systems consist of four main components: fuzzy rule base, fuzzy inference engine, fuzzifier and defuzzifier [11]. The fuzzy rule base is composed of a collection of IF-THEN inference rules: Rl: IF x1 is A l 1, · · · , xn is A l nŁŹTHEN y is G l(l = 1, · · ·p) where Ali(i = 1, · · · , l) and G l(l = 1, · · ·p) are fuzzy sets. The kth element of ∆(x, x(t−τ)) is of the following form: ∆̂k(x, x(t − τ)|θk) = ξ T k (x, x(t − τ))θk, where θk = (θ 1 k , · · · , θ p k )T ∈ Rp, ξT k (x, x(t − τ)) = (ξ1 k , · · · , ξ p k ) ∈ Rp, ξlk = n ∏ i=1 µF l i (xi, xi(t − τ)) / p ∑ l=1 n ∏ i=1 µF l i (xi, xi(t − τ)), µF l i (xi, xi(t−τ)) = µF l i (xi) r ∏ j=1 µF l i (xi(t − τj)), and µF l i (xi)(i = 1, 2, · · · , n) are the membership functions. In this paper, fuzzy logic systems are constructed to eliminate the uncertainties ∆(x, x(t−τ)). The approximation form is given as follows: ∆̂(x, x(t − τ)|Θ) = Ψ(x, x(t − τ))Θ, (14) where Ψ(x, x(t − τ)) = diag[ξT1 (x, x(t − τ)), · · · , ξ T m(x, x(t − τ))], Θ = [θ T 1 , θ T 2 , · · · , θ T m] T . Define the optimal the parameter Θ∗ as Θ∗ = arg min Θ∈Ω1 [ sup x∈U1 |û(x, x(t − τ)|Θ) − ∆(x, x(t − τ))|], (15) where U1 = {x ∈ R n : ‖x‖ ≤ N}, Ω1 = {Θ ∈ R pm: ‖Θ‖ ≤ M}. U1, Ω1 denote the sets of suitable bounds on x,Θ respectively, N, Mare upper bounds. The approximation error for the function ∆(x, x(t − τ))can be expressed as ∆̂(x, x(t − τ)|Θ) − ∆(x, x(t − τ)) = Ψ(x, x(t − τ))Θ̃ + w, (16) where Θ̃ = Θ − Θ∗ the estimation error for Θ, w = [w1, · · · , wm] T is a residual term. Remark 2: In order to guarantee ‖Θ‖ ≤ M, the adaptive law (11) must be modified by the projection algorithm [11] as follows: Θ̇ = { η1Ψ T (x, x(t − τ))B̄T Px̃, if(‖Θ‖ 0 where PΘ[.]=η1Ψ T (x, x(t − τ))B̄T Px̃ − η1 x̃T PB̄ Ψ(x,x(t−τ))Θ ‖Θ‖2 . Fuzzy Robust Tracking Control for Uncertain Nonlinear Time-Delay System 817 4 Stability analysis Substituting (11) into (11) yields ẋ(t) = L ∑ i=1 µi[Aix(t) + r ∑ l=1 Ailx(t − τl)] + L ∑ i=1 L ∑ j=1 µiµjBiKj(x(t) − xr(t)) −B(û(x, x(t − τ)|Θ) − ∆(x, x(t − τ))) + d. (17) Let x̃(t) = [xT (t), xTr (t)] T , andB̄ = [ B T 0 ]T . By using (11) and (17), a new extended closed- loop system is as follows: ˙̃x(t) = L ∑ i=1 L ∑ j=1 µiµj[Āijx̃(t)+ r ∑ l=1 Āilx̃(t − τl)]+B̄(−(û(x, x(t−τ)|Θ)−∆(x, x(t−τ)))+d′, (18) where d′ = [dT , rT (t)]T . When fuzzy logic systems û(x, x(t−τ)|Θ) could eliminate ∆(x, x(t−τ)), the closed-loop system (18) is stable. By denoting w′ = [w̄T , rT (t)]T , w̄ = [0, · · · , d1 − w1, · · · , 0, · · · , dm − wm] T and using (14), the closed-loop system (18) could be rewritten as ˙̃x(t) = L ∑ i=1 L ∑ j=1 µiµj[Āijx̃(t) + r ∑ l=1 Āilx̃(t − τl)] + B̄(−Ψ(x, x(t − τ))Θ̃) + w′. (19) From the above analysis, we have the following conclusion. Theorem 1. Given a matrixQ > 0, scalarsρ > 0,αl(l = 1, · · · , r) > 0, η1 > 0.If there exist matricesP > 0,Kj(j = 1, 2, · · ·, L) such that the inequalities (9) hold. If the updating law for fuzzy logic systems is chosen as (11). Then there exists a controller (11) with the fuzzy state feedback controller (8) and the adaptive compensator (10) such that, while maintaining all the closed-loop signals UUB, the following tracking performance(20) is achieved ∫ T 0 (x(t) − xr(t)) T Q(x(t) − xr(t))dt ≤ x̃ T (0)Px̃(0) + 1 η1 Θ̃T (0)Θ̃(0) + ρ2 ∫ T 0 (w′T w′)dt. (20) Proof: Consider the following Lyapunov-Krasoviskii candidate V = 1 2 x̃T Px̃ + 1 2 r ∑ l=1 ∫ t t−τl αlx̃ T (v)x̃(v)dv + 1 2η1 Θ̃T Θ̃, (21) where V̇ = V̇1 + V̇2, V̇1and V̇2 are given in (22) and (26), respectively. V̇1 = 1 2 ( L ∑ i=1 L ∑ j=1 µiµj[Āijx̃(t) + r ∑ l=1 Āilx̃(t − τl)]) T Px̃(t) + 1 2 x̃T (t)P( L ∑ i=1 L ∑ j=1 µiµj[Āijx̃(t) + r ∑ l=1 Āilx̃(t − τl)]) + 1 2 w′T Px(t) + 1 2 xT (t)Pw′ + 1 2 r ∑ l=1 αlx̃ T (t)x̃(t) − 1 2 r ∑ l=1 αlx̃ T (t − τl)x̃(t − τl) ≤ 1 2 ( L ∑ i=1 L ∑ j=1 µiµj[x̃ T (t)ĀTijPx̃(t)+x̃ T (t)PĀijx̃(t)+ r ∑ l=1 α −1 l x̃T (t)PĀilĀ T ilPx̃(t)+ r ∑ l=1 αlx̃ T (t − τl)x̃(t − τl)] 818 Z.-B. Du, T.-C. Lin, T.-B. Zhao − 1 2 ( 1 ρ Px(t) − ρw′)T ( 1 ρ Px(t) − ρw′) + 1 2 ρ2w′T w′ + 1 2ρ2 x̃T (t)PPx̃(t) + 1 2 r ∑ l=1 αlx̃ T (t)x̃(t) −1 2 r ∑ l=1 αlx̃ T (t − τl)x̃(t − τl) ≤ 1 2 L ∑ i=1 L ∑ j=1 µiµjx̃ T (t)(ĀTijP +PĀij + r ∑ l=1 α −1 l PĀilĀ T ilP + r ∑ l=1 αlI + 1 ρ2 PP)x̃(t)+ 1 2 ρ2w′T w′. (22) Substituting (9) into (22) yields V̇1 ≤ − 1 2 x̃T (t)Q̄x̃(t) + 1 2 ρ2w′T w′. (23) By using (11), V2 = [x̃ T PB̄(−(Ψ(x, x(t − τ))Θ̃) + 1 η1 Θ̃T Θ̇] = 0. (24) From (23)-(24), V̇ ≤ − 1 2 x̃T (t)Q̄x̃(t) + 1 2 ρ2w′T w′. (25) When‖x̃(t)‖ > ρ λmin(Q̄) ‖w′‖,V̇ < 0.Thus, the closed-loop system consisting of (1), (11), (8) and (10) is UUB . ✷ Note that ∫ T 0 (x(t) − xr(t)) T Q(x(t) − xr(t))dt ≤ ∫ T 0 x̃T (t)Q̄x̃(t)dt. Integrating the above equation (25) from t = 0 to T yields (20). 5 Simulation example In this section, we provide an example to verify the effectiveness of the proposed control scheme. Example: Consider the following 2-link manipulator system in [14] q̈(t) + C(q, q̇)q̇(t) + g(q) = B(q)u(t) + r ∑ i=1 ξi(t)q(t − τi) + d′, (26) where C(q, q̇) = H−1(q)C′(q, q̇), g(q) = H−1(q)g′(q), B(q) = H−1(q),d′ = H−1(q)d,q = [q1, q2] T , ξi(t)(i = 1, · · · , r)are uncertain and bounded, and dis the external bounded disturbance. The reference model is as follows: ẋr(t) = Arxr(t) + r(t), (27) whereAr = diag{Ar1, Ar2},Ar1 = Ar2 = [ 0 1 −6 −5 ] , r(t) = [0, r1(t), 0, r2(t)] T , r1(t) = r2(t) = 3 sin(2t). Fuzzy Robust Tracking Control for Uncertain Nonlinear Time-Delay System 819 Step1: Denote x1 = q1, x2 = q̇1, x3 = q2, and x4 = q̇2. Then, (26) can be written as a fourth- dimension system. A nine-rule fuzzy T-S model is used to approximate the nonlinear 2-link manipulator system at x1 = − π 2 , 0, π 2 and x3 = − π 2 , 0, π 2 , where A1 =       0 1 0 0 5.927 −0.001 −0.315 −0.0000084 0 0 0 1 −6.859 0.002 3.155 0.0000062       , A2 =       0 1 0 0 3.0428 −0.0011 −0.1791 −0.0002 0 0 0 1 −3.5436 0.0313 2.5611 0.0000114       , A3 =       0 1 0 0 6.2728 0.003 0.4339 −0.0001 0 0 0 1 −9.1041 0.0158 −1.0574 −0.000032       , A4 =       0 1 0 0 6.4535 0.0017 1.2427 −0.0002 0 0 0 1 −3.1873 0.0306 −5.1911 −0.000018       , A5 =       0 1 0 0 11.1336 0 −1.8145 0 0 0 0 1 −9.0918 0 9.1638 0       , A6 =       0 1 0 0 6.1702 −0.001 1.687 −0.0002 0 0 0 1 −2.3559 0.0314 4.5298 −0.000011       , A7 =       0 1 0 0 6.1206 0.0041 0.6205 0.0001 0 0 0 1 8.8794 0.0193 −1.0119 0.000044       , A8 =       0 1 0 0 3.6421 −0.0018 0.0721 0.0002 0 0 0 1 2.429 −0.0305 2.9832 −0.000019       , A9 =       0 1 0 0 6.2933 −0.0009 0.2188 −0.000012 0 0 0 1 −7.4649 0.0024 3.2693 −0.0000092       , A11 = A21 = A31 = A41 = A51 = A61 = A71 = A81 = A91 =       0 0 0 0 0.01 0 0 0 0 0 0 0 0 0 0 0       , A12 = A22 = A32 = A42 = A52 = A62 = A72 = A82 = A92 =       0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.01 0       , B1 = [ 0 1 0 −1 0 −1 0 2 ]T , B2 = [ 0 0.5 0 0 0 0 0 1 ]T , B3 = [ 0 1 0 1 0 1 0 2 ]T , B4 = [ 0 0.5 0 0 0 0 0 1 ]T , B5 = [ 0 1 0 −1 0 −1 0 2 ]T , B6 = [ 0 0.5 0 0 0 0 0 1 ]T , 820 Z.-B. Du, T.-C. Lin, T.-B. Zhao B7 = [ 0 1 0 1 0 1 0 2 ]T , B8 = [ 0 0.5 0 0 0 0 0 1 ]T , B9 = [ 0 1 0 −1 0 −1 0 2 ]T . The membership functions are adopted as the triangle type. Step 2: On the basis of Theorem1, withα1 = 0.005, α2 = 0.005, andρ = 1,we have K1 = [ -76.9685 -42.9566 -19.6919 -8.9116 6.0025 -0.4619 -51.4252 -25.0336 ] , K2 = [ -77.7828 -42.8754 -13.6211 -5.9413 8.7179 1.2251 -50.6614 -24.6859 ] , K3 = [ -76.8347 -42.9089 -19.8204 -8.9785 5.8595 -0.5257 -51.3739 -25.0109 ] , K4 = [ -77.7828 -42.8754 -13.6211 -5.9413 8.7179 1.2251 -50.6614 -24.6859 ] , K5 = [ -77.7828 -42.8754 -13.6211 -5.9413 8.7179 1.2251 -50.6614 -24.6859 ] , K6 = [ -77.7828 -42.8754 -13.6211 -5.9413 8.7179 1.2251 -50.6614 -24.6859 ] , K7 = [ -79.8424 -43.4072 -6.0780 -2.2626 12.7745 3.6898 -50.2150 -24.4989 ] , K8 = [ -77.7828 -42.8754 -13.6211 -5.9413 8.7179 1.2251 -50.6614 -24.6859 ] , K9 = [ -80.1162 -43.5088 -5.8152 -2.1328 13.0509 3.8147 -50.3242 -24.5472 ] . Step 3: In fuzzy adaptive compensator, the membership functions are selected as µF 1 i (xi) = 1 1 + exp[5(xi + 0.8)] , µF 2 i (xi) = exp[−(xi + 0.6) 2], µF 3 i (xi) = exp[−(xi + 0.4) 2], µF 4 i (xi) = exp[−(xi) 2], µF 5 i (xi) = exp[−(xi − 0.4) 2], µF 6 i (xi) = exp[−(xi − 0.6) 2], µF 7 i (xi) = 1 1 + exp[5(xi − 0.8)] , i = 1, 2, · · · , 4. Step 4: Some parameters are choose as η1 = 10, r = 2, τ1 = 0.5, τ2 = 1, ξ1(t) = 5 + 20sin(5t), andξ1(t) = 1 + 15cos(5t), Θ(0) = [0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2], (x1(0), x2(0), x3(0), x4(0), xr1(0), xr2(0), xr3(0), xr4(0)) = (0.4, 0,−0.4, 0, 0, 0, 0, 0). By using the method in Theorem 1, the tracking performances of x1(t), x2(t), x3(t), x4(t) are shown in Fig.1,and the control efforts u1(t) and u2(t) are given in Fig.2,respectively. Fuzzy Robust Tracking Control for Uncertain Nonlinear Time-Delay System 821 0 2 4 6 8 10 −1 −0.5 0 0.5 1 t(Sec.) S ta te 1 0 2 4 6 8 10 −1 −0.5 0 0.5 1 t(Sec.) S ta te 2 0 2 4 6 8 10 −1 −0.5 0 0.5 1 t(Sec.) S ta te 3 0 2 4 6 8 10 −1 −0.5 0 0.5 1 t(Sec.) S ta te 4 x 1 x r1 x 2 x r2 x 3 x r3 x 4 x r4 Figure 1: The responses of x1,x2,x3,x4, xr1,xr2,xr3andxr4 0 2 4 6 8 10 −80 −60 −40 −20 0 20 40 t(Sec.) C o n tr o l1 0 2 4 6 8 10 −10 0 10 20 30 40 50 60 70 C o n tr o l2 t(Sec.) Figure 2: The control inputs u1,u2 0 2 4 6 8 10 −1 −0.5 0 0.5 1 t(Sec.) S ta te 1 0 2 4 6 8 10 −1 −0.5 0 0.5 1 t(Sec.) S ta te 2 0 2 4 6 8 10 −1 −0.5 0 0.5 1 t(Sec.) S ta te 3 0 2 4 6 8 10 −1 −0.5 0 0.5 1 t(Sec.) S ta te 4 x 1 x r1 x 2 x r2 x 3 x r3 x 4 x r4 Figure 3: The responses of x1,x2,x3,x4, xr1,xr2,xr3andxr4 822 Z.-B. Du, T.-C. Lin, T.-B. Zhao 0 2 4 6 8 10 −1 −0.5 0 0.5 1 t(Sec.) S a te 1 0 2 4 6 8 10 −1 −0.5 0 0.5 1 t(Sec.) S a te 2 0 2 4 6 8 10 −1 −0.5 0 0.5 1 t(Sec.) S a te 3 0 2 4 6 8 10 −1 −0.5 0 0.5 1 t(Sec.) S a te 4 x 1 x r1 x 2 x r2 x 3 x r3 x 4 x r4 Figure 4: The responses of x1,x2,x3,x4, xr1,xr2,xr3andxr4 0 2 4 6 8 10 −0.1 0 0.1 0.2 0.3 0.4 t(Sec.) S ta te 1 0 2 4 6 8 10 −0.8 −0.6 −0.4 −0.2 0 0.2 S ta te 2 t(Sec.) 0 2 4 6 8 10 −0.4 −0.3 −0.2 −0.1 0 0.1 t(Sec.) S ta te 3 0 2 4 6 8 10 −0.5 0 0.5 1 t(Sec.) S ta te 4 x 4 x r4 x 2 x r2 x 1 x r1 x 3 x r3 Figure 5: The responses of x1,x2,x3,x4, xr1,xr2,xr3andxr4 0 2 4 6 8 10 −15 −10 −5 0 5 t(Sec.) C o n tr o l1 0 2 4 6 8 10 −5 0 5 10 15 t(Sec.) C o n tr o l2 u 1 u 2 Figure 6: The control inputs u1,u2 Fuzzy Robust Tracking Control for Uncertain Nonlinear Time-Delay System 823 Whenτ1 = 1,τ2 = 1, simulation results are shown in Fig.3.Whenτ1 = 1,τ2 = 2, simulation results are shown in Fig.4. When r1(t)andr2(t) are square waves having an amplitude ±0.2 with a period of 2π, the tracking performances of x1(t), x2(t), x3(t), x4(t) are shown in Fig. 5, and the control efforts u1(t) and u2(t) are given in Fig.6. 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