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Non-fragile static output feedback control for singular T–S fuzzy delay-dependent systems subject to Markovian jump and actuator saturation
Author’s Accepted Manuscript Non-fragile static output feedback control for singular T-S fuzzy delay-dependent systems subject to Markovian jump and actuator saturation Yuechao Ma, Menghua Chen, Qingling Zhang www.elsevier.com/locate/jfranklin
PII: DOI: Reference:
S0016-0032(16)30126-0 http://dx.doi.org/10.1016/j.jfranklin.2016.04.006 FI2579
To appear in: Journal of the Franklin Institute Received date: 24 November 2015 Revised date: 11 February 2016 Accepted date: 21 April 2016 Cite this article as: Yuechao Ma, Menghua Chen and Qingling Zhang, Nonfragile static output feedback control for singular T-S fuzzy delay-dependent systems subject to Markovian jump and actuator saturation, Journal of the Franklin Institute, http://dx.doi.org/10.1016/j.jfranklin.2016.04.006 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Non-fragile static output feedback control for singular T-S fuzzy delay-dependent systems subject to Markovian jump and actuator saturation Yuechao Maa , Menghua Chena ∗ , Qingling Zhangb a College b College
of Science, Yanshan University, Qinhuangdao, Hebei Province, P.R.China, 066004 of Science, Northeastern University, Shenyang, Liaoning Province, P.R.China, 110004
Abstract In this paper, we consider the problem of non-fragile static output feedback control for singular T-S fuzzy delaydependent systems subject to Markovian jump and actuator saturation. Sufficient conditions are presented to guarantee the resulted closed-loop system to be stochastically admissible and dissipative. Then, the non-fragile fuzzy static output feedback controller can be obtained by solving the LMIs. And the estimation of the largest domain of attraction for the system is formulated and solved as an optimization problem. Finally, some numerical examples are provided to show the effectiveness of the proposed approaches. Key words: singular T-S fuzzy system, non-fragile, dissipative, Markovian jump, static output feedback control
1. Introduction The past decade has witnessed rapidly growing interest in T-S fuzzy models [1], especially the singular T-S fuzzy model. Since it can combine the merits of both fuzzy logic theory and linear systems, and then stability analysis and controller design of the overfall fuzzy systems can be carried out in the Lyapunov function framework. For example, memory dissipative control, memory H∞ control and H∞ filter were studied in [2-4], respectively. In [5], Chadli et al. obtained new stability and stabilization conditions for singular uncertain T-S fuzzy system. H∞ descriptor fault detection filter for T-S fuzzy discrete-time systems was demonstrated in [6]. On the other aspect, delay is often encountered in various engineering systems. In many cases, time delay is a source of instability and performance deterioration. It has been proven that delay-dependent conditions are less conservative than time-independent conditions by many researches, such as [2-4, 6-8]. Markovian jump systems (MJSs) are special stochastic hybrid systems, where the system mode is governed by a continuous Markovian chain taking values in a finite set [9]. And singular (or descriptor) system model has been used to deal with complex practical engineering systems [10]. It has been generally recognized that ? Project supported by the National Nature Science Foundation of China No. 61273004, and the Natural Science Foundation of Hebei province No. F2014203085. ∗ Corresponding author. Email address: chenmodehao [email protected]; telnumber: 86-18637031605; faxnumber: 86-03358057027 (Menghua Chena ). Preprint submitted to
singular Markovian jump systems (sMJSs) have the advantage of better describing physical systems with abrupt variations and has successful applications in control and communication fields, to name a few, [11,12] and the references therein. Using T-S fuzzy model, some nonlinear Markovian jump systems were discussed effectively. In [13], robust finite-time fuzzy H∞ control for uncertain time-delay systems with stochastic jumps was studied. Zhang et al. discussed delay-dependent robust stabilization for uncertain discrete-time fuzzy Markovian jump systems with mode-dependent time delays in [14]. However, in these papers, the relatively small uncertainties in controller implementation were ignored. Since perturbations often appear in the controller gain, which may result from either the actuator degradations or the requirements for readjustment of controller gains. The non-fragile control concept is how to design a feedback controller that will be insensitive to some error in gains of feedback control. Non-fragile robust control [15], and non-fragile passive controller [16] were discussed. However, the uncertain parameter in Bi was ignored in [15,16] and many other articles on non-fragile control. In practice, it is necessary to consider the uncertainty. Dissipative has captured comprehensive attention, since it is a more general criterion comparing with passivity and H∞ performance, see, [17,18]. In [19], the problem of fuzzy dissipative control for nonlinear Markovian jump systems via retarded feedback was investigated. On the other hand, some state variables may be difficult to measure and sometimes have no physical meaning and thus cannot be measured at all. In this situation, the static output feedback control is more suitable for practical application, see, [20]. In [21], Xia et al. discussed the problem of delay-dependent fuzzy static output feedback control for discretetime fuzzy stochastic systems with distributed time-varying delays. Static-output-feedback H∞ control of continuous-time T-S fuzzy affine systems via piecewise Lyapunov functions was studied in [22]. In practice, actuator saturation is very ubiquitous, which is a main cause to result in poor performance of the closed-loop systems and sometimes it may lead the system unstable. Robust H∞ static output feedback stabilization of T-S fuzzy systems subject to actuator saturation was demonstrated in [23]. Zhao and Li [24] discussed the problem of robust stabilization of T-S fuzzy discrete systems with actuator saturation via PDC and non-PDC law. Yang and Tong [25] put forward the problem of output feedback robust stabilization of switched fuzzy systems with time-delay and actuator saturation. To our best knowledge, non-fragile fuzzy dissipative static output feedback control for singular Markovian jump systems subject to actuator saturation has not been fully investigated so far. These motivate the present study. In this paper, we consider the problem of non-fragile fuzzy dissipative static output feedback control for singular Markovian jump systems subject to actuator saturation. Contributions of this paper can be summarized as following aspects. First, some stochastically admissible and dissipative conditions are established on the basis of Lyapunov theory and the reciprocally convex combination lemma. Second, the estimation of the largest domain is estimated. Third, all the conditions are presented in terms of LMIs, then the case can be solved by solving the LMIs optimization problem. Finally, owing to the Lyapunov function, reciprocally convex combination lemma and proper design method of the non-fragile controller, the results in this paper are superior, which has been verified in the examples. Notation: Throughout this paper, Rn denotes the n-dimensional Euclidean space, and Rn×m is the set of n × m real matrices. For A ∈ Rn×m , A−1 and AT denote the matrix inverse and matrix transpose respectively. λmin (A) means the minimal eigenvalue of A. ε (.) stands for the expectation operator. For a real symmetric matrix A ∈ Rn×n , Sym (A) = A + AT , and A > 0(A ≥ 0) means that A is positive defined (positive semi-defined). A+ denotes the generalized inverse matrix of A. k.k stands for the Euclidean norm for a vector and kφ (t)kd = sup kφ (t)k stands for the norm of a function φ (t) ∈ Cn,d . The symbol ∗ −d≤t≤0
means the symmetric term in a symmetric matrix.
2. Problem formulation Fix a probability space (Ω, F, P (rt )), and consider a class of singular Markovian jump system (SMJS), which can be described by the following fuzzy model. Plant rule i: IF a1 (t) is Λi1 and a2 (t) is Λi2 ...ag (t) is Λig , THEN 2
¯i (rt ) sat (u (t)) + B ¯wi (rt ) w (t) , E x˙ (t) = A¯i (rt ) x (t) + A¯τ i (rt ) x (t − τ (t)) + B ¯ i (rt ) sat (u (t)) + D ¯ wi (rt ) w (t) , z (t) = C¯i (rt ) x (t) + C¯τ i (rt ) x (t − τ (t)) + D
(1)
y (t) = Cyi (rt ) x (t) , ∆
x (t) = φ (t) , t ∈ [−τ2 , 0] , i ∈ T = {1, 2, ...r} , where r is the number of If-Then rules, Λij (i ∈ T, j = 1, 2, ...g) are fuzzy sets, a1 (t) , a2 (t) , ..., ag (t) are premise variables. x (t) ∈ Rn is the state vector, w (t) ∈ Rq is a disturbance which belongs to L2 [0, ∞), z (t) ∈ Rp is the controlled output, y (t) ∈ Rm is the measurement output. τ (t) is the interval delay satisfying τ1 ≤ τ (t) ≤ τ2 , τ˙ (t) ≤ µ. u (t) ∈ Rl is the control input, and sat : Rl → Rl is the standard saturation function defined as follows: T
sat (u (t)) = [sat (u1 (t)) , · · · , sat (ul (t))] , without loss of generality, sat (ui (t)) = sign (ui (t)) min {1, |ui (t)|}. Here the notation of sat (·) is abused to denote the scalar values and the vector valued saturation functions. The matrix E ∈ Rn×n may be singular and assume that rank (E) = r ≤ n. φ (t) ∈ Cn,τ2 is a compatible vector valued initial function. {rt , t ≥ 0} is a continuous-time Markovian process with right continuous trajectories taking values in a finite set given ∆ by S = {1, 2, ..., N } with the transition rates matrix (TRM) Π = {πpq } given by π h + o (h) , p 6= q pq Pr {rt+h = q |rt = p } = 1 + π h + o (h) , p = q pq
o(h) h
= 0, and πpq ≥ 0, for q 6= p, is the transition rate from mode p to q at time t + h, PN which satisfies πpp = − q=1,q6=p πpq , for all p ∈ S. ¯i (rt ) = Bi (rt ) + ∆Bi (rt ) , B ¯wi (rt ) = A¯i (rt ) = Ai (rt ) + ∆Ai (rt ) , A¯τ i (rt ) = Aτ i (rt ) + ∆Aτ i (rt ) , B ¯ ¯ ¯ Bwi (rt )+∆Bwi (rt ) , Ci (rt ) = Ci (rt )+∆Ci (rt ) , Cτ i (rt ) = Cτ i (rt )+∆Cτ i (rt ) , Di (rt ) = Di (rt )+∆Di (rt ) , ¯ wi (rt ) = Dwi (rt ) + ∆Dwi (rt ) . For notional simplicity, in the sequel, for each possible rt = p ∈ S, the D matrix Ai (rt ) will be denoted by Api , Aτ i (rt ) will be denoted by Aτ pi and so on. Api , Aτ pi , Bpi , Bwpi , Cpi , Cτ pi , Dpi , Dwpi , Cypi are known mode-dependent constant matrices with appropriate dimensions. And Cypi are row full rank. ∆Api , ∆Aτ pi , ∆Bpi , ∆Bwpi ,∆Cpi , ∆Cτ pi , ∆Dpi , ∆Dwpi are unknown matrices describing the model uncertainties, and are assumed to be certain bound compact sets and of the form: ∆Api ∆Aτ pi ∆Bpi ∆Bwpi H1pi = F1pi (t) [E1pi E2pi E3pi E4pi ] , (2) ∆Cpi ∆Cτ pi ∆Dpi ∆Dwpi H2pi where h > 0, lim
h→0
where H1pi , H2pi , E1pi , E2pi , E3pi , E4pi , are known real constant matrices with appropriate dimensions and F1pi (t) are unknown real and possibly time-varying matrices satisfying T F1pi (t) F1pi (t) ≤ I.
(3)
Using singleton fuzzifier, product inference, and center-average defuzzifier, the global dynamics of the T-S system (1) is described by the convex sum form: E x˙ (t) =
r X
¯pi sat (u (t)) + B ¯wpi w (t) , ai (ε (t)) A¯pi x (t) + A¯τ pi x (t − τ (t)) + B
i=1
z (t) =
r X
¯ pi sat (u (t)) + D ¯ wpi w (t) , ai (ε (t)) C¯pi x (t) + C¯τ pi x (t − τ (t)) + D
i=1
y (t) =
r X
ai (ε (t)) [Cypi x (t)] ,
i=1
x (t) = φ (t) , t ∈ [−τ2 , 0] , 3
(4)
T
where a (t) = [a1 (t) , a2 (t) , ..., ag (t)] , βi (a (t)) =
p Q
Λij (aj (t)) is the membership function of the system r r P P with respect to the ith pant rule. Let λi (a (t)) = βi (a (t)) βi (a (t)), then λi (a(t)) ≥ 0 and λi (a(t)) = j=1
i=1
i=1
1. In the sequel, we denote λi (a(t)) by λi . Remark 1 In [15,16], the uncertain parameters in Bpi were ignored, which is unavoidable in practice. In this paper, it is considered. That is to say, the system in this paper is more general. We consider the following non-fragile mode-dependent static output feedback control law: u (t) =
r X
λi (Kpi + ∆Kpi ) y (t) , p ∈ S, i ∈ T,
(5)
i=1
where Kpi ∈ Rm×n are the control gain matrices to be determined. ∆Kpi are perturbed matrices satisfying
where H3pi , E5pi
∆Kpi = H3pi F2pi (t) E5pi , are known real constant matrices, F2pi (t) are unknown matrices satisfying T F2pi (t) F2pi (t) ≤ I.
(6) (7)
Remark 2 ∆Kpi are the relatively small uncertainties on controller implementation. In practical application, it is necessary to consider the controller non-fragility problems to guarantee the stability of the system. For the rest of this paper, let us recall some import notions and results which have been in [9]. Let P ∈ Rn×n be a symmetric matrix and E T P E ≥ 0, ρ be a scalar. Denote Ω E T P E, ρ = x (t) ∈ Rn : xT (t) E T P Ex (t) ≤ ρ . For matrices Hi , hik is the kth row of the matrix Hi , we define L (Hi ) = {x (t) ∈ Rn : |hik x (t)| ≤ 1, k ∈ [1, l]} .
Thus Ω E T P E, ρ is an ellipsoid and L (Hi ) is a polyhedral consisting of states for which the saturation does not occur. Let D be the set of l × l diagonal matrices whose diagonal elements are either 1 or 0. Suppose each element of D is labeled as Es , s = 1, 2, · · · , η = 2l , and denote Es− = I − Es . Clearly, if Es ∈ D, then Es − ∈ D. Lemma 1 ([26]) Let Fp , Hp ∈ Rp×n . Then for any x (t) ∈ L (Hp ) (p ∈ S), sat (Fp x (t)) ∈ co Es Fp x (t) + Es− Hp x (t) , s = 1, 2, · · · , η ; or, equivalently, sat (Fp x (t)) =
η X
αs Es Fp + Es− Hp x (t) ,
s=1
where co stands for the convex hull, αs for s = 1, 2, · · · , η are some scalars which satisfy 0 ≤ αs ≤ 1 and η P αs = 1. s=1
¯psi = Es K ¯ pi Cypl + Es− Hpi , then From Lemma 1, for any x (t) ∈ L (Hpi ), denote ~ η X ¯psi x (t) . ¯ pi Cypl x (t) = sat K αs ~
(8)
s=1
Then using (8), substituting (5) into (4), for ηt = p ∈ S, yields the following closed-loop system: E x˙ (t) =
η X r X r X r X
¯wpi w (t) , λl λi λj αs A¯pij x (t) + A¯τ pi x (t − τ (t)) + B
i=1 j=1 s=1 l=1
z (t) =
η X r X r X r X
(9)
¯ wpi w (t) , λl λi λj αs C¯pij x (t) + C¯τ pi x (t − τ (t)) + D
i=1 j=1 s=1 l=1
¯psj , C¯pij = C¯pi + D ¯psj . ¯pi ~ ¯ pi ~ where A¯pij = A¯pi + B 4
Lemma 2 [14]Given a set of suited dimension real matrices A, B, X, with BT B ≤ I, then there exists a scalar ε > 0 such that ABX + XT BT AT ≤ εAAT + ε−1 XT X. Lemma 3 [27]For given matrices E, X > 0, Y ,if E T X + Y ΓT is nonsingular, then there exist matrices −1 S > 0, I, such that ES+IKT = E T X + Y ΓT , where X, S ∈ Rn×n ,, Y, I ∈ Rn×(n−r) and Γ, K ∈ Rn×(n−r) are any matrices with full column rank satisfying E T Γ = 0, EK = 0. Lemma 4 [28](Reciprocally convex combination lemma)Let f1 ,f2 ,...,fN :Rm → R have positive values in an open subset D of Rm . Then, the reciprocally convex combination of fi over D satisfies X 1 X X min fi (t) = fi (t) + min gi,j (t), P gi,j (t) αi =1 } i αi {αi |αi >0, i i6=j
i
subject to f (t) g (t) i i,j ∆ ≥0 . gi,j : Rm 7→ R,gj,i (t) = gi,j (t) , gj,i (t) fj (t) Definition 1 ([29]) (i) For given scalar τ2 > 0, the SMJS (9) with w (t) = 0 is said to be regular and impulse-free for any time-delay satisfying τ1 ≤ τ (t) ≤ τ2 , if the pair E, A¯pij and E, A¯pij + A¯τ pi are regular and impulse-free. (ii) The SMJS (9) with w (t) = 0 is stochastically stable for all finite φ (t) defined on t ∈ [−τ2 , 0] and initial mode r0 , if there exists a finite number Φ (φ, τ2 , r0 ) > 0, such that the following inequality holds: Z t 2 lim εkx (t)k dt < Φ (φ, τ2 , r0 ) . t→∞
0
(iii) The SMJS (9) with w (t) = 0 is said to be stochastically admissible if it is regular, impulse-free and stochastically stable. Definition 2 ([18]) Given a scalar α > 0, real symmetric matrices M, L, and matrix N . The closed-loop system (8) is called strictly (M, N, L)−α−dissipative, for any t ≥ 0, with zero initial condition, the following condition is satisfied: ε {hz, M zit + 2hz, N ωit + hω, Lωit } ≥ αhω, ωit (10) Rt T √ ¯ = −M . where ha, bit = 0 a b dt. Without loss of generality, we assume that M < 0 and M Remark 3 From Definition 2, the above strictly (M, N, L) − α−dissipative performance includes H∞ performance that is, when M = −I, N = 0, and L = α + γ 2 I, Eq.(10) corresponds to an H∞ performance requirement, that is ε{hz, zi}t ≤ γ 2 {hw, wi}t . Definition 3 ([30]) Denoting the solution of system (9) with initial condition x0 = φ (t) ∈ Cn,τ2 by ψ (t, x0 ), then the domain of attraction of the origin is n o ∆ ς = φ ∈ Cn,τ2 : lim ψ (t, x0 ) = 0 . t→∞
Remark 4 The estimate of the domain of attraction Xδ ⊂ ς, where n o Xδ = φ ∈ Cn,τ2 : max |φ| ≤ δ1 , max φ˙ ≤ δ2 , with scalars δ1 , δ2 > 0 what will be maximized in the following 3. Main results 3.1. Stochastic admissibility and dissipativity analysis This section gives sufficient conditions for stochastic admissibility and dissipativity for the closed-loop system. 5
Theorem 1 For given scalars 0 ≤ τ1 ≤ τ2 and µ > 0, scalar α > 0, symmetric matrices M < 0, L, and a T matrix N , system (9) is stochastically admissible and (M, N, L) − α−dissipative within ∩N p=1 Ω E Pp E, ρ , if there exist positive-defined matrices Q, Q1 , Q2 , Q3p , U, Up , Pp , and any matrices Mp with appropriate dimensions satisfying
Λ Mp ∗ Λ
≥ 0, matrices Sp , such that for any p ∈ S, i ∈ T, the following inequalities
hold:
¯ pi Ξ
T T ¯ij ¯ij U U V¯ijT −1 ∗ − Up 0 0 τ12 < 0, = −1 2U ∗ ∗ − 2 0 τ12 ∗ ∗ ∗ −I ¯ pi Σ
N X
(11)
πpq Q3q − Q < 0,
(12)
q=1 N X
πpq Uq − U < 0,
(13)
q=1
T ∩N p=1 Ω E Pp E, ρ ⊂ L (Hpi )
(14)
where
¯ pi Σ
=
h
¯ (11) P¯pT A¯τ pi Σ pi
0
∗
¯ (22) Σ pi
¯ (23) Σ pi
∗
∗
¯ (33) Σ pi
∗
∗
∗
∗
∗ i
¯ij = A¯ ¯ ¯ U pij Aτ pi 0 0 Bwpi
0
¯ (15) Σ pi
¯ (25) Σ pi MpT 0 , −Q2 − Λ 0 ¯ (55) ∗ Σ pi ¯ (24) Σ pi
∗ h i ¯ C¯pij M ¯ C¯τ pi 0 0 M ¯D ¯ wpi , , V¯ij = M
N X ¯ (11) = Sym XpT A¯pij + Σ πpq E T Pq E + Q1 + Q3p + τ2 Q, pi q=1
¯ (22) Σ pi
= − (1 − µ) Q3p − 2Λ + MpT + Mp ,
¯ (23) = −Mp + Λ, Σ ¯ (24) = −M T + Λ, Σ ¯ (33) = −Q1 + Q2 − Λ, Σ p pi pi pi T ¯ (15) = XpT B ¯wpi − C¯pij ¯ (25) = −C¯τTpi N T , Σ ¯ (55) = −L + αI − Sym(N D ¯ wpi ), Σ NT , Σ pi pi pi T −1 T τ12 = τ2 − τ1 , Λ = τ12 E Up E, Xp = E T Pp + Sp RT ,
R ∈ Rn×(n−r) is any matrix with full column rank satisfying E T R = 0, and the estimate of the domain is Γδ ≤ ρ, where τ2 Γδ = δ12 λmax E T Pp E + τ1 λmax (Q1 ) + τ12 λmax (Q2 ) + τ2 λmax (Q3p ) + 2 λmax (Q) 2 2 (15) 2 τ23 − τ13 2 τ2 − τ1 T T +δ2 λmax E Up E + λmax E U E 2 6 Proof.Firstly, we prove the system (9) with w (t) = 0 is regular, impulse-free. 6
n×n Since rankE , then R can be rewritten = r < n, there must exist two invertible matrices G and H ∈ R 0 as R = GT , where Φ ∈ R(n−r)×(n−r) . Denote Φ Ir 0 Ap1 Ap2 P P p1 p2 , GA¯pij H = , G−T Pp G−1 = , GEH = 0 0 Ap3 Ap4 Pp3 Pp4 (16) S x (t) p1 1 T H Sp = , Hpi H = [Hpi1 Hpi2 ] , x (t) = H . Sp2 x2 (t)
¯ (11) < 0 by H T and H respectively, we can get that for every p ∈ S, AT ΦS T + Pre- and post-multiplying Σ p4 p2 pi Sp2 ΦT Ap4 < 0, which implies Ap4 is nonsingular and thus the pair (E, A¯pij ) is regular and impulse-free. On the other hand, it can be seen from (11) that, T ¯ (11) P¯ T A¯τ pi 0 I Σ 0 I p pi (22) (23) (24) ¯ ¯ ¯ I I ∗ Σ Σ Σ pi pi pi < 0, ¯ (33) I ∗ ∗ Σ MpT I pi I ∗ ∗ ∗ −Q2 − Λ I and considering Q3p and Q are positive , then, N X πpq E T Pp E < 0, Sym XpT A¯pij + A¯τ pi + q=1
which implies the pair (E, A¯pij + A¯τ pi ) is regular and impulse-free. So, we can get the system (9) is regular T and impulse-free within ∩N p=1 Ω E Pp E, ρ . Next, we will show the stochastic stability. The weak infinitesimal operator ` [.] ([12]) of the stochastic process {xt , rt , t ≥ 0} acting on V (xt , t, rt ) at the point {t, xt , rt = p} is given by `[V ] =
N X ∂V ∂V + x˙ T (t) + πpq V (xt , t, q). ∂t ∂x rt =p q=1
We choose the following stochastic Lyapunov-Krasovskii candidate for this system: V (xt , p, t) =
3 X
Vi (xt , p, t),
i=1
where V1 (xt , p, t) = xT (t) E T Pp Ex (t) , Z t Z t−τ1 V2 (xt , p, t) = xT (s)Q1 x (s) ds + xT (s)Q2 x (s) ds t−τ t−τ Z t1 Z2 0 Z t T + x (s)Q3p x (s) ds + xT (s)Qx (s) dsdθ, t−τ (t) −τ2 t+θ Z −τ1 Z t V3 (xt , p, t) = x˙ T (s)E T Up E x˙ (s) dsdθ −τ t+θ 2 Z −τ Z 0Z t 1 + x˙ T (s) E T U E x˙ (s)dsdvdθ, −τ2
θ
t+v
using the operator ` [.] we can obtain 7
" `[V1 (xt , p, t)] = 2x
T
(t) XpT E x˙ (t)
T
+ x (t)
N X
# T
πpq E Pq E x (t) ,
q=1
`[V2 (xt , p, t)] ≤ xT (t) (Q1 + Q3p + τ2 Q) x (t) + xT (t − τ1 ) (−Q1 + Q2 ) x (t − τ1 ) +xT (t − τ2 ) (−Q2 ) x (t − τ2 ) − (1 − µ) xT (t − τ (t)) Q3p x (t − τ (t)) "N # Z t X T + x (s) πpq Q3q − Q x (s) ds, t−τ2 q=1 2 τ12 T T `[V3 (xt , p, t)] = x˙ (t) E τ12 Up + U E x˙ (t) " 2N # Z t−τ1 Z −τ1 Z t X T x˙ T (s) E T Up E x˙ (s) ds, x˙ (s) E πpq Uq − U E x˙ (s) dsdθ − + −τ2
t+θ
t−τ2
q=1
where Z
t−τ1
−
x˙ T (s) E T Up E x˙ (s) ds
t−τ2 Z t−τ (t)
Z
T
t−τ1
x˙ (s) E Up E x˙ (s) ds − x˙ T (s) E T Up E x˙ (s) ds t−τ t−τ (t) 2 "Z # "Z # "Z # "Z t−τ (t) t−τ (t) t−τ1 T T ≤ −δ1 x˙ (s) ds Λ x˙ (s) ds − δ2 x˙ (s) ds Λ =−
T
t−τ2
t−τ (t)
t−τ2
#
t−τ1
x˙ (s) ds ,
t−τ (t)
τ12 τ12 , δ2 = τ2 − τ (t) τ (t) − τ1 from Lemma4, it can be obtained that, Z t−τ1 − x˙ T (s) E T Up E x˙ (s) ds t−τ2 T Λ Mp x (t − τ (t)) − x (t − τ2 ) x (t − τ (t)) − x (t − τ2 ) . ≤ − ∗ Λ x (t − τ1 ) − x (t − τ (t)) x (t − τ1 ) − x (t − τ (t)) h i Let ξ T (t) = xT (t) xT (t − τ (t)) xT (t − τ1 ) xT (t − τ2 ) wT (t) , from (11-13) and considering Schur complement lemma, it is true that, δ1 =
`[V (xt , p, t)] − z T (t) M z (t) − 2wT (t) N z (t) − wT (t) (L − αI) w (t) r X r 2 X T ¯ pi + V¯ T V¯ij + U ¯ T τ12 Up + τ12 U U ¯ij ξ (t) < 0, ≤ λi λj ξ (t) Σ ij ij 2 i=1 j=1 that is, `[V (xt , p, t)] − z T (t) M z (t) − 2wT (t) N z (t) − wT (t) (L − αI) w (t) < 0. (17) ¯ pi , p ∈ S}, then r1 > 0. It It is easy to see that, when w (t) = 0, ` [V (xt , p, t)] < 0. Set r1 = min{λmin −Ξ is easy to obtained that 2 `V (xt , p, t) ≤ −r1 kx (t)k . (18) By Dynkin’s formula, for each rt ∈ S, t ≥ τ2 , we have from (11) that Z t ε {V (xt , t, rt ) − V (xτ2 , τ2 , rτ2 )} = ε `[V (xs , s, rs )]ds τ2Z (19) t 2 ≤ −r1 ε kx (s)k ds , τ2
it follows that Z
t
ε
2 kx (s)k ds ≤ r1−1 ε {V (xτ2 , τ2 , rτ2 )} .
τ2
8
Following the similar line of the proof of [29], it is clear that there exists a scalar r2 such that: Z τ2 2 2 kx (s)k ds ≤ r2 kφ (s)kτ2 . ε 0
Therefore, by the definitions of V (xt , p, t) and x (t) there always exists a scalar β such that Z t Z τ2 Z t 2 2 2 2 kx (s)k ds ≤ βε kφ (s)kτ2 , kx (s)k ds + ε kx (s)k ds = ε ε τ2
0
0
then Z
t
lim ε
t→∞
0
2 kx (s)k ds ≤ βε kφ (s)kτ2 < Φ (φ, τ2 , r0 ) . 2
According to Definition1, system (9) with w (t) = 0 is stochastically admissible. On the other hand, when 0 6= w (t) ∈ L2 [0, ∞). We introduce the following performance index J = ε {hz, M zit + 2hz, N ωit + hω, Lωit − αhω, ωit } . Integrating both sides of (17) from 0 to t yields Z t [`[V (xs , p, t)]] ds =V (xt , p, t) − V (x0 , r0 , 0) > 0, J> 0
we conclude that Eq. (10) is satisfied. In the end, it is clear to see that xT (t) E T Pp Ex (t) ≤ V (xt , p, t) ≤ V (x0 , p, t) τ22 2 T ≤ max |φ (θ)| λmax E Pp E + τ1 λmax (Q1 ) + τ12 λmax (Q2 ) + τ2 λmax (Q3p ) + λmax (Q) (20) 2 θ∈[−τ2 ,0] 2 τ 2 − τ 2 3 3 τ − τ1 ∆ 2 1 + max φ˙ (θ) λmax E T Up E + 2 λmax E T U E = < φ, φ˙ 2 6 θ∈[−τ2 ,0] For any φ ∈ < φ, φ˙ , form (20), we can further get xT (t) E T Pp Ex (t) ≤ ρ, which means that all trajectories of x (t) which start from the set < φ, φ˙ always stay in the domain Ω E T Pp E, ρ . This completes the proof.
3.2. Non-fragile static output feedback controller design In this section, we will establish LMI conditions to solve the controller synthesis problem based on the results of Theorem 1, which can be easily solved by using the existing solvers such as LMI TOOLBOX in Matlab software. Theorem 2 For given scalars 0 ≤ τ1 < τ2 and µ > 0, α > 0, symmetric matrices M < 0, L and a matrix N , there exist a non-fragile static output feedback controller of the form (5) such that the closed-loop system (9) ˜1, Q ˜ 2 , Q, ˜ U ˜, is admissible and dissipative, if there exist scalars αij > 0, βij > 0, positive-definite matrices Q ˜ M ˜p Λ ˜ 3p , U ˜p , P˜p , and any matrices M ˜ p with appropriate dimensions satisfying ≥ 0, matrices S˜p , Q ˜ pT Λ ˜ M such that for each p ∈ S, i, j, l ∈ T, the following inequalities hold: Ξpii < 0,
(21)
Ξpij + Ξpji < 0, N X
i < j,
˜ 3q − Q ˜ < 0, πpq Q
q=1
9
(22)
N X
˜q − U ˜ < 0, πpq U
(23)
q=1
1 mpik −ρ ≤ 0, ∗ −EYp E T
where
Ξpij
˜ pij O ˜1 Θ
˜3 O
J˜1
J˜3
Υp
(24)
T T T ˜pij ˜pij ψ˜pi U U V˜pij 0 −1 ∗ ψ˜22 0 0 ∗ ∗ −αij 0 0 0 ˜ pij = , ,Θ ∗ ∗ ψ˜33 0 ∗ ∗ ∗ −βij 0 0 −1 ∗ ∗ ∗ ∗ −βij ∗ ∗ ∗ −I 0 ∗ ∗ ∗ ∗ ∗ Γp (11) (12) (15) ψ˜pi ψ˜pi 0 0 ψ˜pi (22) ˜(23) (24) ∗ ψ˜pi ψpi ψ˜pi −YpT CτTpi N T (33) T ˜ ˜ ψ˜pi = ∗ , Mp 0 ∗ ψpi ˜2 − Λ ˜ ∗ ∗ ∗ −Q 0 (55) ∗ ∗ ∗ ∗ ψ˜pi h i ˜pij = A Y + B λ U ¯ A Y 0 0 B pi p pi psj τ pi p wpi , h i ¯ (Cpi Yp + Dpiλ ¯ Cτ pi Yp 0 0 M ¯ Dwpi , V˜pij = M ¯psj ) M
=
∗
−αij
ψ˜22 =
0
0
0
˜p ˜ −Yp − YpT + U −Yp − YpT + U , ψ˜33 = 2 , 2 τ12 τ12
T ˜ 1 = H1pi O 0 0 0
T T T T ¯ − H1pi N T H1pi H1pi H2pi M
T
,
T ˜ 3 = [E1pi Yp + E3piλ O ¯psj E2pi Yp 0 0 E4pi 0 0 0] , h iT T T T T T T J˜1 = Bpi 0 0 0 (−N Dpi ) Bpi Bpi Dpi 0 E3pi Es H3pj , T J˜3 = [E5pj Cypl Yp 0 0 0 0 0 0 0 0 0] , (11) ˜1 + Q ˜ 3p + τ2 Q ˜ + πpp Y T E T , ψ˜pi = Sym (Api Yp + Bpiλ ¯psj ) + Q p (12) (22) ˜ 3p − 2Λ ˜ +M ˜ pT + M ˜ p, ψ˜pi = Aτ pi Yp , ψ˜pi = − (1 − µ) Q (23) ˜ p + Λ, ˜ ψ˜(24) = −M ˜ pT + Λ, ˜ ψ˜(33) = −Q ˜1 + Q ˜ 2 − Λ, ˜ Λ ˜ = τ −1 E U ˜p E T , ψ˜pi = −M 12 pi pi (15) (55) T T ψ˜pi = Bwpi − YpT Cpi NT − λ ¯Tpsj Dpi N T , ψ˜pi = −L + αI − Sym(N Dωi ), √ Ir √ √ √ Υp = πp1 YpT ... πp(p−1) YpT πp(p+1) YpT ... πpN YpT H −T , 0 Ir Γp = −diag [Ir 0] G {EY1 , ..., EYp−1 , EYp+1 , ...EYN } GT , 0
10
˜ ∈ Rn×(n−r) is any matrix with full column rank and satisfies E R ˜ = 0, G, H are nonsingular matrices that R Ir 0 . In this case, the control gain matrices are given by Kpj = Wpj Yp−1 C + , and the make GEH = ypl 0 0 ˜ δ ≤ ρ, where estimate of the domain of attraction is Γ T −1 −1 ˜ −1 −1 ˜ −1 + τ λ Y λ E Y Q Y + τ λ Y Q Y 1 max max 1 12 max 2 p p p p p ˜ δ = δ12 τ2 Γ 2 −1 ˜ −1 −1 ˜ −1 + λmax Yp QYp +τ2 λmax Yp Q3p Yp (25) 2 2 2 3 3 τ − τ τ − τ 1 1 ˜p Y −1 E T + 2 ˜ Y −1 E T +δ22 2 λmax EYp−1 U λmax EYp−1 U p p 2 6 Proof.From Theorem1, it can be obtained easily that ¯ pi = Ξpi + ∆Ξpi < 0, Ξ where
(11) (15) T T T Σpi XpT Aτ pi 0 0 Σpi Σpi Upij Upij Vpij −1 (22) (23) (24) ∗ Σpi Σpi Σpi −CτTpi N T ∗ − Up 0 0 τ12 (33) , Σpi = = ∗ ∗ Σpi MpT 0 −1 2U ∗ 0 ∗ − 2 (44) ∗ ∗ ∗ Σpi 0 τ12 (55) ∗ ∗ ∗ −I ∗ ∗ ∗ ∗ Σpi h i ¯psj Aτ pi 0 0 Bwpi , Upij = Api + Bpi ~ h i ¯psj M ¯ Cpi + Dpi ~ ¯ Cτ pi 0 0 M ¯ Dwpi , Vpij = M
Ξpi
(11)
Σpi
N X ¯psj + = Sym XpT Api + Bpi ~ πpq E T Pq E + Q1 + Q3p + τ2 Q, q=1 (22) Σpi (23)
Σpi (15)
Σpi
(44)
= − (1 − µ) Q3p − 2Λ + MpT + Mp , Σpi (24)
(33)
= −MpT + Λ, Σpi = −Q1 + Q2 − Λ, ¯psj N T , Σ(55) = −L + αI − Sym(N Dwpi ). + Dpi ~ pi
= −Mp + Λ, Σpi
= XpT Bwpi − Cpij
= −Q2 − Λ,
Then from (2), and Lemma 2, there exist scalars αij > 0, such that T
∆Ξpi = O1 O2 O3 + (O1 O2 O3 ) where
−1 ≤ αij O3T O3 + αij O1 O1T ,
T iT T T T ¯ H2pi T , XpT H1pi 0 0 0 − (N H2pi ) (H1pi ) (H1pi ) M ¯psj E2pi 0 0 E4pi 0 0 0 , O2 = F1pi (t) , O3 = E1pi + E3pi ~
O1 =
h
via Schur complement lemma, we can get that
Ξpi O1
O3T
¯ (1) = Ξ < 0. ∗ −α 0 ij pi −1 ∗ ∗ −αij From (5), we can see that ¯ (1) = Ξ(1) + ∆Ξ(1) , Ξ pi pi pi 11
,
(1) ¯psj in Ξ ¯ (1) to ~psj . Considering (6), and Lemma 2, there exist where Ξpi is the matrix changing the item ~ pi scalars βij > 0, such that (1)
T
−1 ∆Ξpi = J1 J2 J3 + (J1 J2 J3 ) ≤ βij J3T J3 + βij J1 J1T ,
where J1 =
h
XpT Bpi
T
T
− (N Dpi )
0 0 0
T
(Bpi )
T
(Bpi )
T
(Dpi )
T
0 (E3pi )
iT
Es H3pj ,
J3 = [E5pj Cypl 0 0 0 0 0 0 0 0 0] , J2 = F2pi (t) . Via Schur complement lemma, we can get that (1) Ξpi J1 J3T (26) ∗ −βij 0 < 0. −1 ∗ ∗ −βij ¯ (11) < 0, we can get Xp = E T Pp + Sp RT T is nonsingular. Using Lemma3, there exists From Theorem 1, Σ pi T ˜ T , where P˜p > 0 and R ˜ ∈ Rn×(n−r) is any matrix with full column rank and Yp = Xp−1 = E P˜p + S˜p R ˜ = 0. It is easy to see that, satisfies E R EYp = YpT E T = E P˜p E T ≥ 0. Denoting H −1 Yp GT =
Yp11 Yp12
(27)
, from (27), it is easy to obtain that Yp12 = 0, and Yp11 is symmetric, Yp21 Yp22 Y 0 p11 , so Yp11 and Yp22 are nonsingular. There, it can be concluded that then we have H −1 Yp GT = Yp21 Yp22 −1 I Y 0 p11 and [Ir 0] GEYq GT r = Yq11 are nonsingular. Then, we have G−T Yp−1 H = −1 −1 −1 −Yp22 Yp21 Yp11 Yp22 0 −1 Ir Ir H −T [Ir 0] GEYq GT [Ir 0] H −1 0 0 −1 I Y 0 r q11 −1 H −1 = H −T Yq11 [Ir 0] H −1 = H −T 0 0 0 −T −1 −1 −T T T T =H H E G G Yp H H = E T Yq−1 = E T Xq Pre- and Post-multiply (21) by diag XpT , ..., XpT , In , ...In , Ir , ...Ir and its transposition and denote λ ¯psj Xp = {z } | {z } | {z } | 4
8
N −1
˜ p = Ω, (Ω ˜ =Q ˜1, Q ˜2, Q ˜ 3p , Q, ˜ U ˜p , U ˜ ). And we have for Z = Up , U , −Z−1 ≤ ~psj , Wpj Xp = Kpj Cypl , XpT ΩX T ˜ −Yp − Yp + Z. Then using Schur complement lemma and above treatment, (21) implies (26). T On the other hand, from ∩N p=1 Ω E Pp E, ρ ⊂ L (Hpi ), and (16), it follows that Hpi2 = 0. Otherwise, let x1 (t) = 0, and |hpi2k x2 (t)| > ρ1/2 , then xT (t) E T Pp Ex (t) = 0, |hpik x (t)| > ρ1/2 , it contradicts that T ∩N p=1 Ω E Pp E, ρ ⊂ L (Hpi ) . Then xT (t) E T Pp Ex (t) = xT1 (t) Pp1 x1 (t) , Hpi x (t) = Hpi1 x1 (t) , (28) 12
xT (t) E T Pp Ex (t) ≤ ρ, Then the condition
∩N p=1 Ω
(29)
T
E Pp E, ρ ⊂ L (Hpi ) is obtained by −1 T hpi1k Pp1 hpi1k ≤
1 , k = 1, 2, · · · , l, ρ
which, by Schur complements, is equivalent to
−
[h
1 ρ T
pi1k
0]
[hpi1k 0] ≤ 0, k = 1, 2, · · · , l Ir 0 Pp1 Pp2 Ir 0 − 0 0 Pp3 Pp4 0 0
(30)
hpi1k is the kth row of Hpi1 . Pre- and post-multiply (30) by diag 1, YpT H −T and its transpose. Denote Hpj Yp = Mpj and consider h i Mpi H = Mpi1 0 , mpik is the kth row of Mpi . Together with (16), we can obtain (24). This completes the proof. Remark 5 Based on Theorem 2 in this paper, LMI conditions are given for singular Markovion jump fuzzy system to obtain the non-fragile static output feedback controller gain. When we set r = 1, the systems in this paper are SMJSs in usual sense. The similar systems were discussed in [31]. However, in [31], some matrices are given as special forms, and the conditions were obtained at the limitation of ε → 0+ . Without considering these techniques, the superiority of this paper can be seen in Example 1. Remark 6 With all the ellipsoids satisfying the set invariance condition of Theorem 1, the ”largest” one will be chosen to obtain the least conservative estimate of the domain of attraction. In the following, we will measure the largeness of the ellipsoids with respect to a shape reference set. δ1 = δ2 is chosen in (25) to simplify the process of maximizing the estimate of domain of attraction. Here the result in [30] is adopted to complete the approximating optimization problem which could be expressed as follows: P1 min r ω1 I E ωι I I ≥ 0, ≥ 0, e ˜ι I Yp I 2Yp − O s.t. T T ω I E ω I E 6 ≥ 0, 7 ≥ 0, E 2Y − U ˜ ˜ E 2Yp − U p p
T
e e e e ˜2 = Q ˜1, O ˜3 = Q ˜2, O ˜4 = Q ˜ 3p , O ˜ 5 = Q, ˜ and where ι = 2, 3, 4, 5, O r = ω1 + τ1 ω2 + τ12 ω3 + τ2 ω4 +
τ 2 − τ12 τ 3 − τ13 τ22 ω5 + 2 ω6 + 2 ω7 . 2 2 6
Then the maximized estimate of domain of saturation is δmax = √1∆ , where ˜ 1 Y −1 + τ12 λmax Y −1 Q ˜ 2 Y −1 ∆ = λmax E T Yp−1 + τ1 λmax Yp−1 Q p p p 2 ˜ 3p Yp−1 + τ2 λmax Yp−1 QY ˜ p−1 +τ2 λmax Yp−1 Q 2 3 3 τ 2 − τ12 ˜p Yp−1 E T + τ2 − τ1 λmax EYp−1 U ˜ Yp−1 E T . + 2 λmax EYp−1 U 2 6 Remark 7 We consider the following singular Markovian system as a special case of the system (4) with r = 1 and without saturation. 13
E x(t) ˙ = Ap x(t) + Aτ p x(t − τ (t)) + Bp u(t) + Bwp w(t) z (t) = Cp x(t) + Cτ p x(t − τ (t)) + Dp u(t) + Dwp w(t)
(31)
x(t) = φ(t), t ∈ [−τ2 , 0]. We give the following corollary to discuss the H∞ performance of system (31), which is a special case of Theorem2. So the result is given directly. Corollary 1 For given scalars 0 ≤ τ1 < τ2 and µ > 0, α > 0, there exist a non-fragile state feedback controller u (t) = (Kp + ∆Kp ) x (t) , p ∈ S, such that the closed-loop system (31) is admissible and satisfying ˜1, Q ˜ 2 , Q, ˜ Q ˜ 3p , U ˜, U ˜p , the H∞ performance ε{hz, zi}t ≤ γ 2 {hw, wi}t , if there exist positive-definite matrices Q ˜ M ˜p Λ ˜ p with appropriate dimensions satisfying ≥ 0, matrices S˜p , such that P˜p , and any matrices M ˜ pT Λ ˜ M for each p ∈ S, the following inequalities hold: T ˜T ˜T ˜ ˜ ψ U U V Υp p p p p ∗ ψ˜22 0 0 0 (32) ∗ ∗ ψ˜33 0 0 < 0, ∗ ∗ ∗ −I 0 ∗ ∗ ∗ ∗ Γp N X
˜ 3q − Q ˜ < 0, πpq Q
(33)
q=1 N X
˜q − U ˜ < 0, πpq U
(34)
q=1
where
ψ˜p =
ψ˜p(11) ψ˜p(12) ∗
0
ψ˜p(22) ψ˜p(23)
0
Bwp
ψ˜p(24)
0
∗
∗
ψ˜p(33)
˜ pT M
0
∗
∗
∗
˜2 − Λ ˜ −Q
0
∗
∗
∗
∗
−γ 2 I
,
h i ˜p = A Y + B W A Y 0 0 B U p p p p τp p wp , h i V˜p = Cp Yp + Dp Wp Cτ p Yp 0 0 Dwp , ψ˜22 =
˜p ˜ −Yp − YpT + U −Yp − YpT + U , , ψ˜33 = 2 2 τ12 τ12
˜1 + Q ˜ 3p + τ2 Q ˜ + πpp YpT E T , ψ˜p(11) = Sym (Ap Yp + Bp Wp ) + Q ˜ 3p − 2Λ ˜ +M ˜ pT + M ˜ p, ψ˜p(12) = Aτ p Yp , ψ˜p(22) = − (1 − µ) Q ˜ p + Λ, ˜ ψ˜p(24) = −M ˜ pT + Λ, ˜ ψ˜p(33) = −Q ˜1 + Q ˜ 2 − Λ, ˜ Λ ˜ = τ −1 E U ˜p E T , ψ˜p(23) = −M 12 √ Ir √ √ √ Υp = πp1 YpT ... πp(p−1) YpT πp(p+1) YpT ... πpN YpT H −T , 0 14
Γp = −diag [Ir 0] G {EY1 , ..., EYp−1 , EYp+1 , ...EYN } GT
Ir 0
.
In this case, for p ∈ S, the state feedback gain are given by Kp = Wp Yp−1 . 4. Numerical example Example 1 Consider the T-S singular Markovian jump time-delay system (31) with two modes with the following parameters as [31] Mode 1 −0.1669 0.0802 1.682 −3 1 0 A11 = 0.3 −2.5 −4 , Aτ 11 = −0.8162 −0.9373 0.5936 , 2.0941 0.6357 0.7902 −0.1 0.3 −3.8 1 1.5 h i B11 = 1 , Bw11 = 0 , C11 = 0.5 −0.1 1 , D11 = 0.1, Dw11 = 0, 1 1 Mode 2
−2.5 0.5 −0.1
0.1053 −0.1948 −0.6855
A21 = 0.1 −3.5 0.3 , Aτ 21 = −0.1586 0.0755 −0.2684 , 0.8709 −0.5266 −1.1883 −0.1 1 −2 −0.6 1.0 h i B21 = 0.5 , Bw21 = 0.5 , C21 = 0 1 0.6 , D21 = 0.3, Dw21 = 0. 0 1.5 The TRM and singular matrix are given by as following: Π=
−0.8 0.8
1 0 0
,E = 0 1 0. 0.3 −0.3 0 0 0
Let µ = 0, τ1 = 0, τ2 = 0.8, the minimum attenuation level γmin obtained using Corollary 1 of this paper is γmin = 0.4788, which is better than γmin = 0.8635 obtained in [31]. It is easy to see that the result in this paper is less conservative than that in [31]. Example 2 We use the proposed design method to deal with the control problem of a computer simulated truck-trailer system with time delay [32]. Here, we consider the reflection of actuator saturation. The modified truck-trailer model with time-varying delay and actuator saturation is as following: v t¯ v t¯ v t¯ x˙ 1 (t) = −a x1 (t) − (1 − a) x1 (t − τ (t)) + sat (u (t)) (L + ∆L (t)) t0 (L + ∆L (t)) t0 (l + ∆l (t)) t0 v t¯ v t¯ x˙ 2 (t) = a x1 (t) + (1 − a) x1 (t − τ (t)) (L +∆L (t)) t0 (L + ∆L (t)) t0 v t¯ v t¯ v t¯ x˙ 3 (t) = sin x2 (t) + a x1 (t) + (1 − a) x1 (t − τ (t)) t0 2 (L + ∆L (t)) t0 2 (L + ∆L (t)) t0 where x1 (t) is the angle difference between truck and trailer, x2 (t) is the angle of trailer, x3 (t) is the vertical position of rear end of trailer. The model parameters are given as l = 2.8, L = 5.5, v = −1.0, t¯ = 2.0, t0 = 15
0.12, d = 0.1/π, and a = 0.75. S = {1, 2, 3} is used to describe the actuator failure. As in [32], the transition probability rate matrix that relates the three failure modes is given as follows −0.35 0.25 0.1 0.29 −0.52 0.23 0.1 0.3 −0.4 Then the model is expressed by the following T-S fuzzy system with two fuzzy rules: x˙ (t) =
2 X
¯pi sat (u (t)) + B ¯wpi w (t) , λi (ε (t)) A¯pi x (t) + A¯τ pi x (t − τ (t)) + B
i=1
z (t) =
2 X
¯ pi sat (u (t)) + D ¯ wpi w (t) , λi (ε (t)) C¯pi x (t) + C¯τ pi x (t − τ (t)) + D
i=1
x (t) = φ (t) , t ∈ [−τ2 , 0] , where 1 Ap1 = av t¯/(Lt0 ) 0 0 , Bwp1 = 0 , 0 0 , Aτ p1 = (1 − a) v t¯/(Lt0 ) 0 (1 − a) v 2 t¯2 (2Lt0 ) v t¯/t0 0 av 2 t¯2 (2Lt0 ) v t¯/t0 0 −av t¯/(Lt0 ) 0 0 − (1 − a) v t¯/(Lt0 ) 0 0 1 Ap2 = av t¯/(Lt0 ) 0 0 , Aτ p2 = (1 − a) v t¯/(Lt0 ) 0 0 , Bwp2 = 0 , adv 2 t¯2 (2Lt0 ) dv t¯/t0 0 (1 − a) dv 2 t¯2 (2Lt0 ) v t¯/t0 0 0 0.5 ∗ v t¯/(lt0 ) v t¯/(lt0 ) 0.75 ∗ v t¯/(lt0 ) B11 = B12 = , , B21 = B22 = , B31 = B32 = 0 0 0 0 0 0 h i h i Cp1 = Cp2 = 0.5 0 0 , Cτ p1 = Cτ p2 = 0 0 0 , Dp1 = Dp2 = 0, Dwp1 = Dwp2 = 0, h iT h i H1p1 = H1p2 = 0.3 0 0 , H2p1 = H2p2 = 0, E1p1 = E1p2 = m 0 0 , h i E2p1 = E2p2 = (1 − m) 0 0 , E3p1 = E3p2 = 0, E4p1 = E4p2 = 0, p ∈ S.
−av t¯/(Lt0 )
0
0
− (1 − a) v t¯/(Lt0 )
0
0
Assume the controller gains have the perturbations with h i H3pi = 0.01, E5pi = 0 0.1 0 , p ∈ S, i = 1, 2. For µ = 0, τ1 = 0, τ2 = 0.2, γ = 1, M = −diag {1, 1, 1} , N = 0, and L = α + γ 2 toolbox, we can obtain h i h K11 = 21.5470 −124.0125 128.6721 , K12 = 42.5982 −100.2258 h i h K21 = 32.4458 −248.3112 246.8135 , K22 = 38.1963 −248.3112 h i h K31 = 59.6950 −388.7384 329.7658 , K32 = 55.6480 −342.8135
I, using the matlab LMI
i 218.1857 , i 246.8135 , i 312.3145 .
For simulation, we choose the fuzzy weighting function be h1 (t) = 1/1 + exp(0.5x1 (t + 1)), h2 (t) = 1−h1 (t), and the disturbance input as w (t) = sin (2t) ∗ exp (−0.05t) ∈ L2 [0, ∞) and the initial condition φT (t) = 16
Fig. 1. The operation modes (Example 2)
Fig. 2. State response of the closed-loop system (Example 2)
iT 0.5π 0.75π 2 , t ∈ [−0.2, 0]. Fig. 1 plots the operation modes. Fig. 2 shows the response of states of the closed-loop systems. From Fig1-2, we can see that, the method proposed in this paper is feasible. Example 3 Consider the following uncertain stochastic fuzzy system with two fuzzy plant rules: IF ai (t) is Λi1 (i = 1, 2) , THEN
h
¯pi u (t) + B ¯wpi w (t) , E x˙ (t) = A¯pi x (t) + A¯τ pi x (t − τ (t)) + B ¯ pi u (t) + D ¯ wpi w (t) , z (t) = C¯pi x (t) + C¯τ pi x (t − τ (t)) + D y (t) = Cypi x (t) , where
0
−0.6 0
0 0 0.01
−0.2
0
0 0
A11 = −0.3 0.8 0 , Aτ 11 = 0 0 0 , B11 = 0 , Bw11 = 0 0 0 , 0 0 0.01 0 0.01 0 0 0.1 0 0 0 0 0 0 0 0 0 0 0.01 0.1 A12 = 0 0 0 , Aτ 12 = 0 0.3 0 , B12 = 0.1 , Bw12 = 0 0 0 , −0.1 0 0.01 0 0 0.01 0 0 0 0 −0.2 −0.5 0 0 0 0 −0.2 0 0 0 A21 = −0.3 0.8 0 , Aτ 21 = 0.02 0.3 0 , B21 = 0 , Bw21 = 0 0.05 0 , 0.1 0.1 0 0 0 0 0 0 0 0 −0.9 0 0.1 −0.5 −0.2 0 −0.2 0 0 0 A22 = −0.3 0.8 0 , Aτ 22 = 0 , B = , B = 0.3 0 0 0 0.05 0 , 22 w22 0.1 0 0 0.1 0 0 0 0 0 0 17
0 0.01 0
0
0
0
−1 1 0
−0.1 1 0
C11 = 0 0 0.1 , C12 = 0 −0.3 0 , Cτ 21 = 1 2 0 Cτ 22 = 1 2 0 , 0.1 0 0 0 0 0.1 0 0 0.1 0 0 0.1 0 0 0 0 00 0 00 0 C22 = 0 0 0 , C21 = 0 0 0 , Cτ 11 = Cτ 12 = 0 −0.3 0 , D11 = 0.01 , 0 0 0.1 0 0 0 0.1 0 0 0.01 0 0 0 0 0 0 D21 = 0 , D12 = −0.01 , D22 = 0 , Dw11 = Dw21 = 0 0 0 , 0 0 0.01 0.01 0 0.05 1 0 0 1 0 1 00 0 Dw12 = Dw22 = 0 0 0 , Cy11 = Cy12 = Cy21 = Cy22 = 0 1 1 , E = 0 1 0 , 0 0 0 1 1 1 0 0 −0.01 −0.03 −0.03 −0.2 −0.2 H111 = 0.1 , H112 = 0 , H121 = 0.1 , H122 = 0 , −0.3 −0.3 0.2 −0.2 −0.3 −0.03 −0.2 −0.2 H211 = 0.1 , H212 = 0.1 , H221 = 0 H222 = 0 , −0.3 −0.3 0.2 −0.2 h i h i h i E111 = 0.5 0.2 0.1 , E112 = 0.15 0.2 0.1 , E121 = E122 = 0.15 0.2 0.2 , h i h i h i E211 = 0 0 0.01 , E212 = −0.35 0 0 , E221 = E222 = 0.05 0.45 0.01 , h i E311 = E312 = 0.1, E321 = E322 = 0.01, E411 = E412 = E421 = E422 = 0 0 0.01 . Assume the controller gains have the perturbations with h i h i h i H3pi = 0.01, p, i = 1, 2, E511 = −3 0 0 , E512 = E522 = −0.03 0 0 , E521 = 0 0.1 0 . For µ = 0.1, τ1 = 0, τ2 = 0.01, α = 1, π11 = −4, π22 = −5, M = −diag {1, 1, 1} , N = I3 , and L = diag {10, 10, 10}, using the matlab LMI toolbox, we can obtain 0.0540 0.0269 0.1308 1.2142 0.6089 4.3457 P1 = 0.0269 0.1075 0.0634 , P2 = 0.6089 0.4021 2.1306 , 0.1308 0.0634 8.1750 4.3457 2.1306 75.9800 the controller gains are given by h i h i K11 = −1.0631 −1.6324 43.8204 , K12 = 5.7075 1.0821 19.2970 , h i h i K21 = 84.0031 −6.2666 8.9983 , K22 = 42.9986 29.6029 17.4221 . h iT The initial state is assumed to be x0 = 1 −2 1.3 , solving the optimization problem in Remark 6, the value of δmax = 0.00474. 18
Fig. 3. The operation modes(Example 3)
Fig. 4. The estimation of attraction of the close-loop system (Example 3)
Fig. 5. The response of the closed-loop system(Example 3)
Fig. 6. The simulation of Γ (t) (Example 3)
Let the fuzzy weighting function be h1 (x1 ) = 1/[1 + exp(0.5(x1 + 1))], h2 (x1 ) = 1 − h1 (x1 ), disturbance input is ω (t) = exp(−t) sin(t). Fig.3 plots the operation modes. Fig.4 plots the estimation of attraction for the non-fragile control law. Then using the non-fragile controller, the simulation result for the state response of the closed-loop system is given in Fig. 5. In the end, we consider the dissipativity of the closed-loop system. Denote ∆
Γ (t) = ε {hz, M zit + 2hz, N ωit + hω, Lωit } − αhω, ωit , where (M, N, L) are given above and α = 1. Fig. 6 plots the response of Γ (t) for resulting closed-loop systems as t ∈ [0, 100]. In is seen from Fig. 6 that Γ (t) ≥ 0 is satisfied, which implies that the resulting closed-loop system is strictly (M, N, L) − α−dissipative. 19
5. Conclusion The problem of non-fragile fuzzy dissipative static output feedback control for singular Markovian jump time delay system subject to actuator saturation has been discussed. Sufficient conditions have been obtained, which guarantee that the closed-loop system is not only stochastic admissible but also dissipative. Then the non-fragile static output feedback controller can be obtained by solving the LMIs. And the largest estimation of the domain of attraction is obtained by solving the LMIs optimization problem. Examples are presented to demonstrate the feasibility and superiority. In this paper, the TRM is supposed all available, however, in some practical application, the TRM is partly unknown. In future work, we will focus on this issue. Acknowledgment This Letter was supported by the National Natural Science Foundation of China No. 61273004, and the Natural Science Foundation of Hebei province No. F2014203085.
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PII: DOI: Reference:
S0016-0032(16)30126-0 http://dx.doi.org/10.1016/j.jfranklin.2016.04.006 FI2579
To appear in: Journal of the Franklin Institute Received date: 24 November 2015 Revised date: 11 February 2016 Accepted date: 21 April 2016 Cite this article as: Yuechao Ma, Menghua Chen and Qingling Zhang, Nonfragile static output feedback control for singular T-S fuzzy delay-dependent systems subject to Markovian jump and actuator saturation, Journal of the Franklin Institute, http://dx.doi.org/10.1016/j.jfranklin.2016.04.006 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Non-fragile static output feedback control for singular T-S fuzzy delay-dependent systems subject to Markovian jump and actuator saturation Yuechao Maa , Menghua Chena ∗ , Qingling Zhangb a College b College
of Science, Yanshan University, Qinhuangdao, Hebei Province, P.R.China, 066004 of Science, Northeastern University, Shenyang, Liaoning Province, P.R.China, 110004
Abstract In this paper, we consider the problem of non-fragile static output feedback control for singular T-S fuzzy delaydependent systems subject to Markovian jump and actuator saturation. Sufficient conditions are presented to guarantee the resulted closed-loop system to be stochastically admissible and dissipative. Then, the non-fragile fuzzy static output feedback controller can be obtained by solving the LMIs. And the estimation of the largest domain of attraction for the system is formulated and solved as an optimization problem. Finally, some numerical examples are provided to show the effectiveness of the proposed approaches. Key words: singular T-S fuzzy system, non-fragile, dissipative, Markovian jump, static output feedback control
1. Introduction The past decade has witnessed rapidly growing interest in T-S fuzzy models [1], especially the singular T-S fuzzy model. Since it can combine the merits of both fuzzy logic theory and linear systems, and then stability analysis and controller design of the overfall fuzzy systems can be carried out in the Lyapunov function framework. For example, memory dissipative control, memory H∞ control and H∞ filter were studied in [2-4], respectively. In [5], Chadli et al. obtained new stability and stabilization conditions for singular uncertain T-S fuzzy system. H∞ descriptor fault detection filter for T-S fuzzy discrete-time systems was demonstrated in [6]. On the other aspect, delay is often encountered in various engineering systems. In many cases, time delay is a source of instability and performance deterioration. It has been proven that delay-dependent conditions are less conservative than time-independent conditions by many researches, such as [2-4, 6-8]. Markovian jump systems (MJSs) are special stochastic hybrid systems, where the system mode is governed by a continuous Markovian chain taking values in a finite set [9]. And singular (or descriptor) system model has been used to deal with complex practical engineering systems [10]. It has been generally recognized that ? Project supported by the National Nature Science Foundation of China No. 61273004, and the Natural Science Foundation of Hebei province No. F2014203085. ∗ Corresponding author. Email address: chenmodehao [email protected]; telnumber: 86-18637031605; faxnumber: 86-03358057027 (Menghua Chena ). Preprint submitted to
singular Markovian jump systems (sMJSs) have the advantage of better describing physical systems with abrupt variations and has successful applications in control and communication fields, to name a few, [11,12] and the references therein. Using T-S fuzzy model, some nonlinear Markovian jump systems were discussed effectively. In [13], robust finite-time fuzzy H∞ control for uncertain time-delay systems with stochastic jumps was studied. Zhang et al. discussed delay-dependent robust stabilization for uncertain discrete-time fuzzy Markovian jump systems with mode-dependent time delays in [14]. However, in these papers, the relatively small uncertainties in controller implementation were ignored. Since perturbations often appear in the controller gain, which may result from either the actuator degradations or the requirements for readjustment of controller gains. The non-fragile control concept is how to design a feedback controller that will be insensitive to some error in gains of feedback control. Non-fragile robust control [15], and non-fragile passive controller [16] were discussed. However, the uncertain parameter in Bi was ignored in [15,16] and many other articles on non-fragile control. In practice, it is necessary to consider the uncertainty. Dissipative has captured comprehensive attention, since it is a more general criterion comparing with passivity and H∞ performance, see, [17,18]. In [19], the problem of fuzzy dissipative control for nonlinear Markovian jump systems via retarded feedback was investigated. On the other hand, some state variables may be difficult to measure and sometimes have no physical meaning and thus cannot be measured at all. In this situation, the static output feedback control is more suitable for practical application, see, [20]. In [21], Xia et al. discussed the problem of delay-dependent fuzzy static output feedback control for discretetime fuzzy stochastic systems with distributed time-varying delays. Static-output-feedback H∞ control of continuous-time T-S fuzzy affine systems via piecewise Lyapunov functions was studied in [22]. In practice, actuator saturation is very ubiquitous, which is a main cause to result in poor performance of the closed-loop systems and sometimes it may lead the system unstable. Robust H∞ static output feedback stabilization of T-S fuzzy systems subject to actuator saturation was demonstrated in [23]. Zhao and Li [24] discussed the problem of robust stabilization of T-S fuzzy discrete systems with actuator saturation via PDC and non-PDC law. Yang and Tong [25] put forward the problem of output feedback robust stabilization of switched fuzzy systems with time-delay and actuator saturation. To our best knowledge, non-fragile fuzzy dissipative static output feedback control for singular Markovian jump systems subject to actuator saturation has not been fully investigated so far. These motivate the present study. In this paper, we consider the problem of non-fragile fuzzy dissipative static output feedback control for singular Markovian jump systems subject to actuator saturation. Contributions of this paper can be summarized as following aspects. First, some stochastically admissible and dissipative conditions are established on the basis of Lyapunov theory and the reciprocally convex combination lemma. Second, the estimation of the largest domain is estimated. Third, all the conditions are presented in terms of LMIs, then the case can be solved by solving the LMIs optimization problem. Finally, owing to the Lyapunov function, reciprocally convex combination lemma and proper design method of the non-fragile controller, the results in this paper are superior, which has been verified in the examples. Notation: Throughout this paper, Rn denotes the n-dimensional Euclidean space, and Rn×m is the set of n × m real matrices. For A ∈ Rn×m , A−1 and AT denote the matrix inverse and matrix transpose respectively. λmin (A) means the minimal eigenvalue of A. ε (.) stands for the expectation operator. For a real symmetric matrix A ∈ Rn×n , Sym (A) = A + AT , and A > 0(A ≥ 0) means that A is positive defined (positive semi-defined). A+ denotes the generalized inverse matrix of A. k.k stands for the Euclidean norm for a vector and kφ (t)kd = sup kφ (t)k stands for the norm of a function φ (t) ∈ Cn,d . The symbol ∗ −d≤t≤0
means the symmetric term in a symmetric matrix.
2. Problem formulation Fix a probability space (Ω, F, P (rt )), and consider a class of singular Markovian jump system (SMJS), which can be described by the following fuzzy model. Plant rule i: IF a1 (t) is Λi1 and a2 (t) is Λi2 ...ag (t) is Λig , THEN 2
¯i (rt ) sat (u (t)) + B ¯wi (rt ) w (t) , E x˙ (t) = A¯i (rt ) x (t) + A¯τ i (rt ) x (t − τ (t)) + B ¯ i (rt ) sat (u (t)) + D ¯ wi (rt ) w (t) , z (t) = C¯i (rt ) x (t) + C¯τ i (rt ) x (t − τ (t)) + D
(1)
y (t) = Cyi (rt ) x (t) , ∆
x (t) = φ (t) , t ∈ [−τ2 , 0] , i ∈ T = {1, 2, ...r} , where r is the number of If-Then rules, Λij (i ∈ T, j = 1, 2, ...g) are fuzzy sets, a1 (t) , a2 (t) , ..., ag (t) are premise variables. x (t) ∈ Rn is the state vector, w (t) ∈ Rq is a disturbance which belongs to L2 [0, ∞), z (t) ∈ Rp is the controlled output, y (t) ∈ Rm is the measurement output. τ (t) is the interval delay satisfying τ1 ≤ τ (t) ≤ τ2 , τ˙ (t) ≤ µ. u (t) ∈ Rl is the control input, and sat : Rl → Rl is the standard saturation function defined as follows: T
sat (u (t)) = [sat (u1 (t)) , · · · , sat (ul (t))] , without loss of generality, sat (ui (t)) = sign (ui (t)) min {1, |ui (t)|}. Here the notation of sat (·) is abused to denote the scalar values and the vector valued saturation functions. The matrix E ∈ Rn×n may be singular and assume that rank (E) = r ≤ n. φ (t) ∈ Cn,τ2 is a compatible vector valued initial function. {rt , t ≥ 0} is a continuous-time Markovian process with right continuous trajectories taking values in a finite set given ∆ by S = {1, 2, ..., N } with the transition rates matrix (TRM) Π = {πpq } given by π h + o (h) , p 6= q pq Pr {rt+h = q |rt = p } = 1 + π h + o (h) , p = q pq
o(h) h
= 0, and πpq ≥ 0, for q 6= p, is the transition rate from mode p to q at time t + h, PN which satisfies πpp = − q=1,q6=p πpq , for all p ∈ S. ¯i (rt ) = Bi (rt ) + ∆Bi (rt ) , B ¯wi (rt ) = A¯i (rt ) = Ai (rt ) + ∆Ai (rt ) , A¯τ i (rt ) = Aτ i (rt ) + ∆Aτ i (rt ) , B ¯ ¯ ¯ Bwi (rt )+∆Bwi (rt ) , Ci (rt ) = Ci (rt )+∆Ci (rt ) , Cτ i (rt ) = Cτ i (rt )+∆Cτ i (rt ) , Di (rt ) = Di (rt )+∆Di (rt ) , ¯ wi (rt ) = Dwi (rt ) + ∆Dwi (rt ) . For notional simplicity, in the sequel, for each possible rt = p ∈ S, the D matrix Ai (rt ) will be denoted by Api , Aτ i (rt ) will be denoted by Aτ pi and so on. Api , Aτ pi , Bpi , Bwpi , Cpi , Cτ pi , Dpi , Dwpi , Cypi are known mode-dependent constant matrices with appropriate dimensions. And Cypi are row full rank. ∆Api , ∆Aτ pi , ∆Bpi , ∆Bwpi ,∆Cpi , ∆Cτ pi , ∆Dpi , ∆Dwpi are unknown matrices describing the model uncertainties, and are assumed to be certain bound compact sets and of the form: ∆Api ∆Aτ pi ∆Bpi ∆Bwpi H1pi = F1pi (t) [E1pi E2pi E3pi E4pi ] , (2) ∆Cpi ∆Cτ pi ∆Dpi ∆Dwpi H2pi where h > 0, lim
h→0
where H1pi , H2pi , E1pi , E2pi , E3pi , E4pi , are known real constant matrices with appropriate dimensions and F1pi (t) are unknown real and possibly time-varying matrices satisfying T F1pi (t) F1pi (t) ≤ I.
(3)
Using singleton fuzzifier, product inference, and center-average defuzzifier, the global dynamics of the T-S system (1) is described by the convex sum form: E x˙ (t) =
r X
¯pi sat (u (t)) + B ¯wpi w (t) , ai (ε (t)) A¯pi x (t) + A¯τ pi x (t − τ (t)) + B
i=1
z (t) =
r X
¯ pi sat (u (t)) + D ¯ wpi w (t) , ai (ε (t)) C¯pi x (t) + C¯τ pi x (t − τ (t)) + D
i=1
y (t) =
r X
ai (ε (t)) [Cypi x (t)] ,
i=1
x (t) = φ (t) , t ∈ [−τ2 , 0] , 3
(4)
T
where a (t) = [a1 (t) , a2 (t) , ..., ag (t)] , βi (a (t)) =
p Q
Λij (aj (t)) is the membership function of the system r r P P with respect to the ith pant rule. Let λi (a (t)) = βi (a (t)) βi (a (t)), then λi (a(t)) ≥ 0 and λi (a(t)) = j=1
i=1
i=1
1. In the sequel, we denote λi (a(t)) by λi . Remark 1 In [15,16], the uncertain parameters in Bpi were ignored, which is unavoidable in practice. In this paper, it is considered. That is to say, the system in this paper is more general. We consider the following non-fragile mode-dependent static output feedback control law: u (t) =
r X
λi (Kpi + ∆Kpi ) y (t) , p ∈ S, i ∈ T,
(5)
i=1
where Kpi ∈ Rm×n are the control gain matrices to be determined. ∆Kpi are perturbed matrices satisfying
where H3pi , E5pi
∆Kpi = H3pi F2pi (t) E5pi , are known real constant matrices, F2pi (t) are unknown matrices satisfying T F2pi (t) F2pi (t) ≤ I.
(6) (7)
Remark 2 ∆Kpi are the relatively small uncertainties on controller implementation. In practical application, it is necessary to consider the controller non-fragility problems to guarantee the stability of the system. For the rest of this paper, let us recall some import notions and results which have been in [9]. Let P ∈ Rn×n be a symmetric matrix and E T P E ≥ 0, ρ be a scalar. Denote Ω E T P E, ρ = x (t) ∈ Rn : xT (t) E T P Ex (t) ≤ ρ . For matrices Hi , hik is the kth row of the matrix Hi , we define L (Hi ) = {x (t) ∈ Rn : |hik x (t)| ≤ 1, k ∈ [1, l]} .
Thus Ω E T P E, ρ is an ellipsoid and L (Hi ) is a polyhedral consisting of states for which the saturation does not occur. Let D be the set of l × l diagonal matrices whose diagonal elements are either 1 or 0. Suppose each element of D is labeled as Es , s = 1, 2, · · · , η = 2l , and denote Es− = I − Es . Clearly, if Es ∈ D, then Es − ∈ D. Lemma 1 ([26]) Let Fp , Hp ∈ Rp×n . Then for any x (t) ∈ L (Hp ) (p ∈ S), sat (Fp x (t)) ∈ co Es Fp x (t) + Es− Hp x (t) , s = 1, 2, · · · , η ; or, equivalently, sat (Fp x (t)) =
η X
αs Es Fp + Es− Hp x (t) ,
s=1
where co stands for the convex hull, αs for s = 1, 2, · · · , η are some scalars which satisfy 0 ≤ αs ≤ 1 and η P αs = 1. s=1
¯psi = Es K ¯ pi Cypl + Es− Hpi , then From Lemma 1, for any x (t) ∈ L (Hpi ), denote ~ η X ¯psi x (t) . ¯ pi Cypl x (t) = sat K αs ~
(8)
s=1
Then using (8), substituting (5) into (4), for ηt = p ∈ S, yields the following closed-loop system: E x˙ (t) =
η X r X r X r X
¯wpi w (t) , λl λi λj αs A¯pij x (t) + A¯τ pi x (t − τ (t)) + B
i=1 j=1 s=1 l=1
z (t) =
η X r X r X r X
(9)
¯ wpi w (t) , λl λi λj αs C¯pij x (t) + C¯τ pi x (t − τ (t)) + D
i=1 j=1 s=1 l=1
¯psj , C¯pij = C¯pi + D ¯psj . ¯pi ~ ¯ pi ~ where A¯pij = A¯pi + B 4
Lemma 2 [14]Given a set of suited dimension real matrices A, B, X, with BT B ≤ I, then there exists a scalar ε > 0 such that ABX + XT BT AT ≤ εAAT + ε−1 XT X. Lemma 3 [27]For given matrices E, X > 0, Y ,if E T X + Y ΓT is nonsingular, then there exist matrices −1 S > 0, I, such that ES+IKT = E T X + Y ΓT , where X, S ∈ Rn×n ,, Y, I ∈ Rn×(n−r) and Γ, K ∈ Rn×(n−r) are any matrices with full column rank satisfying E T Γ = 0, EK = 0. Lemma 4 [28](Reciprocally convex combination lemma)Let f1 ,f2 ,...,fN :Rm → R have positive values in an open subset D of Rm . Then, the reciprocally convex combination of fi over D satisfies X 1 X X min fi (t) = fi (t) + min gi,j (t), P gi,j (t) αi =1 } i αi {αi |αi >0, i i6=j
i
subject to f (t) g (t) i i,j ∆ ≥0 . gi,j : Rm 7→ R,gj,i (t) = gi,j (t) , gj,i (t) fj (t) Definition 1 ([29]) (i) For given scalar τ2 > 0, the SMJS (9) with w (t) = 0 is said to be regular and impulse-free for any time-delay satisfying τ1 ≤ τ (t) ≤ τ2 , if the pair E, A¯pij and E, A¯pij + A¯τ pi are regular and impulse-free. (ii) The SMJS (9) with w (t) = 0 is stochastically stable for all finite φ (t) defined on t ∈ [−τ2 , 0] and initial mode r0 , if there exists a finite number Φ (φ, τ2 , r0 ) > 0, such that the following inequality holds: Z t 2 lim εkx (t)k dt < Φ (φ, τ2 , r0 ) . t→∞
0
(iii) The SMJS (9) with w (t) = 0 is said to be stochastically admissible if it is regular, impulse-free and stochastically stable. Definition 2 ([18]) Given a scalar α > 0, real symmetric matrices M, L, and matrix N . The closed-loop system (8) is called strictly (M, N, L)−α−dissipative, for any t ≥ 0, with zero initial condition, the following condition is satisfied: ε {hz, M zit + 2hz, N ωit + hω, Lωit } ≥ αhω, ωit (10) Rt T √ ¯ = −M . where ha, bit = 0 a b dt. Without loss of generality, we assume that M < 0 and M Remark 3 From Definition 2, the above strictly (M, N, L) − α−dissipative performance includes H∞ performance that is, when M = −I, N = 0, and L = α + γ 2 I, Eq.(10) corresponds to an H∞ performance requirement, that is ε{hz, zi}t ≤ γ 2 {hw, wi}t . Definition 3 ([30]) Denoting the solution of system (9) with initial condition x0 = φ (t) ∈ Cn,τ2 by ψ (t, x0 ), then the domain of attraction of the origin is n o ∆ ς = φ ∈ Cn,τ2 : lim ψ (t, x0 ) = 0 . t→∞
Remark 4 The estimate of the domain of attraction Xδ ⊂ ς, where n o Xδ = φ ∈ Cn,τ2 : max |φ| ≤ δ1 , max φ˙ ≤ δ2 , with scalars δ1 , δ2 > 0 what will be maximized in the following 3. Main results 3.1. Stochastic admissibility and dissipativity analysis This section gives sufficient conditions for stochastic admissibility and dissipativity for the closed-loop system. 5
Theorem 1 For given scalars 0 ≤ τ1 ≤ τ2 and µ > 0, scalar α > 0, symmetric matrices M < 0, L, and a T matrix N , system (9) is stochastically admissible and (M, N, L) − α−dissipative within ∩N p=1 Ω E Pp E, ρ , if there exist positive-defined matrices Q, Q1 , Q2 , Q3p , U, Up , Pp , and any matrices Mp with appropriate dimensions satisfying
Λ Mp ∗ Λ
≥ 0, matrices Sp , such that for any p ∈ S, i ∈ T, the following inequalities
hold:
¯ pi Ξ
T T ¯ij ¯ij U U V¯ijT −1 ∗ − Up 0 0 τ12 < 0, = −1 2U ∗ ∗ − 2 0 τ12 ∗ ∗ ∗ −I ¯ pi Σ
N X
(11)
πpq Q3q − Q < 0,
(12)
q=1 N X
πpq Uq − U < 0,
(13)
q=1
T ∩N p=1 Ω E Pp E, ρ ⊂ L (Hpi )
(14)
where
¯ pi Σ
=
h
¯ (11) P¯pT A¯τ pi Σ pi
0
∗
¯ (22) Σ pi
¯ (23) Σ pi
∗
∗
¯ (33) Σ pi
∗
∗
∗
∗
∗ i
¯ij = A¯ ¯ ¯ U pij Aτ pi 0 0 Bwpi
0
¯ (15) Σ pi
¯ (25) Σ pi MpT 0 , −Q2 − Λ 0 ¯ (55) ∗ Σ pi ¯ (24) Σ pi
∗ h i ¯ C¯pij M ¯ C¯τ pi 0 0 M ¯D ¯ wpi , , V¯ij = M
N X ¯ (11) = Sym XpT A¯pij + Σ πpq E T Pq E + Q1 + Q3p + τ2 Q, pi q=1
¯ (22) Σ pi
= − (1 − µ) Q3p − 2Λ + MpT + Mp ,
¯ (23) = −Mp + Λ, Σ ¯ (24) = −M T + Λ, Σ ¯ (33) = −Q1 + Q2 − Λ, Σ p pi pi pi T ¯ (15) = XpT B ¯wpi − C¯pij ¯ (25) = −C¯τTpi N T , Σ ¯ (55) = −L + αI − Sym(N D ¯ wpi ), Σ NT , Σ pi pi pi T −1 T τ12 = τ2 − τ1 , Λ = τ12 E Up E, Xp = E T Pp + Sp RT ,
R ∈ Rn×(n−r) is any matrix with full column rank satisfying E T R = 0, and the estimate of the domain is Γδ ≤ ρ, where τ2 Γδ = δ12 λmax E T Pp E + τ1 λmax (Q1 ) + τ12 λmax (Q2 ) + τ2 λmax (Q3p ) + 2 λmax (Q) 2 2 (15) 2 τ23 − τ13 2 τ2 − τ1 T T +δ2 λmax E Up E + λmax E U E 2 6 Proof.Firstly, we prove the system (9) with w (t) = 0 is regular, impulse-free. 6
n×n Since rankE , then R can be rewritten = r < n, there must exist two invertible matrices G and H ∈ R 0 as R = GT , where Φ ∈ R(n−r)×(n−r) . Denote Φ Ir 0 Ap1 Ap2 P P p1 p2 , GA¯pij H = , G−T Pp G−1 = , GEH = 0 0 Ap3 Ap4 Pp3 Pp4 (16) S x (t) p1 1 T H Sp = , Hpi H = [Hpi1 Hpi2 ] , x (t) = H . Sp2 x2 (t)
¯ (11) < 0 by H T and H respectively, we can get that for every p ∈ S, AT ΦS T + Pre- and post-multiplying Σ p4 p2 pi Sp2 ΦT Ap4 < 0, which implies Ap4 is nonsingular and thus the pair (E, A¯pij ) is regular and impulse-free. On the other hand, it can be seen from (11) that, T ¯ (11) P¯ T A¯τ pi 0 I Σ 0 I p pi (22) (23) (24) ¯ ¯ ¯ I I ∗ Σ Σ Σ pi pi pi < 0, ¯ (33) I ∗ ∗ Σ MpT I pi I ∗ ∗ ∗ −Q2 − Λ I and considering Q3p and Q are positive , then, N X πpq E T Pp E < 0, Sym XpT A¯pij + A¯τ pi + q=1
which implies the pair (E, A¯pij + A¯τ pi ) is regular and impulse-free. So, we can get the system (9) is regular T and impulse-free within ∩N p=1 Ω E Pp E, ρ . Next, we will show the stochastic stability. The weak infinitesimal operator ` [.] ([12]) of the stochastic process {xt , rt , t ≥ 0} acting on V (xt , t, rt ) at the point {t, xt , rt = p} is given by `[V ] =
N X ∂V ∂V + x˙ T (t) + πpq V (xt , t, q). ∂t ∂x rt =p q=1
We choose the following stochastic Lyapunov-Krasovskii candidate for this system: V (xt , p, t) =
3 X
Vi (xt , p, t),
i=1
where V1 (xt , p, t) = xT (t) E T Pp Ex (t) , Z t Z t−τ1 V2 (xt , p, t) = xT (s)Q1 x (s) ds + xT (s)Q2 x (s) ds t−τ t−τ Z t1 Z2 0 Z t T + x (s)Q3p x (s) ds + xT (s)Qx (s) dsdθ, t−τ (t) −τ2 t+θ Z −τ1 Z t V3 (xt , p, t) = x˙ T (s)E T Up E x˙ (s) dsdθ −τ t+θ 2 Z −τ Z 0Z t 1 + x˙ T (s) E T U E x˙ (s)dsdvdθ, −τ2
θ
t+v
using the operator ` [.] we can obtain 7
" `[V1 (xt , p, t)] = 2x
T
(t) XpT E x˙ (t)
T
+ x (t)
N X
# T
πpq E Pq E x (t) ,
q=1
`[V2 (xt , p, t)] ≤ xT (t) (Q1 + Q3p + τ2 Q) x (t) + xT (t − τ1 ) (−Q1 + Q2 ) x (t − τ1 ) +xT (t − τ2 ) (−Q2 ) x (t − τ2 ) − (1 − µ) xT (t − τ (t)) Q3p x (t − τ (t)) "N # Z t X T + x (s) πpq Q3q − Q x (s) ds, t−τ2 q=1 2 τ12 T T `[V3 (xt , p, t)] = x˙ (t) E τ12 Up + U E x˙ (t) " 2N # Z t−τ1 Z −τ1 Z t X T x˙ T (s) E T Up E x˙ (s) ds, x˙ (s) E πpq Uq − U E x˙ (s) dsdθ − + −τ2
t+θ
t−τ2
q=1
where Z
t−τ1
−
x˙ T (s) E T Up E x˙ (s) ds
t−τ2 Z t−τ (t)
Z
T
t−τ1
x˙ (s) E Up E x˙ (s) ds − x˙ T (s) E T Up E x˙ (s) ds t−τ t−τ (t) 2 "Z # "Z # "Z # "Z t−τ (t) t−τ (t) t−τ1 T T ≤ −δ1 x˙ (s) ds Λ x˙ (s) ds − δ2 x˙ (s) ds Λ =−
T
t−τ2
t−τ (t)
t−τ2
#
t−τ1
x˙ (s) ds ,
t−τ (t)
τ12 τ12 , δ2 = τ2 − τ (t) τ (t) − τ1 from Lemma4, it can be obtained that, Z t−τ1 − x˙ T (s) E T Up E x˙ (s) ds t−τ2 T Λ Mp x (t − τ (t)) − x (t − τ2 ) x (t − τ (t)) − x (t − τ2 ) . ≤ − ∗ Λ x (t − τ1 ) − x (t − τ (t)) x (t − τ1 ) − x (t − τ (t)) h i Let ξ T (t) = xT (t) xT (t − τ (t)) xT (t − τ1 ) xT (t − τ2 ) wT (t) , from (11-13) and considering Schur complement lemma, it is true that, δ1 =
`[V (xt , p, t)] − z T (t) M z (t) − 2wT (t) N z (t) − wT (t) (L − αI) w (t) r X r 2 X T ¯ pi + V¯ T V¯ij + U ¯ T τ12 Up + τ12 U U ¯ij ξ (t) < 0, ≤ λi λj ξ (t) Σ ij ij 2 i=1 j=1 that is, `[V (xt , p, t)] − z T (t) M z (t) − 2wT (t) N z (t) − wT (t) (L − αI) w (t) < 0. (17) ¯ pi , p ∈ S}, then r1 > 0. It It is easy to see that, when w (t) = 0, ` [V (xt , p, t)] < 0. Set r1 = min{λmin −Ξ is easy to obtained that 2 `V (xt , p, t) ≤ −r1 kx (t)k . (18) By Dynkin’s formula, for each rt ∈ S, t ≥ τ2 , we have from (11) that Z t ε {V (xt , t, rt ) − V (xτ2 , τ2 , rτ2 )} = ε `[V (xs , s, rs )]ds τ2Z (19) t 2 ≤ −r1 ε kx (s)k ds , τ2
it follows that Z
t
ε
2 kx (s)k ds ≤ r1−1 ε {V (xτ2 , τ2 , rτ2 )} .
τ2
8
Following the similar line of the proof of [29], it is clear that there exists a scalar r2 such that: Z τ2 2 2 kx (s)k ds ≤ r2 kφ (s)kτ2 . ε 0
Therefore, by the definitions of V (xt , p, t) and x (t) there always exists a scalar β such that Z t Z τ2 Z t 2 2 2 2 kx (s)k ds ≤ βε kφ (s)kτ2 , kx (s)k ds + ε kx (s)k ds = ε ε τ2
0
0
then Z
t
lim ε
t→∞
0
2 kx (s)k ds ≤ βε kφ (s)kτ2 < Φ (φ, τ2 , r0 ) . 2
According to Definition1, system (9) with w (t) = 0 is stochastically admissible. On the other hand, when 0 6= w (t) ∈ L2 [0, ∞). We introduce the following performance index J = ε {hz, M zit + 2hz, N ωit + hω, Lωit − αhω, ωit } . Integrating both sides of (17) from 0 to t yields Z t [`[V (xs , p, t)]] ds =V (xt , p, t) − V (x0 , r0 , 0) > 0, J> 0
we conclude that Eq. (10) is satisfied. In the end, it is clear to see that xT (t) E T Pp Ex (t) ≤ V (xt , p, t) ≤ V (x0 , p, t) τ22 2 T ≤ max |φ (θ)| λmax E Pp E + τ1 λmax (Q1 ) + τ12 λmax (Q2 ) + τ2 λmax (Q3p ) + λmax (Q) (20) 2 θ∈[−τ2 ,0] 2 τ 2 − τ 2 3 3 τ − τ1 ∆ 2 1 + max φ˙ (θ) λmax E T Up E + 2 λmax E T U E = < φ, φ˙ 2 6 θ∈[−τ2 ,0] For any φ ∈ < φ, φ˙ , form (20), we can further get xT (t) E T Pp Ex (t) ≤ ρ, which means that all trajectories of x (t) which start from the set < φ, φ˙ always stay in the domain Ω E T Pp E, ρ . This completes the proof.
3.2. Non-fragile static output feedback controller design In this section, we will establish LMI conditions to solve the controller synthesis problem based on the results of Theorem 1, which can be easily solved by using the existing solvers such as LMI TOOLBOX in Matlab software. Theorem 2 For given scalars 0 ≤ τ1 < τ2 and µ > 0, α > 0, symmetric matrices M < 0, L and a matrix N , there exist a non-fragile static output feedback controller of the form (5) such that the closed-loop system (9) ˜1, Q ˜ 2 , Q, ˜ U ˜, is admissible and dissipative, if there exist scalars αij > 0, βij > 0, positive-definite matrices Q ˜ M ˜p Λ ˜ 3p , U ˜p , P˜p , and any matrices M ˜ p with appropriate dimensions satisfying ≥ 0, matrices S˜p , Q ˜ pT Λ ˜ M such that for each p ∈ S, i, j, l ∈ T, the following inequalities hold: Ξpii < 0,
(21)
Ξpij + Ξpji < 0, N X
i < j,
˜ 3q − Q ˜ < 0, πpq Q
q=1
9
(22)
N X
˜q − U ˜ < 0, πpq U
(23)
q=1
1 mpik −ρ ≤ 0, ∗ −EYp E T
where
Ξpij
˜ pij O ˜1 Θ
˜3 O
J˜1
J˜3
Υp
(24)
T T T ˜pij ˜pij ψ˜pi U U V˜pij 0 −1 ∗ ψ˜22 0 0 ∗ ∗ −αij 0 0 0 ˜ pij = , ,Θ ∗ ∗ ψ˜33 0 ∗ ∗ ∗ −βij 0 0 −1 ∗ ∗ ∗ ∗ −βij ∗ ∗ ∗ −I 0 ∗ ∗ ∗ ∗ ∗ Γp (11) (12) (15) ψ˜pi ψ˜pi 0 0 ψ˜pi (22) ˜(23) (24) ∗ ψ˜pi ψpi ψ˜pi −YpT CτTpi N T (33) T ˜ ˜ ψ˜pi = ∗ , Mp 0 ∗ ψpi ˜2 − Λ ˜ ∗ ∗ ∗ −Q 0 (55) ∗ ∗ ∗ ∗ ψ˜pi h i ˜pij = A Y + B λ U ¯ A Y 0 0 B pi p pi psj τ pi p wpi , h i ¯ (Cpi Yp + Dpiλ ¯ Cτ pi Yp 0 0 M ¯ Dwpi , V˜pij = M ¯psj ) M
=
∗
−αij
ψ˜22 =
0
0
0
˜p ˜ −Yp − YpT + U −Yp − YpT + U , ψ˜33 = 2 , 2 τ12 τ12
T ˜ 1 = H1pi O 0 0 0
T T T T ¯ − H1pi N T H1pi H1pi H2pi M
T
,
T ˜ 3 = [E1pi Yp + E3piλ O ¯psj E2pi Yp 0 0 E4pi 0 0 0] , h iT T T T T T T J˜1 = Bpi 0 0 0 (−N Dpi ) Bpi Bpi Dpi 0 E3pi Es H3pj , T J˜3 = [E5pj Cypl Yp 0 0 0 0 0 0 0 0 0] , (11) ˜1 + Q ˜ 3p + τ2 Q ˜ + πpp Y T E T , ψ˜pi = Sym (Api Yp + Bpiλ ¯psj ) + Q p (12) (22) ˜ 3p − 2Λ ˜ +M ˜ pT + M ˜ p, ψ˜pi = Aτ pi Yp , ψ˜pi = − (1 − µ) Q (23) ˜ p + Λ, ˜ ψ˜(24) = −M ˜ pT + Λ, ˜ ψ˜(33) = −Q ˜1 + Q ˜ 2 − Λ, ˜ Λ ˜ = τ −1 E U ˜p E T , ψ˜pi = −M 12 pi pi (15) (55) T T ψ˜pi = Bwpi − YpT Cpi NT − λ ¯Tpsj Dpi N T , ψ˜pi = −L + αI − Sym(N Dωi ), √ Ir √ √ √ Υp = πp1 YpT ... πp(p−1) YpT πp(p+1) YpT ... πpN YpT H −T , 0 Ir Γp = −diag [Ir 0] G {EY1 , ..., EYp−1 , EYp+1 , ...EYN } GT , 0
10
˜ ∈ Rn×(n−r) is any matrix with full column rank and satisfies E R ˜ = 0, G, H are nonsingular matrices that R Ir 0 . In this case, the control gain matrices are given by Kpj = Wpj Yp−1 C + , and the make GEH = ypl 0 0 ˜ δ ≤ ρ, where estimate of the domain of attraction is Γ T −1 −1 ˜ −1 −1 ˜ −1 + τ λ Y λ E Y Q Y + τ λ Y Q Y 1 max max 1 12 max 2 p p p p p ˜ δ = δ12 τ2 Γ 2 −1 ˜ −1 −1 ˜ −1 + λmax Yp QYp +τ2 λmax Yp Q3p Yp (25) 2 2 2 3 3 τ − τ τ − τ 1 1 ˜p Y −1 E T + 2 ˜ Y −1 E T +δ22 2 λmax EYp−1 U λmax EYp−1 U p p 2 6 Proof.From Theorem1, it can be obtained easily that ¯ pi = Ξpi + ∆Ξpi < 0, Ξ where
(11) (15) T T T Σpi XpT Aτ pi 0 0 Σpi Σpi Upij Upij Vpij −1 (22) (23) (24) ∗ Σpi Σpi Σpi −CτTpi N T ∗ − Up 0 0 τ12 (33) , Σpi = = ∗ ∗ Σpi MpT 0 −1 2U ∗ 0 ∗ − 2 (44) ∗ ∗ ∗ Σpi 0 τ12 (55) ∗ ∗ ∗ −I ∗ ∗ ∗ ∗ Σpi h i ¯psj Aτ pi 0 0 Bwpi , Upij = Api + Bpi ~ h i ¯psj M ¯ Cpi + Dpi ~ ¯ Cτ pi 0 0 M ¯ Dwpi , Vpij = M
Ξpi
(11)
Σpi
N X ¯psj + = Sym XpT Api + Bpi ~ πpq E T Pq E + Q1 + Q3p + τ2 Q, q=1 (22) Σpi (23)
Σpi (15)
Σpi
(44)
= − (1 − µ) Q3p − 2Λ + MpT + Mp , Σpi (24)
(33)
= −MpT + Λ, Σpi = −Q1 + Q2 − Λ, ¯psj N T , Σ(55) = −L + αI − Sym(N Dwpi ). + Dpi ~ pi
= −Mp + Λ, Σpi
= XpT Bwpi − Cpij
= −Q2 − Λ,
Then from (2), and Lemma 2, there exist scalars αij > 0, such that T
∆Ξpi = O1 O2 O3 + (O1 O2 O3 ) where
−1 ≤ αij O3T O3 + αij O1 O1T ,
T iT T T T ¯ H2pi T , XpT H1pi 0 0 0 − (N H2pi ) (H1pi ) (H1pi ) M ¯psj E2pi 0 0 E4pi 0 0 0 , O2 = F1pi (t) , O3 = E1pi + E3pi ~
O1 =
h
via Schur complement lemma, we can get that
Ξpi O1
O3T
¯ (1) = Ξ < 0. ∗ −α 0 ij pi −1 ∗ ∗ −αij From (5), we can see that ¯ (1) = Ξ(1) + ∆Ξ(1) , Ξ pi pi pi 11
,
(1) ¯psj in Ξ ¯ (1) to ~psj . Considering (6), and Lemma 2, there exist where Ξpi is the matrix changing the item ~ pi scalars βij > 0, such that (1)
T
−1 ∆Ξpi = J1 J2 J3 + (J1 J2 J3 ) ≤ βij J3T J3 + βij J1 J1T ,
where J1 =
h
XpT Bpi
T
T
− (N Dpi )
0 0 0
T
(Bpi )
T
(Bpi )
T
(Dpi )
T
0 (E3pi )
iT
Es H3pj ,
J3 = [E5pj Cypl 0 0 0 0 0 0 0 0 0] , J2 = F2pi (t) . Via Schur complement lemma, we can get that (1) Ξpi J1 J3T (26) ∗ −βij 0 < 0. −1 ∗ ∗ −βij ¯ (11) < 0, we can get Xp = E T Pp + Sp RT T is nonsingular. Using Lemma3, there exists From Theorem 1, Σ pi T ˜ T , where P˜p > 0 and R ˜ ∈ Rn×(n−r) is any matrix with full column rank and Yp = Xp−1 = E P˜p + S˜p R ˜ = 0. It is easy to see that, satisfies E R EYp = YpT E T = E P˜p E T ≥ 0. Denoting H −1 Yp GT =
Yp11 Yp12
(27)
, from (27), it is easy to obtain that Yp12 = 0, and Yp11 is symmetric, Yp21 Yp22 Y 0 p11 , so Yp11 and Yp22 are nonsingular. There, it can be concluded that then we have H −1 Yp GT = Yp21 Yp22 −1 I Y 0 p11 and [Ir 0] GEYq GT r = Yq11 are nonsingular. Then, we have G−T Yp−1 H = −1 −1 −1 −Yp22 Yp21 Yp11 Yp22 0 −1 Ir Ir H −T [Ir 0] GEYq GT [Ir 0] H −1 0 0 −1 I Y 0 r q11 −1 H −1 = H −T Yq11 [Ir 0] H −1 = H −T 0 0 0 −T −1 −1 −T T T T =H H E G G Yp H H = E T Yq−1 = E T Xq Pre- and Post-multiply (21) by diag XpT , ..., XpT , In , ...In , Ir , ...Ir and its transposition and denote λ ¯psj Xp = {z } | {z } | {z } | 4
8
N −1
˜ p = Ω, (Ω ˜ =Q ˜1, Q ˜2, Q ˜ 3p , Q, ˜ U ˜p , U ˜ ). And we have for Z = Up , U , −Z−1 ≤ ~psj , Wpj Xp = Kpj Cypl , XpT ΩX T ˜ −Yp − Yp + Z. Then using Schur complement lemma and above treatment, (21) implies (26). T On the other hand, from ∩N p=1 Ω E Pp E, ρ ⊂ L (Hpi ), and (16), it follows that Hpi2 = 0. Otherwise, let x1 (t) = 0, and |hpi2k x2 (t)| > ρ1/2 , then xT (t) E T Pp Ex (t) = 0, |hpik x (t)| > ρ1/2 , it contradicts that T ∩N p=1 Ω E Pp E, ρ ⊂ L (Hpi ) . Then xT (t) E T Pp Ex (t) = xT1 (t) Pp1 x1 (t) , Hpi x (t) = Hpi1 x1 (t) , (28) 12
xT (t) E T Pp Ex (t) ≤ ρ, Then the condition
∩N p=1 Ω
(29)
T
E Pp E, ρ ⊂ L (Hpi ) is obtained by −1 T hpi1k Pp1 hpi1k ≤
1 , k = 1, 2, · · · , l, ρ
which, by Schur complements, is equivalent to
−
[h
1 ρ T
pi1k
0]
[hpi1k 0] ≤ 0, k = 1, 2, · · · , l Ir 0 Pp1 Pp2 Ir 0 − 0 0 Pp3 Pp4 0 0
(30)
hpi1k is the kth row of Hpi1 . Pre- and post-multiply (30) by diag 1, YpT H −T and its transpose. Denote Hpj Yp = Mpj and consider h i Mpi H = Mpi1 0 , mpik is the kth row of Mpi . Together with (16), we can obtain (24). This completes the proof. Remark 5 Based on Theorem 2 in this paper, LMI conditions are given for singular Markovion jump fuzzy system to obtain the non-fragile static output feedback controller gain. When we set r = 1, the systems in this paper are SMJSs in usual sense. The similar systems were discussed in [31]. However, in [31], some matrices are given as special forms, and the conditions were obtained at the limitation of ε → 0+ . Without considering these techniques, the superiority of this paper can be seen in Example 1. Remark 6 With all the ellipsoids satisfying the set invariance condition of Theorem 1, the ”largest” one will be chosen to obtain the least conservative estimate of the domain of attraction. In the following, we will measure the largeness of the ellipsoids with respect to a shape reference set. δ1 = δ2 is chosen in (25) to simplify the process of maximizing the estimate of domain of attraction. Here the result in [30] is adopted to complete the approximating optimization problem which could be expressed as follows: P1 min r ω1 I E ωι I I ≥ 0, ≥ 0, e ˜ι I Yp I 2Yp − O s.t. T T ω I E ω I E 6 ≥ 0, 7 ≥ 0, E 2Y − U ˜ ˜ E 2Yp − U p p
T
e e e e ˜2 = Q ˜1, O ˜3 = Q ˜2, O ˜4 = Q ˜ 3p , O ˜ 5 = Q, ˜ and where ι = 2, 3, 4, 5, O r = ω1 + τ1 ω2 + τ12 ω3 + τ2 ω4 +
τ 2 − τ12 τ 3 − τ13 τ22 ω5 + 2 ω6 + 2 ω7 . 2 2 6
Then the maximized estimate of domain of saturation is δmax = √1∆ , where ˜ 1 Y −1 + τ12 λmax Y −1 Q ˜ 2 Y −1 ∆ = λmax E T Yp−1 + τ1 λmax Yp−1 Q p p p 2 ˜ 3p Yp−1 + τ2 λmax Yp−1 QY ˜ p−1 +τ2 λmax Yp−1 Q 2 3 3 τ 2 − τ12 ˜p Yp−1 E T + τ2 − τ1 λmax EYp−1 U ˜ Yp−1 E T . + 2 λmax EYp−1 U 2 6 Remark 7 We consider the following singular Markovian system as a special case of the system (4) with r = 1 and without saturation. 13
E x(t) ˙ = Ap x(t) + Aτ p x(t − τ (t)) + Bp u(t) + Bwp w(t) z (t) = Cp x(t) + Cτ p x(t − τ (t)) + Dp u(t) + Dwp w(t)
(31)
x(t) = φ(t), t ∈ [−τ2 , 0]. We give the following corollary to discuss the H∞ performance of system (31), which is a special case of Theorem2. So the result is given directly. Corollary 1 For given scalars 0 ≤ τ1 < τ2 and µ > 0, α > 0, there exist a non-fragile state feedback controller u (t) = (Kp + ∆Kp ) x (t) , p ∈ S, such that the closed-loop system (31) is admissible and satisfying ˜1, Q ˜ 2 , Q, ˜ Q ˜ 3p , U ˜, U ˜p , the H∞ performance ε{hz, zi}t ≤ γ 2 {hw, wi}t , if there exist positive-definite matrices Q ˜ M ˜p Λ ˜ p with appropriate dimensions satisfying ≥ 0, matrices S˜p , such that P˜p , and any matrices M ˜ pT Λ ˜ M for each p ∈ S, the following inequalities hold: T ˜T ˜T ˜ ˜ ψ U U V Υp p p p p ∗ ψ˜22 0 0 0 (32) ∗ ∗ ψ˜33 0 0 < 0, ∗ ∗ ∗ −I 0 ∗ ∗ ∗ ∗ Γp N X
˜ 3q − Q ˜ < 0, πpq Q
(33)
q=1 N X
˜q − U ˜ < 0, πpq U
(34)
q=1
where
ψ˜p =
ψ˜p(11) ψ˜p(12) ∗
0
ψ˜p(22) ψ˜p(23)
0
Bwp
ψ˜p(24)
0
∗
∗
ψ˜p(33)
˜ pT M
0
∗
∗
∗
˜2 − Λ ˜ −Q
0
∗
∗
∗
∗
−γ 2 I
,
h i ˜p = A Y + B W A Y 0 0 B U p p p p τp p wp , h i V˜p = Cp Yp + Dp Wp Cτ p Yp 0 0 Dwp , ψ˜22 =
˜p ˜ −Yp − YpT + U −Yp − YpT + U , , ψ˜33 = 2 2 τ12 τ12
˜1 + Q ˜ 3p + τ2 Q ˜ + πpp YpT E T , ψ˜p(11) = Sym (Ap Yp + Bp Wp ) + Q ˜ 3p − 2Λ ˜ +M ˜ pT + M ˜ p, ψ˜p(12) = Aτ p Yp , ψ˜p(22) = − (1 − µ) Q ˜ p + Λ, ˜ ψ˜p(24) = −M ˜ pT + Λ, ˜ ψ˜p(33) = −Q ˜1 + Q ˜ 2 − Λ, ˜ Λ ˜ = τ −1 E U ˜p E T , ψ˜p(23) = −M 12 √ Ir √ √ √ Υp = πp1 YpT ... πp(p−1) YpT πp(p+1) YpT ... πpN YpT H −T , 0 14
Γp = −diag [Ir 0] G {EY1 , ..., EYp−1 , EYp+1 , ...EYN } GT
Ir 0
.
In this case, for p ∈ S, the state feedback gain are given by Kp = Wp Yp−1 . 4. Numerical example Example 1 Consider the T-S singular Markovian jump time-delay system (31) with two modes with the following parameters as [31] Mode 1 −0.1669 0.0802 1.682 −3 1 0 A11 = 0.3 −2.5 −4 , Aτ 11 = −0.8162 −0.9373 0.5936 , 2.0941 0.6357 0.7902 −0.1 0.3 −3.8 1 1.5 h i B11 = 1 , Bw11 = 0 , C11 = 0.5 −0.1 1 , D11 = 0.1, Dw11 = 0, 1 1 Mode 2
−2.5 0.5 −0.1
0.1053 −0.1948 −0.6855
A21 = 0.1 −3.5 0.3 , Aτ 21 = −0.1586 0.0755 −0.2684 , 0.8709 −0.5266 −1.1883 −0.1 1 −2 −0.6 1.0 h i B21 = 0.5 , Bw21 = 0.5 , C21 = 0 1 0.6 , D21 = 0.3, Dw21 = 0. 0 1.5 The TRM and singular matrix are given by as following: Π=
−0.8 0.8
1 0 0
,E = 0 1 0. 0.3 −0.3 0 0 0
Let µ = 0, τ1 = 0, τ2 = 0.8, the minimum attenuation level γmin obtained using Corollary 1 of this paper is γmin = 0.4788, which is better than γmin = 0.8635 obtained in [31]. It is easy to see that the result in this paper is less conservative than that in [31]. Example 2 We use the proposed design method to deal with the control problem of a computer simulated truck-trailer system with time delay [32]. Here, we consider the reflection of actuator saturation. The modified truck-trailer model with time-varying delay and actuator saturation is as following: v t¯ v t¯ v t¯ x˙ 1 (t) = −a x1 (t) − (1 − a) x1 (t − τ (t)) + sat (u (t)) (L + ∆L (t)) t0 (L + ∆L (t)) t0 (l + ∆l (t)) t0 v t¯ v t¯ x˙ 2 (t) = a x1 (t) + (1 − a) x1 (t − τ (t)) (L +∆L (t)) t0 (L + ∆L (t)) t0 v t¯ v t¯ v t¯ x˙ 3 (t) = sin x2 (t) + a x1 (t) + (1 − a) x1 (t − τ (t)) t0 2 (L + ∆L (t)) t0 2 (L + ∆L (t)) t0 where x1 (t) is the angle difference between truck and trailer, x2 (t) is the angle of trailer, x3 (t) is the vertical position of rear end of trailer. The model parameters are given as l = 2.8, L = 5.5, v = −1.0, t¯ = 2.0, t0 = 15
0.12, d = 0.1/π, and a = 0.75. S = {1, 2, 3} is used to describe the actuator failure. As in [32], the transition probability rate matrix that relates the three failure modes is given as follows −0.35 0.25 0.1 0.29 −0.52 0.23 0.1 0.3 −0.4 Then the model is expressed by the following T-S fuzzy system with two fuzzy rules: x˙ (t) =
2 X
¯pi sat (u (t)) + B ¯wpi w (t) , λi (ε (t)) A¯pi x (t) + A¯τ pi x (t − τ (t)) + B
i=1
z (t) =
2 X
¯ pi sat (u (t)) + D ¯ wpi w (t) , λi (ε (t)) C¯pi x (t) + C¯τ pi x (t − τ (t)) + D
i=1
x (t) = φ (t) , t ∈ [−τ2 , 0] , where 1 Ap1 = av t¯/(Lt0 ) 0 0 , Bwp1 = 0 , 0 0 , Aτ p1 = (1 − a) v t¯/(Lt0 ) 0 (1 − a) v 2 t¯2 (2Lt0 ) v t¯/t0 0 av 2 t¯2 (2Lt0 ) v t¯/t0 0 −av t¯/(Lt0 ) 0 0 − (1 − a) v t¯/(Lt0 ) 0 0 1 Ap2 = av t¯/(Lt0 ) 0 0 , Aτ p2 = (1 − a) v t¯/(Lt0 ) 0 0 , Bwp2 = 0 , adv 2 t¯2 (2Lt0 ) dv t¯/t0 0 (1 − a) dv 2 t¯2 (2Lt0 ) v t¯/t0 0 0 0.5 ∗ v t¯/(lt0 ) v t¯/(lt0 ) 0.75 ∗ v t¯/(lt0 ) B11 = B12 = , , B21 = B22 = , B31 = B32 = 0 0 0 0 0 0 h i h i Cp1 = Cp2 = 0.5 0 0 , Cτ p1 = Cτ p2 = 0 0 0 , Dp1 = Dp2 = 0, Dwp1 = Dwp2 = 0, h iT h i H1p1 = H1p2 = 0.3 0 0 , H2p1 = H2p2 = 0, E1p1 = E1p2 = m 0 0 , h i E2p1 = E2p2 = (1 − m) 0 0 , E3p1 = E3p2 = 0, E4p1 = E4p2 = 0, p ∈ S.
−av t¯/(Lt0 )
0
0
− (1 − a) v t¯/(Lt0 )
0
0
Assume the controller gains have the perturbations with h i H3pi = 0.01, E5pi = 0 0.1 0 , p ∈ S, i = 1, 2. For µ = 0, τ1 = 0, τ2 = 0.2, γ = 1, M = −diag {1, 1, 1} , N = 0, and L = α + γ 2 toolbox, we can obtain h i h K11 = 21.5470 −124.0125 128.6721 , K12 = 42.5982 −100.2258 h i h K21 = 32.4458 −248.3112 246.8135 , K22 = 38.1963 −248.3112 h i h K31 = 59.6950 −388.7384 329.7658 , K32 = 55.6480 −342.8135
I, using the matlab LMI
i 218.1857 , i 246.8135 , i 312.3145 .
For simulation, we choose the fuzzy weighting function be h1 (t) = 1/1 + exp(0.5x1 (t + 1)), h2 (t) = 1−h1 (t), and the disturbance input as w (t) = sin (2t) ∗ exp (−0.05t) ∈ L2 [0, ∞) and the initial condition φT (t) = 16
Fig. 1. The operation modes (Example 2)
Fig. 2. State response of the closed-loop system (Example 2)
iT 0.5π 0.75π 2 , t ∈ [−0.2, 0]. Fig. 1 plots the operation modes. Fig. 2 shows the response of states of the closed-loop systems. From Fig1-2, we can see that, the method proposed in this paper is feasible. Example 3 Consider the following uncertain stochastic fuzzy system with two fuzzy plant rules: IF ai (t) is Λi1 (i = 1, 2) , THEN
h
¯pi u (t) + B ¯wpi w (t) , E x˙ (t) = A¯pi x (t) + A¯τ pi x (t − τ (t)) + B ¯ pi u (t) + D ¯ wpi w (t) , z (t) = C¯pi x (t) + C¯τ pi x (t − τ (t)) + D y (t) = Cypi x (t) , where
0
−0.6 0
0 0 0.01
−0.2
0
0 0
A11 = −0.3 0.8 0 , Aτ 11 = 0 0 0 , B11 = 0 , Bw11 = 0 0 0 , 0 0 0.01 0 0.01 0 0 0.1 0 0 0 0 0 0 0 0 0 0 0.01 0.1 A12 = 0 0 0 , Aτ 12 = 0 0.3 0 , B12 = 0.1 , Bw12 = 0 0 0 , −0.1 0 0.01 0 0 0.01 0 0 0 0 −0.2 −0.5 0 0 0 0 −0.2 0 0 0 A21 = −0.3 0.8 0 , Aτ 21 = 0.02 0.3 0 , B21 = 0 , Bw21 = 0 0.05 0 , 0.1 0.1 0 0 0 0 0 0 0 0 −0.9 0 0.1 −0.5 −0.2 0 −0.2 0 0 0 A22 = −0.3 0.8 0 , Aτ 22 = 0 , B = , B = 0.3 0 0 0 0.05 0 , 22 w22 0.1 0 0 0.1 0 0 0 0 0 0 17
0 0.01 0
0
0
0
−1 1 0
−0.1 1 0
C11 = 0 0 0.1 , C12 = 0 −0.3 0 , Cτ 21 = 1 2 0 Cτ 22 = 1 2 0 , 0.1 0 0 0 0 0.1 0 0 0.1 0 0 0.1 0 0 0 0 00 0 00 0 C22 = 0 0 0 , C21 = 0 0 0 , Cτ 11 = Cτ 12 = 0 −0.3 0 , D11 = 0.01 , 0 0 0.1 0 0 0 0.1 0 0 0.01 0 0 0 0 0 0 D21 = 0 , D12 = −0.01 , D22 = 0 , Dw11 = Dw21 = 0 0 0 , 0 0 0.01 0.01 0 0.05 1 0 0 1 0 1 00 0 Dw12 = Dw22 = 0 0 0 , Cy11 = Cy12 = Cy21 = Cy22 = 0 1 1 , E = 0 1 0 , 0 0 0 1 1 1 0 0 −0.01 −0.03 −0.03 −0.2 −0.2 H111 = 0.1 , H112 = 0 , H121 = 0.1 , H122 = 0 , −0.3 −0.3 0.2 −0.2 −0.3 −0.03 −0.2 −0.2 H211 = 0.1 , H212 = 0.1 , H221 = 0 H222 = 0 , −0.3 −0.3 0.2 −0.2 h i h i h i E111 = 0.5 0.2 0.1 , E112 = 0.15 0.2 0.1 , E121 = E122 = 0.15 0.2 0.2 , h i h i h i E211 = 0 0 0.01 , E212 = −0.35 0 0 , E221 = E222 = 0.05 0.45 0.01 , h i E311 = E312 = 0.1, E321 = E322 = 0.01, E411 = E412 = E421 = E422 = 0 0 0.01 . Assume the controller gains have the perturbations with h i h i h i H3pi = 0.01, p, i = 1, 2, E511 = −3 0 0 , E512 = E522 = −0.03 0 0 , E521 = 0 0.1 0 . For µ = 0.1, τ1 = 0, τ2 = 0.01, α = 1, π11 = −4, π22 = −5, M = −diag {1, 1, 1} , N = I3 , and L = diag {10, 10, 10}, using the matlab LMI toolbox, we can obtain 0.0540 0.0269 0.1308 1.2142 0.6089 4.3457 P1 = 0.0269 0.1075 0.0634 , P2 = 0.6089 0.4021 2.1306 , 0.1308 0.0634 8.1750 4.3457 2.1306 75.9800 the controller gains are given by h i h i K11 = −1.0631 −1.6324 43.8204 , K12 = 5.7075 1.0821 19.2970 , h i h i K21 = 84.0031 −6.2666 8.9983 , K22 = 42.9986 29.6029 17.4221 . h iT The initial state is assumed to be x0 = 1 −2 1.3 , solving the optimization problem in Remark 6, the value of δmax = 0.00474. 18
Fig. 3. The operation modes(Example 3)
Fig. 4. The estimation of attraction of the close-loop system (Example 3)
Fig. 5. The response of the closed-loop system(Example 3)
Fig. 6. The simulation of Γ (t) (Example 3)
Let the fuzzy weighting function be h1 (x1 ) = 1/[1 + exp(0.5(x1 + 1))], h2 (x1 ) = 1 − h1 (x1 ), disturbance input is ω (t) = exp(−t) sin(t). Fig.3 plots the operation modes. Fig.4 plots the estimation of attraction for the non-fragile control law. Then using the non-fragile controller, the simulation result for the state response of the closed-loop system is given in Fig. 5. In the end, we consider the dissipativity of the closed-loop system. Denote ∆
Γ (t) = ε {hz, M zit + 2hz, N ωit + hω, Lωit } − αhω, ωit , where (M, N, L) are given above and α = 1. Fig. 6 plots the response of Γ (t) for resulting closed-loop systems as t ∈ [0, 100]. In is seen from Fig. 6 that Γ (t) ≥ 0 is satisfied, which implies that the resulting closed-loop system is strictly (M, N, L) − α−dissipative. 19
5. Conclusion The problem of non-fragile fuzzy dissipative static output feedback control for singular Markovian jump time delay system subject to actuator saturation has been discussed. Sufficient conditions have been obtained, which guarantee that the closed-loop system is not only stochastic admissible but also dissipative. Then the non-fragile static output feedback controller can be obtained by solving the LMIs. And the largest estimation of the domain of attraction is obtained by solving the LMIs optimization problem. Examples are presented to demonstrate the feasibility and superiority. In this paper, the TRM is supposed all available, however, in some practical application, the TRM is partly unknown. In future work, we will focus on this issue. Acknowledgment This Letter was supported by the National Natural Science Foundation of China No. 61273004, and the Natural Science Foundation of Hebei province No. F2014203085.
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