Dev. Chem Eng. Mineral Process., 9(IR),pp.175-182, 2001. Robust Sampled-Data Control for Fuzzy Uncertain Systems L.-S. Hu', H.-H. Shao, Y.-X. Sun' Department of Automation, Shanghai Jiaotong University, Shanghai 200030, P. R. China #National Laboratory of Industrial Control Technology, Zhejiang University, Hangzhou 310027, P. R. China Sampled-data control which is capable of stabilizing general nonlinear systems with the intersample behavior taken into account axe of great current interest. In this paper, the Thkagi-Sugeno fuzzy models were considered for representing the nonlinear plants. For the Thkagi-Sugeno fuzzy model and the Takagi-Sugeno uncertain fuzzy model, the fuzzy sampled-data control and the robust sampleddata control were considered in this paper. The robust sampled-data control for the TakagiSugeno uncertain fuzzy model with the constraints on the states was also developed. The results were described as the matrix inequalities, which could be solved by LMITool. Finally, a numerical example based on the missile autopilot model was considered, the result showed the effectiveness of the proposed procedures. Introduction Mathematical models of the system under consideration are the basis of the control design methodologies. In most cases, the industrial process presented is highly nonlinear, we usually need to linearize the system on the nominal operating point so that we are able to proceed with the control design using linear system theory. However, for systems with shifting or multiple operating conditions, the traditional mathematical model may not be sufficient to represent the dynamic behavior. As each local model is valid only for a certain range of operating conditions, so these results can only guarantee the local stability of nonlinear systems. Recently, a nonlocal approach, the so-called Takagi-Sugeno (TS)type fuzzy model, which is conceptually simple and straightforward is proposed for nonlinear systems design via fuzzy control [l,21. The Takagi-Sugeno fuzzy model, proposed by Takagi and Sugeno [l]is based on a fuzzy partition of input-output space. In each fuzzy subspace a linear input-output relation is formed. The output of fuzzy reasoning is given by the aggregation of the values inferred by some implications that were applied to an input. There are a number of works aimed to analyze and synthesize fuzzy control systems based on the Takagi-Sugeno fuzzy model (see [3, 41 and refer therein). *Author for correspondence (email: lshu@maill.sjt u .edu. cn). 175 L-S. Hu, H.-H. Shao and Y.-X. Sun We purpose to study the design procedure of sampled-data control for the Takagi-Sugeno fuzzy model in this paper. Sampled-data control is a direct design procedure of a digital controller for the continuous time systems, which takes account of the effects of the intersample behavior of the system and has no degrade of closed-loop performance. Although there is much research on H2 and H , sampled-data control for linear systems [2], a few results of sampleddata control for the nonlinear systems are obtained. Hagiwara et al. [5] and Okuyama and Takemori [6] dealt with stability analysis of sampled-data systems with a sector nonlinearity. Rui et al. [7] dealt with sampled-data control for a class of nonlinear systems. In this paper, the sampled-data control for the nonlinear systems described as Takagi-Sugenofuzzy models was considered. For the Takagi-Sugeno fuzzy model and uncertain fuzzy model, the fuzzy sampleddata control and the fuzzy robust sampled-data control were considered. The results were described as the matrix inequalities. The fuzzy robust sampled-data control for the Takagi-Sugeno uncertain fuzzy model with the hard constraints on the states was also developed. The result was also described as the matrix inequality, which could be solved by LMITool. Finally, to demonstrate the design procedure, we applied the design procedure to a nonlinear missile autopilot problem [3]. Simulation result shows the effectiveness of the proposed approach. This paper is organized as follows. First, we present some preliminaries on the Takagi-Sugenofuzzy model. Then, we present the main results of fuzzy sampleddata control and robust sampled-data control and constrained robust sampleddata control for the Takagi-Sugenofuzzy model. After that, a numerical example based on the missile autopilot model was considered to show the effectiveness of the proposed procedures. Finally, we give our conclusions. Preliminary Discussion In this section, the nonlinear plant is represented as the following Takagi-Sugeno fuzzy model, and its ith Plant Rule is I f zl(t) is Fjl and - . . and z p ( t ) i s Fip Then k ( t ) = Aiz(t) + B+(t). . (1) i = 1,2, . . .,T , where F,j is the fuzzy set and T is the number of If-Then rules. zj(t), j = 1 , 2 , . - -, p are the premise variables. Given a pair of (z(t),u(t)), the final output of the fuzzy systems is inferred as follows + k ( t )= . . . (2) Xi(z(t))(Aiz(t) BiU(t)), where ~ ( t=)( z l ( t ) ,za(t),. . . ,z p ( t ) ) .Xi(z(t)) = xi==, *), hi * where haMt)) = *=I II;=lF,j(zj(t)) for all t , satisfy X i ( z ( t ) ) = 1, X i ( z ( t ) ) 20, i = 1,2,...,r. F i j ( z j ( t ) )is the grade of membership of z j ( t ) in Fij. Traditionally, the parallel distributed compensation can be employed to design each local fuzzy control rule so as to compensate each local rule of a fuzzy system (as in many reports). Except for the discreteness of the fuzzy sampled-data controller, its ith Control Rule (3) has the same structure as Plant rule i (1)of the fuzzy model If 176 zl(tk) is Fil and . and zp(tk) is FiP Then G i [ t k ] = Fi[tk]z(tk). .. (3) Robust Sampled-DataControlfor Fuuy Uncertain Systems for i = 1,2,. .. ,T . Then the fuzzy sampled-data controller is inferred as follows = EL1Xi(z(tk))Gi[tk], with Sa[tk]= F i [ t k ] ~ ( t kLet ) . Z ( t ) = ( Z T ( t ) ,uT(t))T, and use the zero order holder, u ( t ) = ii[tk),for t E (tk,&+I],k = 0,1,2,-.-,we obtain the following fuzzy hybrid system i ( t )= c;=, Ai(.z(t>)Ai5(t), t E (tk,tk+l], c;=lAi(Z(tk))(A+ BFi[tk])qtk), qt;> = [ Ai = A* Bi 1, [ I 0 [ . . . (4) 1, o], B = pi[tk[tkl= [ & [ t k ] 0 1 . The fuzzy sampled-data controller design is to determine the local sampleddata feedback gains Fi[tk]in the consequent parts. The feedback gains kftk] are determined by an LMI-based design technique presented in Theorem 1 in the next section. However, as the reason presented in the following, the main purpose of this paper is to present the sampled-data control, with the form where A = u(t>= f i [ t k ] , f i [ t k ] = F[tk]Z(tk), t E (tk, t k + l ] , for the fuzzy uncertain systems and the ones with state constraints. . .. ( 5 ) f i z z y Sampled-Data Control In this section, we first present fuzzy sampled-data control for the system of equation (1),then consider sampled-data control for the fuzzy uncertain system and the one with state constraints. - Theorem 1 For given positive scalars pk 5 1, k = 0,1,2 - -,and the sampling period T,,the system (1) is stabilizable by the sampled-data fuzzy controller (3), if there exist symmetric piecewise continuous matrix functions P[t]= P [ t i ]+ (t - t b ) Y [ t ~ >]0, &I [t]= Ql[tt] + (t- tk)Z[tk]2 0 , for t E ( t k , t k + 1 ] , the semipositive symmetric matrices Q&], and the controller gain Fi[tk],i = 1,2, ' - .,r, satisfying the following matrix inequalities Y[tk]f P[t]A¶+ Apqt] fori < j , 2Y[tk]4- P[t](Ai4-Aj) for t + (T - l)Ql[t] 5 0, + (Ai + Aj)TP[t]- 2Q1[t]5 0, E ( t k , tk+l], and Proof. From Corollary 1 of [8] and Theorem 1 of [9],the proof is trivial. I77 L-S. Hu. H.-H. S ~ L Wand Y.-X.Sun Along the same lines of Theorem 1, robust fuzzy sampled-data control for the uncertain fuzzy model can be obtained. However, in the following, we presented robust sarnpled-data control and constrained robust sampled-data control with the form ( 5 ) for the uncertain fuzzy systems to improve the precision of control. Robust Sampled-data Control Instead of the system (l),we consider the following uncertain fuzzy model ) is Fil and ... and zp (t )i s FipThen { I5 (f tz)=l( t(Ai + A A i )z(t )+ (Bi + ABi)u(t), . , . (6) i = 1,2,. . - ,T , where Ai, Bi are matrices with appropriate dimensions, AAi and ABi are real valued continuous uncertain matrix functions specified as [ AAi ABi ] = DiAi [ Eia Eib 3, where Di, Ei, and Eib are real matrices with appropriate dimensions, Aj are Lebesgue measurable unknown matrix functions satisfying the following bounds ATAi 5 I. Similar to ( 2 ) , we obtain k ( t ) = C61Xi(z(t))((Ai+ A A i ) z ( t )+ (Bi+ ABi)u(t)). . . . (7) If usmg the sampled-data controller ( 5 ) to stabiliie the plant (7), the closedloop system is described as k ( z ( t ) ) ( A+ i AAi)z(t),tE tk+1], { i ( t )=-C&1 (A + k = 0,1,2, - . , (tk, + BP[tk])E(tk), . . . (8) Theorem 2 For given positive scalars / l k 5 1, k = 0,1,2. ., and the sampling period TB,the system (6) is stabilizable by the sampled-data controller (S), if there exists symmetric piecewise continuous matrix function P[t]= P [ t i ]+ (t t k ) Y [ t k ] > 0, 91[t]= & l [ t ; ] + ( t - t k ) 2 [ t k ] 2 0, f o r t E (tk,tk+l]t =0 ,1 ,2 ,'.-, positive scalars ( i , ( l i , <2i, C k , and the controller gain F[tk] satisfying the followang matrix inequalities . . . (9) 5 0,i< j , . . . (10) . .. (11) 178 Robust Sampled-Data Controlfor Fuuy Uncertain Systems Proof. The proof is trivial. We present the outline of the proof. It is easy to know that the system ( 8 ) is equivalent to i ( t )= fr CI=1C,'=iAi(t(t))Aj(Z(t))(Ai +A& + Aj + AAj)Z(t), for t E (h,tk+i],and z(t;) = ( A+ BF[tk])Z(tk), k = 0,1,2,.. .. Choose the Lyapunov function candidate as V ( t ,5 ) = ZT(t)P[t]Z(t), which is right continuous on (tk,tk+l],we obtain c;==, V ( t , Z )= ZT(t){Y[tk] + c;=, Xi(Z(t))Aj(t(t)) (P[t](Ai + A& + Aj + AAj) + (Ai + AAi + Aj + AAj)TP[t])}Z(t) = xi==, A?(t(t))ZT(t)(Y[tk] P[t](Ai AA,) +(Ai + AAi)TP[t])Z(t)2 &(t(t))Aj(t(t))ZT(t)(Y[tk] +iP[t](Ai AAi Aj + AAj) + $(Ai+ AAi + Aj + AAj)TP[tJ)Z(t). 3 + &j + + + + z:<j If there exist the semipositive symmetric matrix Ql[t], positive scalars tli, and ck, such that (10)holds for i < j,then we obtain from Corollary 1 of [8] V(t,Z)5 +(T Ci=1A : ( ~ ( t ) ) Z ~ ( t ) ( Y+[P[t](A, t k ] + AAi) + (Ai + AAi)TP[t] - 1)Qi[t])Z(t) - 2ck x:<j Ai(.Z(t))Aj(Z(t))ZT(t)Z(t), for t E (tk,tk+lI. It is easy to know if the condition (9) holds, then V ( t , E ) 5 -Ck for t E (tk,tk+l].For the given positive scalars pk, we obtain, if the condition (11) holds, then V ( t ; , Z ( t t ) )- p k V ( t k , Z ( t k ) ) 5 0. Furthermore from Theorem 1 of 191, we know the Theorem 2 holds. Constrained Robust Sampled-Data Control In this section, we consider the robust sampled-data control for the system (6) with the hard constraints on the states. For given positive scalars 6 and 0 with 6 < 0,the state constraints are described as Ix(t)l < 0,t 2 0, for 1x01 < 6. Let .=[; ;]. Theorem 3 For given time-domain performance indexes 6 and 0 with 6 < 0, positive scalars pk 5 1, C(k 5 1, k = 0 , 1 , 2 , - . - ,and the sampling period T,, the system (6) with 1x01 < 6 , is stabilizable by the sampled-data controller (S), and satisfies Ix(t)l < r, t 2 0 , if there exist symmetric piecewise continuous matrix function ~ 1 [ = t ]Pl[t;] ( t - tk)Yl[tk]> 0, P2[t]= Pz[t;] + ( t - tk)Y2[tk]> 0, Q[t]= + ( t - tk)Z[tk]2 0, Qi[t]= Qi[t;] + ( t - tk)Zi[tkI 2 0, for t E (tk,tk+l],k = 0 , 1 , 2 , . . . , let P[t]= Pl[t]+ P2[t],Y[tk]= Yl[tk]+ Y2[tk], positive scalars cli, Q , &it &j, J l l i , E 2 1 j , Ck, the positive scalars a l , a2 and b satisfying Q[tt] a1J2 + + a2b2 < bu2, 179 L.-S. Hu, H.-H. Shao and Y.-X. Sun and the controller gain F[tk]satisfying the following matrix inequalities . . . (13) P2[t]2 bCTC, . . . (17) Proof. Choose the Lyapunov function candidate as V ( t ,Z) = Z*(t>P[t]Z(t), Vl(t,Z)= ZT(t)P1[t]Z(t). If there exist the semipositive symmetric matrix Q[t], Q l [ t ]positive , scalars <I,, ci, &i, &, & j , Ck such that the conditions (13) hold for a < j , and the conditions (12) hold, then V ( t , Z ) 5 0, and Vl(t,Z)5 -Ck li512, for t E ( t k , t k + l ] . For the given positive scalars p k 5 1, p i 5 1, if the conditions (14) hold, then V(tl,Z(tkf))- pkV(tk,Z(tk)) 5 0, K(tl,Z(tk+)) pf’V1(tk,S(tk))5 0. From Theorem 1 of [9],we h o w the Theorem 3 holds. Remark 1 The conditions listed in Theorem 1, Theorem 2 and Theorem 3 are not the linear matrix inequalities exactly, even if we specified Fi[tk] or F[tk]. 180 Robust Sampled-Data Controlfor Fuuy Uncertain Systems However PI and P2, jkom their definitions, are piecewise linear continuous functions. So we only need to check the conditions on two points of the sampling intervals ( t k , &+I], for k = 0,1,2,.-. In such cases, these conditions are the linear matrix inequalities if we specified F,[tk] or F [ t k ] . After specifying all the parameters, such as pk, Ck etc., then using Matlab LMITool, optimizing Fi[tk] or F[tk] at each sampling intervals by other optimal procedures, we can solve these mat* inequalities. Numerical Examples Example 1. The missile autopilot model [3] is the complex nonlinear one. We propose the robust control design based on the Takagi-Sugenofuzzy model as an alternative approach to the missile guidance problem. This model represents a missile traveling at Mach 3 at an altitude of 20,000ft..The nonliiear dynamic model is as follows where a angle of attack, q pitch rate, W weight, V speed, IvD pitch moment of inertia, Q dynamic pressure, S reference area, d reference diameter, Z = C,QS normal force, M = C,QSd pitch moment. The normal force and pitch moment aerodynamic coefficients C, and Cm are approximated by C, = +=(a) + b,d, C, = 4m(a) b,6, where 6 is fin deflection. b, = -0.034(1 + Ab,), b, = -0.206(1+ Abm), 4z(a) = 0.000103~~~ - 0-00945ala1 - 0.170(1+ Adz)al) #,(a) = 0.000215cU3- 0.00195ala1 0.051(1+ A4,)a. It is noted that there exist uncertain parameter variations which are denoted by Ab,, Ab,, Ad, and A&,. These parameter variations may be due to modeling error, system uncertainty, and external disturbance. We assume that these variations are within 30%, i.e., lAb,(t)l 5 0.3, lAb,(t)l 5 0.3, lA$,(t)J5 0.3,and lA#,(t)l 5 0.3, V t 2 0. We linearize the original system around the following angle of attack, a = 0, 30°,-30’. Let x1 denote the angle of attack, 5 2 denote the pitch rate q and u denote the fin deflection 6. Let x = ( x I , x ~ then ) ~ , the system was described as the following 2-rule Takagi-Sugeno fuzzy model + + I” 51 is Small Then k ( t )= Alx(t)+ Blu(t), ..- (18) If 5 2 is Large Then z(t)= Azs(t)+ BZu(t), -O.SOS(l + Arb,) 1 -0.121(1+ Ab,) [ 32.404(1+ A&) 0 ] ’ B1 = [ -130.888(1+ -1.112(1+ A$z) -0.105(1+ Ab,) [ -216.3464(1+ A&) 0 1, B2 = [ -130.888(1+ Ab,) 1, where A1 = Abm) 1 1, A2 = and lAbz(t)l 5 0.3,lAb,(t)l 5 0.3,lA+,(t)l 5 0.3, lA$m(t)l5 0.3,V t 2 0. Triangular membership functions are adopted in this model. It was noted that the 2-rule TakagiSugeno fuzzy model has lower natural frequency than the original missile system. The robust sampled-data controller was designed to make the missiles to follow certain trajectories with the same structure as (18). Choose the sampling pe181 L-S. Hu. H.-H. Shao and Y.-X. Sun 1, riod T, = 5ms and F[O] = [ 10 0.5 pk = 0.95, for simplicity. We apply the robust sampled-data controller, designed using Theorem 2, to the original missile system. The angle of attack follows the desired trajectory after 1.5 seconds, which is superior to that of [3] (4 seconds). These show the effectiveness of proposed method. Conclusions In this paper, the Takagi-Sugeno fuzzy models are considered for representing nonlinear plants. For the Takagi-Sugenofuzzy model and uncertain fuzzy model, the fuzzy sampled-data control and the robust sampled-data control were considered in the paper. The results were described as the matrix inequalities. The robust sampled-data control for the Takagi-Sugeno uncertain fuzzy model with the constraints on the states was also developed. The result was also described as the matrix inequalities, which could be solved by LMITool. Finally, a numerical example was considered, the result shows the effectiveness of the proposed procedures. References 1. Takagi, T., and Sugeno, M., 1985. Fuzzy identification of systems and its applications t o modeling and control. IEEE Trans. Sys., Man, Cybern, 15, 116-132. 2. Wang, H. O., Tanaka, K., and Griffin, M. F., 1996. An approach to fuzzy control of nonlinear systems: stability and design issues. IEEE Trans. Fuzzy Sys., 4(1), 1423. 3. Lee, T. S., Chen, Y.H., and Chuang, J . , 1998. Fuzzy modeling and uncertainty-based control for nonlinear systems. Proc. of American Control Conference, Philadelphia. 4. Chen, T., and Francis, B., 1995. Optimal Sampled-data Control Systems. Springer. 5. Hagiwara, T., Kuroda, G., and Araki, M., 1996. Stability of sampled-data systems with a timeinvariant memoryless sector nonlinearity. Proc. of the 35th Conference Decision and Control, 1264-1265. 6. Okuyama, Y., and Takemori, F., 1996. Robust stability evalution of sampled-data control systems with a sector nonlinearity. IFAC, 13th Triennial World Conference, 41-46. 7. Rui, C., Kolmanovsky, I., and McClamroch, N. H., 1997. Hybrid control for stabilization of class of cascade nonlinear systems. Proc. of American Control Conference. 8. Tanaka, K., Ikeda, T., and Wang, H., 1996. Design of fuzzy control systems based on relaxed LMI stability conditions. Proc. of the 35th Conference on Decision and Control, 598-603. 9. Hu, L., Shao, H., and Sun, Y.,1999. Constrained robust sampled-data control for a class of nonlinear uncertain systems. IFAC, 14th Triennial World Congress, 521-526. Received: 28 October 1999; Accepted after revision: 12 May 2000. 182

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