H Filter/Controller Design for Discrete-time Takagi-Sugeno Fuzzy Systems with Time Delays

Transcription

1 H Filter/Controller Design for Discrete-time Takagi-Sugeno Fuzzy Systems with Time Delays Yu-Cheng Lin and Ji-Chang Lo Department of Mechanical Engineering National Central University, Chung-Li, Taiwan Abstract Unlike existing procedure, a simple transformation converting delay systems into non-delay systems to examine the stability of discrete-time fuzzy systems is proposed in this paper And the advantage of the proposed method is that it can be specialized to the existing scheme This method can be extended to deal with both control and filtering problems, therefore two sufficient conditions assuring the stability of filtering error system and closed-loop control system respectively are addressed in terms of LMIs Also, H performance is considered in both design problems Lastly, a filter design example illustrates the proposed design methodology Keywords: H control; H filtering; Time delays; Takagi-Sugeno fuzzy model (T-S fuzzy model); Linear matrix inequality (LMI) 1 Introduction During the last few years, time-delay systems have attracted a great deal of attentions due to the fact that it is closer to reality in applications And there have been many significant accomplishments both in filtering and control problems such as [1]-[6] In addition to control issues, recently, more and more experts interest in the problems of robust H filtering/control for discrete-time linear systems In these robust H filtering/control problems, the noise signals are assumed to be arbitrary but energy bounded deterministic signals, and the goal is to design a filter or controller that ensures the stability of filtering error systems and closed-loop control systems, respectively, stable [4],[7]-[10] However, those methods are too simplified in view of demands of physical industries with the progressing of time They should be discussed in the context of nonlinear systems Thanks for the rising of Takagi-Sugeno fuzzy model, we can use many linear local models to stand for a very complex nonlinear system And there have many significant works addressing the stability of fuzzy systems Apart from stability issues, performance is also an important topic needed to be discussed There are some literature dealing with H performance such as [8],[11]-[15] Inspired by current developments, this paper deals with H filtering/control problems for the discretetime fuzzy systems with multiple time delays in the states With the proposed design methodology for a stable full-order fuzzy filter and a state feedback controller, the filtering error system and closed-loop control system, respectively, is asymptotically stable for all nonzero w L 2 [0, ] and achieves a prescribed H performance It deserves to mention that, with a simple mathematical manipulations, the delay systems can be transformed into non-delay systems and the Lyapunov candidate function can be chosen easily as a popular linear quadratic form The structure of this article is as follows Section 2 sets up a discrete-time fuzzy system with multiple time delays, and the most commonly-accepted way to examine the stability of above delay system is also introduced Furthermore, a transformation from delay systems to non-delay systems is stated to show that the new method is more relaxed than the old one Besides, we construct H constraint in this section, too Section 3 reveals two sufficient conditions for the quadratic stability of the given fuzzy system satisfying disturbance attenuation requirement in control and filtering syntheses, respectively The validity of the proposed approach are illustrated by an example involving H filter design problem in Section 4 Finally, this paper concludes with some concise notes in Section 5 Notations: The notation used throughout the paper is as follows The superscript denotes transpose of a matrix Two symmetric matrices X, Y such that X > Y (respectively, X Y )meansthatx Y is positive definite (positive semidefinite) The indexes i, j 1, 2,, n; q 1, 2,, m Besides, the notation is used to indicate terms that can be induced by symmetry For example, A + BC + A + C B (A + BC)+( ), MPM MP( ) and A B A B C B C p 1

2 2 Problem Formulations Consider the inferred discrete-time T-S fuzzy systems with multiple time delays, which can represent filtering error dynamics in filtering problems or closed-loop systems in control problems as follows σx A 0µµ x + m q1 A qµµx q + B µµ w z C 0µµ x + m q1 C qµµx q + D µµ w (1) x ϕ Φ ϕ, ϕ T σx, x, x q, w, z, x ϕ and Φ ϕ stand for x(k+1), x(k), x(k q), w(k), z(k), x(ϕ) andφ(ϕ) respectively x is a state variable, z denotes controlled output vector, w represents an exogenous signals of energy-bounded and Φ ϕ is a given initial value sequence over ϕ T T [ m, m +1,, 0] is the time sequence And A0µµ A qµµ B µµ C 0µµ C qµµ D µµ A0ij A µ i µ qij B ij j C 0ij C qij D ij i,j1 In addition to the stability issue, the performance of a system is also an important topic Therefore, here is the H performance criterion with a prescribed level of disturbance attenuation ρ>0 z z <ρ 2 k0 k0 w w (2) under zero initial condition and w L 2 [0, ] The most common technique to discuss the stability issues of these time-delay systems is choosing a Lyapunov candidate function as V (k, m) x (k)px(k)+ m k 1 q1 ϱk q x (ϱ)s q x(ϱ) (3) for the delay-system (1) directly Therefore we have the following result Lemma 1 Given a constant ρ>0as a disturbance attenuation level The delay-system (1), assuming m 2, is asymptotically stable and achieves H criterion (2) with all nonzero w L 2 if the following matrix inequality holds P + S 1 + S 2 0 S S A 0µµ A 1µµ A 2µµ B µµ P 1 C 0µµ C 1µµ C 2µµ D µµ 0 I < 0 (4) Proof: Choose a Lyapunov candidate function as (3) and derive the following inequality V (k +1, 2) V (k, 2) + z z ρ 2 w w<0 (5) Utilizing Schur complement, LHS of (5) yields x A 0µµ C 0µµ x 1 A 1µµ C 1µµ P x 2 A 2µµ C ( ) 2µµ 0 I w B µµ D µµ P 2 q1 S q 0 S S 2 ( ) (6) ρ 2 I Recalling Schur complement, the sufficient condition (4)isequivalentto(6)< 0 Obviously, (4) guarantees (5) being satisfied Besides, H criterion (2) is also achieved by summing up (5) from k 0to Therefore, according to Hamilton-Jacoby condition, delaysystem (1) of m 2 is asymptotically stable and achieves H performance Actually, there exists a better way to treat the stability issues of delay systems than above method In the following, a transformation from delay systems into non-delay systems is introduced with a new state defined x(k) [ x (k) x (k 1) x (k m) ] Thus, a non-delay fuzzy system is derived from (1) as follows σ x Āµµ x + B µµ w z C µµ x + D µµ w x 0 Φ (7) 0 σ x, x, x 0 and Φ 0 stand for x(k +1), x(k), x(0) and Φ(0) respectively, and A0µµ A Ā µµ 1µµ A mµµ I 0 B µµ [ B µµ 0 0 ] C µµ C 0µµ C 1µµ C mµµ D µµ D µµ Φ 0 [ Φ (0) Φ ( 1) Φ ( m) ] (8) Note that the non-delay system (7) is equivalent to the original system (1), thus the stability of delay system (1) can be examined via checking the stability of nondelay system (7) Here is a sufficient condition assuring the stability of non-delay system (7) (or delay system (1)) Lemma 2 Given a constant ρ>0 as a disturbance attenuation level The non-delay system (7), assuming m 2, is asymptotically stable and achieves H p 2

3 criterion (2) with all nonzero w L 2 if the following matrix inequality holds Ā µµ Bµµ Q 1 < 0 (9) C µµ D µµ 0 I Proof: Since the underlying system has no delay terms, the simplest and standard Lyapunov candidate function is chosen as V (k, m) x Q x (10) So it follows V (k +1, 2) V (k, 2) + z z ρ 2 w w (11) ( x Ā µµ C µµ Q ( ) w 0 I [ Q 0 ρ 2 I B µµ D µµ ]) ( ) (12) Applying Schur complement again, the sufficient condition (9) is obtained easily from (12) Apparently, the satisfaction of (9) implies Hamilton-Jacoby condition (11)< 0, hence the stability of non-delay fuzzy system (7) is assured from Lyapunov theory This completes the proof Why is the second method more relaxed than existing one? Now we will show that the sufficient condition (4), the most adopted method in delay systems, is only a special case of sufficient condition (9) Let Q Q 1 Q 2 Q 4 Q 2 Q 3 Q 5 Q 4 Q 5 Q 6 and note that the sufficient condition (4) is equivalent to sufficient condition (9) when one utilizes congruence transformation, sets Q 1 P, Q 3 S 1 + S 2, Q 6 S 2 and Q 2 Q 4 Q 5 0 and applies Schur complement Therefore we have the fact that the satisfaction of Lemma 1 implies existence of Lemma 2 So, it is worth emphasizing again that the second method dealing with the stability issues of delay-systems can be specialized to the existing approach Therefore the proposed method is more relaxed than the existing method Corollary 1 In fact, the above results can be extended to multiple delay-term cases and the implications between Lemma 1 and 2 also hold in multiple delay-term cases To see when these two lemmas are equivalent, recalling (3) and (10) yields V (k, m) and x Px + P x m q1 x q ( m ) S p x q pq m q1 S q m qm S q x, V (k, m) x Q x Q 0,0 x Q 1,0 Q 1,1 x Q m,0 Q m,1 Q m,m x x x 1 x 2 x m Observing above two matrices, it is clear that Lemma 1 equals to a special case of Lemma 2whenQ 0,0 P, Q 1,1 m q1 S q, Q 2,2 m q2 S q,, Q m,m m qm S q and others being zero 3 H Filter/Controller Design With fundamentals mentioned in previous section, the new method is adopted to design an H filter and a state feedback controller Consider an inferred fuzzy system with multiple time delays as follows σx A 0µ x + m q1 A qµx q + B 1µ w + B 2µ u y C 0µ x + m q1 C qµx q + D µ w z E 0µ x + m q1 E qµx q + F µ w x ϕ φ ϕ, ϕ T (13) Y rµ µ i Y ri, Y rµ [A 0µ,A qµ,b 1µ,,E qµ,f µ ], i1 and y y(k) andu u(k) stand for system output signals and force input, respectively φ ϕ φ(ϕ) isa given initial value sequence for delay system (13) over ϕ T Similarly, a filter is needed to well estimate systemcontrolledoutputandisstatedbelow { σxf A fµ x f + B fµ y (14) z f C fµ x f + D fµ y σx f and z f stand for x f (k +1) and z f (k) respectively Y fµ µ j Y fj, Y fj [A fj,b fj,c fj,d fj ], j1 p 3

4 and A fi, B fi, C fi and D fi are filter gain variables Besides, x f (k) 0, k 0 B 2µ is set to be zero for the filtering problems Introducing an augmented new state vector as x [x x f ] and taking the error vector as z z z f, we have filtering error dynamics of the form (1) [ ] A A 0µµ 0µ 0 [ B fµ C 0µ A fµ ] A A qµµ qµ 0 B fµ C qµ 0 B B µµ 1µ B fµ D µ C 0µµ [ (15) ] E 0µ D fµ C 0µ C fµ C qµµ [ E qµ D fµ C qµ 0 ] D µµ F µ D fµ D µ Φ ϕ [ φ ϕ 0 ] So, the new method applicable to designing an H filter for non-delay system (7) (or delay system (1)) is constructed in the following theorem Theorem 1 Given a constant ρ>0 as a desired H norm bound constraint The filtering error fuzzy system (7) with (8) and (15) is asymptotically stable and achieves a presetting disturbance attenuation level ρ if there exist symmetric matrix Q>0, andvariablesc fj, D fj, G j and H j such that the following linear matrix inequalities hold Υ 1 Υ 2 Q Υ 3 Υ 4 0 I < 0 (16) Q 1,1 Q 1,2 Q 1,2m+2 Q 2,1 Q 2,2 Q 2,2m+2 Q (17) Q 2m+2,1 Q 2m+2,2 Q 2m+2,2m+2 Q 2,r Q r,2 Q 0, r 1, 2,, 2m +2 Besides, Υ 1 is shown in (18) on the top of next page, and Q 1,1 B 1i + H j D i Υ 2 Q 0 B 1i + H j D i Q 3,1 B 1i + H j D i Q 2m+2,1 B 1i + H j D i Υ 3 [ E 0i D fj C 0i C fj E 1i D fj C 1i 0 E mi D fj C mi 0 ] Υ 4 F i D fj D i Moreover, the stable filter gains are A fj Q 1 0 G j, B fj Q 1 0 H j, C fj, D fj Proof: Selecting a Lyapunov candidate function as V (k, m) x Q x for error system (7) and taking account of (8) and (15), Lemma 2 yields the following BMI condition QĀµµ Q B µµ Q C µµ D µµ 0 I µ i µ j i,j1 QĀij Q B ij Q C ij D ij 0 I < 0 (19) Following the same lines of Lemma 2, letting Q be (17) and recalling (8) and (15), it is easily obtained the sufficient condition (16) from (19) with notations Q 0 A fj G j and Q 0 B fj H j Now we have shown that the new method, transformation of a delay system into a non-delay system, can be used in a filtering problem In the next, we will show that the new method can also be used in control issues With the same plant addressed in (13) and considering the PDC state feedback control law as u(k) K cµ x(k) (20) K cµ µ j K cj, j1 the closed-loop delay system, therefore, is stated in (1) A 0µµ A 0µ + B 2µ K cµ A qµµ A qµ B µµ B 1µ C 0µµ E 0µ (21) C qµµ E qµ D µµ F µ Φ φ Note that y is omitted in (13) and z in (1) equals to z in (13) The objective in control problem is to stabilize (13) and make controlled output z(k) satisfying H criterion (2) via 20 Thus, we conclude a sufficient condition for controller syntheses as follows Theorem 2 Given a constant ρ>0 as a desired H norm bound constraint The fuzzy system (7) with (8) and (21) is quadratically stabilizable and achieves a presetting disturbance attenuation level ρ if there exist symmetric matrix R>0 and variable L j such that the following linear matrix inequalities hold R Ψ 1 Ψ 2 R 0 Ψ 3 F i 0 I < 0 (22) p 4

5 Υ 1 Q 1,1 A 0i + H j C 0i + Q 1,3 G j + Q 1,4 Q 1,1 A 1i + H j C 1i + Q 1,5 Q 1,6 Q 0 A 0i + H j C 0i + Q 0 G j + Q 0 Q 0 A 1i + H j C 1i + Q 0 Q 0 Q 3,1 A 0i + H j C 0i + Q 3,3 G j + Q 3,4 Q 3,1 A 1i + H j C 1i + Q 3,5 Q 3,6 Q 2m+2,1 A 0i + H j C 0i + Q 2m+2,3 G j + Q 2m+2,4 Q 2m+2,1 A 1i + H j C 1i + Q 2m+2,5 Q 2m+2,6 Q 1,1 A 2i + H j C 2i + Q 1,7 Q 1,8 Q 1,1 A mi + H j C mi 0 Q 0 A 2i + H j C 2i + Q 0 Q 0 Q 0 A mi + H j C mi 0 Q 3,1 A 2i + H j C 2i + Q 3,7 Q 3,8 Q 3,1 A mi + H j C mi 0 Q 2m+2,1 A 2i + H j C 2i + Q 2m+2,7 Q 2m+2,8 Q 2m+2,1 A mi + H j C mi 0 (18) and R Ψ 1 R 0 R 0 R 0 R 0 R 0 R 1,1 R 1,2 R 1,m R 0 R 2,1 R 2,2 R 2,m R 0 R m,1 R m,2 R m,m R r,s R s,r, r, s 1, 2,,m, R 0 A 0i + L j B 2i + m q1 R 0A qi R 0 A 0i + L j B 2i + m q1 R q,1a qi R 0 A 0i + L j B 2i + m q1 R q,m A qi R 0 R 0 R 0 R 0 R 1,1 R m 1,1 R 0 R 1,m R m 1,m Ψ 2 [ B 1i 0 0 ] R 0 E 0i + m q1 R 0E qi R 0 E 0i Ψ 3 + m q1 R q,1e qi R 0 E 0i + m q1 R q,m E qi Therefore, we have the stabilizing controller gains as K cj R 1 0 L j (23) Proof: Choosing the same Lyapunov function as in Theorem 1 for closed-loop system (7) with (8) and (21), we have Ā µµ Bµµ Q 1 C µµ Dµµ 0 I R Ā µµ R B µµ R C µµ R D µµ 0 I µ i µ j i,j1 R Ā ij R Bij R C ij R Dij 0 I < 0 (24) R Q 1 Similar to the lines mentioned in the proof of Theorem 1, we obtain the sufficient condition (22) from (24) with K cj R 0 L j denoted 4 Numerical Simulation A numerical example is presented for illustrating the filter design technique proposed in Section 3 Consider the following two-rule T-S fuzzy system Plant Rule i, i 1, 2: IF x 1 (k) isf1, i THEN σx A 0i x + 3 q1 A qix q + B 1i w y C 0i x + 3 q1 C qix q + D i w z E 0i x + 3 q1 E qix q + F i w and here is the fuzzy rules F1 1 (x 1(k)) { sin(x1(k)) x 1(k), x 1 (k) 0, 0, x 1 (k) 0, F1 2 (x 1 (k)) 1 F1 1 For the sake of simplicity, the delay time is set as m 3 In order to fit the form addressed in (13), some mathematical adjustments are needed And here are the system matrices A p 5

6 A A 11 A 12 A 21 A 22 0 A A B 11 B C 01 C 02 [ ] C 11 C 12 C 21 C 22 0 C 31 C 32 [ ] D 1 D 2 F 1 F 2 01 E 01 E 02 [ ] E 11 E 12 E 21 E 22 0 E 31 E 32 [ ] Also, here is two-rule fuzzy filter and its rules as follows Filter Rule j, j 1, 2: IF x f1 (k) isf j 1 {, THEN σxf A fj x + B fj y z f C fj x + D fj y fuzzy rules F j 1 F 1 i with replacing x 1 (k) by x f1 (k) Moreover, define the initial value function φ(ϕ) as a random signal when ϕ 0 From Theorem 1, to determine the quadratic stability of the above timedelay system, a feasible solution to a set of 4 LMIs (16) with a given disturbance attenuation level ρ 1is P G 1 G H 1 H Furthermore, solutions of other variables are omitted to save space Using A fj Q 1 0 G j, B fj Q 1 0 H j, the filter gains are Af1 B f1 Af2 B f2 C f1 D f1 C f2 D f Simulations were also conducted with external disturbance w being a random signal over [ 1, 1] for k 30 and zero for k>30 Fig 1 gives the state response under initial condition being x 0 [ 4, 10, 5] And Fig 2 shows the trajectories of controlled output, estimated output, and estimated error It also shows that the estimated output z f (k) can always converge to the controlled output z(k), and the delay fuzzy system is stable with the required level of disturbance attenuation 5 Conclusion In this paper, a new method to examine the stability of discrete-time delay fuzzy systems is proposed We also show the old method, the prevailing way being used, is only a special case of the new method It has been shown that the proposed method is more relaxed than the existing results Moreover, this method can be extended to both control and filtering syntheses, and two sufficient conditions are stated in terms of LMI, respectively With a full-order fuzzy filter and state feedback control law, the filtering error systems and closed-loop control systems are ensured to be globally stable and achieve H performance requirement, respectively In the end, an example verifying the proposed method in filter design problems is illustrated using an LMI approach However, although the method, transforming delay systems into non-delay system, can simplify the complexity of Lyapunov function, it is not suitable for the largest delay time m is huge In this case, the size of matrix Āµµ of (7) is huge and the CPU time of calculation is in great demand Acknowledgement This work was supported in part by the National Science Council of the ROC under grant NSC E References [1] CE de Souza, RM Palhares, and PLD Peres, Robust H filtering design for uncertain linear systems with multiple time-varying state delays, IEEE Trans Signal Processing, vol 49, no 3, pp , Mar 2001 p 6

7 [2] JH Kim, DC Oh, and HB Park, Guaranteed cost and H filtering for time delay systems, in ProctheAmerContrConf, Arlington, VA, 2001, pp [3] A Fattouh, O Sename, and JM Dion, H observer design for time-delay systems, in Proc 37th Conf on Decision and Control, Tampa, FL, 1998, pp [4] RM Palhares, CE de Souza, and PLD Peres, Robust H filtering for uncertain discrete-time state-delayed systems, IEEE Trans Signal Processing, vol 49, no 8, pp , Aug 2001 [5] E Fridman and U Shaked, A new H filter design for linear time delay systems, IEEE Trans Signal Processing, vol 49, no 11, pp , Nov 2001 [6] YS Suh, Stability of discrete state delay systems, in Proc 40th IEEE Conf on Decision and Control, Orlando, FL, 2001, pp [7] RM Palhares and PLD Peres, LMI approach to the mixed H 2 /H filtering design for discretetime uncertain systems, IEEE Trans Aerospace and Electronic Systems, vol 37, no 1, pp , Jan 2001 [8] B Hassibi and T Kailath, On nonlinear filters for mixed H 2 /H estimation, in Proc of 1997 Amer Contr Conf, Albuquerque, New Mexico, 1997, pp [9] CS Tseng and BS Chen, H fuzzy estimation for a class of nonlinear discrete-time dynamic systems, IEEE Trans Signal Processing, vol 49, no 11, pp , Nov 2001 [10] JC Geromel and MC de Oliveira, H 2 and H robust filtering for convex bounded uncertain system, IEEE Trans Automat Contr, vol 46, no 1, pp , Jan 2001 [11] YY Cao and PM Frank, Robust H disturbance attenuation for a class of uncertain discretetime fuzzy systems, IEEE Trans Fuzzy Syst, vol 8, no 4, pp , Aug 2000 [12] BS Chen, CS Tseng, and HJ Uang, Mixed H 2 /H fuzzy output feedback control design for nonlinear dynamic systems: an LMI approach, IEEE Trans Fuzzy Syst, vol 8, no 3, pp , June 2000 [13] YY Cao and PM Frank, Stability analysis and synthesis of nonlinear time-delay systems via linear Takagi-Sugeno fuzzy model, Fuzzy Sets and Systems, vol 124, pp , 2001 [15] HD Tuan, P Apkarian, T Narikiyo, and Y Yamamoto, Parameterized linear matrix inequality techniques in fuzzy control system design, IEEE Trans Fuzzy Syst, vol 9, no 2, pp , Apr Time step (k) Figure 1: Trajectories of system states Controlled Output Estimated Output Estimated Error Time step (k) Figure 2: Trajectories of controlled output, estimated output and estimated error x 1 x 2 x 3 [14] AW Pila, U Shaked, and CE de Souza, H filtering for continuous-time linear systems with delay, IEEE Trans Automat Contr, vol 44, no 7, pp , 1999 p 7