- Email: [email protected]

R. Sakthivel, Peng Shi, A. Arunkumar, K. Mathiyalagan

PII: DOI: Reference:

S0165-0114(15)00514-X http://dx.doi.org/10.1016/j.fss.2015.10.007 FSS 6929

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Fuzzy Sets and Systems

Received date: Revised date: Accepted date:

19 January 2015 15 October 2015 19 October 2015

Please cite this article in press as: R. Sakthivel et al., Robust reliable H∞ control for fuzzy systems with random delays and linear fractional uncertainties, Fuzzy Sets and Systems (2015), http://dx.doi.org/10.1016/j.fss.2015.10.007

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1

Robust reliable H∞ control for fuzzy systems with random delays and linear fractional uncertainties R. Sakthivel1∗ , Peng Shi2 , A. Arunkumar3 and K. Mathiyalagan3 1 2

Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea

School of Electrical and Electronic Engineering, The University of Adelaide, SA 5005, Australia and College of Engineering and Science, Victoria University, Melbourne 8001, Australia 3

Department of Mathematics, Anna University, Regional Centre, Coimbatore- 641 047, India

Abstract This article studies the reliable robust stabilization problem for a class of uncertain Takagi-Sugeno (TS) fuzzy systems with time-varying delays. The delay factor is assumed to be random delay which belongs to a given interval and parameter uncertainties are considered with linear fractional transformation form. By implementing a proper novel Lyapunov functional together with linear matrix inequality (LMI) approach, a new set of delay-dependent sufﬁcient conditions is derived to guarantee the asymptotic stability of TS fuzzy system with a prescribed H∞ performance index. Further, a reliable robust H∞ control design with an appropriate gain matrix has been derived to achieve the robust asymptotic stability for uncertain TS fuzzy system. Further, Schur complement and Jensen’s integral inequality are used to simplify the derivation in the main results. The set of sufﬁcient conditions are established using the relationship among the random time-varying delay and its lower and upper bounds, which can be easily solved by MATLAB LMI toolbox. Finally, an illustrative example based on the truck-trailer model is provided to show the effectiveness of the proposed new design technique.

Keywords: Fuzzy system; Random time-varying delay; Actuators fault; Reliability; H∞ control. I. I NTRODUCTION In recent years, fuzzy systems in the form of well-known Takagi-Sugeno (TS) model [30] have become the most recognized modeling framework. In particular, Takagi-Sugeno fuzzy model has been recognized as a popular and powerful tool in approximating and describing complex nonlinear systems. Many issues related to the stability and stabilization of TS fuzzy systems via LMI approach have been reported in [6], [13], [23], [32], [16], [17]. Recently, problem of robust H∞ control for nonlinear uncertain stochastic Takagi-Sugeno fuzzy systems with time delays has been studied in [27]. In particular, time delay often appears in many practical systems such as chemical processes, *-Corresponding author, e-mail: [email protected]

2

communication networks, economics, biological and mechanical systems [33], [45], [47], [14]. Also, the existence of time delay is frequently a source of instability and poor performance of control systems. As a result, there are many issues for time delay fuzzy systems such as stability analysis, control synthesis, the ﬁltering problem and so on [28], [29], [35], [44]. More recently, Chen et al [7] investigated the problem of stability analysis and stabilization for two-dimensional discrete fuzzy systems via Lyapunov approach together with linear matrix inequality technique. On the other hand, failures of control components such as sensors and actuators are often occur in realistic dynamical systems. But when failure occurs, the conventional controller will become conservative and may not satisfy certain control performance indexes. Moreover, the reliable control maintains an acceptable stability performance for the closed-loop systems in the case of actuator or sensor failures [11], [19], [21], [26], [9], [24]. Feng and Lam [10] studied the problem of reliable dissipative control for a continuous-time singular Markovian system with actuator failure, where a set of sufﬁcient conditions is established in terms of LMIs, which guarantees a singular Markovian system to be stochastically admissible and dissipative. The issue of optimal robust H∞ reliable control with circular disk pole constraints for switched systems with actuator faults and arbitrary switching rules has been discussed [12]. Yang and Gao [41] obtained the reliable feedback controller design for spacecraft rendezvous with parameter uncertainties and limited-thrust by using Lyapunov approach and linear matrix inequality technique. In recent years, robust H∞ control problem for dynamical systems have received much attention because it is closely related to the capability of disturbance rejection in the dynamical systems [8], [22], [31], [40], [25], [15], [46]. Liu and Chiang [20] proposed an observer-based output tracking H∞ control via virtual desired reference model for a class of nonlinear systems with time-varying delay. The problem of robust fuzzy H∞ ﬁltering for a class of uncertain TS fuzzy nonlinear discrete-time Markov jump systems with nonhomogeneous jump transition probabilities has been studied in [42]. A delay-dependent robust stabilization and H∞ control for uncertain stochastic Takagi-Sugeno fuzzy systems with discrete interval and distributed time-varying delays is discussed in [2]. Zhu et al., [48] obtained a less conservative H∞ stabilization criterion in terms of linear matrix inequalities for nonuniform sampling fuzzy systems by using an appropriate Lyapunov-Krasovskii functional (LKF) combined with Jensen’s integral inequality. More recently, robust H∞ ﬁltering problem for Takagi-Sugeno sampled-data fuzzy systems with uncertain parameters is investigated in [43], where a set of sufﬁcient conditions with H∞ disturbance attenuation is given in terms of linear matrix inequalities to achieve the robust stability. Very recently, the reliable H∞ control problem has been discussed for discrete-time Takagi-Sugeno fuzzy systems with inﬁnite-distributed delay and actuator faults in [36], where a set of sufﬁcient conditions has been established to guarantee the desired stability and H∞ performance index and also the explicit expression of the desired controller has also been characterized in terms of the solution to a set of LMIs. Moreover, the time delay in a real system often exists in a stochastic way [37], and its probabilistic characteristic

3

can be calculated by statistical methods, so the random delay on TS fuzzy systems cannot be ignored. On the other hand, it is unavoidable to include parameter uncertainties in practical control systems due to modeling errors, measurement errors, approximations and so on. So it is important to study the problems of stability and stabilization for TS fuzzy systems with parameter uncertainties [1], [34]. Recently, a new class of uncertainty in the linear fractional form is considered in [3], [9], [38] which can include the norm bounded uncertainties as a special case. Balasubramaniam et al [3] discussed the problem robust stability analysis of Markovian jumping uncertain stochastic systems with interval time-varying delays with linear fractional transformation (LFT) uncertainties. To the best of author’s knowledge, up to now no work has been reported in the present literature regarding the reliable robust stabilization problem for a class of TS fuzzy systems with random delays and linear fractional transformation uncertainties. Motivated by this observation, in this paper we deal with the problem of reliable H∞ control for a class of TS fuzzy systems with random time-varying delays and linear fractional uncertainties. Based on a properly constructed LKF involving both lower and upper bounds of the delays with linear matrix inequality approach, a robust reliable H∞ controller is designed for obtaining the required result. Further, the result is extended to investigate the robust

stabilization of uncertain random delayed TS fuzzy systems with linear fractional uncertainties via a reliable robust H∞ control. The contribution of this paper lies in following aspects:

* Designing a reliable H∞ control such that the resulting closed loop system is robustly asymptotically stable with a disturbance attenuation level . * The effects of both variation range and probability distribution of time-delays are taken into account. * The results are expressed in the form of convex optimization problem to design the appropriate control. It should be pointed out that the obtained results are more general because it can guarantee the required result whether or not the actuator meets failure. Finally, a numerical example with simulation result based on truck-trailer model is provided to demonstrate the effectiveness and applicability of the proposed method. II. P ROBLEM FORMULATION AND PRELIMINARIES In this section, we start by introducing some notations and basic results which will be used in this paper. The superscripts T and (−1) denote matrix transposition and matrix inverse respectively; Rn×n represents the n × ndimensional Euclidean space; P > 0 means that P is real, symmetric and positive deﬁnite; Z≥0 denotes the set including zero and positive integers; I and 0 denote the identity and zero matrix with compatible dimensions respectively; diag{·} denotes the block-diagonal matrix; we use an asterisk (∗) to represent a term that is induced by symmetry.

4

Takagi-Sugeno fuzzy dynamical system can describe a wide class of nonlinear systems, and it is used in many ﬁelds of science and engineering. Consider the following uncertain TS fuzzy system with time-varying delays: Plant Rule : IF {1 (t) is M1η } , {2 (t) is M2η } ,. . . , {p (t) is Mpη } THEN x(t) _ = Aη (t)x(t) + Adη (t)x(t − (t)) + B η

(t)uf (t)

⎫ ⎪ + w(t) ⎪ ⎪ ⎪ ⎬

z(t) = Cη x(t)

⎪ ⎪ ⎪ ⎪ ⎭

x(t) = (t); for every t ∈ [−M ; 0];

(1)

where x(t) ∈ Rp is the state vector; uf (t) ∈ Rq is the control input of actuator fault; Mjη ; (j = 1; 2; : : : ; p; = 1; 2; : : : ; r) are fuzzy sets, {1 (t); 2 (t); : : : ; p (t)}T are the premise variable vectors; r is the number of IF-

THEN rules; w(t) ∈ Rs is the disturbance input vector; z(t) ∈ Rm is the output vector. Further, Aη (t) = η (t) = Bη + Bη (t), where Aη ; Adη ; Bη and Cη are known apAη + Aη (t); Adη (t) = Adη + Adη (t) and B

propriate dimension matrices and Aη (t); Adη (t) and Bη (t) are time-varying matrices representing parametric uncertainties as follows

Aη (t) Adη (t) Bη (t) = Mη η (t) N1η N2η N3η ;

(2)

where N1η ; N2η ; N3η and Mη are known constant matrices of appropriate dimensions. The linear fractional form was proposed in [18], which can include the norm bounded uncertainties as a special case. The class of parameter uncertainties η (t) that satisfy η (t) = [I − Fη (t)J]−1 Fη (t) is said to be admissible, where J is also a known matrix satisfying I − JJ T > 0 and F (t) is an unknown time-varying matrix with Lebesgue measurable elements bounded by FηT (t)Fη (t) ≤ I ([9], [38]). Let η ((t)) be the normalized membership function of the inferred fuzzy set η ((t)). The defuzziﬁed system can be written as x(t) _ = z(t) =

r η=1 r

f η ((t)) Aη (t)x(t) + Adη (t)x(t − (t)) + B η (t)u (t) + w(t) ; (3) η ((t))Cη x(t);

η=1

where η ((t)) =

βη (ξ(t)) r βη (ξ(t))

with η ((t)) =

p i=1

η=1

Miη (i (t)) in which Miη (i (t)) is the grade of the membership

function of i (t) in Miη . Also, we assume η ((t)) ≥ 0, = 1; 2; : : : ; r, η ((t)) ≥ 0, = 1; 2; : : : ; r,

r

r

η ((t)) > 0, and η ((t)) satisfy

η=1

η ((t)) = 1 for any (t).

η=1

Our main aim here is to design a control law with minimum H∞ performance index such that the closed loop

5

TS fuzzy system is robustly stable. For this, we deﬁne the control law with actuator failures in the following form uf (t) = Gη u(t);

(4)

where, Gη is the actuator fault matrix deﬁned as follows [12]; Gη = diag {g1η ; g2η ; : : : ; gmη } ; 0 ≤ g kη ≤ gkη ≤ g kη ≤ 1;

(5)

where g kη and g kη ; k = 1; 2; : : : ; m; = 1; 2; · · · ; r are given constants. Also, we deﬁne G0η = diag {g10η ; g20η ; : : : gm0η }, gk0η =

g kη +g kη , 2

G1η = diag {g11η ; g21η ; : : : gm1η }, gk1η =

g kη −g kη . 2

Then the matrix Gη can be written as

Gη = G0η + η = G0η + diag {1η ; : : : ; mη } ; |kη | ≤ gk1η ; (k = 1; · · · ; m; = 1; · · · ; r):

(6)

Remark 2.1: In (5), parameters g kη and g kη characterize the admissible failures of the signal from the controller. Obviously, when g kη = g kη = 0, denotes that the actuator completely fails (case of the outage). When 0 ≤ g kη < g kη ≤ 1, it corresponds to the case of partial failure. If g kη = g kη = 1, then actuator is normal.

The following fuzzy control rule is deﬁned as: Control Rule : IF {1 (t) is M1η } , {2 (t) is M2η } ,. . . , {p (t) is Mpη } THEN u(t) = Kη x(t);

(7)

Then, the overall fuzzy controller is described as: u(t) =

r

η ((t))Kη x(t);

(8)

η=1

where Kη ( = 1; 2; · · · ; r) are the control gains. Substituting uf (t) for u(t) in (3), and considering (8) and (4), the resulting closed-loop fuzzy systems can be described as: x(t) _ =

r r

η ((t))j ((t)) (Aη (t) + B η (t)Gη Kj )x(t) + Adη (t)x(t − (t)) + w(t) :

(9)

η=1 j=1

The time delay in above systems is assumed to be a random one and satisfy the following assumptions: Assumption (H1) The time delay (t) is bounded such that m ≤ (t) ≤ M and its probability distribution can be observed, i.e., suppose (t) takes values in [m : 0 ] or (0 : M ] and Prob{ (t) ∈ [m : 0 ]} = 0 or Prob{ (t) ∈ (0 : M ]} = 1 − 0 ; where m ; 0 and M are integers satisfying m ≤ 0 < M , and 0 ≤ 0 ≤ 1. It is noted that, in practice, some values of the delay are very large but the probabilities of the delays taking such large

6

values are very small. Taking this point into consideration, a scalar 0 satisfying m ≤ 0 < M is introduced here. In order to describe the probability distribution of the time delays, deﬁne the following two sets D1 = {t| (t) ∈ [m : 0 ]} and D2 = {t| (t) ∈ (0 : M ]} :

Moreover, we deﬁne two mapping functions as follows ⎧ ⎧ ⎨ (t); t ∈ D1 ⎨ (t); t ∈ D2 1 (t) = 2 (t) = ⎩ ; ⎩ ; t ∈ D2 ; t ∈ D1 : m 0

(10)

(11)

where m = [m : 0 ] and 0 = (0 : M ]. Assumption (H2) Further, the time-varying delays 1 (t) and 2 (t) satisfying the condition m ≤ 1 (t) ≤ 0 ;

_1 (t) ≤ 1 ;

0 < 2 (t) ≤ M ;

_2 (t) ≤ 2 ;

(12)

where m ; 0 ; M ; 1 and 2 are positive constants. For more details about probability distribution delay related issue, one can refer to the paper [4]. Further, it follows from (10) that D1 ∪ D2 = Z≥0 ; D1 ∩ D2 = , where is the empty set. It is easy to check that t ∈ D1 implies the event (t) ∈ [m ; 0 ] occurs and t ∈ D2 implies the event (t) ∈ (0 ; M ] occurs. Deﬁne a Bernoulli distributed stochastic variable ⎧ ⎨ 1; (t) = ⎩ 0;

t ∈ D1

(13)

t ∈ D2

with Prob{(t) = 1} = Prob{ (t) ∈ [m , 0 ]} = E[(t)] = 0 and Prob{(t) = 0} = Prob{ (t) ∈ (0 , M ]} = 1 − E[(t)] = 1 − 0 . Further, it is easy to see that E[(t) − 0 ] = 0 and E[((t) − 0 )2 ] = 0 (1 − 0 ).

By introducing the random delay, the uncertain TS fuzzy system (9) can be written in the following form r r x(t) _ = η ((t))j ((t)) (Aη (t) + B η (t)Gη Kj )x(t) + (t)Adη (t)x(t − 1 (t)) η=1 j=1

(14) +(1 − (t))Adη (t)x(t − 2 (t)) + w(t) : The following deﬁnition and lemmas are needed to prove the main results. Deﬁnition.1: Fuzzy system (14) is said to be robustly stable with disturbance attenuation if for all w(t) ∈ L2 [0; ∞], the response z(t) under the zero initial condition, i.e., (t) = 0 satisﬁes

E

∞ 0

T

z (t)z(t)dt

≤

2

∞ 0

wT (t)w(t)dt:

Lemma.1:[4] Given constant matrices 1 ; 2 and 3 with appropriate dimensions, where 1 = T1 > 0 and

7

⎡ ⎣ 2 = T2 > 0 then 1 + T3 −1 2 3 < 0 if and only if

1

T3

⎤ ⎦ < 0:

−2 S = S T > 0 and scalars m < (t) < M , for vector T t−τm t−τm x(s)ds _ S x(s)ds _ . t−τM t−τM

3 n×n R ,

Lemma.2:[9] Given a positive deﬁnite matrix S ∈ t−τ 1 function x(t), we have − t−τMm x_ T (s)S x(s)ds _ ≤ − τM −τ m

Lemma.3: [9] Given matrices = T , S and N of appropriate dimensions, the inequality + S(t)N + N T T (t)S T < 0

⎤

⎡ ⎢ ⎢ T holds for F (t) such that F T (t)F (t) ≤ I if and only if for some > 0, ⎢ ⎢ S ⎣ N III. R ELIABLE H∞

S

N T

−I

J T

J

−I

⎥ ⎥ ⎥ < 0. ⎥ ⎦

CONTROL DESIGN

In this section, we focus on the problem of robust reliable stabilization of TS fuzzy system with known as well as unknown actuator failure. The main aim of this section is to obtain conditions for the existence of a stabilizing state feedback reliable H∞ control such that the resulting closed-loop system is robustly asymptotically stable with given disturbance attenuation level > 0: First we consider the nominal form of the closed-loop fuzzy system (14) as follows

x(t) _ = η ((t))j ((t)) (Aη + Bη Gη Kj )x(t) + (t)Adη x(t − 1 (t)) η=1 j=1

+(1 − (t))Adη x(t − 2 (t)) + w(t) : r r

(15)

Theorem 1: Assume that the conditions (H1) and (H2) hold. For the given positive scalars 1 and 2 with known actuator failure parameter matrix Gη , the closed-loop fuzzy system with random delay (15) is asymptotically stable with a given disturbance attenuation level of > 0 if there exist positive deﬁnite symmetric matrices ~ i ; i = 1; · · · ; 6; R ~ i ; S~i ; i = 1; 2; 3; 4 and any matrix Yj with appropriate dimension such that the following X; Q

LMIs hold for ; j = 1; 2; : : : ; r ; ⎡ ~ ηj = ⎣

~ (m×n)ηj

~T η1

∗

−I

⎤ ⎦ < 0;

m; n = 1; 2; · · · ; 14;

(16)

8

where ~1 + Q ~2 + Q ~3 + Q ~4 + Q ~5 + Q ~ 6 + (0 − m )R ~ 1 + 0 R ~ 3 + M R ~ (1,1)ηj = Q ~ 2 + (M − 0 )R ~4 1 ~ 1 ~ ~ 1 ~ (1,4)ηj = 1 S~4 ; ~ (1,7)ηj = X + XATη T + YjT GTη B T ; S2 − S4 ; (1,3)ηj = S~2 ; 0 M 0 M 1 1 1 ~3 − ~ (2,3)ηj = ~ (3,3)ηj = −(1 − 1 )Q ~1 − = −Q (2S~1 + S~2 ) S~1 ; S~1 ; 0 − m 0 − m 0 − m 1 1 ~ (3,5)ηj = ~ (3,7)ηj = 0 XAT ; ~ (4,4)ηj = −(1 − 2 )Q ~4 (S~1 + S~2 ); − S~2 ; dη 0 0 − m 1 1 ~ ~ 1 1 ~ (4,6)ηj = − (2S~3 + S~4 ) − (S~3 + S~4 ); S4 ; (4,5)ηj = S~T ; 0 − m M M − 0 3 0 − m 1 1 ~ (5,5)ηj = −Q ~2 − Q ~6 − S~3 ; = XATdη (1 − 0 ); (S~1 + S~2 ) − 0 − m M − 0 −

~ (2,2)ηj

~ (4,7)ηj

1 ~ (7,7)ηj = (0 − m )S~1 + 0 S~2 + (M − 0 )S~3 + M S~4 (S~3 + S~4 ); M − 0 ~1; ~ (7,14)ηj = X; ~ (8,8)ηj = − 1 (R ~1 + R ~ 2 ); ~ (9,9)ηj = − 1 R − 2X; 0 − m 0 − m 1 ~ ~ 1 1 ~3 + R ~ 4 ); ~ (12,12)ηj = − ~3; =− R (R R 2 ; (11,11)ηj = − 0 M − 0 M − 0 1 ~ ~ ~ η1 = C X 0 =− R4 ; (14,14)ηj = − 2 ; η 13n M

~ (6,6)ηj = −Q ~5 −

~ (10,10)ηj ~ (13,13)ηj

and the other parameters are zero. In this case, the controller gain matrix in (4) is given by Kj = Yj X −1 . Proof. In order to obtain the required result, we construct the following LKF for nominal system (15) which involves the lower and upper bounds of the delay: 7

V (t; x(t)) =

Vi (t; x(t));

(17)

i=1

where V1 (t; x(t)) = xT (t)P x(t); t T x (s)Q1 x(s)ds + V2 (t; x(t)) =

t−τ1 (t) t

xT (s)Q4 x(s)ds +

V3 (t; x(t)) =

t−τ2 (t) −τm t

t

t

T

t−τ0 t

x (s)Q2 x(s)ds +

t−τM

t−τm t

xT (s)Q3 x(s)ds;

xT (s)Q5 x(s)ds +

0

t−τ0

xT (s)Q6 x(s)ds;

t

xT (s)R1 x(s)dsd + xT (s)R2 x(s)dsd; t+θ −τ t+θ −τ −τ0 0 t 00 t xT (s)R3 x(s)dsd + xT (s)R4 x(s)dsd; V5 (t; x(t)) = t+θ −τ t+θ −τ −τMm t 0M t x_ T (s)S1 x(s)dsd _ + x_ T (s)S2 x(s)dsd; _ V6 (t; x(t)) = t+θ −τ0 t+θ −τ0 −τ0 t 0 t T x_ (s)S3 x(s)dsd _ + x_ T (s)S4 x(s)dsd: _ V7 (t; x(t)) = V4 (t; x(t)) =

−τM

t+θ

−τM

t+θ

9

By calculating the derivatives V_ (t; x(t)) along the trajectories of system (15) and taking the mathematical expectation, we can obtain _ ; E V_ 1 (t; x(t)) = E 2xT (t)P x(t) E V_ 2 (t; x(t)) ≤E xT (t) (Q1 + Q2 + Q3 ) x(t) − (1 − 1 )xT (t − 1 (t))Q1 x(t − 1 (t))

T T − x (t − 0 )Q2 x(t − 0 ) − x (t − m )Q3 x(t − m ) ; E V_ 3 (t; x(t)) ≤E xT (t) (Q4 + Q5 + Q6 ) x(t) − (1 − 2 )xT (t − 2 (t))Q4 x(t − 2 (t))

T T − x (t − M )Q5 x(t − M ) − x (t − 0 )Q6 x(t − 0 ) ; t−τm E V_ 4 (t; x(t)) = E (0 − m )xT (t)R1 x(t) − xT (s)R1 x(s)ds + 0 xT (t)R2 x(t) t−τ0

t − xT (s)R2 x(s)ds ; t−τ0 t−τ0 T _ E V5 (t; x(t)) = E (M − 0 )x (t)R3 x(t) − xT (s)R3 x(s)ds + M xT (t)R4 x(t) t−τM

t T − x (s)R4 x(s)ds ; t−τM t−τm _ − x_ T (s)S1 x(s)ds _ + 0 x_ T (t)S2 x(t) _ E V_ 6 (t; x(t)) = E (0 − m )x_ T (t)S1 x(t) t−τ0

t T − x_ (s)S2 x(s)ds _ ; t−τ0 t−τ0 T _ E V7 (t; x(t)) = E (M − 0 )x_ (t)S3 x(t) _ − x_ T (s)S3 x(s)ds _ + M x_ T (t)S4 x(t) _ t−τM

t T − x_ (s)S4 x(s)ds _ : t−τM

(18)

(19)

(20)

(21)

(22)

(23)

(24)

10

Also, it follows from (17)-(24) that

_ E V (t; x(t)) ≤ E 2xT (t)P x(t) _ + xT (t) Q1 + Q2 + Q3 + Q4 + Q5 + Q6 + (0 − m )R1 ! + 0 R2 + (M − 0 )R3 + M R4 x(t) − (1 − 1 )xT (t − 1 (t))Q1 x(t − 1 (t)) ! T − x (t − 0 ) Q2 + Q6 x(t − 0 ) − xT (t − m )Q3 x(t − m ) − (1 − 2 )xT (t − 2 (t))Q4 x(t − 2 (t)) − xT (t − M )Q5 x(t − M ) ! T _ + x_ (t) (0 − m )S1 + 0 S2 + (M − 0 )S3 + M S4 x(t) t−τm t t−τ0 T T − x (s)R1 x(s)ds − x (s)R2 x(s)ds − xT (s)R3 x(s)ds t−τ t−τ t−τ t 0 t−τ0m tM − xT (s)R4 x(s)ds − x_ T (s)S1 x(s)ds _ − x_ T (s)S2 x(s)ds _ t−τM t−τ0 t−τ0

t−τ0 t T T − x_ (s)S3 x(s)ds _ − x_ (s)S4 x(s)ds _ :

(25)

t−τM

t−τM

By using the time-varying delay deﬁned in (12), the integrations in (25) can be written as − −

t−τm

t−τ0 t−τm t−τ0

x (s)R1 x(s)ds = −

t−τ1 (t)

T

x_ T (s)S1 x(s)ds _ =−

t−τ0 t−τ1 (t) t−τ0

x (s)R1 x(s)ds − T

x_ T (s)S1 x(s)ds _ −

t−τm

t−τ1 (t) t−τm t−τ1 (t)

xT (s)R1 x(s)ds;

(26)

x_ T (s)S1 x(s)ds; _

(27)

By applying Lemma 2 to each integrals in above equations, we can obtain the following inequalities; −

t−τ1 (t)

t−τ0 t−τm

1 x (s)R1 x(s)ds ≤ − 0 − m T

T

t−τ1 (t)

x(s)ds t−τ0 t−τm

R1

t−τ1 (t)

x(s)ds

T 1 − x (s)R1 x(s)ds ≤ − x(s)ds R1 x(s)ds ; 0 − m t−τ1 (t) t−τ1 (t) t−τ1 (t) t−τ1 (t) T t−τ1 (t) t−τ1 (t) 1 T − x_ (s)S1 x(s)ds _ ≤− x(s)ds _ S1 x(s)ds _ 0 − m t−τ0 t−τ0 t−τ0 T 1 ≤− x(t − 1 (t)) − x(t − 0 ) S1 x(t − 1 (t)) − x(t − 0 ) ; 0 − m t−τm T t−τm t−τm 1 − x_ T (s)S1 x(s)ds _ ≤− x(s)ds _ S1 x(s)ds _ 0 − m t−τ1 (t) t−τ1 (t) t−τ1 (t) T 1 ≤− x(t − m ) − x(t − 1 (t)) S1 x(t − m ) − x(t − 1 (t)) : 0 − m T

(28)

t−τ0 t−τm

(29)

(30)

(31)

11

On the other hand, for any matrices P1 of appropriate dimensions the following equalities hold, E

r r

" η ((t))j ((t)) 2x_ T (t)P1 (Aη + Bη Gη Kj )x(t) + (t)Adη x(t − 1 (t))

η=1 j=1

+(1 − (t)Adη x(t − 2 (t)) + w(t) − x(t) _

#

= 0:

(32)

By applying similar arguments of (26)-(31) to all the integral terms in (25) and adding (32), we obtain

r r _ η ((t))j ((t)) 2xT (t)(P + (Aη + Bη Gη Kj )T P1T )x(t) _ + E V (t; x(t)) ≤ E η=1 j=1

$ % x (t) Q1 + Q2 + Q3 + Q4 + Q5 + Q6 + (0 − m )R1 + 0 R2 + (M − 0 )R3 + M R4 x(t) $ % −(1 − 1 )xT (t − 1 (t))Q1 x(t − 1 (t)) − xT (t − 0 ) Q2 + Q6 x(t − 0 ) T

−xT (t − m )Q3 x(t − m ) − xT (t − M )Q5 x(t − M ) − (1 − 2 )xT (t − 2 (t))Q4 x(t − 2 (t)) $ % +2x_ T (t)P1 0 Adη x(t − 1 (t)) + x_ T (t) (0 − m )S1 + 0 S2 + (M − 0 )S3 + M S4 − 2P1 1 T T ×x(t) _ + 2x_ (t)P1 (1 − 0 )Adη x(t − 2 (t)) + 2x_ (t)P1 w(t) − x(t − 1 (t)) 0 − m T T 1 x(t − m ) − x(t − 1 (t)) −x(t − 0 ) (S1 + S2 ) x(t − 1 (t)) − x(t − 0 ) − 0 − m T 1 S1 x(t − m ) − x(t − 1 (t)) − x(t) − x(t − 1 (t)) S2 x(t) − x(t − 1 (t)) 0 T 1 − x(t − 2 (t)) − x(t − M ) (S3 + S4 ) x(t − 2 (t)) − x(t − M ) M − 0 T 1 1 − x(t − 0 ) − x(t − 2 (t)) S3 x(t − 0 ) − x(t − 2 (t)) − x(t) M − 0 M T t−τ1 (t) T $ % 1 x(s)ds R1 + R2 −x(t − 2 (t)) S4 x(t) − x(t − 2 (t)) − 0 − m t−τ0

t−τ1 (t)

x(s)ds −

t−τ0

t

1 0 − m T x(s)ds R2

1 − 0 t−τ1 (t) t−τ2 (t) x(s)ds − t−τM

−

1 M

t t−τ2 (t)

T

t−τm t−τ1 (t)

x(s)ds

t

R1

x(s)ds

t−τ1 (t) t−τ2 (t)

1 x(s)ds − M − 0 t−τ1 (t) t−τ0 T x(s)ds R3

1 M − 0 T x(s)ds R4

t−τ2 (t)

t−τ2 (t)

x(s)ds

T x(s)ds

t−τM t−τ0

t−τ2 (t)

t

t−τm

$

R3 + R4

%

x(s)ds

:

(33)

In order to study the H∞ performance of the system, we introduce the following relation J(t) = E = E

0

0

∞"

2

#

z (t)z(t) − w (t)w(t) dt ;

∞

T

T

∀t > 0;

_ z (t)z(t) − w (t)w(t) + V (t; x(t)) + V (0; x(0)) − V (∞; x(∞)) dt : T

2

T

(34)

12

Under the zero initial condition, using (33) in (34) gives J(t) ≤ E

r r

η ((t))j ((t))

η=1 j=1

∞ 0

T

(t)ηj (t)dt

;

(35)

t−τ (t) where T (t) = xT (t) xT (t − m ) xT (t − 1 (t)) xT (t − 2 (t))xT (t − 0 ) xT (t − M ) x_ T (t) t−τ01 xT (s)ds t−τm T t t−τ2 (t) T t−τ0 t T T T T t−τ1 (t) x (s)ds t−τ1 (t) x (s)ds t−τM x (s)ds t−τ2 (t) x (s)ds t−τ2 (t) x (s)ds w (t) and ⎡ ηj = ⎣

(m×n)ηj

Tη1

∗

−I

⎤ ⎦;

m; n = 1; 2; · · · ; 14

(36)

with (1,1)ηj = Q1 + Q2 + Q3 + Q4 + Q5 + Q6 + (0 − m )R1 + 0 R2 + (M − 0 )R3 1 1 1 1 S2 − S4 ; (1,3)ηj = S2 ; (1,4)ηj = S4 ; 0 M 0 M 1 1 = P + ATη P1T + KjT GTη BηT P1T ; (2,2)ηj = −Q3 − S1 ; (2,3)ηj = S1 ; 0 − m 0 − m 1 1 1 = −(1 − 1 )Q1 − (2S1 + S2 ) − S2 ; (3,5)ηj = (S1 + S2 ); 0 − m 0 0 − m 1 1 = 0 ATdη P1T ; (4,4)ηj = −(1 − 2 )Q4 − S4 − (2S3 + S4 ); M M − 0 1 1 = S T ; (4,6)ηj = (S3 + S4 ); (4,7)ηj = ATdη (1 − 0 )P1T ; M − 0 3 M − 0 1 1 1 = −Q2 − Q6 − (S1 + S2 ) − S3 ; (6,6)ηj = −Q5 − (S3 + S4 ); 0 − m M − 0 M − 0

+ M R4 − (1,7)ηj (3,3)ηj (3,7)ηj (4,5)ηj (5,5)ηj

(7,7)ηj = (0 − m )S1 + 0 S2 + (M − 0 )S3 + M S4 − 2P1 ; (7,14)ηj = P1 ; 1 1 1 (R1 + R2 ); (9,9)ηj = − R1 ; (10,10)ηj = − R2 ; 0 − m 0 − m 0 1 1 1 =− (R3 + R4 ); (12,12)ηj = − R3 ; (13,13)ηj = − R4 ; M − 0 M − 0 M = − 2 ; η1 = Cη 013n :

(8,8)ηj = − (11,11)ηj (14,14)ηj

In order to obtain the reliable H∞ feedback control gain matrix, take P1 = P , where is the designing parameter and let T = diag {X; : : : ; X} ∈ R13×13 : Pre- and post- multiplying (36) by diag {T; I},where X = P −1 and letting & i = XQi X; i = 1; · · · ; 6; R &i = XRi X; S&i = XSi X; i = 1; · · · ; 4 and Yj = Kj X , we can obtain LMI (16). Q & ηj < 0 then we can get J(t) ≤ 0, that is In the view of LMI (16), if

# E z (t)z(t) − w (t)w(t) dt ≤ 0; 0

∞ ∞ T 2 E z (t)z(t)dt ≤ wT (t)w(t)dt: 0

∞"

T

2

T

0

(37)

13

Hence, based on the Deﬁnition 1, we can conclude that the closed-loop fuzzy system (15) with known actuator failure matrix Gη is asymptotically stable with a given performance attenuation level > 0. The proof is completed.

Now, we design the robust reliable H∞ controller for the TS fuzzy system (14) with LFT uncertainty based on the results in Theorem 1. Theorem 2: Under the assumption (H1), for the given positive scalars 1 and 2 with known actuator failure parameter matrix Gη , the uncertain closed-loop fuzzy random delay system (14) is robustly asymptotically stable with a given disturbance attenuation level of > 0 and for any time-varying delay 1 (t) and 2 (t) satisfying (12) ~ i ; i = 1; · · · ; 6; R ~ i ; S~i ; i = 1; 2; 3; 4, any matrix Yj with if there exist positive deﬁnite symmetric matrices X; Q

appropriate dimension and the scaler 1 such that the following LMIs hold for ; j = 1; 2; : : : ; r; ⎡ ⎤ T T ~ ~ ~ ⎢ ηj η1 η2 ⎥ ⎢ ⎥ ⎥ ~ηj = ⎢ ⎢ ∗ −1 1 J ⎥ < 0; ⎣ ⎦ ∗ ∗ −1

(38)

T T T T ~ ~ where η1 = 1 Mη 0 1 Mη 1 Mη 010n , η2 = N1η X + N3η Yj 0 0 N2η X (1 − 0 )N2η X 010n and other parameters are deﬁned as in Theorem 1. Moreover, the controller gain matrix in (4) can be obtained by Kj = Yj X −1 .

Proof: The proof of this Theorem 2 follows immediately from Theorem 1 by replacing Aη ; Adη ; Bη ; with (Aη + Mη η (t)N1η ); (Adη + Mη η (t)N2η ) and (Bη + Mη η (t)N3η ) and applying Lemma 3 in the resulting

inequality, we can obtain the LMI (38). The proof is completed.

Next, we design the robust reliable H∞ controller when the actuator failure matrix Gη is unknown but satisfy the constraints (5)-(6). Theorem 3: Let the conditions (H1) and (H2) hold. For the given positive scalars 1 and 2 with unknown actuator fault matrix Gη , the uncertain closed-loop fuzzy random delay system (14) is robustly asymptotically stable with a given disturbance attenuation level of > 0 if there exist positive deﬁnite symmetric matrices ~ i ; i = 1; · · · ; 6; R ~ i ; S~i ; i = 1; 2; 3; 4, any matrix Yj with appropriate dimension and the scaler i ; i = 1; 2 X; Q

such that the following LMIs hold for ; j = 1; 2; : : : ; r; ⎡ ' ~ T ⎢ ηj Bηj 1 Mη ⎢ ⎢ ∗ −~ I 0 ⎢ ⎢ ⎢ ∗ ∗ −1 ⎢ ⎣ ∗ ∗ ∗

⎤ T YjT GT1η N 3η ⎥ ⎥ ⎥ 0 ⎥ ⎥ < 0; ⎥ 0 ⎥ ⎦ −1

(39)

14

~ηj = where B

η Yj B

, ~ = diag{2 , 2 } and the other parameters are deﬁned as in Theorem 2.

Proof: If the actuator failure matrix Gη is unknown, with the use of (6), the LMI conditions in (38) for the design of reliable state feedback controller can be obtained as

ηj η = where B

T η Yj + M η Fη (t)N 3η η Yj + Y T η N T; η + B T F T (t)M = 'ηj + YjT η B η j 3η η η

GT1η BηT

013n

; Yj =

06n Yj

07n

η = ;M

06n Mη 07n

3η = ; N

(40)

N3η 013n

and

'ηj is obtained by replacing Gη by G0η in ~ηj . Further, it follows from Lemma 3 and (40) that

ηj

−1 T T T T T 2 T = 'ηj + −1 2 Yj Yj + 2 Bη G1η Bη + 1 Mη Mη + 1 N3η η Yj Yj η N3η :

(41)

Then by using Lemma 1, we obtain that (41) is equivalent to LMI (39). Hence uncertain closed-loop fuzzy system with random delay (14) is robustly asymptotically stable. This completes the proof. Remark 3.1: The scalar > 0 can be used as an optimization variable to obtain a reduction in the H∞ disturbance attention level bound. Thus, the minimum H∞ disturbance attention level bound with admissible controllers can be readily found by solving the given LMIs. Remark 3.2: In general, the outputs from the failed actuators may have arbitrary signals different from normal controller outputs, and these signals will act on the plant as an unexpected control inputs. In order to maintain the stability of closed-loop system by reducing the effects of actuator failures, it is desirable to use feedbacks. There are many attempts made to overturn the signals on the system outputs caused by faulty actuators as well as disturbance inputs below a given level (see [10], [11], [12], [17] and references therein.) IV. N UMERICAL E XAMPLE In this section, we provide an illustrative example with simulation result which is based on the truck-trailer model to demonstrate the applicability of the proposed reliable controller. First, we consider the result for nominal case and further it is extended to the uncertain case. To provide a realistic framework for the simulation result, we borrow the truck-trailer model parameters provided in [5], [39]: v t v t x1 (t) + u(t); Lt0 lt0 v t x1 (t); x_ 2 (t) = Lt0 $ % v t v t sin x2 (t) + x1 (t) : x_ 3 (t) = t0 2L x_ 1 (t) = −

15

Fig. 1: Truck trailer model.

We assume that the system x1 (t) is perturbed by time-varying delay and the delayed model can be represented in dimensionless variables in the following form v t v t v t x1 (t) − (1 − c) x1 (t − (t)) + u(t); Lt0 Lt0 lt0 v t v t x1 (t) − (1 − c) x1 (t − (t)); x_ 2 (t) = −c Lt0 Lt0 $ % v t v t v t x_ 3 (t) = sin x2 (t) + c x1 (t) + (1 − c) x1 (t − (t)) ; t0 2L 2L x_ 1 (t) = −c

(42)

where x1 (t) is the angle difference between truck and trailer (rad); x2 (t) the angle of trailer (rad); x3 (t) the vertical position of rear of trailer (m); The model parameters are given as t = 2s, l = 2:8m, L = 5:5m, v = −1:0, c = 0:7m=s and t0 = 0:5. $ vt¯ % $ vt¯ % x1 (t) + (1 + c) 2L x1 (t − (t)). Under the condition −179:4270o < (t) < 179:4270o , Let (t) = x2 (t) + c 2L

the nonlinear term sin((t)) can be represented as follows; sin((t)) = h1 ((t)) · (t) + h2 ((t)) · g · (t);

where g =

10−2 π

with h1 ((t)), h2 ((t)) ∈ [0; 1] and h1 ((t)) + h2 ((t)) = 1. By solving the equations, the

membership functions h1 ((t)) and h2 ((t)) are obtained as follows: ⎧ ⎨ h1 ((t)) =

⎩

sin(θ(t))−¯ g ·θ(t) θ(t)·(1−¯ g) ;

1;

(t) = 0 (t) = 0

⎧ ⎨ h2 ((t)) =

⎩

θ(t)−sin(θ(t)) θ(t)·(1−¯ g) ;

0;

(t) = 0 (t) = 0:

It can be seen that h1 ((t)) = 1 and h2 ((t)) = 0 when (t) is about 0 rad, and h1 ((t)) = 0 and h2 ((t)) = 1 when (t) is about and − rad. To demonstrate the results in Theorems.1 and 2, we assume the delay in (42) as randomly occurring and satisfy Assumptions (HI) and (H2). Then, system (42) can be represented by the following TS fuzzy model:

16

Plant Rule 1:

IF

1 (t) is M11

THEN

x(t) _ =A1 (t)x(t) + (t)Ad1 (t)x(t − 1 (t)) + (1 − (t)Ad1 )(t)x(t − 2 (t)) + B1 (t)uf (t) + w(t)

Plant Rule 2:

IF

2 (t) is M22

THEN

x(t) _ =A2 (t)x(t) + (t)Ad2 (t)x(t − 1 (t)) + (1 − (t)Ad2 )(t)x(t − 2 (t)) + B2 (t)uf (t) + w(t);

where x(t) = [x1 (t) x2 (t) x3 (t)]T , ⎡ v t¯ 0 0 ⎢ −c Lt0 ⎢ v t¯ A1 = ⎢ 0 0 ⎢ c Lt0 ⎣ 2 ¯2 v t v t¯ −c 2Lt Lt0 0 0 ⎡ v t¯ ⎢ −(1 − c) Lt0 0 ⎢ v t¯ Ad2 = ⎢ 0 ⎢ (1 − c) Lt0 ⎣ 2 ¯2 v t 0 −(1 − c) g¯2Lt 0

⎤

⎡

⎥ ⎢ ⎥ ⎢ ⎥ ; A2 = ⎢ ⎥ ⎢ ⎦ ⎣ ⎤

v t¯ −c Lt 0 ¯

⎤ 0

vt c Lt 0

0

v t¯ −c g¯2Lt 0

g¯v t¯ t0

2 2

0 ⎥ ⎥ ⎥ 0 ⎥ ; B1 = B2 = ⎦ 0

v t¯ lt0

⎡

v t¯ c) Lt 0

0 ⎥ ⎢ −(1 − ⎥ ⎢ ⎢ v t¯ 0 ⎥ ⎥ ; Ad1 = ⎢ (1 − c) Lt0 ⎦ ⎣ v 2 t¯2 −(1 − c) 2Lt 0 0

⎤ 0

0 ⎥ ⎥ 0 0 ⎥ ⎥; ⎦ 0 0

0 0

; C1 = C2 = I:

Case 1: Nominal system- Consider the actuator failure matrix G1 = G2 = 0:7I . For the time-varying delays πt satisfying 1 (t) = 0:01 + 0:002 sin( πt 2 ), 2 (t) = 0:15 + 0:08 sin( 2 ), with given values of 1 = 2 = 0:7, = 0:5,

0 = m + 0:0001 and 0 = 0:5,

by solving the LMI in Theorem 1 using the Matlab LMI Tool box, we can obtain feasible solutions. The calculated upper bound of time delay M for different values of is given in Table I. It is observed from Table I that the upper bound M increases as increases. Secondly, the guaranteed H∞ performance index for various values of TABLE I: Delay upper bound M for various in Case 1

M

0.5 0.094

0.55 0.104

0.6 0.113

0.7 0.129

0.8 0.143

0.9 0.156

upper bound M is presented in Table II. For instance, if we take = 0:7, we have M = 0:129 (see Table I) for TABLE II: Minimum for different values of M in Case 1 M

0.05 0.3082

0.06 0.3550

0.07 0.4037

0.1 0.5740

0.13 0.8091

0.15 1.0799

the nominal system (15), the corresponding gain matrices of the state feedback reliable H∞ controller are given by K1 = K2 =

3:5667 −2:5876 0:1061 4:5750 −4:8833 1:7959

;

:

17

x1(t)

4 a) 2

Without Failure With Failure

0 −2 −4

x2(t)

b)

0

5

10

15

20

25

30

35

40

45

50

5 Without Failure With Failure 0

−5

0

5

10

15

20

25

30

35

40

45

50

c) 40 Without Failure With Failure

x3(t)

20 0 −20

d)

0 19 x 10

5

10

15

20

25

30

35

40

45

50

40

45

50

1

x(t)

State responses without control 0 −1

0

5

10

15

20

25 Time ’t’

30

35

Fig. 2: State trajectories in case 1 when M = 0:129 It is noted that, Fig.2 represents the state trajectories of the closed and open loop fuzzy system. In Fig.2 (a), (b) & (c) presents the trajectories of the system with control and (d) represents the state trajectories of the system without control. For the simulation purpose, we consider the disturbance input w(t) =

0.05 20−t2

and the initial condition

x(0) = [−0:5 0:75 − 10]T :

Case 2: Uncertain system - To design a robust reliable H∞ controller for the uncertain TS fuzzy system, we choose the following parameter values for uncertain matrices in addition to the values in case 1: M1 = M2 = 0:2I; J = 0:5; N11 = N12 = 0:1I; N21 = N22 = 0:3I; T N31 = N32 = 0:06 0:08 0:07 : πt For the time-varying delays satisfying 1 (t) = 0:01 + 0:02 sin( πt 2 ), 2 (t) = 0:12 + 0:05 sin( 2 ), by solving the LMI

conditions in Theorem 2, we present the results of maximum allowable delay bound M for different in Table III. Also, more computational results are given in Table IV which provides the minimum guaranteed H∞ performance level for different values of M . Further, when the disturbance attenuation level = 0:7, we get time delay upper

18

bound M = 0:125, the corresponding state feedback controller gains are given by K1 =

K2 =

3:6311 −2:8757 0:2005 4:1358 −3:7490 1:4265

;

:

TABLE III: Delay upper bound M for various in Case 2

M

0.5 0.093

0.55 0.102

0.6 0.111

0.7 0.125

0.8 0.138

0.9 0.149

TABLE IV: Minimum for different values of M in Case 2 M

0.05 0.3087

a)

0.06 0.3565

0.07 0.4056

0.1 0.5834

0.13 0.8557

0.15 1.2591

4 Without Failure With Failure

x1(t)

2 0 −2

b)

5

x2(t)

0

0

5

10

15

20

25

30

35

40

45

50

Without Failure With Failure

−5 0

5

10

15

20

25

30

35

40

45

50

x3(t)

c) 20 Without Failure With Failure

10 0

d)

1

x(t)

−10

0

0 19 x 10

5

10

15

20

25

30

35

40

45

50

40

45

50

State responses without control

−1

0

5

10

15

20

25 Time ’t’

30

35

Fig. 3: State trajectories in case 2 for M = 0:125 To this end, we present the simulation results for case 2 in Fig.3 by taking the disturbance input and initial condition are same as in case 1. Fig.3(d) represents the state trajectories of the open loop fuzzy system in the presence of uncertainties when M = 0:125. In order to show the effectiveness of the proposed controller design, the simulation result of the closed-loop fuzzy system in the presence of uncertainties are given in Fig.3(a), (b) & (c). From Fig.3(a), (b) & (c), it is observed that the obtained controller makes the state responses of the closed-loop fuzzy system converge to zero. The result reveals that the uncertain fuzzy system is robustly asymptotically stable.

19

Case 3: Now, we consider the unknown actuator fault case, that is, the actuator fault matrix Gη satisﬁes 0:6I ≤ Gη ≤ 0:9I; = 1; 2. With the same parameters considered in case 1 and case 2, by solving the LMI conditions

in Theorem 3 we can obtain feasible solution. Further in Table’s V & VI, we present the results of maximum allowable delay and minimum guaranteed H∞ performance level . TABLE V: Delay upper bound M for various in Case 3

M

0.5 0.092

0.55 0.101

0.6 0.109

0.7 0.124

0.8 0.135

0.9 0.145

TABLE VI: Minimum for different values of M in Case 3 M

0.05 0.3092

0.06 0.3573

0.07 0.4074

0.1 0.5922

0.13 0.8938

0.15 1.3638

It is observed that the considered nominal and uncertain fuzzy systems are stabilized through the appropriate obtained feedback reliable H∞ controller. More precisely, from Fig.2(d) and Fig.3(d), we can see that the open loop fuzzy system with and without uncertainty is unstable. Moreover, from Fig.2(a), (b), (c) and Fig.3(a), (b), (c), it is concluded that the state trajectories of closed-loop system converges to the equilibrium point which demonstrates the effectiveness of our obtained reliable controller. It is noted that the considered fuzzy system is asymptotically stable even if the fault value occurs in range of interval. Thus, the obtained controller is an effective one to stabilize the considered fuzzy system. V. C ONCLUSION In this paper, the problem of reliable H∞ control for a class of uncertain TS fuzzy systems has been investigated with random time-varying delays and actuators faults. Using proper LKF, sufﬁcient conditions has been obtained to ensure the closed-loop fuzzy system is robustly asymptotically stable with a given H∞ performance index > 0. It should be mentioned that the obtained robust stabilization conditions are established in terms of LMIs which can be easily solved via available MATLAB LMI tool box. Finally, a numerical example based on truck-trailer model has been given to illustrate the effectiveness of the proposed reliable controller. Further, the fuzzy systems with stochastic effect governed by Poisson process or Levy noise is an untreated topic. Our future work will focus on improving the existing techniques and ﬁnding new methods to deal the above probabilistic issues.

R EFERENCES [1] W. Assawinchaichote, Further results on robust fuzzy dynamic systems with LIM D-stability constraints, Int. J. Appl. Math. Comput. Sci., Vol.24, pp.785-794, 2014..

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[2] P.Balasubramaniam and T.Senthilkumar, Delay-dependent robust stabilization and H∞ control for uncertain stochastic T-S fuzzy systems with discrete interval and distributed time-varying delays, International Journal of Automation and Computing, Vol.10, pp.18-31, 2013. [3] P. Balasubramaniam, R. Krishnasamy and R. Rakkiyappan, Delay-interval-dependent robust stability results for uncertain stochastic systems with Markovian jumping parameters, Nonlinear Analysis: Hybrid Systems, Vol.5, pp.681-691, 2011. [4] H.Bao and J.Cao, Delay-distribution-dependent state estimation for discrete-time stochastic neural networks with random delay, Neural Networks Vol.24, pp.19-28, 2011. [5] Y.Y.Cao and P.M.Frank, Stability analysis and synthesis of nonlinear time-delay systems via linear Takagi-Sugeno fuzzy models, Fuzzy Sets Syst., vol. 124, pp. 213-229, 2001. [6] M. Chadli, H. R. Karimi and P. Shi, On stability and stabilization of singular uncertain Takagi-Sugeno fuzzy systems, Journal of the Franklin Institute, Vol.351, pp.1453-1463, 2014. [7] X.Chen, J.Lam, H.Gao and S.Zhou, Stability analysis and control design for 2-D fuzzy systems via basis-dependent Lyapunov functions, Multidimensional Systems and Signal Processing, Vol.24, pp.395-415, 2013. [8] X.H.Chang and G.H.Yang, A descriptor representation approach to observer-based H∞ control synthesis for discrete-time fuzzy systems, Fuzzy Sets and Systems, Vol.185, pp.38-51, 2011. [9] Z.Feng and J.Lam, Integral partitioning approach to robust stabilization for uncertain distributed time-delay systems, International Journal of Robust and Nonlinear Control, Vol.22, pp.676-689, 2012. [10] Z.Feng and J.Lam, Reliable dissipative control for singular Markovian systems, Asian Journal of Control, Vol.15, pp.901-910, 2013. [11] Y.Xu, Y.Li and S.Tong, Fuzzy adaptive actuator failure compensation dynamic surface control of multi-input and multi-output nonlinear systems, International Journal of Innovative Computing, Information and Control, vol.9, no.12, pp.4875-4888, 2013. [12] H.Hu, B.Jiang and H.Yang, Robust H∞ reliable control for uncertain switched systems with circular disk pole constraints, Journal of the Franklin Institute, Vol.350, pp.802-817, 2013. [13] S.Jafarzadeh and M.S.Fadali, On the stability and control of continuous-time TSK fuzzy systems, IEEE Transactions on Cybernetics, Vol.43, pp.1073-1087, 2013. [14] H. Li, Y. Gao, L. Wu and H.K. Lam, Fault detection for T-S fuzzy time-delay systems: Delta operator and input-output methods, IEEE Transactions on Cybernetics, Vol.45, pp.229-241, 2015. [15] L. Li, Q. Zhang and B. Zhu, H∞ fuzzy control for nonlinear time-delay singular Markovian jump systems with partly unknown transition rates, Fuzzy Sets and Systems, Vol.254, pp.106-125, 2014. [16] H. Li, Z. Chen, Y. Sun and H.R. Karimi, Stabilization for a class of nonlinear networked control systems via polynomial fuzzy model approach, Complexity, (2014) DOI: 10.1002/cplx.21579. [17] H. Li, H. Liu, H. Gao and P. Shi, Reliable fuzzy control for active supension systems with atuator delay and fault, IEEE Transactions on Fuzzy Systems, Vol.20, pp.342-357, 2012. [18] T.Li, L.Guo and C.Sun, Robust stability for neural networks with time-varying delays and linear fractional uncertainties, Neurocomputing, Vol.71, pp.421-427, 2007. [19] F.Liu and Z.Dziong, Reliable decentralized control of fuzzy discrete-event systems and a test algorithm, IEEE Transactions on Cybernetics, Vol. 43, pp.321-331, 2013. [20] P.Liu and T.S.Chiang, H∞ output tracking fuzzy control for nonlinear systems with time-varying delay, Applied Soft Computing, Vol.12, pp.2963-2972, 2012. [21] M.S.Mahmoud, Reliable decentralized control of interconnected discrete delay systems, Automatica, Vol.48, pp.986-990, 2012. [22] R.Sakthivel, K.Mathiyalagan and S.Marshal Anthoni, Robust H∞ control for uncertain discrete-time stochastic neural networks with time-varying delays, IET Control Theory & Applications, Vol.6, pp. 1220-1228, 2012.

21

[23] R.Sakthivel, K.Mathiyalagan and S.Marshal Anthoni, Design of a passiﬁcation controller for uncertain fuzzy Hopﬁeld neural networks with time-varying delays, Physica Scripta, Vol.84, pp.045024, 2011. [24] R. Sakthivel, P. Vadivel, K. Mathiyalagan and A. Arunkumar, Fault-distribution dependent reliable H∞ control for TS fuzzy systems, Journal of Dynamic Systems, Measurement and Control, Vol.136, pp.021021-11, 2014. [25] Y. Wang and J. Fei, Adaptive fuzzy sliding mode control for PMSM position regulation system, International Journal of Innovative Computing, Information and Control, vol.11, no.3, pp.881-891, 2015. [26] J. Wang, H∞ fault-tolerant controller design for networked control systems with time-varying actuator faults, International Journal of Innovative Computing, Information and Control, vol.11, no.4, pp.1471-1481, 2015. [27] T.Senthilkumar and P.Balasubramaniam, Robust H∞ control for nonlinear uncertain stochastic T-S fuzzy systems with time delays, Applied Mathematics Letters, Vol.24, pp.1986-1994, 2011. [28] X.Su, P.Shi, L.Wu and S.Nguang, Induced l2 ﬁltering of fuzzy stochastic systems with time-varying delays, IEEE Transactions on Cybernetics, vol.43, no.4, pp.1251-1264, 2013. [29] X.Su, P.Shi, L.Wu, Y.D.Song, A novel approach to ﬁlter design for T-S fuzzy discrete-time systems with time-varying delay, IEEE Transactions on Fuzzy Systems, Vol.20, pp. 1114-1129, 2012. [30] T.Takagi and M.Sugeno, Fuzzy identiﬁcation of systems and its applications to modeling and control, IEEE Transactions on System Man Cybernetics, Vol.15, pp. 116-132, 1985. [31] N.T.Thanh and V.N.Phat, Decentralized H∞ control for large-scale interconnected nonlinear time-delay systems via LMI approach, Journal of Process Control, Vol.22, pp.1325-1339, 2012. [32] P.Vadivel, R.Sakthivel, K.Mathiyalagan and P.Thangaraj, Robust stabilization of nonlinear uncertain Takagi-Sugeno fuzzy systems by H∞ control, IET Control Theory & Applications, Vol.6, pp. 2556-2566, 2012. [33] Y.C.Wang, H.G.Zhang, X.Y.Wang and D.S.Yang, Networked synchronization control of coupled dynamic networks with time-varying delay, IEEE Transactions on System, Man and Cybernetics-Part B , Vol.40, pp.1468-1479, 2010. [34] J. W. Wang, H. N. Wu, L. Guo and Y. S. Luo, Robust H∞ fuzzy control for uncertain nonlinear Markovian jump systems with time-varying delay, Fuzzy Sets and Systems, Vol.212, pp.41-61, 2013. [35] H.Wang, L.Y.Peng, H.H.Ju and Y.L.Wang, H∞ state feedback controller design for continuous-time T-S fuzzy systems in ﬁnite frequency domain, Information Sciences, Vol.223, pp.221-235, 2013. [36] Z.G.Wu, P.Shi, H.Su and J.Chu, Reliable H∞ control for discrete-time fuzzy systems with inﬁnite-distributed delay, IEEE Transactions on Fuzzy Systems, Vol.20, pp.22-31, 2012. [37] A.Ray, Output feedback control under randomly varying distributed delays, Journal of Guid- ance, Control and Dynamics, Vol.17, pp.701-711, 1994. [38] L.El Ghaoui and G.Scorletti, Control of rational systems using linear fractional representations and linear matrix inequalities, Automatica, Vol.32, pp.1273-1284, 1996. [39] H.N.Wu and H.X.Li, New approach to delay-dependent stability analysis and stabilization for continuous-time fuzzy systems with time-varying delay, IEEE Transactions on Fuzzy Systems, Vol.15, pp.482-493, 2007. [40] H.Yang, P.Shi, J.Zhang and J.Qiu, Robust H∞ control for a class of discrete time fuzzy systems via delta operator approach, Information Sciences, Vol.184, pp. 230-245, 2012. [41] X.Yang and H.Gao, Robust reliable control for autonomous spacecraft rendezvous with limited-thrust, Aerospace Science and Technology, Vol.24, pp. 161-168, 2013. [42] Y.Yin, P.Shi, F.Liu and K.L.Teo, Fuzzy model-based robust H∞ ﬁltering for a class of nonlinear nonhomogeneous Markov jump systems, Signal Processing, Vol.93, pp. 2381-2391 2013. [43] J.Yoneyama, Robust H∞ ﬁltering for sampled-data fuzzy systems, Fuzzy Sets and Systems, Vol.217, pp. 110-129, 2013.

22

[44] J.Zhang, P.Shi and Y.Xia, Fuzzy delay compensation control for T-S fuzzy systems over network, IEEE Transactions on Cybernetics, vol. 43, no. 1, pp. 259-268, 2013. [45] H.Zhang and Y.Wang, Stability analysis of Markovian jumping stochastic Cohen-Grossberg neural networks with mixed time delays, IEEE Trans. on Neural Networks, Vol. 19, pp. 366-370, 2008. [46] S.Zhang, Z. Wang, D. Ding and H. Shu, H∞ fuzzy control with randomly occurring inﬁnite distributed delays and channel fadings, IEEE Transactions on Fuzzy Systems, Vol.22, pp.189-200, 2014. [47] Q.Zhou, P.Shi, H.Liu and S.Xu, Neural-network-based decentralized adaptive output-feedback control for large-scale stochastic nonlinear systems, IEEE Transactions on Systems, Man and Cybernetics, Part B: Cybernetics, vol.42, no.6, pp.1608-1619, 2012. [48] X.L.Zhu, B.Chen, Y.Wang and D.Yue, H∞ stabilization criterion with less complexity for nonuniform sampling fuzzy systems, Fuzzy Sets and Systems, Vol.225, pp.58-73, 2013.

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