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1 Applied Mathematical Sciences, Vol. 7, 2013, no. 27, HIKARI Ltd, Output Feedback Control Design with Guaranteed Cost for Fuzzy Bilinear Time-Delay Systems Ryutaro Takada, Yuzu Uchida and Jun Yoneyama Department of Electrical Engineering and Electronics Aoyama Gakuin University Fuchinobe, Chuo-ku, Sagamihara, Kanagawa , Japan Copyright c 2013 Ryutaro Takada et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract This paper is concerned with observer-based guaranteed cost control design of fuzzy bilinear systems with time-varying delay. First, we consider the state feedback control design by assuming a special form of a fuzzy controller. A delay-dependent condition, which is less conservative than delay-independent one, for guaranteed cost control design of fuzzy bilinear systems with time-varying delays is given in terms of linear matrix inequality (LMI). Next, we consider a fuzzy observer design for the same class of systems, and propose an observer design method based on another delay-dependent condition. Then, we show that state feedback controller and observer make a stabilizing observer-based output feedback controller with guaranteed cost. Finally, we give a numerical example to illustrate our design procedures and to show the effectiveness of our approach. Mathematics Subject Classification: 93E20 Keywords: Guaranteed cost control; Output feedback control; Fuzzy systems; Bilinear systems; Time-delay 1 Introduction In real life, almost all physical systems are nonlinear and they often appear in engineering systems. The Takagi-Sugeno fuzzy system has been used for

2 1304 R. Takada, Y. Uchida and J. Yoneyama nonlinear systems control because it can approximate nonlinear systems, and even describe a certain class of nonlinear systems exactly([7], [8]). Time-delay also occurs in many physical systems and system formulations such as transportation systems, networked control systems and telecommunication systems. Thus, it is important to analyze and synthesize nonlinear systems with timedelay. As an extension of system representation, fuzzy bilinear system and fuzzy bilinear time-delay system were considered in [1], [3], [4], and [9]. The results in [3] and [4] considered bilinear systems with no time-delay, and gave stability conditions for such systems. The paper [9] is the extension to the stability for fuzzy bilinear systems with time-delay. However, the result in [9] gives only stability conditions for the closed-loop fuzzy bilinear time-delay systems with given controllers, and their given controllers are state feedback controllers. It did not propose even state feedback controller design method. The paper [6] discussed a delay-dependent control design method for a fuzzy bilinear system with time-delay, but it only considered a stabilizing state feedback controller with guaranteed cost. In practical systems control, the state feedback control is not always feasible because all the state variables of the system are not always available. Then, the research has extended to an output feedback control design of fuzzy bilinear time-delay systems in [11] where the state feedback control and observer design have been proposed, but the design conditions in [11] were based on delay-independent conditions, which are generally conservative. Therefore, we need an output feedback control design based on less conservative delay-dependent conditions. In this paper, we consider an observer-based guaranteed cost control design for fuzzy bilinear time-delay systems. We give delay-dependent stability conditions via LMI, and propose a control design method for a stabilizing output feedback controller with guaranteed cost of fuzzy bilinear systems with timevarying delay. First, we assume a special form of a state feedback controller([3], [4], and [9]), and then propose a state feedback controller design method. Next, we propose an observer design method. By considering the error system which results from the original fuzzy bilinear system and its observer, we give a delay-dependent stability condition for the error system. Our design method chooses observer gain matrices such that the error system is asymptotically stable. Then, we show that an output feedback controller which combines state feedback controller and observer becomes a stabilizing output feedback controller with guaranteed cost. In other words, we establish the separation principle for fuzzy bilinear time-delay systems. Finally, we give an illustrative example to show our design procedure and the effectiveness of our design approach.

3 Guaranteed cost control for fuzzy bilinear systems Fuzzy Bilinear Systems In this section, we introduce Takagi-Sugeno fuzzy bilinear systems with timevarying delay. Consider the Takagi-Sugeno fuzzy model, described by the following IF-THEN rules: IF ξ 1 is M i1 and and ξ p is M ip, THEN ẋ(t) =A i x(t)+a di x(t τ(t)) + u(t)n i x(t) +u(t)n di x(t τ(t)) + B i u(t), y(t) =C i x(t)+c di x(t τ(t)), x(θ) =φ(t), θ [ τ M, 0], i =1,,r. where x(t) R n is the state, u(t) Ris the control input and y(t) R q is the measurement output. The matrices A i,a di,n i,n di,b i and C i are constant matrices of appropriate dimensions. The time-delay τ(t) may be unknown but is assumed to be bounded. It is also assumed to be a smooth function such that τ(t) τ M, τ(t) d<1 where τ M and d are known constant numbers. of IF-THEN rules. r is the number of IF-THEN rules. M ij are fuzzy sets and ξ 1,,ξ p are premise variables. We set ξ =[ξ 1 ξ p ] T. The premise variable ξ(t) is assumed to be measurable. Then, the state and output equations are described by where ẋ(t) = y(t) = λ i (ξ){a i x(t)+a di x(t τ(t)) + u(t)n i x(t) +u(t)n di x(t τ(t)) + B i u(t)}, (1) λ i (ξ){c i x(t)+c di x(t τ(t))}. (2) λ i (ξ) = β i(ξ), β i (ξ) = β i (ξ) p M ij (ξ j ) j=1 and M ij ( ) is the grade of the membership function of M ij. We assme β(ξ(t)) 0,,,r, r β i (ξ(t)) > 0 for any ξ(t). Hence λ i (ξ(t)) satisfy λ i (ξ(t)) 0,,,r, r λ i (ξ(t)) = 1 for any ξ(t). The following lemmas are given to prove our main results in the later sections. Lemma 2.1 ([5]) Given any matrices X, Y and Z > 0 with appropriate dimensions. Then, we have X T Y + X T Y X T ZX + Y T Z 1 Y.

4 1306 R. Takada, Y. Uchida and J. Yoneyama Lemma 2.2 ([2]) For any given matrices X i, Y i, i =1,,r and S>0 with appropriate dimensions, we have 2 λ i (ξ)λ j (ξ)x i T SY j λ i (ξ)(xi T SX i + Yi T SY i ), j=1 2 λ i (ξ)λ j (ξ)λ k (ξ)λ l (ξ)x ij T SY kl j=1k=1l=1 λ i (ξ)λ j (ξ)(xijsx T ij + Yij T SY ij ). j=1 Lemma 2.3 ([10]) λ i (ξ(t))λ j (ξ(t))φ ij < 0 j=1 holds if the followings are satisfied: Φ ii < 0, i =1,,r, 2 r 1 Φ ii +Φ ij +Φ ji < 0, i,j =1,,r, i j. 3 Guaranteed Cost Control We first show a design method of the state feedback controller for the fuzzy system (1). We employ the following rules for a controller: IF ξ 1 is M i1 and and ξ p is M ip, ρk i x(t) THEN u(t) =, i =1,,r. 1+x T (t)ki T K i x(t) where ρ is a given scalar, and K i, i =1,,r are a constant matrices to be determined. Then, the natural choice of a controller is given by ρλ i (ξ)k i x(t) u(t) = 1+x T (t)ki T K i x(t) = λ i (ξ)ρ sin θ i (t) where sin θ i (t) = = λ i (ξ)ρk i x(t) cos θ i (t). (3) K i x(t) 1+x T (t)k T i K i x(t), cos θ i (t) = 1 1+x T (t)k T i K i x(t), θ i [ π 2, π 2 ].

5 Guaranteed cost control for fuzzy bilinear systems 1307 The closed-loop system (1) with (3) is given by ẋ(t) = λ i (ξ)λ j (ξ){(a i + ρn i sin θ j + ρb i K j cos θ j )x(t) j=1 +(A di + ρn di sin θ j )x(t τ(t))}. (4) Our problem is to find a state feedback controller (3) such that the closed-loop system (4) is asymptotically stable, and the following cost function J is smaller than some number. J = 0 {x T (t)ux(t)+u T (t)tu(t)}dt < J where U and T are semi-positive and positive definite matrices, respectively, and J is a positive scalar. If there exists such a controller, the system is said to be stabilized with guaranteed cost. Based on the results of [6], we can show a design method of a stabilizing state feedback controller with guaranteed cost. Since the proof of Theorem 3.1 is very similar to that of Theorem 2 in ([6]), it is omitted here. Theorem 3.1 Given scalars ρ and ε > 0. A controller (3) stabilizes the fuzzy system (1) with guaranteed cost if there exist matrices X 0, P > 0, Q >0, S >0 and Gi,,,r such that X = Ξ ii = ˆΞ ij = X 11 X12 X13 X 12 T X 22 X23 X 13 T X 23 T ε P 0, (5) T Ξ 11i Ξ 12i Ξ 13ii 0 PA i 0 Ξ T T 12i Ξ 22 0 Ξ 24i 0 PA di Ξ T 13ii 0 Ξ Ξ T 24i 0 Ξ < 0, i =1,,r,(6) A i P Ξ56 Ξ 56 0 A di P 0 0 Ξ56 4Ξ 56 ˆΞ 11ij ˆΞ 12ij ˆΞ 13ij 0 PA T i 0 PA T j 0 ˆΞ T 12ij ˆΞ 22 0 ˆΞ24ij 0 PA T di 0 PA T dj ˆΞ T 13ij 0 ˆΞ ˆΞT 24ij 0 ˆΞ A i P (r 1) r+1 Ξ 56 r 1 r+1 Ξ A di P 0 0 r 1 r+1 Ξ 56 4(r 1) r+1 Ξ A j P Ξ56 Ξ 56 0 A dj P Ξ56 4Ξ 56 < 0, i, j =1,,r, i j (7)

6 1308 R. Takada, Y. Uchida and J. Yoneyama where β 1 = ετ M,β 2 =1+3ετ M ρ 2,β 3 =1+4ετ M ρ 2 and Ξ 11i = PA T i + A i P + ρ 2 P τ 1 d M X11 + X 13 + X 13 T + Ū, Ξ 12i = A di P + τm X12 X 13 + X 23 T, Ξ 13ii = [ ] ρn di P G T i Bi T PN i T ρg T i, Ξ 13ij = [ ] ρn di P G T j Bi T PN i T ρg T j, Ξ 22 = ( Q S)+τ M X22 X 23 X 23 T, Ξ 24i = ρ PN di T, Ξ 33 = diag [ S β 1 2 P β3 1 P T ] 1, Ξ 44 = (4β 1 ) 1 P, Ξ 56 = (11β 1 ) 1 P, ˆΞ 11ij = r+1ξ r 1 11i +Ξ 11j, ˆΞ 12ij = r+1 r 1 12i +Ξ 12j, ˆΞ 13ij = [ ] Ξ 13ii Ξ 13ij Ξ 13ji, ˆΞ 22 = 2r r 1 22, ˆΞ 24ij = [ ρ PN di T ρ PN ] dj T, ˆΞ 33 = diag [ r 1 ] 2 33 Ξ 33 Ξ 33, ˆΞ 44 = diag [ r 1 r+1 Ξ ] 44 Ξ 44. In this case, control gain matrices K j can be obtained by K j = G j P 1, j =1,,r. (8) 4 Observer In real life, all the states of the system are not always observed. Hence, we need to estimate the unobserved states by an observer. Here we define a fuzzy observer and make an analysis of the error system, from which we provides a design method of a fuzzy observer for the fuzzy bilinear time-delay system (1) and (2). 4.1 Error System Analysis Let us consider the following rules for an observer-based controller: IF ξ 1 is M i1 and and ξ p is M ip, THEN ˆx(t) =A iˆx(t)+a diˆx(t τ(t)) + u(t)n iˆx(t)+u(t)n diˆx(t τ(t)) +B i u(t)+h(ξ)(y(t) C iˆx(t) C diˆx(t τ(t))), ρk iˆx(t) u(t) = 1+ˆx T (t)ki T K iˆx(t) = ρ sin ˆθ i (t) = ρk iˆx(t) cos ˆθ i (t), i =1,,r

7 Guaranteed cost control for fuzzy bilinear systems 1309 where H(ξ) = λ i (ξ)h i, sin ˆθ i (t) = cos ˆθ i (t) = K iˆx(t) 1+ˆx T (t)k T i K iˆx(t), 1 1+ˆx T (t)k T i K iˆx(t), ˆθi [ π 2, π 2 ]. and H i,,,rare constant matrices to be determined. Then, the overall observer-based controller is given by ˆx(t) = λ i (ξ){a iˆx(t)+a diˆx(t τ(t)) + u(t)n iˆx(t)+u(t)n diˆx(t τ(t)) +B i u(t)+h(ξ)(y(t) C iˆx(t) C diˆx(t τ(t)))}, (9) u(t) = ρλ i (ξ)k iˆx(t) 1+ˆx T (t)ki T K iˆx(t) = λ i (ξ)ρ sin ˆθ i (t) = λ i (ξ)ρk iˆx(t) cos ˆθ i (t). (10) By defining the error e(t) =x(t) ˆx(t), we have the error system from (1), (2), (9) and (10): ė(t) = λ i (ξ)λ j (ξ){(a i H j C i + ρn i sin ˆθ j )e(t) j=1 +(A di H j C di + ρn di sin ˆθ j )e(t τ(t))}. (11) Thus, our problem is to find an observer (9) such that the error system (11) is asymptotically stable. The following theorem gives a condition such that the error system (11) is asymptotically stable. Theorem 4.1 Given scalar ρ and observer gain matrices H i, i =1,,r, the error system (11) is asymptotically stable if there exist matrices P 2 > 0, Q 2 > 0, S 2 > 0 and R 2 > 0 such that X 2 = Φ ij = X 11 X 12 X 13 X12 T X 22 X 23 X13 T X23 T R 2 Φ 11ij Φ 12ij ρp 2 N di Φ T 12ij Φ 22ij 0 ρndi T P 2 0 S 2 0, (12) <0, i,j=1,,r (13)

8 1310 R. Takada, Y. Uchida and J. Yoneyama where Φ 11ij = (A i H j C i ) T P 2 + P 2 (A i H j C i )+Ni T P 2 N i + ρ 2 P d 2 +6τ M A T i R 2A i +6τ M ρ 2 Ni T R 2N i +6τ M Ci T HT j R 2H j C i +τ M X 11 + X 13 + X13 T, Φ 12ij = P 2 A di P 2 H j C di + τ M X 12 X 13 + X23 T, Φ 22ij = (Q 2 S 2 )+6τ M A T dir 2 A di +6τ M ρ 2 NdiR T 2 N di +6τ M Cdi T HT j R 2H j C di + τ M X 22 X 23 X23 T. Proof: We consider the following Lyapunov-Krasovskii functional candidate: V (e t )=V 1 (e)+v 2 (e t )+V 3 (e t ) where e t = e(t + s), τ M s 0 and V 1 (e) = e T (t)p 2 e(t), 1 t V 2 (e t ) = e T (s)q 2 e(s)ds, 1 d t τ(t) V 3 (e t ) = 0 τ M t t+θ ė T (s)r 2 ė(s)dsdθ with P 2 > 0, Q 2 > 0, R 2 > 0 to be determined. We differentiate V 1 (e), V 2 (e t ) and V 3 (e t ) with respect to t along the solution of the closed-loop system (11), respectively. First, we calculate V 1 (e): V 1 (e) = 2e T (t)p 2 ė(t) = λ i (ξ)λ j (ξ)e T (t)2p 2 (A i H j C i + ρn i sin ˆθ j )e(t) j=1 +2 λ i (ξ)λ j (ξ)e T (t)p 2 (A di H j C di + ρn di sin ˆθ j )e(t τ(t)) j=1 for some S 2 > 0 where by applying Lemma 2.1, we calculate 2e T (t)p 2 (A di H j C di + ρn di sin ˆθ j )e(t τ(t)) e T (t)ρ 2 P 2 N di S2 1 NdiP T 2 e(t)+2e T (t)p 2 (A di H j C di )e(t τ(t)) +e T (t τ(t))s 2 e(t τ(t)), (A i H j C i + ρn i sin ˆθ j ) T P 2 + P 2 (A i H j C i + ρn i sin ˆθ j ) = (A i H j C i ) T P 2 + P 2 (A i H j C i )+ρni T P 2 sin ˆθ j + ρp 2 N i sin ˆθ j (A i H j C i ) T P 2 + P 2 (A i H j C i )+Ni T P 2N i + ρ 2 P 2 sin 2 ˆθj (A i H j C i ) T P 2 + P 2 (A i H j C i )+Ni T P 2 N i + ρ 2 P 2 Hence, we have V 1 (e) λ i (ξ)λ j (ξ){e T (t)[(a i H j C i ) T P 2 + P 2 (A i H j C i ) j=1 +Ni T P 2N i + ρ 2 P 2 ρ 2 P 2 N di S2 1 di T P 2]e(t) +2e T (t)p 2 (A di H j C di )e(t τ(t))} + e T (t τ(t))s 2 e(t τ(t)).

9 Guaranteed cost control for fuzzy bilinear systems 1311 Next, we obtain V 2 (e t ): V 2 (e t ) = 1 1 d et (t)q 2 e(t) 1 τ(t) 1 d et (t τ(t))q 2 e(t τ(t)) 1 1 d et (t)q 2 e(t) e T (t τ(t))q 2 e(t τ(t)). Furthermore, using Lemma 2.2, we calculate V 3 (e t ): t V 3 (e t )=τ M ė T (t)r 2 ė(t) ė T (s)r 2 ė(s)ds t τ M τ M λ i (ξ)λ j (ξ){(a i H j C i + ρn i sin ˆθ j )e(t) j=1 +(A di H j C di + ρn di sin ˆθ j )e(t τ(t))} T R 2 +(A di H j C di + ρn di sin ˆθ j )e(t τ(t))} {(A i H j C i + ρn i sin ˆθ j )e(t) t t τ M where each term can be expanded by the use of Lemma 2.1 as ė T (s)r 2 ė(s)ds e T (t)(a i H j C i + ρn i sin ˆθ j ) T R 2 (A i H j C i + ρn i sin ˆθ j )e(t) = e T (t)(a T i R 2 A i +2ρA T i R 2 H j C i +2ρA T i R 2 N i sin ˆθ j + C T i H T j R 2 H j C i 2ρC T i H T j R 2 N i sin ˆθ j + ρ 2 N T i R 2 N i sin 2 ˆθj )e(t) e T (t)(3a T i R 2 A i +3C T i H T j R 2 H j C i +3ρ 2 N T i R 2 N i sin 2 ˆθj )e(t), 2e T (t)(a i H j C i + ρn i sin ˆθ j ) T R 2 (A di H j C di + ρn di sin ˆθ j )e(t τ(t)) e T (t)(3a T i R 2A i +3Ci T HT j R 2H j C i +3ρ 2 Ni T R 2N i sin 2 ˆθj )e(t) +e T (t τ(t))(3a di R 2 A di +3CdiH T j T R 2 H j C di +3ρ 2 NdiR T 2 N di sin 2 ˆθj )e(t τ(t)) and e T (t τ(t))(a di H j C di + ρn di sin ˆθ j ) T R 2 (A di H j C di + ρn di sin ˆθ j )e(t τ(t)) 3e T (t τ(t))(a di R 2 A di + C T i HT j R 2H j C i + ρ 2 N T di R 2N di sin 2 ˆθj )e(t τ(t)). Hence, we have V 3 (e t ) τ M λ i (ξ)λ j (ξ) j=1 {e T (t)(6a T i R 2A i +6Ci T HT j R 2H j C i +6ρ 2 Ni T R 2N i sin 2 ˆθj )e(t) +e T (t τ(t))(6a di R 2 A di +6Cdi T HT j R 2H j C di t +6ρ 2 NdiR T 2 N di sin 2 ˆθj )e(t τ(t))} ė T (s)r 2 ė(s)ds. t τ M

10 1312 R. Takada, Y. Uchida and J. Yoneyama For X 2 0 as in (12), we have T t e(t) X 11 X 12 X 13 e(t) 0 e(t τ(t)) X12 T X 22 X 23 e(t τ(t)) ds t τ(t) ė(s) X13 T XT 23 R 2 ė(s) e T (t)(τ M X 11 + X 13 + X13)e(t)+2e T T (t)(τ M X 12 X 13 + X23)e(t T τ(t)) t +e T (t τ(t))(τ M X 22 X 23 X23 T )e(t τ(t)) + ė T (s)r 2 ė(s)ds. t τ M (14) Considering V (e t ) and (14), we obtain V (e t ) + (14) [ ] e(t) T [ ˆΞ11ij ˆΞ12ij λ i (ξ)λ j (ξ) e(t τ(t)) j=1 ˆΞ T 12ij ˆΞ 22ij ] [ e(t) ] e(t τ(t)) (15) where ˆΞ 11ij =(A i H j C i ) T P 2 + P 2 (A i H j C i )+Ni T P 2 N i + ρ 2 P d Q 2 +ρ 2 P 2 N di S2 1 Ndi T P 2 +6A T i R 2A i +6Ci T HT j R 2H j C i +6ρ 2 Ni T R 2N i sin 2 ˆθj + τ M X 11 + X 13 + X13 T, ˆΞ 12ij = P 2 (A di H j C di )+τ M X 12 X 13 + X23 T, ˆΞ 22ij = (Q 2 S 2 )+6A T dir 2 A di +6CdiH T j T R 2 H j C di +6ρ 2 NdiR T 2 N di sin 2 ˆθj +τ M X 22 X 23 X23 T. If ˆΞ ij = [ ] ˆΞ11ij ˆΞ12ij ˆΞ T 12ij ˆΞ 22ij < 0, i,j =1,,r, (16) we have V (e t ) < 0 from (15). It follows from Schur complement formula that (16) can be written as (13). 4.2 Observer Design Based on the conditions of Theorem 4.1, we propose a design method of an observer-based controller in the following theorem. Theorem 4.2 Given scalars ρ and ε>0. (9) and (10) become an observerbased controller for the fuzzy bilinear time-delay system (1) and (2) if there exist matrices P 2 > 0, Q 2 > 0, S 2 > 0 and L j,j=1,,r such that X 11 X 12 X 13 X 2 = X12 T X 22 X 23 0, (17) X13 T XT 23 εp 2

11 Guaranteed cost control for fuzzy bilinear systems 1313 where Φ ii = Φ 11i Φ 12i ρp 2 N di C T i L T i 0 Φ T 12i Φ 22i 0 0 C T di LT i ρndi T P 2 0 S L i C i 0 0 (6τ M ε) 1 P L i C di 0 0 (6τ M ε) 1 P 2 <0,,,r, (18) Φ 11ij Φ12ij Φ13ij Φ14ij 0 Φ T Φ 12ij 22ij 0 0 Φ25ij Φ ij = Φ T 13ij 0 Φ <0, i,j=1,,r, i j (19) Φ T 14ij 0 0 Φ ΦT 25ij 0 0 Φ44 Φ 11i = A T i P 2 + P 2 A i Ci T LT i L i C i + Ni T P 2N i + ρ 2 P Q 1 d 2 +6τ M εa T i P 2 A i +6τ M ερ 2 Ni T P 2 N i + τ M X 11 + X 13 + X13, T Φ 12i = P 2 A di L i C di + τ M X 12 X 13 + X23 T, Φ 22i = (Q 2 S 2 )+6τ M εa T dip 2 A di +6τ M ερ 2 N T dip 2 N di +τ M X 22 X 23 X T 23, Φ 11ij = r+1 r 1 (AT i P 2 + P 2 A i + N T i P 2N i +6τ M εa T i P 2A i +6τ M ερ 2 N T i P 2N i ) +A T j P 2 + P 2 A j + N T j P 2 N j +6τ M εa T j P 2 A j +6τ M ερ 2 N T j P 2 N j + 2r r 1 (ρ2 P Q 1 d 2 + τ M X 11 + X 13 + X13) T 2 r 1 (CT i LT i + L i C i ) Ci T LT j L jc i Cj T LT i L i C j, Φ 12ij = r+1 P r 1 2A di + P 2 A dj 2 L r 1 ic di L j C di L i C dj + 2r (τ r 1 MX 12 X 13 + X23 T ), Φ 13ij =[ρp 2 N di ρp 2 N dj ], Φ 14ij =[C T i LT i C T i LT j C T j LT i ], Φ 22ij = r+1 r 1 (6τ MεA T di P 2A di +6τ M ερ 2 N T di P 2N di ) +6τ M εa T djp 2 A dj +6τ M ερ 2 N T dj P 2 N dj + 2r { (Q r 1 2 S 2 )+τ M X 22 X 23 X23 T }, Φ 25ij =[Cdi T LT i Cdi T LT j Cdj T LT i Φ ], 33 = diag[ r 1 S r+1 2 S 2 ], Φ 44 = diag[ r 1(6τ 2 Mε) 1 P 2 (6τ M ε) 1 P 2 (6τ M ε) 1 P 2 ]. In this case, observer gain matrices H j can be obtained by H j = P 1 2 L j, j =1,,r. (20) Proof: Let R 2 = εp 2 where ε>0 is a given scalar. We define L j = P 2 H j in (13). Then, we get Φ 11ij Φ 12ij ρp 2 N di Φ ij = Φ T 12ij Φ 22ij 0 <0, i,j=1,,r ρndi T P 2 0 S 2

12 1314 R. Takada, Y. Uchida and J. Yoneyama where Φ 11ij = A T i P 2 + P 2 A i Ci T LT j L jc i + Ni T P 2N i + ρ 2 P 2 +6τ M εa T i P 2 A i +6τ M ερ 2 Ni T P 2 N i +6τ M εci T L T j P2 1 L j C i d Q 2 + τ M X 11 + X 13 + X13 T, Φ 12ij = P 2 A di L j C di + τ M X 12 X 13 + X23, T Φ 22ij = (Q 2 S 2 )+6τ M εa T di P 2A di +6τ M ερ 2 Ndi T P 2N di +6τ M εcdil T T j P2 1 L j C di + τ M X 22 X 23 X23, T It follows from Lemma 2.3 that Φ ii < 0,,,r, 2 r 1 Φ ii +Φ ij +Φ ji < 0, i,j=1,,r, i j. Finally, H j can be calculated via definition of L j. 5 Output Feedback Stabilization Here, we show that an output feedback controller resulting from a state feedback controller obtained from Theorem 3.1 and an observer from Theorem 4.2 becomes a stabilizing controller with guaranteed cost for the fuzzy bilinear time-delay system (1) and (2). Consider the state feedback controller (10) with the estimate ˆx, and the estimate of the state is calculated by the observer (9). Then, the closed-loop system (1) and (2) with (9) and (10) is given by where x c (t) =[x T (t) ẋ c (t) = λ i (ξ)λ j (ξ){âijx c (t)+âdijx c (t τ(t))} (21) j=1 e T (t)] T and [ Ai + ρn  ij = i sin ˆθ j + ρb i K j cos ˆθ j ρb i K j cos ˆθ ] j 0 A i H j C i + ρn i sin ˆθ, j [ Adi + ρn  dij = di sin ˆθ ] j 0 0 A di H j C di + ρn di sin ˆθ. j Our final problem is to show that if conditions in Theorems 3.1 and 4.2 are satisfied, the closed-loop system (21) is stable with guaranteed cost. In other words, the output feedback controller (9) and (10) stabilizes the fuzzy bilinear time-delay system (1) and (2) with guaranteed cost.

13 Guaranteed cost control for fuzzy bilinear systems 1315 Theorem 5.1 Suppose that there exist X, X2, P > 0, P2 > 0, Q > 0, Q 2 > 0, S > 0, S2 > 0 such that conditions (5)-(7) and (17)-(19) are satisfied, respectively. Then, an output feedback controller (9) and (10) stabilizes the fuzzy bilinear time-delay system (1) and (2) with guaranteed cost. In this case, gain matrices K j, H j, j =1,,r are given by (8) and (20), respectively. Proof: We first note that if there exists P >0 which satisfies (5)-(7), then we can show as in [6] that there exists P>0which satisfies [ ] Λ11ij Λ Λ ij = 12i Λ T 12i Λ < 0 (22) 22i where Λ ij = A T i P + PA i + ρ 2 P + ρ 2 PN di S 1 N T dip d Q +4τ MA T i RA i +N T i (P +4τ Mρ 2 R)N i + K T j BT i (P +3τ Mρ 2 R)B i K j + U +ρ 2 K T j TK j + τ M X 11 + X 13 + X T 13, Λ 12i = PA di + τ M A T i RA di + τ M X 12 X 13 + X T 23, Λ 22i = (Q S)+3τ M A T di RA di +4τ M ρ 2 N T di RN di + τ M X 22 X 23 X T 23. Consider the following Lyapunov-Krasovskii functional candidate: [ ] P 0 V (x ct )=x T c (t) x 0 αp c (t)+ 1 [ ] t Q 0 x T c 2 1 d (s) x t τ(t) 0 αq c (s)ds 2 0 [ ] t εp 0 + ẋ T c (s) ẋ τ M t θ 0 αεp c (s)dsdθ 2 where x ct = x c (t + s), τ M s 0 and P, P 2, Q and Q 2 satisfy conditions (5)-(7), (17)-(19), and α>0 is some scalar. Following the similar lines of proofs of Theorems 3.1 and 4.2, we calculate the derivative of V (x ct ): [ ] V (x ct ) λ i (ξ)λ j (ξ)ζ T Λij Δ ij (t) Δ T j=1 ij αˆξ ζ(t). ij where ζ(t) =[x T (t) x T (t τ(t)) e T (t) e T (t τ(t))] T and Λ ij and ˆΞ ij are given in (22) and (16), respectively. Δ ij is given by [ ρp Bi K Δ ij = j cos ˆθ ] j If condition (5)-(7) and (17)-(19) are met, we have Λ ij < 0 and ˆΞ ij < 0 for all i, j. Since we can always choose α>0 such that αˆξ ij < Δ T ijλ 1 ij Δ ij, i,j =1,,r,

14 1316 R. Takada, Y. Uchida and J. Yoneyama with that α we have by Schur complement formula [ Λij Δ ij Δ T ij αˆξ ij ] < 0. It follows that V (x ct ) < 0. This proves the stability of the closed-loop system (21). Finally, gain matrices are calculated by (8) and (20). 6 Examples We consider a fuzzy bilinear time-delay system with the following if-then rules: IF ξ 1 is big, THEN ẋ(t) =A 1 x(t)+a d1 x(t τ(t)) + u(t)n 1 x(t) +u(t)n d1 x(t τ(t)) + B 1 u(t), y(t) =C 1 x(t)+c d1 x(t τ(t)), IF ξ 1 is small, THEN ẋ(t) =A 2 x(t)+a d2 x(t τ(t)) + u(t)n 2 x(t) +u(t)n d2 x(t τ(t)) + B 2 u(t), y(t) =C 2 x(t)+c d2 x(t τ(t)) where λ 1 (x 1 )= 1 1+e x 1, λ 2(x 1 )=1 λ 1 (x 1 ) and [ ] [ ] [ ] [ ] A 1 =,A =,A d1 =,A 520 d2 =, 025 [ ] [ ] [ ] [ ] N 1 = N 2 =,N 0 3 d1 =,N d2 =,B =, 0 [ ] 1 B 2 =,C 1 1 = [ 10 ],C 2 = [ 1.10 ],C d1 = [ 0.80 ],C d2 = [ ]. Theorem 3.1 gives the following control gain matrices in (3) and/or (10). K 1 = [ ],K 2 = [ ]. Theorem 4.2 gives the following observer gain matrices in (9). H 1 = [ ] ,H = [ ] Theorem 5.1 proves the observer-based controller (9) and (10) with the above gain matrices become a stabilizing output feedback controller for the system (1) with guaranteed cost.

15 Guaranteed cost control for fuzzy bilinear systems Conclusions In this paper, stability conditions and output feedback stabilization with guaranteed cost of fuzzy bilinear systems with time-varying delays have been considered. It was shown that if there exist stabilizing state feedback controller and observer, then an output feedback controller resulting from the same state feedback controller and observer becomes a stabilizing controller with guaranteed cost. In other words, we establish the separation of state feedback control and observer design for fuzzy bilinear time-delay systems. References [1] Z.-F. Gao, J. Chen and F. Liu: Guaranteed cost control for fuzzy bilinear systems with uncertain paramenters, 2010 International Conference on Electrical Engineering and Automatic Control, pp , [2] X.P. Guan and C.L. Chen: Delay-dependent guaranteed cost control for T-S fuzzy systems with time-delay, IEEE Transactions on Fuzzy Systems, Vol. 12, pp , [3] T.-H.S. Li, S.-H. Tsai, J.-Z. Lee, M.-Y. Hsiao and C.-H. Chao: Robust H fuzzy control for a class of uncertain discrete fuzzy nonlinear systems, IEEE Transactions on Fuzzy Systems, Man, and Cybernetics, Part B, Vol.28, pp , [4] T.-H.S. Li and S.-H. Tsai: T-S Fuzzy bilinear model and fuzzy controller design for a class of nonlinear systems, IEEE Transactions on Fuzzy Systems, Vol. 15, pp , [5] M. S. Mahmoud: Robust Control and Filtering for Time-Delay Systems, Marcel Dekker, Inc., New York, [6] R. Takada, Y. Uchida and J.Yoneyama: Guaranteed Cost Control Design for Bilinear Fuzzy Systems, to appear in IEEJ Transactions on Electronics, Information and Systems, Vol. 132, No. 12, [7] T. Takagi and M. Sugeno: Fuzzy identification of systems and its applications to modeling and control, IEEE Transaction on Systems, Man, Cybernetics, Vol. 15, pp , [8] K. Tanaka and H.O. Wang: Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach, Wiley Inter-Science, 2000.

16 1318 R. Takada, Y. Uchida and J. Yoneyama [9] S.-H. Tsai and T.-H.S. Li: Robust fuzzy control of a class of fuzzy bilinear systems with time-delay, Chaos, Solitons and Fractals, Vol. 39, pp , [10] H.D Tuan, P. Apkarian, T. Narikiyo and Y. Yamamoto: Parameterized linear matrix inequality techniques in fuzzy control system design, IEEE Transactions on Fuzzy Systems, Vol. 9, pp , [11] J. Yoneyama, Output feedback control design for Takagi-Sugeno fuzzy bilinear time-delay systems, 2010 IEEE Conference on Systems, Man and Cybernetics, pp , Received: November, 2012