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Delay-dependent fuzzy static output feedback control for discrete-time fuzzy stochastic systems with distributed time-varying delays

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This paper is concerned with the delay-dependent H∞ fuzzy static output feedback control scheme for discrete-time Takagi–Sugeno (T–S) fuzzy stochastic systems with distributed time-varying delays. To …

This paper is concerned with the delay-dependent H∞ fuzzy static output feedback control scheme for discrete-time Takagi–Sugeno (T–S) fuzzy stochastic systems with distributed time-varying delays. To begin with, the T–S fuzzy stochastic system is transformed to an equivalent switching fuzzy stochastic system. Then, based on novel matrix decoupling technique, improved free-weighting matrix technique and piecewise Lyapunov–Krasovskii function (PLKF), a new delay-dependent H∞ fuzzy static output feedback controller design approach is first derived for the switching fuzzy stochastic system. Some drawbacks existing in the previous papers such as matrix equalities constraint, coordinate transformation, the same output matrices, diagonal structure constraint on Lyapunov matrices and BMI problem have been eliminated. Since only a set of LMIs is involved, the controller parameters can be solved directly by the Matlab LMI toolbox. Finally, two examples are provided to illustrate the validity of the proposed method.

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  • 1. ISA Transactions 51 (2012) 702–712 Contents lists available at SciVerse ScienceDirect ISA Transactions journal homepage: www.elsevier.com/locate/isatrans Delay-dependent fuzzy static output feedback control for discrete-time fuzzy stochastic systems with distributed time-varying delays$ ZhiLe Xia a,b,n, JunMin Li a, JiangRong Li a a b School of Science, Xidian University, ShanXi, Xi’an 710071, China School of Mathematics and Information Engineering, Taizhou University, ZheJiang, Taizhou 317000, China a r t i c l e i n f o a b s t r a c t Article history: Received 18 April 2010 Received in revised form 19 June 2012 Accepted 20 June 2012 Available online 12 July 2012 This paper is concerned with the delay-dependent H1 fuzzy static output feedback control scheme for discrete-time Takagi–Sugeno (T–S) fuzzy stochastic systems with distributed time-varying delays. To begin with, the T–S fuzzy stochastic system is transformed to an equivalent switching fuzzy stochastic system. Then, based on novel matrix decoupling technique, improved free-weighting matrix technique and piecewise Lyapunov–Krasovskii function (PLKF), a new delay-dependent H1 fuzzy static output feedback controller design approach is first derived for the switching fuzzy stochastic system. Some drawbacks existing in the previous papers such as matrix equalities constraint, coordinate transformation, the same output matrices, diagonal structure constraint on Lyapunov matrices and BMI problem have been eliminated. Since only a set of LMIs is involved, the controller parameters can be solved directly by the Matlab LMI toolbox. Finally, two examples are provided to illustrate the validity of the proposed method. & 2012 ISA. Published by Elsevier Ltd. All rights reserved. Keywords: Piecewise Lyapunov–Krasovskii function (PLKF) Linear matrix inequalities (LMIs) Fuzzy static output feedback Switching fuzzy stochastic system Delay-dependent 1. Introduction The well-known Takagi–Sugeno (T–S) fuzzy model [1] has been recognized as a popular and powerful tool in approximating complex nonlinear systems. As a consequence, the study of T–S fuzzy systems has attracted an increasing interest in the past decades (see [2–6] for more details). Moreover, system state variables are often not fully available for practical systems. Some state variables may be difficult/costly to measure and sometimes have no physical meaning and thus cannot be measured at all. It may be possible to use an observer to estimate unknown states, but this approach not only requires more hardware resources, but also makes the dimension of the system increase greatly. In this situation, the static output feedback (SOF) control is more suitable for practical application. Syrmos et al. [20] show that any dynamic output feedback [29] problem can be transformed into a SOF problem, but the converse is not true. Recently, the fuzzy SOF control for T–S fuzzy systems has drawn intensive attention [18,19,21,23–28]. As stated in [16,17,25], the fuzzy SOF controller design for T–S fuzzy system $ This work was supported by the National Natural Science Foundation of China (60974139) and Fundamental Research Funds for the Central Universities (72103676). n Corresponding author at: School of Mathematics and Information Engineering, Taizhou University, ZheJiang, Taizhou 317000, China. Tel.: þ 86 13736515790. E-mail address: zhle_xia@yahoo.com.cn (Z. Xia). is not easy, due to the fact that many rules interference effects are increased. Moreover, the fuzzy SOF control design becomes much more difficult and complex than state feedback one because it belongs to a nonlinear matrix inequalities problem. In [19], the fuzzy SOF control law was given in terms of bilinear matrix inequalities which cannot be solved with ease by a convex optimization algorithm. An iterative linear matrix inequality algorithm was proposed in [21,27] and an optimization technique was investigated in [26]. The problem of designing robust SOF controllers for linear discrete-time systems with time-varying polytopic uncertainties was studied in [22]. In [23], three drawbacks existing in the previous papers such as coordinate transformation, the same output matrices and BMI problem have been eliminated. But linear matrix equalities (LMEs) (i.e. MC yi ¼ C yi P, i ¼ 1, . . . ,r) were included in [23]. By using coordination transformation, a less conservative result was derived in [24] by removing the constraint that the considered Lyapunov matrix is diagonal. In [24], all the local output matrices were assumed to be the same (i.e. C yi ¼ C 2 , i ¼ 1, . . . ,r). If the local output matrices are not the same, coordinate transformation is almost impossible. Hybrid approaches for regional T–S fuzzy SOF controller design was considered in [25]. The authors in [28] investigated the reliable mixed L2 =H1 fuzzy SOF control for nonlinear systems with sensor faults. On the other hand, time delay often occurs in many dynamic systems, such as rolling mill systems, biological systems, metallurgical processing systems, network systems, and so on. Their 0019-0578/$ - see front matter & 2012 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2012.06.011
  • 2. Z. Xia et al. / ISA Transactions 51 (2012) 702–712 existence is frequently a cause of instability and poor performance. The T–S fuzzy model was first used to deal with the stability analysis and control synthesis of nonlinear time delay systems in [7]. After that, many people devoted a great deal of effort to both theoretical research and implementation techniques for T–S fuzzy systems with time delays [8,9]. However, these results rely on the existence of a common positive definite matrix P for all linear models, which in general leads to a conservative result. To reduce this conservatism, the piecewise Lyapunov function approach [10,11] and the fuzzy Lyapunov function approach [12,13] have been proposed. The delay-independent SOF stabilization for T–S fuzzy system with interval time-delay was investigated in [18] by using the common Lyapunov function approach, which typically led to conservative results. Moreover, as well known as, the delay-independent results [18] are generally more conservative than the delay-dependent ones [8], especially when the size of time delay is small [10–13]. It should be noted that all the above-mentioned results are in the deterministic setting. It is well known that stochastic disturbance often exist in many practical systems. Their existence is a source of instability of control systems, so the investigation on stability analysis and control design of stochastic systems has received increasing attention in recent years [31,32,37,38]. It should be pointed out that, so far, there have been only a few papers that have taken the stochastic phenomena into account in the fuzzy systems [30–37,39]. Research in this area should be interesting yet challenging as it involves the combination of two classes of important systems, namely, stochastic systems and fuzzy systems [37]. The problem of delay-dependent robust fuzzy control for stochastic fuzzy systems with parameter uncertainties and time-varying delay was studied in [31]. Robust stability analysis for uncertain stochastic fuzzy systems with time delay was investigated in [32], and some delay-independent stability conditions were derived. The authors in [33] investigated the passivity and passification problems of the stochastic discretetime T–S fuzzy systems with delay. A sliding mode control approach was proposed in [35] to investigate the fuzzy control problem for uncertain stochastic systems. The problem of stabilization for T–S fuzzy stochastic delay systems was considered in [36,39]. The filtering problem was investigated in [30,34,37]. In addition, the distributed delays in discrete-time systems [40–43] are seldom found in the literature when compared with the distributed delay in continuous-time systems that are described in the form of either finite or infinite integral [44]. As pointed in [40,41], a large-scale system network usually has a spatial nature due to the presence of an amount of parallel pathways of a variety of subsystem sizes and lengths, which gives rise to possible distributed delays for discrete-time systems. With the increasing application of digitalization, distributed delays may emerge in a discrete-time manner, and therefore, it becomes desirable to study the discrete-time systems with distributed delays. The research on discrete-time fuzzy systems with distributed delays remains as an open topic for further investigation. To the best of the authors’ knowledge, there have not been any reported results on the fuzzy SOF control for discrete-time fuzzy stochastic system with distributed time-varying delay. Motivated by the aforementioned discussion, we investigate the H1 fuzzy SOF control for discrete-time T–S fuzzy stochastic systems with distributed time-varying delays. By using improved free-weighting technique and piecewise Lyapunov–Krasovskii function, a new delay-dependent H1 fuzzy SOF controller design approach is derived. Adopting a new matrix decoupling technique, some drawbacks existing in the previous papers such as coordinate transformation, the same output matrices, the diagonal constraint condition on Lyapunov matrices, matrix equalities constraint and BMI problem have been eliminated. Since only a 703 set of LMIs is involved, the fuzzy SOF controller parameters can be solved directly. Finally, two examples are given to illustrate the validity of the proposed method. Notation. For a real matrix S, He fSg denotes S þ ST . The symmetric elements of the symmetric matrix will be denoted by n. Efng stands for the mathematical expectation of the stochastic process or vector. I and 0 represent, respectively, identity matrix and zero matrix. We first introduce the following lemmas, which are crucial to the development of our main results. Lemma 1. For the real positive-definite symmetric matrix P1, matrices A1, A2, B1, B2, D1 ,D2 , D3 and P2 with compatible dimensions, the following inequalities are equivalent, where aI are extra slack nonsingular matrix: " T # A1 P 1 A1 þ BT P 1 B1 þAT A2 þ BT B2 ÀHe fAT P 2 g þ D1 D2 1 2 2 1 ðaÞ o0, n D3 2 0 0 0 ÀaB1 n P1 À2aI 0 0 ÀP 2 þ aA1 n n ð1À2aÞI 0 ÀaA2 n n n ð1À2aÞI n n n n ÀaB2 D1 n ðbÞ 6 6 6 6 6 6 6 6 6 4 P1 À2aI n n n n 0 3 7 0 7 7 0 7 7 7 o0: 0 7 7 D2 7 5 D3 Proof. See the Appendix. Remark 1. By the introduction of the auxiliary slack matrix variable aI, the matrices P1, P2 and A1 , A2 , B1 , B2 are decoupled. This novel technique is proposed in this paper to transform the nonlinear matrix inequalities (30) into a set of linear matrix inequalities (LMIs), which can be easily solved by LMI Toolbox in Matlab. Lemma 2 (Wei et al. [40,41]). Let M A RnÂn be a positive semidefinite matrix, xðmÞ A Rn , we have !T ! t t t X X 1 X xT ðmÞMxðmÞ r À xðmÞ M xðmÞ : À t m¼1 m¼1 m¼1 Lemma 3 (Tian et al. [46] and Lin et al. [47]). Let L, L1 A RpÂp be symmetric matrices. Then L þ tðtÞL1 o0 holds for all tðtÞ A ½t1 , t2 Š if and only if L þ tn L1 o0, n ¼ 1; 2: 2. Problem formulation Consider the following discrete-time Takagi–Sugeno (T–S) fuzzy stochastic system with distributed time-varying delays: Plant Rule i: IF y1 ðtÞ is mi1 and y and yp ðtÞ is mip THEN " # tÀ1 X xðnÞ þ Bi oðtÞ þB1i uðtÞ xðt þ1Þ ¼ Ai xðtÞ þ Adi n ¼ tÀtðtÞ " tÀ1 X þ M i xðtÞ þ M di # xðnÞ þ B2i uðtÞ vðtÞ, n ¼ tÀtðtÞ " zðtÞ ¼ C i xðtÞ þ C di " tÀ1 X # xðnÞ þ Di oðtÞ þ D1i uðtÞ n ¼ tÀtðtÞ þ N i xðtÞ þ Ndi tÀ1 X n ¼ tÀtðtÞ # xðnÞ þ D2i uðtÞ vðtÞ,
  • 3. 704 Z. Xia et al. / ISA Transactions 51 (2012) 702–712 yðtÞ ¼ Gi xðtÞ, t ¼ Àt2 ,Àt2 þ1, . . . ,0, i ¼ 1; 2, . . . ,r, xðtÞ ¼ fðtÞ, ð1Þ where r is the number of IF–THEN rules; yj and mij ðj ¼ 1; 2, . . .Þ are the premise variables and the fuzzy sets, respectively; xðtÞ A Rn , yðtÞ A Rn1 , zðtÞ A Rn2 and fðtÞ are the state, the measured output, the controlled output to be estimated, and the initial condition, respectively; oðtÞ is the disturbance input which belongs to l2 ½0,1Þ; tðtÞ denotes distributed time-varying delays and satisfies 0 o t1 r tðtÞ r t2 ; Ai , Adi , Bi , M i , M di , C i , C di , Di , Ni , Ndi , B1i , B2i , D1i , D2i and Gi are all constant matrices with appropriate dimensions; v(t) is a real scalar stochastic process with Efv2 ðtÞg ¼ 1, EfvðtÞg ¼ 0, EfvðlÞvðmÞg ¼ 0 ðl amÞ: The final output of the switching fuzzy stochastic system (3) is inferred as follows: (" # bðjÞ tÀ1 X X xðt þ1Þ ¼ hjk Ajk xðtÞ þ Adjk xðnÞ þ Bjk oðtÞ þ B1jk uðtÞ " tÀ1 X þ M jk xðtÞ þ M djk # ) xðnÞ þ B2jk uðtÞ vðtÞ , n ¼ tÀtðtÞ (" bðjÞ X zðtÞ ¼ hjk tÀ1 X C jk xðtÞ þ C djk # xðnÞ þDjk oðtÞ þ D1jk uðtÞ n ¼ tÀtðtÞ k¼1 " tÀ1 X þ N jk xðtÞ þ Ndjk ð2Þ Remark 2. As mentioned in introduction, the research on discrete-time fuzzy systems with distributed delays remains as an open topic for further investigation. In [40,41,45], the authors introduced the distributed delays in the form of constant delay P1 PÀ1 t ¼ Àd hðxðk þ tÞÞ or infinite delay t ¼ 1 xðtÀtÞ. The authors in [42,43] investigated the distributed time-varying delays in the PÀ1 form of m ¼ ÀdðkÞ f ðxðk þmÞÞ, where nonlinear function f ðÞ was assumed to satisfy sector-bounded condition. For the sake of simplicity, this paper studies the distributed time-varying delay PtÀ1 in the form of m ¼ tÀtðtÞ xðmÞ and a new delay-dependent H 1 fuzzy static output feedback controller design approach is proposed for discrete-time T–S fuzzy stochastic system. In contrast to aforementioned results, two new techniques, improved freeweighting technique and new matrix decoupling technique, are developed. n ¼ tÀtðtÞ k¼1 # ) xðnÞ þD2jk uðtÞ vðtÞ , n ¼ tÀtðtÞ bðjÞ X yðtÞ ¼ hjk Gjk xðtÞ, k¼1 xðtÞ ¼ fðtÞ, t ¼ Àt2 ,Àt2 þ 1, . . . ,0, yðtÞ A Oj , j ¼ 1, . . . ,s, ð4Þ where Qp mjkl ðyl ðtÞÞ : m ðy ðtÞÞ l ¼ 1 jkl l hjk ¼ hjk ðyðtÞÞ ¼ PbðjÞ l ¼ 1p Q k¼1 Considering the fuzzy system (3) in each subregion, we choose the following fuzzy SOF controller: Region Rule j: IF yðtÞ A Oj THEN Local Plant Rule k IF y1 ðtÞ is mjk1 , . . . , and yp ðtÞ is mjkp THEN uðtÞ ¼ ÀF jk yðtÞ, k ¼ 1, . . . , bðjÞ, j ¼ 1, . . . ,s, ð5Þ As illustrated in [11], we will define open subregions as Op ðp ¼ 1, . . . ,sÞ in the state-space. The corresponding close subregions are defined as O p , which satisfy where Fjk is a local controller gain to be designed. The final output of (5) is inferred by O p O q ¼ @Ou , p a q, p,q ¼ 1, . . . ,s, i ¼ 1; 2, . . . ,r, i uðtÞ ¼ À n ¼ tÀtðtÞ tÀ1 X þ M jk xðtÞ þM djk # tÀ1 X # þ Njk xðtÞ þ N djk tÀ1 X xðnÞ þ D2jk uðtÞ vðtÞ, yðtÞ ¼ Gjk xðtÞ, t ¼ Àt2 ,Àt2 þ1, . . . ,0, i ¼ 1; 2, . . . ,r, xðt þ1Þ ¼ bðjÞ bðjÞ bðjÞ XX X hjk hjl hjm ½f jklm xðtÞ þ g jklm xðtÞvðtÞŠ k¼1l¼1m¼1 9f jklm xðtÞ þ g jklm xðtÞvðtÞ, zðtÞ ¼ bðjÞ bðjÞ bðjÞ XX X hjk hjl hjm ½f zjklm xðtÞ þ g zjklm xðtÞvðtÞŠ k¼1l¼1m¼1 9f zjklm xðtÞ þg zjklm xðtÞvðtÞ, t ¼ Àt2 ,Àt2 þ1, . . . ,0, j ¼ 1, . . . ,s, yðtÞ A Oj , ð7Þ Definition 1 (Gao et al. [38]). The closed-loop system (7) is said to be mean-square stable if, under oðtÞ ¼ 0, for any E 40, there is 2 a dðEÞ 4 0 such that Ef9xðtÞ9 g o E,t 4 0 when supÀt2 r s r 0 2 2 Ef9xðsÞ9 g o dðEÞ. In addition, if limt-1 Ef9xðtÞ9 g ¼ 0 for any initial conditions, then it is said to be mean-square asymptotically stable. # n ¼ tÀtðtÞ xðtÞ ¼ fðtÞ, j ¼ 1, . . . ,s, yðtÞ A Oj : g jklm ¼ ½M jk ÀB2jk F jl Gjm where f jklm ¼ ½Ajk ÀB1jk F jl Gjm Adjk Bjk Š, Mdjk 0Š, f zjklm ¼ ½C jk ÀD1jk F jl Gjm C djk Djk Š, g zjklm ¼ ½Njk ÀD2jk F jl Gjm PtÀ1 T T T Ndjk 0Š, xðtÞ ¼ ½xT ðtÞ n ¼ tÀtðtÞ x ðnÞ o ðtÞŠ . xðnÞ þ Djk oðtÞ þ D1jk uðtÞ n ¼ tÀtðtÞ hjl hjm F jl Gjm xðtÞ, l¼1m¼1 Applying the fuzzy SOF controller (6) to the global fuzzy system (4), we have the following closed-loop system: T n ¼ tÀtðtÞ zðtÞ ¼ C jk xðtÞ þC djk bðjÞ bðjÞ X X ð6Þ xðtÞ ¼ f ðtÞ, xðnÞ þ B2jk uðtÞ vðtÞ, hjl F jl yðtÞ ¼ À l¼1 @Ou i ¼ fyðtÞ9hi ðyðtÞÞ ¼ 1; 0 r hi ðyðtÞ þ EÞ o 1,80 o 9E9 51g, u is where the set of face indexes of the polyhedral hull @Oi ¼ [@Ou , i P hi ðyðtÞÞ ¼ oi ðyðtÞÞ= r ¼ 1 oi ðyðtÞÞ, oi ðyðtÞÞ ¼ Pp¼ 1 mij ðyj ðtÞÞ, yðtÞ ¼ i j ½y1 ðtÞ, . . . , yp ðtÞŠ. Next, we follow the idea of [14] to rewrite the system (1) to be an equivalent discrete-time switching fuzzy stochastic system as the following form: Region Rule j: IF yðtÞ A Oj THEN Local Plant Rule k IF y1 ðtÞ is mjk1 , . . ., and yp ðtÞ is mjkp THEN # tÀ1 X xðnÞ þBjk oðtÞ þB1jk uðtÞ xðt þ 1Þ ¼ Ajk xðtÞ þ Adjk bðjÞ X ð3Þ where Oj denotes the jth subregion, s is the number of subregions partitioned on the state space, and bðjÞ is the number of rules in the subregion Oj . Definition 2 (Gao et al. [38]). Given g 40, the closed-loop system (7) is said to be mean-square asymptotically stable with H1 performance g if it is mean-square asymptotically stable and, under zero-initial conditions, for all nonzero disturbance oðtÞ A l2 ½0,1Þ,
  • 4. Z. Xia et al. / ISA Transactions 51 (2012) 702–712 satisfies EfJzðtÞJ2 g o gJoðtÞJ2 , ð8Þ where 8 !1=2 9 X = 1 , zT ðtÞzðtÞ EfJzðtÞJ2 g9E : t¼0 ; f jklm ¼ ½aj Ajk ÀB1jk U jl Gjm aj Adjk aj Bjk Š, gjklm ¼ ½aj M jk ÀB2jk U jl Gjm aj M djk aj Bjk Š, f zjklm ¼ ½aj C jk ÀD1jk U jl Gjm aj C djk aj Djk Š, gzjklm ¼ ½aj Njk ÀD2jk U jl Gjm aj N djk 0Š: 1 X JoðtÞJ2 9 705 oT ðtÞoðtÞÞ1=2 : Moreover, the switching fuzzy SOF controllers are given by t¼0 ð9Þ The main purpose of this paper is to design fuzzy SOF controller of the form (6) such that the closed-loop system (7) is mean-square asymptotically stable with an H1 performance g. F jl ¼ aÀ1 U jl , j l ¼ 1, . . . , bðjÞ, j ¼ 1, . . . ,s: Proof. Let ZðnÞ ¼ xðn þ1ÞÀxðnÞ: ð14Þ Then, I ZðtÞ ¼ xðt þ 1ÞÀxðtÞ9f jklm xðtÞ þ g jklm xðtÞvðtÞ, 3. Main results In this section, the delay-dependent H1 fuzzy SOF controller design approach is presented for the T–S fuzzy stochastic systems described in the previous section. Let the subregion transition from Oj to Oi be denoted by O ¼ fðj,iÞ9yðtÞ A Oj , yðt þ 1Þ A Oi g. Here, i may be equal to j in O, when yðtÞ and yðt þ1Þ are in the same subregion. Consequently, we have the following result. Theorem 1. Given a constant g 40, the closed-loop system (7) is mean-square asymptotically stable with H1 performance g, if there exist a set of positive-definite symmetric matrices Pj, Z 1 , Z 2 , Q 1 , Q 2 , Q 3 , matrices Ujl, Tj, Wj, Sj, Xj, Yj and the nonsingular matrices aj I, j¼1,y,s, l ¼ 1, . . . , bðjÞ, such that the following LMIs are satisfied: Pn o0, l ¼ 1, . . . , bðjÞ, ðj,iÞ A O, n ¼ 1; 2, ijlll Xj Tj n Z1 # Yj Sj n 4 0, Z2 ð11Þ V 2 ðtÞ ¼ t2 X tÀ1 X m ¼ 1 n ¼ tÀm V 3 ðtÞ ¼ tðtÞ X tÀ1 X t2 X xT ðnÞQ 1 xðnÞ þ ZT ðnÞZ 2 ZðnÞ, mÀ1 X tÀ1 X xT ðsÞQ 1 xðsÞ, m ¼ t1 þ 1 n ¼ 1 s ¼ tÀn xT ðmÞQ 2 xðmÞ þ tÀ1 X xT ðmÞQ 3 xðmÞ, yðtÞ A Oj , m ¼ tÀt2 where Pj, Q 1 ,Q 2 ,Q 3 , Z1 and Z2 are the real symmetric positive definite matrices to be determined. Then along the solution of the closed-loop system (7), we have # Xj þ Y j Wj Z1 þ Z2 ¼ EfDV 1 ðtÞ9xðtÞg þ EfDV 2 ðtÞ9xðtÞg þ EfDV 3 ðtÞ9xðtÞg þ EfDV 4 ðtÞ9xðtÞg, where 40, ð13Þ EfDV 1 ðtÞ9xðtÞg ¼ Ef½xT ðt þ 1ÞP i xðt þ 1ÞÀxT ðtÞPj xðtÞŠ9xðtÞg T T T ¼ Ef½x ðtÞf jklm Pi f jklm xðtÞ þ2vT ðtÞx ðtÞg T P i f jklm xðtÞ jklm where 2 6 6 6 6 n Pijklm ¼ 6 6 6 6 6 4 P1 À2aj I n 0 P1 À2aj I n n 0 0 ð1À2aj ÞI n n n 0 0 0 ð1À2aj ÞI n n n n n n n D1 n Àgjklm ÀP2 þ f jklm Àf zjklm Àgzjklm n 0 0 0 0 3 7 7 7 7 7 7, 7 7 D2 7 5 D3 T þ vT ðtÞx ðtÞg T Pi g jklm xðtÞvðtÞÀxT ðtÞPj xðtÞŠ9xðtÞg, jklm ð16Þ È EfDV 2 ðtÞ9xðtÞg ¼ E ZT ðtÞðt2 Z 1 þ t21 Z 2 ÞZðtÞ À tÀ1 X ZT ðnÞZ 1 ZðnÞÀ n ¼ tÀt2 with ¼E t21 ¼ t2 Àt1 , n ¼ 1; 2, P1 ¼ Pi þ t2 Z 1 þ t21 Z 2 , P2 ¼ ðt2 Z 1 þ t21 Z 2 Þ½I 0 0Š, 2 D11 D12 D22 D1 ¼ 6 n 4 n n 0 3 0 7 5, Àg2 I 0 Sj D2 ¼ B 0 @ 0 ÀW j 1 C 0 A, 0 D3 ¼ diagfÀQ 2 ,ÀQ 3 g, D11 ¼ t2 Q 1 þ 1t21 ðt2 þ t1 À1ÞQ 1 þ Q 2 þQ 3 þ t2 X j þ t21 Y j 2 þ He fT j g þ t2 Z 1 þ t21 Z 2 ÀPj , 1 tn ðW j ÀT j ÀSj Þ, D22 ¼ À 1 tn Q 1, tÀt1 À1 X n ¼ tÀt2 ( D12 ¼ tÀ1 X m ¼ t1 þ 1 n ¼ tÀm m ¼ 1 n ¼ tÀm tÀ1 X t2 X ZT ðnÞZ 1 ZðnÞ þ J9EfVðt þ1Þ9xðtÞgÀVðtÞ ¼ Ef½Vðt þ 1ÞÀVðtÞŠ9xðtÞg ð12Þ n 4 0, V 1 ðtÞ ¼ xT ðtÞPj xðtÞ, m ¼ tÀt1 m ¼ 1, . . . , bðjÞÀ2, k ¼ m þ 1, . . . , bðjÞÀ1, l ¼ k þ1, . . . , bðjÞ, ðj,iÞ A O, n ¼ 1; 2, VðtÞ ¼ V 1 ðtÞ þ V 2 ðtÞ þ V 3 ðtÞ þ V 4 ðtÞ, V 4 ðtÞ ¼ Pn þ Pn þ Pn þ Pn þ Pn þ Pn o 0, ijklm ijkml ijlmk ijlkm ijmlk ijmlk # where f ¼ f jklm À½I 0 0Š. Consider a discrete-time piecewise Lyapunov–Krasovskii function candidate for the closed-loop system (7) as following: ð10Þ Pn þ Pn þ Pn o0, k ¼ 1, . . . , bðjÞÀ1, ijkll ijlkl ijllk l ¼ kþ 1, . . . , bðjÞ, ðj,iÞ A O, n ¼ 1; 2, ð15Þ I jklm tðtÞ X ZT ðtÞðt2 Z 1 þ t21 Z 2 ÞZðtÞÀ tðtÞ X tÀt1 À1 X ZT ðnÞ m ¼ 1 n ¼ tÀt2 1 Z ZðnÞ tðtÞ 2 È ¼ E ZT ðtÞðt2 Z 1 þ t21 Z 2 ÞZðtÞ tÀ1 X ZT ðnÞ 1 tðtÞ Z 1 ZðnÞ ) # xðtÞ tðtÞ X tÀt1 À1 X 1 1 Z 1 ZðnÞÀ Z ZðnÞ ZT ðnÞ tðtÞ tðtÞ 2 m ¼ 1 n ¼ tÀm m ¼ 1 n ¼ tÀm ) # tðtÞ X X tÀmÀ1 1 ðZ 1 þ Z 2 ÞZðnÞ xðtÞ ZT ðnÞ À tðtÞ m ¼ 1 n ¼ tÀt2 nh I I T ¼ E x ðtÞðf jklm ÞT ðt2 Z 1 þ t21 Z 2 Þf jklm xðtÞ À tðtÞ X ) m ¼ 1 n ¼ tÀt2 À # ZT ðnÞZ 2 ZðnÞ xðtÞ tÀ1 X T ZT ðnÞ I þ2vT ðtÞx ðtÞg T ðt2 Z 1 þ t21 Z 2 Þf jklm xðtÞ jklm
  • 5. 706 Z. Xia et al. / ISA Transactions 51 (2012) 702–712 T þ vT ðtÞx ðtÞg T ðt2 Z 1 þ t21 Z 2 Þg jklm xðtÞvðtÞ jklm tðtÞ X X tÀt1 À1 1 1 Z 1 ZðnÞÀ Z ZðnÞ ZT ðnÞ tðtÞ tðtÞ 2 m ¼ 1 n ¼ tÀm m ¼ 1 n ¼ tÀm ) # tðtÞ X X tÀmÀ1 1 T ð17Þ ðZ þ Z ÞZðnÞ xðtÞ , Z ðnÞ À tðtÞ 1 2 m ¼ 1 n ¼ tÀt À tðtÞ X tÀ1 X # tðtÞ tÀ1 tÀ1 X 1 1 X X w1 ¼ 2x ðtÞT j xðtÞÀ xðmÞÀ ZðnÞ ¼ 0, tðtÞ m ¼ tÀtðtÞ tðtÞ m ¼ 1 n ¼ tÀm T ZT ðnÞ ð21Þ w2 ¼ 2xT ðtÞW j EfDV 3 ðtÞ9xðtÞg ¼ E ð22Þ tðt þ 1Þ X t X T x ðnÞQ 1 xðnÞ tðtÞ X tÀ1 X t2 X tÀ1 X m ¼ t1 þ 1 n ¼ tÀm þ 1 ( ¼E 1 t21 ðt2 þ t1 À1ÞxT ðtÞQ 1 xðtÞ 2 ) # xT ðnÞQ 1 xðnÞ xðtÞ tðt þ 1Þ X tÀ1 X T x ðnÞQ 1 xðnÞ þ tðt þ 1Þ X m ¼ 1 n ¼ t þ 1Àm À tðtÞ X tÀ1 X ð23Þ On the other hand, for any appropriately dimensioned matrices X j ¼ X T Z 0 and Y j ¼ Y T Z0, the following hold: j j T x ðnÞQ 1 xðnÞÀ T x ðtÞQ 1 xðtÞ tðtÞ X T x ðtÀmÞQ 1 xðtÀmÞ xT ðnÞQ 1 xðnÞÀ 1 À x ðmÞQ 1 xðmÞ r À tðtÞ m ¼ tÀtðtÞ x ðmÞ Q 1 m ¼ tÀtðtÞ X j xðtÞ X j xðtÞ tðtÞ X tÀt1 À1 X tðtÞ X xT ðtÞ m ¼ 1 n ¼ tÀt2 tðtÞ X tÀt1 À1 X xT ðtÞ tÀmÀ1 X xT ðtÞ 1 tðtÞ 1 tðtÞ 1 tðtÞ Y j xðtÞ Y j xðtÞ Y j xðtÞ ð25Þ From (16)–(25) and considering the closed-loop system (7), we obtain J þ EfzT ðtÞzðtÞgÀg2 wT ðtÞwðtÞ ¼ J þ EfzT ðtÞzðtÞg ð18Þ tÀ1 X ! xðmÞ : ð20Þ From (14), we have Àg2 wT ðtÞwðtÞ þ w1 þ w2 þ w3 þ w4 þ w5 1 T ¼ z ðtÞðXijkl þ CÞzðtÞ tðtÞ !T ! # tðtÞ tÀ1 xðtÞ xðtÞ Xj T j 1 X X À ZðnÞ n Z1 tðtÞ m ¼ 1 n ¼ tÀm ZðnÞ !T ! # tðtÞ tÀt1 À1 xðtÞ xðtÞ Y j Sj 1 X X À Â ZðnÞ ZðnÞ n Z2 tðtÞ m ¼ 1 n ¼ tÀm !T ! # tðtÞ tÀmÀ1 xðtÞ xðtÞ Xj þ Y j Wj 1 X X À ZðnÞ n Z1 þ Z2 tðtÞ m ¼ 1 n ¼ tÀt2 ZðnÞ 1 T 9z ðtÞ Xijklm þ C zðtÞÀUðxðtÞ, ZðnÞÞ tðtÞ ð26Þ T xðmÞÀ m ¼ tÀtðtÞ tðtÞ X tÀ1 X ZðnÞ, m ¼ 1 n ¼ tÀm xðmÞÀtðtÞxðtÀt2 ÞÀ tðtÞ X tÀmÀ1 X ZðnÞ, m ¼ 1 n ¼ tÀt2 m ¼ tÀtðtÞ 0 ¼ tðtÞxðtÀt1 ÞÀ 1 tðtÞ X j xðtÞ ¼ 0: EfDV 4 ðtÞ9xðtÞg ¼ Ef½xT ðtÞðQ 2 þQ 3 ÞxðtÞÀxT ðtÀt1 ÞQ 2 xðtÀt1 Þ 0¼ 1 tðtÞ xT ðtÞY j xðtÞÀ m ¼ 1 n ¼ tÀt2 m ¼ tÀtðtÞ ÀxT ðtÀt2 ÞQ 3 xðtÀt2 ÞŠ9xðtÞg: xT ðtÞ m ¼ 1 n ¼ tÀm ð19Þ tÀ1 X xT ðtÞ n ¼ tÀt2 À T tÀ1 X 1 tðtÞ ð24Þ tÀt1 À1 X w5 ¼ xT ðmÞQ 1 xðmÞ ! tÀ1 X xT ðtÞ ¼ 0, m ¼ tÀtðtÞ T 0 ¼ tðtÞxðtÞÀ tÀmÀ1 X m ¼ 1 n ¼ tÀt2 Using Lemma 2, we have tÀ1 X tðtÞ X À 1 þ t21 ðt2 þ t1 À1ÞxT ðtÞQ 1 xðtÞ 2 # ) t2 tÀ1 X X xT ðnÞQ 1 xðnÞ xðtÞ À m ¼ t1 þ 1 n ¼ tÀm þ 1 1 ¼ E t2 xT ðtÞQ 1 xðtÞ þ t21 ðt2 þ t1 À1ÞxT ðtÞQ 1 xðtÞ 2 ) # tÀ1 X xT ðmÞQ 1 xðmÞ xðtÞ : À m ¼ tÀtðtÞ tÀ1 X tðtÞ X ¼ t21 xT ðtÞY j xðtÞÀ m ¼ 1 n ¼ t þ 1Àm tÀ1 X m ¼ 1 n ¼ tÀm m¼1 tÀ1 X tðtÞ X m ¼ 1 n ¼ tÀt2 ¼ t2 xT ðtÞX j xðtÞÀ m ¼ 1 n ¼ t þ 1Àm À xT ðtÞX j xðtÞÀ n ¼ tÀt2 1 þ t21 ðt2 þ t1 À1ÞxT ðtÞQ 1 xðtÞ 2 ) # t2 tÀ1 X X T x ðnÞQ 1 xðnÞ xðtÞ À m ¼ t1 þ 1 n ¼ tÀm þ 1 ( t2 tÀ1 X X rE xT ðnÞQ 1 xðnÞ þ t2 xT ðtÞQ 1 xðtÞ tÀ1 X tÀ1 X w4 ¼ m¼1 m ¼ 1 n ¼ t þ 1Àm t1 X tðtÞ tÀ1 X X 1 1 X tÀt1 À1 xðmÞÀ ZðnÞ ¼ 0: tðtÞ m ¼ tÀtðtÞ tðtÞ m ¼ 1 n ¼ tÀm xT ðnÞQ 1 xðnÞ: þ m ¼ 1 n ¼ tÀm À # w3 ¼ 2xT ðtÞSj xðtÀt1 ÞÀ m ¼ 1 n ¼ t þ 1Àm À tðtÞ tÀ1 X X 1 1 X tÀmÀ1 xðmÞÀxðtÀt2 ÞÀ ZðnÞ ¼ 0, tðtÞ m ¼ tÀtðtÞ tðtÞ m ¼ 1 n ¼ tÀt 2 2 ( # tÀ1 X m ¼ tÀtðtÞ xðmÞÀ tðtÞ X tÀt1 À1 X ZðnÞ: m ¼ 1 n ¼ tÀm Then, the following equations hold for any matrices T j ,W j ,Sj with appropriate dimensions: where zðtÞ ¼ ½x ðtÞ,xT ðtÀt1 Þ,xT ðtÀt2 ÞŠT , and 2 0 D11 T 6 B 6 X þf zjklm f zjklm þg T g zjklm þ @ n zjklm Xijklm ¼ 6 6 n 4 n 2 0 6n 6 6 C¼6n 6 6 4n n with 3 W j ÀT j ÀSj 0 0 0 ÀQ 1 0 0 n 0 0 n n 0 n n n 07 7 7 0 7, 7 7 05 0 0 0 n 1 3 0 C A Àg2 I D2 7 0 D3 7 7, 7 5
  • 6. Z. Xia et al. / ISA Transactions 51 (2012) 702–712 T T with X ¼ f jklm P1 f jklm þ g T P1 g jklm ÀHe ff jklm P2 g: jklm Then ð28Þ By Lemma 3, (28) is equivalent to ð29Þ which can be rewritten as follows: # T X þf zjklm f zjklm þg T g zjklm þ D1 D2 zjklm D3 n Using Lemma 1, we have 2 P1 À2aj I 0 0 6 n P1 À2aj I 0 6 6 6 n n ð1À2aj ÞI 6 6 6 n n n 6 6 n n n 4 n o 0: ð30Þ 0 Àaj g jklm 0 ÀP2 þ aj f jklm 0 ð1À2aj ÞI Àaj f zjklm Àaj g zjklm n D1 n n n 0 3 7 0 7 7 7 0 7 7 o 0, 0 7 7 7 D2 5 D3 hjk hjl hjm 0 n P1 À2aj I n n 0 0 ð1À2aj ÞI n n n n n 0 0 0 ð1À2aj ÞI Àaj g jklm ÀP2 þ aj f jklm Àaj f zjklm Àaj g zjklm n n n D1 n n n n 0 0 0 0 7 7 7 7 7 7 o0: 7 7 D2 7 5 D3 hjk hjl hjm Pn ijklm k¼1l¼1m¼1 ¼ bðjÞ X 3 hjk Pn þ ijkkk k¼1 þ bX ðjÞÀ1 bðjÞ X 2 hjk hjl ðPn þ Pn þ Pn Þ ijkll ijlkl ijllk bX bX ðjÞÀ2 ðjÞÀ1 bðjÞ X hjk hjl hjm k ¼ 1 l ¼ kþ1 m ¼ lþ1 ðPn þ Pn þ Pn þ Pn þ Pn þ Pn Þ: ijklm ijkml ijlkm ijlmk ijmkl ijmlk C9wðtÞ ¼ 0 o0: 1 C9 ÞFðtÞÀUðxðtÞ, ZðnÞÞ, J9wðtÞ ¼ 0 ¼ F ðtÞðXijkl 9wðtÞ ¼ 0 þ tðtÞ wðtÞ ¼ 0 P where FðtÞ ¼ ½xT ðtÞ tÀ1 tÀtðtÞ xT ðnÞxT ðtÀt1 ÞxT ðtÀt2 ÞŠT and n¼ 2 3 Sj ÀW j 6 Lijkl 7 0 0 Xijkl 9wðtÞ ¼ 0 ¼ 4 5, T D3 n ð34Þ Then, we can conclude that the closed-loop system (7) with wðtÞ ¼ 0 is mean-square asymptotically stable by following the same lines as in the proof of Theorem 1 in [38]. Now, to establish the H1 performance for the closed-loop system (7), consider the following index: ( ) 1 X T 2 T ½e ðtÞeðtÞÀg o ðtÞoðtÞŠ : ð36Þ J2 ¼ E t¼0 Under zero initial condition and (27), we have ( ) 1 X ½ÀDVðtÞŠ ¼ EfÀVð1Þ þ Vð0Þg ¼ EfÀVð1Þg o0, J2 r E t¼0 3 W j ÀT j ÀSj 0 0 ÀQ 1 0 n 0 07 7 7: 05 n n 0 4. Simulation In this section, two examples are used to verify the performance of the proposed SOF controller. Example 1. Nowadays, it is often to consider the cases of monitoring and control through networks. Due to the existence of transmission delay in networks for different environments. Consider system (1) with the following parameters: 2 3 2 3 0:12 À0:01 0:03 0:11 À0:15 0:03 6 0:05 0:14 À0:08 7 6 0 À0:1 À0:07 7 A1 ¼ 4 5, A2 ¼ 4 5, 0:11 À0:6 À0:01 0 0:01 0 0:01 0:13 0:09 À0:21 2 3 0:06 0 0:05 6 7 0:02 0 Ad2 ¼ 4 0 5, 0:001 0 À0:005 3 À0:03 7 5, ð32Þ If (10), (11) and (12) hold, P o0 which implies that (27) holds. Therefore, when assuming the zero disturbance input, from (16)-(25), we obtain n ! 0 : 0 ð35Þ B1 ¼ ½0:14 0:25 0:31Š, 0 6n C9wðtÞ ¼ 0 ¼ 6 6 4n D11 7 I 5þ n 0 J9wðtÞ ¼ 0 o 0: 0:07 2 0:02 6 Ad1 ¼ 4 0 0 k ¼ 1 l ¼ kþ1 2 3 3 Let F jl ¼ aÀ1 U jl , we obtain j bðjÞ bðjÞ bðjÞ XX X 1 tðtÞ The proof is completed. P1 À2aj I P¼ 0 EfJeðtÞJ2 g o gJoðtÞJ2 : k¼1l¼1m¼1 6 6 6 6 6 6 6 6 6 4 2 i I 6 40 0 which means that which is equivalent to 2 0 Xijkl 9wðtÞ ¼ 0 þ ð31Þ bðjÞ bðjÞ bðjÞ XX X I Xþf T T zjklm f zjklm þ g zjklm g zjklm From (13) and (34), we have C o 0, n ¼ 1; 2, n ! 0 h By Schur complement and Lemma 3, LMI (30) implies 1 Xijklm þ C o 0: tðtÞ tn 0 0 ð27Þ If (13) hold and Xijklm þ I Lijkl ¼ J þ EfzT ðtÞzðtÞgÀg2 oT ðtÞoðtÞ o0: 1 707 ð33Þ D1 ¼ ½0:1 À0:3 0:4Š, 2 0:61 0:33 B2 ¼ ½0:22 0:18 0:37Š, D2 ¼ ½0:3 À0:2 0:4Š, 3 2 0 0:37 tðtÞ ¼ 3þ ðÀ1Þt , 3 6 7 6 7 0 5, B11 ¼ 4 0:02 0:34 5, B12 ¼ 4 0:91 0:16 0:06 0 0:28 2 3 2 0:08 0 0:13 0 À0:31 6 7 6 0:51 0 5, M 2 ¼ 4 0 0:42 M1 ¼ 4 0 2 0 0 0:15 0 0:39 3 2 0:03 0 3 0 0:02 0:03 6 7 6 0:24 5, M d1 ¼ 4 0 D21 ¼ 4 0 0:32 0 0 2 3 0 0 0 6 7 0 5, Md2 ¼ 4 0 À0:02 0:02 7 0 5, 0:01 0 0 0:01 0 0 3 7 0 5, 0:11
  • 7. 708 Z. Xia et al. / ISA Transactions 51 (2012) 702–712 0:61 0:33 3 2 0 0:31 3 6 7 6 0 7 B21 ¼ 4 0:02 0:34 5, B22 ¼ 4 0:35 5, 0:16 0:06 0 0:28 2 3 2 3 0:02 0 0:1 0 0 0:03 6 0 7 6 0 0:05 0:3 0 5, C 2 ¼ 4 0 7, C1 ¼ 4 5 0 0 0:08 0 0 À0:02 2 3 0:01 0 0:003 6 0:01 0 7 C d1 ¼ 4 0 5, 0:02 2 0:03 6 C d2 ¼ 4 0 0 2 0 À0:03 0 0 À0:03 0 0:13 3 3 2 0:11 6 D11 ¼ 4 0 0:15 0:02 7, 5 0:04 2 0:05 0 3 0 À0:12 7, 5 0:01 6 Nd2 ¼ 4 G1 ¼ 0:002 0 0 0:013 0:021 0 0:16 0:31 0:19 0:31 3 0:018 0 7 5, 2 0:41 6 D22 ¼ 4 0:22 0:7 0:001 ! 0:68 0:27 , G2 ¼ À0:35 0:63 h1 Membership functions 0.6 0.4 Ω2 Ω1 0 3 0 π/18 0 0:001 3 3 0 0:33 7 5, 2 0:09 0:31 0:17 0:42 0:41 ! : 1 0 −1 −2 −3 0 20 80 100 Suppose the unknown disturbance input v(t) to be random noise, as is shown in Fig. 3. Let g ¼ 3:18, if we use a common Lyapunov matrix P instead of piecewise Lyapunov matrix Pi ði ¼ 1; 2,3Þ in LMIs (10)-(12), no feasible solution can be found for switching fuzzy SOF controller design. However, based on Theorem 1, we can obtain feasible solution as follows: 2:0863 6 0:2038 P1 ¼ 4 0:3742 0.6 0.5 0:2038 3:9144 2:7773 6 P2 ¼ 4 0:1060 1:1422 0.3 0.2 1 π/3 1.5 π/2 2:3044 6 P3 ¼ 4 0:0486 0:4124 |x1| F 11 ¼ Fig. 1. The membership functions. 3 À0:2419 7 5, 4:0821 0:1060 1:1422 5:4118 3 À0:2770 7, 5 À0:2770 4:6235 0:0486 0:4124 2 0.1 0:3742 À0:2419 2 0.4 0.5 40 60 Time(Sec) Fig. 3. The unknown disturbance input v(t). 2 π/18 1.5 π/2 7 5, 0.7 0 1 π/3 0.5 Fig. 2. The partition of subspaces Oi , i ¼1,2,3. 3 0 0.8 0 Ω3 |x1| h2 0.9 ∂Ω1 3 ∂Ω1 2 0.2 The membership functions are shown in Fig. 1. As illustrated in [8,11], we divide the state space into three subregions. The membership functions hji and partition of subregions are stated in Fig. 2. The system matrices are A11 ¼ A21 ¼ A1 , A22 ¼ A31 ¼ A2 , Ad11 ¼ Ad21 ¼ Ad1 , Ad22 ¼ Ad31 ¼ Ad2 , B11 ¼ B21 ¼ B1 , B21 ¼ B31 ¼ B2 , B111 ¼ B121 ¼ B11 , B122 ¼ B131 ¼ B12 , M 11 ¼ M 21 ¼ M1 , M 22 ¼ M 31 ¼ M2 , M d11 ¼ M d21 ¼ Md1 , Md22 ¼ M d31 ¼ Md2 , B211 ¼ B221 ¼ B21 , B222 ¼ B231 ¼ B22 , C 11 ¼ C 21 ¼ C 1 , C 22 ¼ C 31 ¼ C 2 , C d11 ¼ C d21 ¼ C d1 , C d22 ¼ C d31 ¼ C d2 , D11 ¼ D21 ¼ D1 , D22 ¼ D31 ¼ D2 , D111 ¼ D121 ¼ D11 , D122 ¼ D131 ¼ D12 , N 11 ¼ N 21 ¼ N1 , N 22 ¼ N 31 ¼ N 2 , Nd11 ¼ N d21 ¼ Nd1 , Nd22 ¼ Nd31 ¼ N d2 , D211 ¼ D221 ¼ D21 , D222 ¼ D231 ¼ D22 , G11 ¼ G21 ¼ G1 , G22 ¼ G31 ¼ G2 . The membership functions are h11 [ h21 ¼ h1 , h22 [ h31 ¼ h2 . 1 h31 h22 ∂Ω1 ∂Ω2 1 1 0.8 0 6 0 7 N1 ¼ 6 0 0:14 0 7 D12 ¼ 4 0:2 5, 4 5, 0 0:17 0:02 0 À0:13 2 3 2 0:02 0 0:21 0:001 0:013 6 0 7, Nd1 ¼ 6 0:04 0:002 N2 ¼ 4 À0:13 0:04 5 4 0:15 0 0:11 0 0:015 2 Membership functions 0 h21 h11 1 v(t) 2 3:4077 À0:1979 0:0828 À0:9788 3 7 À0:1979 5, 3:0487 ! 1:0874 , 0:0050 F 21 ¼ À0:7167 À0:4143 ! À0:0785 , 1:3926
  • 8. Z. Xia et al. / ISA Transactions 51 (2012) 702–712 ! À0:5569 , 0:6786 0:6118 À0:9752 F 31 ¼ À1:6217 À1:3740 0.6 ! 0:4802 : 1:1860 0.4 0.2 y1(t) F 22 ¼ Based on the proposed fuzzy SOF controller (6), the state responses of the closed-loop system (7) are stated in Figs. 4–6 with the initial conditions xðtÞ ¼ ½p=8,Àp=5, p=3ŠT ðt ¼ À4, . . . ,0Þ and the disturbance oðtÞ ¼ 0:1 Á cos t Á eÀ0:05t . Figs. 7–11 show the corresponding output responses. 0 −0.2 −0.4 Example 2. In this example, the proposed fuzzy SOF control technique is applied to backing up control of a truck–trailer. We use the following modified truck–trailer model formulated in [48] and assume that the states are perturbed by distributed timevarying delay and stochastic disturbances. X tÀ1 vt vt x1 ðtÞ þ ð1Àl1 Þ 1À x1 ðnÞ x1 ðt þ 1Þ ¼ l1 1À L L n ¼ tÀtðtÞ ! ! vt vt uðtÞ þ sinðWðtÞÞ þ uðtÞ vðtÞ, þ 0:01oðtÞ þ l l 50 Time(Sec) 100 0.4 y2(t) 0.2 0 −0.2 0.6 x1(t) 0 Fig. 7. Response curve of measured output y1 ðtÞ. 0.8 −0.4 0.4 −0.6 0.2 0 −0.2 709 0 20 40 60 Time(Sec) 80 100 Fig. 8. Response curve of measured output y2 ðtÞ. 0 50 Time(Sec) 100 0.25 0.2 Fig. 4. Response curve of state x1 ðtÞ. 0.15 Z1(t) 0.2 x2(t) 0 0.05 0 −0.2 −0.05 −0.4 −0.1 −0.6 −0.8 0.1 0 50 Time(Sec) 20 40 60 Time(Sec) x2 ðt þ 1Þ ¼ l2 x2 ðtÞ þ ð1Àl2 Þ tÀ1 X x2 ðnÞ þ n ¼ tÀtðtÞ 1.5 1 80 100 Fig. 9. Response curve of controlled output z1 ðtÞ. 100 Fig. 5. Response curve of state x2 ðtÞ. 0 x3 ðt þ 1Þ ¼ l3 x3 ðtÞ þ ð1Àl3 Þ tÀ1 X vt x1 ðtÞ, L x3 ðnÞ þ vt sinðWðtÞÞ, x3(t) n ¼ tÀtðtÞ 0.5 zðtÞ ¼ À 0 vt vt x1 ðtÞ þsinðWðtÞÞ þ uðtÞ, L l yðtÞ ¼ GxðtÞ, −0.5 −1 0 50 Time(Sec) Fig. 6. Response curve of state x3 ðtÞ. 100 ð37Þ where x1 ðtÞ is the angle difference of truck and trailer, x2 ðtÞ is the angle of trailer, x3 ðtÞ is the vertical position of rear end of trailer, oðtÞ ¼ 0:1eÀt þsinðtÞ is the external disturbance, and v(t) is stochastic disturbance. l is the length of truck, L is the length of trailer, t is sampling time, v is the constant speed of backing
  • 9. 710 Z. Xia et al. / ISA Transactions 51 (2012) 702–712 2 0.05 6 6 A11 ¼ A22 ¼ 6 6 4 Z2(t) 0 −0.05 l1 1À vt L −0.15 50 Time(Sec) l3 0 vt L 2 2 6 Ad11 ¼ Ad12 ¼ Ad21 ¼ Ad22 ¼ 6 4 2 0.1 l3 p 2 3 7 7 0 7, 7 5 vt 10À2 100 Fig. 10. Response curve of controlled output z2 ðtÞ. 0 l2 v t 10À2 2Lp 3 7 7 0 7, 7 5 vt −0.2 0 0 l2 2 2 v t 2L l 1À vt L 6 1 6 A12 ¼ A21 ¼ 6 6 4 0 vt L 2 −0.1 l1 1À vt L 0:01 0 0 3 0 1Àl2 0 0 0 1Àl3 Z3(t) 2 −0.1 2 vt 2L 1 6 M11 ¼ M 22 ¼ 4 0 50 Time(Sec) 100 2 6 M12 ¼ M 21 ¼ 4 0 3 0 3 0 7 0 5, 0 0 0 Fig. 11. Response curve of controlled output z3 ðtÞ. 0 vt 10À2 2Lp 10À2 0 0 p 0 3 7 0 5, 0 0 0 # ! vt vt10À2 vt 10À2 1 0 , C 12 ¼ C 21 ¼ À 0 , C 11 ¼ C 22 ¼ À 2L L 2Lp p up, and WðtÞ ¼ x2 ðtÞ þ vt x1 ðtÞ: 2L D111 ¼ D112 ¼ D121 ¼ D122 ¼ The model parameters are given as l ¼ 2:8 m, L ¼ 5:5 m, v ¼ À1:0 m=s, t ¼ 2:0 s, l1 ¼ l2 ¼ l3 ¼ 0:8, tðtÞ ¼ 2:5 þ 0:5nðÀ1Þt , and G ¼ diagfÀ1,À1,À1g. Under the condition À179:42701 o WðtÞ o 179:42701, the nonlinear system (37) can be exactly represented by the following switching fuzzy system: bðjÞ X xðt þ 1Þ ¼ ( hjk Ajk xðtÞ þ Adjk tÀ1 X xðnÞ 3 217:9 À295:5 7 5, 1016:3 F 12 ¼ ½0:4926 0:2187 À0:2072Š, F 22 ¼ ½1:0882 À0:2040 0:0789Š: Under the initial conditions fðÀ3Þ ¼ ½11; 4,À9ŠT , fðÀ2Þ ¼ ½12; 3, 6Š , fðÀ1Þ ¼ ½À13,À4,À3ŠT and fð0Þ ¼ ½2,0:1,À10ŠT , the simulation results for the system (38) with stochastic disturbances are shown in Fig. 12. hjk ½C jk xðtÞ þ D1jk uðtÞŠ, hjk Gjk xðtÞ, Letting g ¼ 1:6 and using Theorem 1, we obtain 2 3 2 384:7 À35:8 151 566:5 85:3 6 7 6 193:2 P 1 ¼ 4 À35:8 202:7 À332 5, P 2 ¼ 4 85:3 151 À332 1066:7 217:9 À295:5 T k¼1 bðjÞ X G11 ¼ G12 ¼ G21 ¼ G22 ¼ diagfÀ1,À1,À1g: F 21 ¼ ½1:1493 0:0937 À0:0230Š, ! ' þ Bjk oðtÞ þ B1jk uðtÞ þ ½M jk xðtÞ þ B2jk uðtÞŠvðtÞ , bðjÞ X vt , l F 11 ¼ ½0:2721 À0:8158 0:0526Š, n ¼ tÀtðtÞ k¼1 yðtÞ ¼ vt 6 l 7 B111 ¼ B112 ¼ B121 ¼ B122 ¼ B211 ¼ B212 ¼ B221 ¼ B222 ¼ 4 0 5, −0.2 zðtÞ ¼ 7 7, 5 6 7 B11 ¼ B12 ¼ B21 ¼ B22 ¼ 4 0 5, 0 0 −0.3 3 ð38Þ 5. Conclusion k¼1 where xðtÞ ¼ ½x1 ðtÞ x2 ðtÞ x3 ðtÞŠT and hjk ðj ¼ 1; 2, bð1Þ ¼ bð2Þ ¼ 2Þ are the membership functions with 8 À2 sinðWðtÞÞÀWðtÞÁ10 =p , WðtÞ a 0, WðtÞÁð1À10À2 =pÞ h11 ¼ h22 ¼ : 1, WðtÞ ¼ 0, h12 ¼ h21 ¼ 8 WðtÞÀsinðWðtÞÞ , WðtÞÁð1À10À2 =pÞ : 0, WðtÞ a 0, WðtÞ ¼ 0, Based on the switching fuzzy stochastic system, piecewise Lyapunov–Krasovskii function, and state transitions between all possible subregions, a new delay-dependent H1 fuzzy SOF controller design approach is proposed for discrete-time fuzzy stochastic systems with distributed time-varying delays. In contrast to the existing results, two new techniques, improved free-weighting technique and new matrix decoupling technique, are developed. If the conditions are feasible, the controller parameters can be easily constructed by solving a set of LMIs. The theoretic results obtained in the paper have
  • 10. 4 2 5 0 −2 0 −5 −4 −10 0 20 40 Time(Sec) 60 80 0 20 40 Time(Sec) 60 80 0 20 40 Time(Sec) 60 80 0 20 40 Time(Sec) 60 80 2 x2(t) 4 10 x1(t) 20 0 −10 −20 0 −2 0 20 40 Time(Sec) 60 −4 80 10 5 5 z(t) 10 x3(t) 711 10 u(t) Stochastic disturbances v(t) Z. Xia et al. / ISA Transactions 51 (2012) 702–712 0 −5 −10 0 −5 0 20 40 Time(Sec) 60 80 −10 Fig. 12. Control results of trucker–trailer. some potential applications, such as image processing via a large-scale system networks, which gives rise to possible distributed time-varying delays for discrete-time systems. Two examples are presented to demonstrate the validity of the proposed approach. Appendix A. Proof of Lemma 1 where 2 6 6 6 6 S0 ¼ 6 6 6 6 6 4 S0 S2 ST 0 2 S3 o 0, S1 P1 0 0 P1 0 0 0 0 0 ÀP 2 0 0 0 0 I 0 0 I 0 0 0 ÀP T 2 0 0 D1 0 DT 2 0 0 0 ÀaI 0 0 0 ÀaI 0 6 6 6 6 S2 ¼ 6 6 6 6 4 0 0 ÀaI 0 0 0 0 0 0 0 0 0 0 3 0 7 7 7 0 7 7, ÀaI 7 7 7 0 5 S3 ¼ diagfI,I,I,I,I,Ig: 0 Then we choose the orthogonal complement of S1 as # ÀBT AT ÀAT ÀBT I 0 1 2 1 2 ST ¼ , 1? 0 0 0 0 0 I Motivated by [11,15], we can rewrite the inequality (b) as #T # # S3 S1 2 ðA:1Þ 3 0 0 7 7 7 0 7 7 , 0 7 7 7 D2 7 5 D3 which satisfies S1 S1? ¼ 0. Moreover, ½ST , S1? Š is of column full 1 rank. Then it follows that (A.1) is equivalent to the following matrix inequality: 2 I 6 60 40 S1 ¼ 6 6 0 0 0 0 B1 I 0 0 I 0 0 ÀA1 A2 0 0 I B2 0 3 7 07 7, 07 5 0 ST 1? S3 S1 #T S0 S2 ST 0 2 # # S3 S o 0, S1 1? which can be further reduced to ST S0 S1? o 0: 1? ðA:2Þ
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