Disturbance rejection for singular Markovian jump systems with time-varying delay and nonlinear uncertainties

Nonlinear Analysis: Hybrid Systems 33 (2019) 130–142

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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs

Disturbance rejection for singular Markovian jump systems with time-varying delay and nonlinear uncertainties ∗

∗∗

S. Mohanapriya a , R. Sakthivel b,c , , O.M. Kwon d , , S. Marshal Anthoni a a

Department of Mathematics, Anna University Regional Campus, Coimbatore 641046, India Department of Applied Mathematics, Bharathiar University, Coimbatore 641046, India c Department of Mathematics, Sungkyunkwan University, Suwon 440-746, South Korea d School of Electrical Engineering, Chungbuk National University, 1 Chungdae-ro, Cheongju 28644, South Korea b

article

info

Article history: Received 6 June 2018 Accepted 16 February 2019 Available online xxxx Keywords: Singular Markovian jumping systems Robust modified repetitive controller Aperiodic disturbances Unknown nonlinear uncertainty

a b s t r a c t This paper employs an active disturbance rejection technique for a class of singular Markovian jump systems with external disturbances, time-varying delay and unknown nonlinear uncertainty. Based on the improved equivalent-input-disturbance approach, a robust modified repetitive controller is proposed to the considered system to solve the periodic output tracking problem. An appropriate Lyapunov–Krasovskii functional is selected to guarantee the mean-square asymptotic admissibility of the system under study. Moreover, an explicit expression of the proposed controller is presented which has the capability of forcing the system output to exactly track any given periodic reference signal. Finally, the obtained results are validated through simulations by considering a DC motor driving model to demonstrate the effectiveness of the proposed control design technique. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Singular systems, which are known as semi-state systems or descriptor systems, have some interesting features unlike classical physical systems for instance, impulse terms in system states and non-causality. Also, singular systems has the ability to accurately describe many real-time systems such as power systems, electrical circuits and so on. Owing to this fact, many important results concerned with stability and stabilization of singular systems have been investigated [1–4]. Further, for the past two decades, the study of singular systems with time-varying delays have become an active research topic [5,6]. It is well known that the time-varying delay is an unavoidable key source for system instability and poor performance [7–13]. The robust stabilization of nonlinear singular discrete time systems has been discussed in [6]. On the other hand, Markovian jumping system is an important hybrid system, which has wide applications in many practical systems including manufacturing control systems, economical systems and network-based control systems [14–20]. Over the past couple of decades, the analysis and control design of singular Markovian jumping systems (mean-square) with time delay have received great attention and some fruitful works related to this topic have been reported (see [21–26] and references therein). On the other hand, output tracking control is an important issue in control engineering problems and many authors have developed various control techniques for output tracking problems, for instance, output feedback control, fuzzy logic control, neural network-based control and so on [27–29]. In many industrial problems, it is an essential task to track the specified periodic reference signals and to reject the periodic disturbances. It is worth mentioning that modified repetitive ∗ Corresponding author at: Department of Applied Mathematics, Bharathiar University, Coimbatore 641046, India. ∗∗ Corresponding author. E-mail addresses: [email protected] (R. Sakthivel), [email protected] (O.M. Kwon). https://doi.org/10.1016/j.nahs.2019.02.010 1751-570X/© 2019 Elsevier Ltd. All rights reserved.

S. Mohanapriya, R. Sakthivel, O.M. Kwon et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 130–142

131

control (MRC) strategy has widely been employed to force the output of the considered control system to follow the periodic reference without steady-state error (see [30–32] and references therein). Moreover, the system uncertainty and aperiodic disturbance signals are commonly encountered in practical control systems due to some measurement errors, component failures and so on [33–37]. Recently, equivalent-input-disturbance technique has been developed in [38] and effectively used for various practical control systems to attenuate the effect of the external disturbances [39–43]. The key role of this approach is that the estimated disturbance on the control input channel provides the same effect on the output as actual disturbances do. Very recently, an improved equivalent-input-disturbance (IEID) technique has been introduced in [44] to enhance the disturbance-rejection performance for a strictly proper plant with a state delay and disturbances. To be precise, the IEID block contains an estimator to suppress the effects of external disturbances and nonlinear uncertainty without knowing prior information about them. Another active disturbance rejection method with the use of an disturbance observer has been designed in [45] to estimate the external disturbances and modelling uncertainties. Compared with this approach, the IEID approach could estimate the complete influence of the lumped disturbance on the system output through a full-order state observer, where it is not necessary to know the prior information about the disturbances. It should be mentioned that, IEID approach has been addressed to attenuate the external disturbances, in which the influence of the lumped disturbance can be estimated from measurable variables and the estimated lumped disturbance can be imposed in the control input channel. In particular, the combination of state observer and IEID estimator provide accurate estimation of lumped disturbance. Moreover, the design of parameters in state observer and IEID estimator do not depend on the information of external disturbances while, the controller design based on disturbance observer (DO) depends on the information of disturbances. Moreover, we have illustrated this concept in the simulation part. Most of the existing results on tracking control problems via equivalent-input-disturbance based repetitive controller are available only for deterministic systems and up to now, no work has been reported on tracking control problems for singular Markovian jump models via IEID-based robust modified repetitive controller, which motivates this study. The main contributions of this paper are summarized as follows:

• It is the first attempt to consider the periodic output tracking problem for a class of SMJSs with external disturbances, unknown nonlinear uncertainties and time-varying delay via an IEID-based modified repetitive controller.

• A rigorous asymptotic mean-square admissibility criterion for the system under study is obtained, wherein the combination of the optimization technique and the Schur complement together with free-weighting matrix approach are employed substantially to simplify the derivation of main results.

• Satisfactory tracking performance and rejection of external lumped disturbances are achieved through the proposed repetitive control technique. The result describes that the proposed IEID-based robust modified repetitive controller for SMJSs is conceptually simple and attractive due to its better performance. 2. Problem formulation and preliminaries Let us consider a finite state Markov process {σt , t ≥ 0} which is continuous and takes values from M = {1, 2, . . . , N } on the probability space and the transition probability matrix Π = {πij } (i, j ∈ M) described by P {σt +h = j|σt = i} = { πij h + o(h), i ̸ = j, o(h) where h > 0 and limh→0 h = 0, πij > 0, i ̸ = j, is the transition rate from mode i at time 1 + πii h + o(h), i = j, ∑ t to mode j at time t + h and πii = − j̸=i πij < 0 for each i ∈ M. In this paper, we consider the continuous SMJSs with time-varying delays represented by the following differential equations:

E x˙ (t) = A(σt )x(t) + Ad (σt )x(t − h(t)) + B(σt )u(t) + H(t , x(t)) + Bd (σt )wd (t) y(t) = C (σt )x(t) x(0) = φ (t), ∀t ∈ [−hM , 0],

(1)

where x(t) ∈ Rn is the state vector; u(t) ∈ Rm is the control input vector; y(t) ∈ Rq is the output vector; H(t , x(t)) is the unknown nonlinear uncertainty; wd (t) ∈ Rp is the disturbance input belongs to the space of square-integrable functions L2 [0, ∞) and φ (t) is the initial state of the system. Further, A(σt ), Ad (σt ), B(σt ), Bd (σt ) and C (σt ) are known matrix functions of random process σt := σ (t) and E ∈ Rn×n is a singular matrix with rank(E) = r < n. Moreover, for any two positive scalars ˙ ≤ µ < 1. For convenience, in the sequel, hM and µ, time-varying delay h(t) satisfies the condition 0 ≤ h(t) ≤ hM , where h(t) for each i = σt , σt ∈ M, matrix A(σt ) will be denoted by Ai and so on. Here, the objective of the output feedback modified repetitive controller is to track the reference signal with the output of the SMJSs (1). The corresponding tracking error is given as e(t) = y(t) − r(t). In addition, the block diagram shown in Fig. 1 demonstrates an overview of the considered IEID-based robust modified repetitive control system (MRCS). Fig. 1 mainly contains two inner-blocks, namely, MRC block and IEID estimator along with the output feedback control input. The MRC block acts as an eliminator of periodic disturbance and the desired tracking performance is achieved through this MRC block whose dynamics is based on the internal model principle. Further, IEID estimator is employed to compensate the disturbance effect on the output. In particular to enhance the tracking performance, MRC block contains a low-pass filter Mr (s), which is ωr and satisfies |Mr (jω)| ≈ 1, ∀ ω ∈ [0, ω ¯ c ], where ωr and ωc are the cutoff angular frequency of Mr (s) defined as Mr (s) = s+ω r and the highest angular frequency of r(t), respectively. The state-space representation of MRC is given by x˙ r (t) = − ωr xr (t) + ωr xr (t − T ) + ωr e(t),

(2)

132

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Fig. 1. Configuration of IEID-based MRCS.

yr (t) =e(t) + xr (t − T ), where xr (t) ∈ R is the state of MRC, T is the period of the reference signal and yr (t) is the output of the MRC. In order to estimate the states of the system (1), we assume that the system is both controllable and observable. The modified full-order state observer of SMJSs (1) can be written as follows: E x˙ˆ (t) = Avi xˆ (t) + Adi xˆ (t − h(t)) + Bi ub (t) + Li (y(t) − yˆ (t)), yˆ (t) = Ci xˆ (t),

(3)

where xˆ (t) is the estimation of the state x(t), Li and Avi are the observer gain matrix of full column rank and variant system matrix, respectively. Moreover, the feedback control law ub (t) can be denoted by ub (t) = KRi yr (t) + KYi y(t). Assume that the external disturbance wd (t) and nonlinear uncertainty H(t , x(t)) acting on the system (1) are considered as lumped disturbance Dli such that ∥wd (t)∥∞ < dM , where dM is unknown positive real number. Therefore, for the equivalent-inputdisturbance wde (t), SMJSs (1) can be described by E x˙ (t) = Ai x(t) + Adi x(t − h(t)) + Bi [u(t) + wde (t)] , ye (t) = Cx(t),

(4)

where ye (t) is the measured output. Let KDi ∈ Rn×n be the disturbance estimator gain and define the state estimation error as ∆x(t) = x(t) − xˆ (t). Then, the equation (4) can be rewritten as E x˙ˆ (t) = Ai xˆ[(t) + Adi xˆ (t − h(t)) + Bi u(t) + Bi wde (t) − (KDi − I)Li Ci ∆x(t) ] + Ai ∆x(t) + Adi ∆x(t − h(t)) − E ∆x˙ (t) + (KDi − I)Li Ci ∆x(t) .

(5)

Let us assume that B∆wd (t) = Ai ∆x(t) + Adi ∆x(t − h(t)) − E ∆x˙ (t) + (KDi − I)Li Ci ∆x(t) and denote w ˆ d (t) = wde (t) + ∆wd (t). Then (5) can be written as

ˆ d (t) . E x˙ˆ (t) = Ai xˆ (t) + Adi xˆ (t − h(t)) − (KDi − I)Li Ci ∆x(t) + Bi u(t) + w

[

]

(6)

It follows from (3), p5e6, the optimal estimated disturbance w ˆ d (t) can be calculated by

[ ] [ ] ˆ w ˆ d (t) = B+ Avi − Ai xˆ (t) + B+ (7) i i KDi Li Ci x(t) − x(t) + ub (t) − u(t), ( ) − 1 T where B+ BTi . Moreover, in order to remove the measurement noise, the estimated disturbance w ˆ d (t) is passed i = Bi Bi through the low-pass filter W (s), where W (s) satisfies |W (jω)| ≈ 1, ∀ ω ∈ [0, ω ˜ c ], and ω˜ c denotes the highest angular frequency. The state-space representation of the low-pass filter W (s) is given by x˙ m (t) = Am xm (t) + Bm w ˆ d (t) w ˜ d (t) = Cm xm (t),

(8)

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133

where xm (t) is state of the filter and w ˜ d (t) is the noiseless estimated disturbance input. Hence, the enhanced control input of closed-loop control system is given by u(t) = ub (t) − w ˜ d (t).

(9)

Combining the singular full-order state observer and the IEID estimator to the robust MRCS, we can have the closedloop singular Markovian jumping MRCS as shown in Fig. 1. Note that the controller gain of robust MRC is independent of exogenous reference signal. It should be noted that in this study, in Fig. 1 MRC improves the periodic tracking performance and IEID improves the disturbance rejection performance. Since, in the IEID-based MRCS approach, the stability of the closedloop augmented system is independent of the external signals. For sake of simplicity, we assumed that reference r(t) and disturbance Dli as zero. Let ξ (t) = [xT (t) ∆xT (t) xTm (t) xTr (t)]T be the state vector of the augmented closed-loop system. Further, based on the above discussions, we can have

⎧ E x˙ (t) ⎪ ⎨ E ∆x˙ (t) ⎪ ⎩ x˙ m (t) x˙ R (t)

= = = =

¯ A ( i x(t) + A) di x(t − (h(t)) + Bi )KPi Ci x(t) − Cm xm (t) + KRi xr (t − T ) , Ai − Avi x(t) + Avi − Li Ci ∆x(t) − Bi Cm xm (t) + Adi ∆x(t − h(t)), + (Am + Bm Cm )xm (t) + Bm B+ i (Avi − Ai ) (x(t) − ∆x(t)) + Bm Bi KDi Li Ci ∆x(t), −ωr xr (t) + ωr xr (t − T ) − ωr Ci x(t).

(

)

(10)

where K¯ Pi = KYi − KRi . Thus, the IEID-based closed-loop augmented MRCS can be formulated as

¯ rci ξ (t − T ), E¯ ξ˙ (t) = A¯i ξ (t) + A¯di ξ (t − h(t)) + M

(11)

where Ai + Bi K¯ Pi Ci Ai (− Avi ) ⎢ A¯i = ⎣ Avi − Ai Bm B+ i −Ci ωr

⎡

0 Avi − Li Ci ( ) + Avi − Ai Bm B+ i KDi Li Ci − Bm Bi 0

⎡

¯ rci A¯di = diag Adi , Adi , 0, 0 , M

{

}

0 ⎢ 0 =⎣ 0 0

0 0 0 0

0 0 0 0

−Bi Cm −Bi Cm Am + Bm Cm

0 0 0

0

−ωr

⎤ ⎥ ⎦,

⎤

Bi KRi 0 ⎥ , and E¯ = diag {E , E , I , I } . 0 ⎦

ωr

3. Main results In this section, we first concentrate our attention on the problems of asymptotic mean-square admissibility of SMJSs (1) with u(t) = 0. Second, IEID-based robust MRC is designed for the augmented MRCS (11). Theorem 3.1. Given constants hM > 0 and µ > 0, the SMJSs (1) with u(t) = 0 is asymptotically mean-square admissible, if there exist symmetric matrices Pi , Q > 0, R > 0, Z > 0 and appropriate matrix L such that the following LMIs hold for every i ∈ M: E T Pi = Pi E ≥ 0,

(12)

Ξ < 0, [ ]

(13)

Z

∗

L Z

> 0,

(14)

where Ξ = [Ξ ]4×4 and

Ξ1,1 =Pi Ai + ATi Pi + Q + R − E T ZE +

N ∑

πij E T Pj ,

j=1

Ξ1,2 =Pi Adi + E ZE − E LE , Ξ1,3 = E LE , Ξ1,4 = hM ATi Z , T

T

T

Ξ2,2 = − (1 − µ)Q − E T ZE + E T LE − EZE T + ELE T , Ξ2,3 = E T ZE − E T LE , Ξ2,4 = hM ATdi Z , Ξ3,3 = − R − E T ZE , Ξ3,4 = 0, Ξ4,4 = −Z and ∗ denotes the corresponding structures in the symmetric matrix. Proof. First, we have to prove the regularity and impulse free of system (1)]with u(t) = 0. Since rank(E) = r < n, [ Ir 0 there exist two nonsingular matrices G and M such that GEM = . Let us consider the following GAi M = 0 0

[

A1i A3i

A2i A4i

]

, G−T Pi M =

[

P1i P3i

P2i P4i

]

. From the above expressions and (12) it can be obtained that P2i = 0. Assume

134

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that (13) holds for Q > 0, R > 0 and Z > 0. Then, pre and post-multiplying Ξ1,1 < 0 by M T and M, respectively, we have T A4i + AT4i P4i < 0 which implies A4i is nonsingular and the pair (E , Ai ) is regular and impulse free. Hence, by Definitions 2.1, P4i

2.2 of regular and impulse free condition in [5], SMJSs (1) with u(t) = 0 is regular and impulse free for every i ∈ M. Next we will prove that SMJSs (1) with u(t) = 0 is mean-square asymptotically stable. Consider the following Lyapunov–Krasovskii functional for the system (1) with u(t) = 0 for every i ∈ M in the form V (t , x(t)) = xT (t)E T Pi x(t) +

∫t t −h(t)

∫t

xT (s)Qx(s)ds +

t −hM

xT (s)Rx(s)ds + hM

Let the infinitesimal operator L of V (t , x(t)) be L [V (t , x(t))] = limh→0+

1 h

∫0 −hM

∫t t +θ

x˙ T (s)E T ZE x˙ (s)dsdθ.

(15)

E(V (t + h, x(t + h))|x(t) ) − V (t , x(t)) . Then, by

[

]

calculating L [V (t , x(t))] and taking the mathematical expectation, we can get

⎡ ⎤ N ∑ E{LV (t , x(t))} ≤ 2xT (t)PiT E x˙ (t) + xT (t) ⎣ πij E T Pj + Q + R⎦ x(t) − (1 − µ)xT (t − h(t))Qx(t − h(t)) j=1 T

T

− x (t − hM )Rx(t − hM ) + x˙ (t)E

T

(h2M Z )E x(t)

˙

∫

t

x˙ T (s)E T ZE x˙ (s)ds.

− hM

(16)

t −hM

By applying Lemma 1 of [23], there exist a matrix L such that the integral term in (16) satisfy the following inequality:

∫

t

x˙ T (s)E T ZE x˙ (s)ds ≤ Ψ T (t)E T Ω1 E Ψ (t),

− hM

(17)

t −hM

⎡

⎤ −Z Z −L L where Ψ (t) = xT (t) xT (t − h(t)) xT (t − hM ) and Ω1 = ⎣ ∗ −2Z + L + LT Z − L ⎦. ∗ ∗ −Z T It follows from (16) and[(17) that E{LV (t , x(t)) ]T } ≤ [Ψ (t)Ω2 Ψ (t), ] [Z ] hM Ai hM Adi 0 and Ξ1,1 , Ξ1,2 , Ξ1,3 , Ξ2,2 , Ξ2,3 , Ξ3,3 are defined in where Ω2 = [Ξ ]3×3 + hM Ai hM Adi 0 (14). If the LMIs (12)–(14) hold, we can get that Ω2 < 0 and hence E{LV (t , x(t))} < 0 which implies that E{V˙ (t , x(t))} < 0. Hence, E{V (t , x(t))} > 0 is decreasing monotonically and from (15) it follows that E{∥x(t)∥2 } → 0, as t → ∞. Therefore, by the Definition 8.2 of [5], it is concluded that SMJSs (1) with u(t) = 0 is mean-square asymptotically stable which implies that SMJSs (1) with u(t) = 0 is mean-square asymptotically admissible for every i ∈ M. The proof is completed. ■ [

]T

Next, based on Theorem 3.1, we present the designing controller parameters for the SMJSs (1). In order to achieve the desired tracking performance between the output of the closed-loop system (1) and the reference input, it is sufficient to show that the augmented system (11) is mean-square asymptotically admissible. Theorem 3.2. For any given positive constants α , β , γ , hM , µ and ϵl , (l = 1, 2, . . . , 8), the closed-loop augmented system (11) is mean-square asymptotically admissible, if there exist scalars σi > 0, positive symmetric matrices Xqi , X¯ 1i , X¯ 2i , Yq , Qq , Rq , Sq , Zq ˆ Zˆ and appropriate matrices Wli , (l = 1, 2, 3, 4, 5, 6) and Nˆ such that the following inequalities hold for (q = 1, 2, 3, 4), Xi , S, every i ∈ M: X1i E T = EX1i ≥ 0, X2i E T = EX2i ≥ 0, Ci X1i = X¯ 1i Ci , Ci X2i = X¯ 2i Ci ,

α X1i E T < σi I , β X2i E T < σi I , [ ] Zˆ Nˆ ˆS < ϵ¯1 Xi , Zˆ < ϵ¯2 Xi , > 0, ∗ Zˆ ⎡ ¯ ¯ 12 Φ ¯ 13 Φ ¯ 14 hM Φ ¯ 15 Φ11 Φ ¯ ¯ ¯ 25 ⎢ ∗ Φ22 Φ23 0 hM Φ ⎢ ¯ 33 ⎢ ∗ ∗ Φ 0 0 ⎢ ¯ 45 ⎢ ∗ ∗ ∗ −Y hM Φ ⎢ Xi ⎢ ∗ ∗ ∗ − ⎣ ∗ ϵ¯2 ∗ ∗ ∗ ∗ ∗

(18) (19)

(ϵ¯1 = diag{ϵ1 , . . . , ϵ4 }, ϵ¯2 = diag{ϵ5 , . . . , ϵ8 }),

¯ 16 Φ 0 0 0 0

¯ 66 Φ

(20)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ < 0, ⎥ ⎥ ⎦

(21)

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135

where

⎧ ⎡ ⎤ ⎡ ⎤ ϕ1,5 0 0 0 ϕ1,1 ϕ1,2 ϕ1,3 −αωr X1i CiT ⎪ ⎪ ⎪ ⎪ 0 0 ⎥ ¯ ⎢ 0 ϕ2,6 ⎥ ∗ ϕ ϕ 0 ⎪ ¯ 11 = ⎢ ⎪ Φ ⎣ ∗ ∗2,2 ϕ2,3 ⎦ , Φ12 = ⎣ 0 0 Z − N ⎦, ⎪ 0 0 ⎪ 3 3 3,3 ⎪ ⎪ ⎪ 0 0 0 Z4 − N4 ∗ ∗ ∗ ϕ4,4 ⎪ ⎪ ⎪ ¯ 13 = {EN1 E T , EN2 E T , N3 , N4 }, ⎪ Φ ⎪ ⎪ ⎡ ⎤ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ϕ1,17 ⎪ ϕ1,18 ϕ1,19 −αωr X1i C T ⎥ ⎪ ⎢ ⎥ ⎪ ⎪ ⎢ ⎥ ⎡ ⎤ ⎪ ⎪ ⎢ ⎥ 0 0 0 γ B W ⎪ i 3i ⎪ ⎢ ⎥ ⎪ ⎪ 0 0 0 0 0 ϕ2,18 ϕ2,19 0 ⎢ ⎥ ⎢ ⎥ ⎪ ¯ ¯ ⎪ , Φ15 = ⎢ ⎥, ⎦ ⎨ Φ14 = ⎣ 0 0 0 0 ⎢ ⎥ ⎢ ⎥ 0 0 0 γ ωr Y4 T T T T ⎢ ⎥ ⎪ −X3i Cm Bi −X3i Cm Bi ϕ3,19 0 ⎪ ⎢ ⎥ ⎪ ⎪ ⎣ ⎦ ⎪ ⎪ ⎪ ⎪ ⎪ ∗ ∗ −X4i ωr ⎪ ⎪ ] ∗ [√ √ √ ⎪ ¯ T T T ⎪ π X . . . π X . . . π X , X = {α X , β X , X , X }, Φ = ⎪ i1 ij iN i 1i 2i 3i 4i 16 i i i{ j̸ =i ⎪ ⎪ ⎪ T T ⎪ − (1 − µ )Q + E(N + N − 2Z ⎪ h h h )E , for h = 1, 2 h ¯ 22 = diag{ψ1 , ψ2 , ψ3 , ψ4 }, ψh = ⎪ , Φ ⎪ T ⎪ − 2Z , for h = 3, 4 − (1 − µ )Q + N + N h h h ⎪ h ⎪ ⎪ T T T T ¯ 23 = {E(Z1 − N1 )E , E(Z2 − N2 )E , Z3 − N3 , Z4 − N4 }, Φ ¯ 25 = diag{α X1 Ad , β X2 Ad , 0, 0}, ⎪Φ ⎪ ⎪ ⎪ ¯ 33 = diag{−EZ1 E T − R1 , −EZ2 E T − R2 , −Z3 − R3 , −Z4 − R4 }, Y = diag{Y1 , Y2 , Y3 , γ Y4 }, ⎪Φ ⎪ ⎪ ⎪ ¯ ¯T ¯ ⎪ ⎪ ⎩ Φ45 = Φ14 , Φ66 = diag{H1 , . . . , Hj , . . . , HN }j̸=i , Hj = diag{−2α X1j + σj I , −2β X2j + σj I , −X3j , −X4j }, and ⎧ ⎪ ϕ1,1 = α Ai X1i + α X1i ATi + α Bi W1i Ci + α CiT W1iT BTi + Q1 + R1 + ϵ1 α X1i − EZ1 E T + απii X1i E T , ⎪ ⎪ ⎪ ⎪ T T ⎪ ⎪ ϕ1,2 = α X1i ATi − α W4iT , ϕ1,3 = −α (X1i ATi − W4iT )B+ ⎪ i Bm − Bi Cm X3i , ⎪ ⎪ ⎪ ⎪ ϕ2,2 = β (W5i + W5iT − W2i Ci − CiT W2iT ) + Q2 + R2 + ϵ2 X2i β − EZ2 E T + βπii X2i E T , ⎪ ⎪ ⎪ ⎨ T T T T ϕ2,3 = −Bi Cm X3i + β (ATi X2iT − W5iT )B+ i Bm + β Ci W6i , ⎪ ⎪ ϕ3,3 = 2(Am + Bm Cm )X3i + Q3 + R3 + ϵ3 X3i − Z3 + πii X3i , ϕ4,4 = −2β X4 ωr + Q4 + R4 ⎪ ⎪ ⎪ ⎪ ⎪ + ϵ4 X4i − Z4 + πii X4i , ϕ1,5 = α Adi X1i + E(Z1 − N1 )E T , ϕ2,6 = β Adi X2i + E(Z2 − N2 )E T , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ϕ1,17 = α X1i ATi + α CiT W1iT BTi , ϕ1,18 = α (X1iT AT1i − W4iT ), ϕ1,19 = −α (X1iT AT1i − W4iT )B+T BTm , ⎪ ⎪ ⎩ T T T T T T T ϕ2,18 = −CiT W2iT β + β W5iT , ϕ2,19 = β (ATi X2iT − W5iT )B+ i Bm + β Ci W6i , ϕ3,19 = X3i (Am + Cm Bm ). Moreover, the gain matrices of the desired observer and the desired controller can be calculated by K¯ Pi = W1i X¯ 1i−1 , Li = 1 ¯ −1 + W2i X¯ 2i−1 , KRi = W3i Y4−1 , Avi = W4i X1i−1 , KDi = Bi B− m W6i X2i Li . Proof. Consider the following Lyapunov–Krasovskii functional for the system (11) for every i ∈ M in the form V (t , ξ (t)) =ξ T (t)E¯ T P¯ i ξ (t) +

∫

t

ξ T (s)Q¯ ξ (s)ds + t −h(t)

∫

0

∫

t

ξ T (s)R¯ ξ (s)ds +

t

∫

t −hM

ξ T (s)S¯ ξ (s)ds t −T

t

+ hM −hM

∫

ξ˙ T (s)E¯ T Z¯ E¯ ξ˙ (s)dsdθ.

t +θ

where P¯ i = diag{ α1 P¯ 1i , β1 P¯ 2i , P¯ 3i , P¯ 4i }, Q¯ = diag{Q¯ 1 , Q¯ 2 , Q¯ 3 , Q¯ 4 }, R¯ = diag{R¯ 1 , R¯ 2 , R¯ 3 , R¯ 4 }, S¯ = diag{S¯1 , S¯2 , S¯3 , γ1 S¯4 } and Z¯ =

¯ Q¯ , R, ¯ S¯ and Z¯ are the symmetric positive definite matrices. Now, from (16) and (17), applying diag{Z¯1 , Z¯2 , Z¯3 , Z¯4 }, P¯ i E, infinitesimal operator and taking mathematical expectation of V (t , ξ (t)), we can get

⎡ ⎤ N ∑ E{LV (t , x(t))} ≤ 2ξ (t)T P¯ iT E¯ ξ˙ (t) + ξ T (t) ⎣ πij E¯ T P¯ j + Q¯ + R¯ ⎦ ξ (t) − (1 − µ)ξ T (t − h(t))Q¯ ξ (t − h(t)) j=1

− ξ T (t − hM )R¯ ξ (t − hM ) + ξ˙ T (t)E¯ T (h2M Z¯ )E¯ ξ˙ (t) − hM

∫

t

ξ˙ T (s)E¯ T Z¯ E¯ ξ˙ (s)ds. t −hM

(22)

136

S. Mohanapriya, R. Sakthivel, O.M. Kwon et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 130–142

Then from Lemma 1 of [23], there exist a matrix N¯ such that the integral term in (22) satisfy the following inequality: t

∫

ξ˙ T (s)E¯ T Z¯ E¯ ξ˙ (s)ds ≤ ηT (t)E¯ T Ω3 E¯ η(t),

− hM

(23)

t −hM

−Z¯ Z¯ − N¯ N¯ T [ T ] ¯ ¯ ¯ ¯ T ⎢ ∗ −2Z + N + N Z − N¯ where η(t) = ξ (t) ξ T (t − h(t)) ξ T (t − hM ) ξ T (t − T ) and Ω3 = ⎣ ∗ ∗ −Z ∗ ∗ ∗ It follows from (22) and (23) that E{LV (t , x(t))} ≤ ηT (t)E¯ T Ω4 E¯ η(t), [ ] [ ] [ ][ ] ¯ ¯ rci T Z¯ ¯ rci and hM A¯i hM A¯di 0 hM M where Ω4 = Ξ + hM A¯i hM A¯di 0 hM M 4×4 ⎧ N ∑ ⎪ ⎪ ⎪ Ξ¯ 1,1 = P¯i A¯i + A¯Ti P¯ i + Q¯ + R¯ + S¯ − E¯ T Z¯ E¯ + πij E¯ T P¯ j , ⎪ ⎪ ⎨ j=1 ¯ rci , Ξ¯ 1,2 = P¯ i A¯di + E¯ T Z¯ E¯ − E¯ T N¯ E¯ , Ξ¯ 1,3 = E¯ T N¯ E¯ , Ξ¯ 1,4 = P¯ i M ⎪ ⎪ ⎪ T ¯¯ T ¯ ¯ T T ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯T ¯ ¯ ¯T ¯ ¯ ¯ ¯ ⎪ Ξ = − (1 − µ ) Q − E Z E + E N E − E Z E + E N E , Ξ 2,3 = E Z E − E N E , ⎪ ⎩ 2 ,2 Ξ¯ 3,3 = −R¯ − E¯ T Z¯ E¯ , Ξ¯ 4,4 = −S¯ and remaining terms are 0.

⎡

⎤

0 0 ⎥ ⎦. 0 0

(24)

In Theorem 3.1, it is shown that the system (1) is asymptotically mean-square admissible, therefore, the augmented ¯ S, ¯ Z¯ closed-loop system (11) is mean-square asymptotically admissible, if there exist positive symmetric matrices P¯ i , Q¯ , R, ¯ and appropriate matrices N such that the following inequalities hold for every i ∈ M: E¯ T P¯ i =

P¯ i E¯ ≥ 0, Ω5 < 0,

[

Z¯

∗

N¯ Z¯

]

> 0,

(25)

[ ] Ξ¯ 5×5 with Ξ¯ 1,1 , Ξ¯ 1,2 , Ξ¯ 1,3 , Ξ¯ 1,4 , Ξ¯ 2,2 , Ξ¯ 2,3 , Ξ¯ 3,3 and Ξ¯ 4,4 are defined in (24) and Ξ¯ 1,5 = hM A¯Ti Z¯ , ¯ Trci Z¯ and Ξ¯ 5,5 = −Z¯ . We know that Ω5 < 0 if and only if LV (t , x(t)) < 0. = hM A¯Td Z¯ , Ξ¯ 4,5 = hM M

where Ω5 =

Ξ¯ 2,5

i

Note that the inequalities in (25) are not strict LMIs, to change the matrix inequality constraints into LMIs, let us choose the ¯ i < ϵ¯1 Xi , −Z¯ −1 < − Xi , following inequalities S¯ < ϵ¯1 P¯ i , Z¯ < ϵ¯2 P¯ i , then for any positive scalars ϵ¯q (q = 1, 2) we can have Xi SX ϵ¯ 2

where Xi = P¯ i−1 . We can employ the congruent transformation by choosing Ci Xqi = X¯ qi Ci , Xqi E T = EXqi (q = 1, 2), ¯ i, K¯ Pi X¯ 1i = W1i , Li X¯ 2i = W2i , K¯ Ri Y4 = W3i , Avi X1i = W4i , Bm B+T KDi Li X¯ 2i = W6i , Avi X2i = W5i , Y = S¯ −1 , Qˆ = Xi Q¯ Xi , Rˆ = Xi RX ¯ i , Zˆ = Xi Z¯ Xi , Nˆ = Xi NX ¯ i , where Qˆ = diag {Q1 , Q2 , Q3 , Q4 }, Rˆ = diag {R1 , R2 , R3 , R4 }, Sˆ = diag {S1 , S2 , S3 , S4 }, Sˆ = Xi SX Zˆ = diag {Z1 , Z2 , Z3 , Z4 } and Nˆ = diag {N1 , N2 , N3 , N4 }. Also, from (19) we have E T

P1i

< σi

T P1i P1i

α

α

, ET

P2i

β

< σi

T P2i P2i

for every

β β } i ∈ M. Denote X = diag Xi , Xi , Xi , Y , Z¯ −1 . Pre- and -post multiplying Ω5 by X and considering the above congruent ( T )−1 ( T )−1 P P P P transformation and applying Schur complement with the inequality − σj α1j α1j < −2α X1j + σj I, − σj β2j β2j <

{

α

−2β X2j + σj I for every j ∈ M, the matrix inequality Ω5 < 0 implies (18)–(21) hold. Then the augmented closed-loop system (11) is mean-square asymptotically admissible. Thus the proof is completed. ■ Remark 3.3. It should be pointed out that Theorem 3.2 provides a sufficient condition for the mean-square asymptotic admissibility of a closed-loop augmented MRCS in Fig. 1. The selection of α mainly deals K¯ Pi , the choice of β mainly deals KDi , Li and the selection of γ deals KRi . In particular, tuning parameters α and γ regulate the accurate tracking precision and tuning parameter β regulates the better disturbance rejection performance. Further, the tuning parameters not only used to deal the gain values but also produce a feasible solution of LMI (21). The suitable choice of adjusting parameters α , β and

γ can be calculated by minimizing the cost function Jc =

[∑ c

q=1

∫ qT (q−1)T

e2 (t)dt

] 21

(c is the number of periods), under the

constraints (18)–(21). Supposing that the LMI (21) is not solvable, by tuning the values of α , β and γ , it is possible to obtain a feasible solution. Remark 3.4. It should be noted that the conditions presented in (18) are not strict LMIs. To convert these conditions into LMIs, we can apply optimization technique [46] as [EXqi − XqiT E T ]T [EXqi − XqiT E T ] ≤ ρ I and [Ci Xqi − X¯ qi Ci ][Ci Xqi − X¯ qi Ci ]T ≤ ρ I (q = 1, 2) for a given small scalar ρ > 0. By using Schur complement, we can have the following LMIs for every i ∈ M

[

−ρ I EXqi − XqiT E T

∗ −I

]

< 0,

[

−ρ I Ci Xqi − X¯ qi Ci

∗ −I

]

<0

(26)

Remark 3.5. It should be mentioned that if KDi = I for every i ∈ M, then IEID estimator will be reduced to the EID estimator. In IEID estimator, KDi is inserted to adjust the value of ∆x(t) such that a better estimation of lumped disturbance Dli is obtained as well as a better tracking performance is achieved.

S. Mohanapriya, R. Sakthivel, O.M. Kwon et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 130–142

137

Fig. 2. Block diagram of a DC motor [47].

Remark 3.6. It should be pointed that the construction of Lyapunov–Krasovskii functional containing integral terms with delay interval limits plays a vital role in reducing conservatism for control system during stability and stabilization performance analysis. Also, it should be pointed out that the free-weighting-matrix based integral-inequality techniques can also be used to reduce conservatism for control systems since they avoid both the use of a model transformation and the technique of bounding cross terms. However, the introduction of Lyapunov–Krasovskii functional with triple integrals and free-matrix-based integral inequality can also increase the computational burden in the derivation of the main result. So, there should be a trade-off between the computational burden and less conservatism. Remark 3.7. It should be mentioned that the proposed IEID based modified repetitive control can be used to deal with a special case of singular Markovian jump systems with actuator fault as described in [14,15]. Since, it seems a practical issue to compensate the effect of actuator or sensor faults in such kind of Markovian jumping networked systems [14,15]. Moreover, the reliability of aforementioned Markovian jumping control systems can be improved by incorporating the combination of state observer with IEID low-pass filter in the proposed controller design. In particular, IEID approach is utilized to obtain the perfect estimation of the effects of actuator faults in the proposed controller design. Robust MRC algorithm based on IEID approach: Step 1 Let the time constant T be the period of the reference input and external disturbances and the cutoff angular frequency of the robust MRC is ωr . Step 2 Select Am , Bm and Cm for the low-pass filter (8) in such a way that |W (jω)| ≈ 1 holds. Step 3 The LMIs (19)–(21) and (26) can be solvable for tuning α , β and γ . Step 4 Calculate gain matrices K¯ Pi , KYi , KRi , Li , Avi and KDi for every i ∈ M from Theorem 3.2. 4. Simulation verification In this section, we provide a numerical example for SMJSs (1) to demonstrate the effectiveness and applicability of the proposed results. The numerical example shows the effectiveness of IEID-based robust MRC against the external disturbance and nonlinear uncertainties. Consider the DC motor driving model as shown in Fig. 2. The switching signal is focused by a Markov jump {σt , t ≥ 0} which is continuous and takes values in M = {1, 2}. The dynamics of DC motor model [47] can be described by the following singular model: E x˙ (t) = Ai x(t) + Bu(t), y(t) = Cx(t), where x(t) = [ω(t) i(t)]T is the state and y(t) = i(t) is the output; Ai = b

J

(27)

[ Bi −J i

Kt Ji

Kw

R

]

,B=

[ ]

[

0 1 ,E= 1 0

]

[ 0 ,C= 0 0

]

1 ;

Bi = BM + ci2 , Ji = JM + ci2 , and ω(t) is the rotational speed [rad/s]; i(t) is the armature current [A]; u(t) is the voltage n n [V]; Kt is the torque constant [Nm/A]; Kw is the electro-motive force constant [Vs/rad]; R is the resistance of armature coil [ohm]; JM , Jci are the moment of inertia [Kgm2 ] of motor and loads, respectively; BM and bci are the damping ratio of motor and loads, respectively; n is the gear ratio. Also, we borrow the remaining values of system parameters from [47] for the simulation purposes as R = 1 [ohm], Kt = 3 [Nm/A], Kw = 1 [Vs/rad], n = 10 and JM = 0.5 [kgm2 ], Jc1 = 50 [kgm2 ], Jc2 = 150 [kgm2 ], BM = 1, bc1 = 100, bc2 = 240. The elements of the transition probability matrix are chosen as π11 = −0.0193, π12 = 0.0193, π21 = 0.0307 and π22 = −0.0307. Without loss of generality time delay introduced as

138

S. Mohanapriya, R. Sakthivel, O.M. Kwon et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 130–142

Fig. 3. Tracking performance of the closed-loop system (17) with and without IEID.

in [47], then the singular time-delay system (1) has the following parameters

[ E =

1 0

]

[

0 −2 , A1 = 0 1

[ ] Bd1 = Bd2 =

3 −1.7 , A2 = 1 1

]

[ 0 , and C1 = C2 = 0 1

[

0 1.5 , Ad1 = Ad2 = 0.5 1

]

[

]

[ ]

0 0 , B1 = B2 = , 0 1

1 .

]

(28)

Moreover, let us take the reference input r(t), the disturbance wd (t) and the nonlinear uncertainty H(t , x(t)) as follows r(t) = 1 − 2|nint (t /4) − t /4|[A] ∀ t > 0, where nint is the nearest integer function,

wd (t) =

{

153 sin(π t) + 153 sin(3π t) − 763 tanh(t − 29) + 763 tanh(t − 30), 0,

15 ≤ t ≤ 45 , elsewhere.

H(t , x(t)) = x1 (t) sin(t). For given scalars α = 0.001, β = 10, γ = 0.01, ϵ1 = 0.001, ϵ2 = 3.2, ϵ3 = 1.5, ϵ4 = 0.009, ϵ5 = 0.001, ϵ6 = 2, ϵ7 = 1.999, ϵ8 = 0.0006, µ = 0.7, ρ = 0.00001, hM = 0.3937s, σ1 = 0.8541, σ2 = 0.6778 and h(t) = 0.19685 + 0.19685 sin (14t ) with the parameters of repetitive control system as T = 4s, ωr = 200 Hz and IEID filter parameters as Am = −251, Bm = 250, Cm = 1, by solving the LMIs in Theorem 3.2, we can obtain a set of feasible solutions. Furthermore, the output feedback control, observer and estimator [repetitive controller for ] [ gains for IEID-based ] −2.0001 3.0246 −494.9 , , Av1 = SMJSs are obtained as follows KY1 = −7.3742, KR1 = 273.2815, L1 = 1.0000 1.0051 2720.9

[ KD1 =

0.0 −0.1758

]

, KY2 = −56.9191, KR2 = 239.6815, L2 =

[

2619.7 −6093.4

] , Av2 =

[

−1.7 1.0000

1.5182 1.0053

] ,

0.0000 . The simulation results of the system with nonlinear uncertainty for tracking performance, 0.8446 control performance, estimated disturbances and tracking error via IEID-based repetitive controller are presented in Figs. 3– 5. In particular, the initial conditions for the singular and observer systems are chosen as φ (t) = [0.3 0]T and φˆ (t) = [0.3 0]T respectively. It is clear from Fig. 3 that the closed-loop system output IEID-based repetitive controller exactly tracks the reference signal without steady-state error. It should be mentioned that the selection of high cut-off angular frequency of low-pass filters Mr (s) and W (s) in repetitive control block and IEID block respectively plays an important role and yields better tracking and disturbance estimation performances. It is easy to see from Fig. 5 that the proposed IEID-based robust MRC estimates the external disturbance without steadystate error. Also, it is observed that due to gain matrix KDi , IEID approach not only increases the disturbance estimation performance but also provides better tracking performance. Fig. 6 provides the state vector xr (t) and the output vector yr (t) of robust MRC with and without IEID. From Fig. 6, it is easy to seen that the repetitive controller state and output trajectories performance are highly improved in the presence of IEID block. Furthermore, in order to emphasize the performance of the IEID-based MRCS against the nonlinear uncertainty, the disturbance and the nonlinear uncertainty are chosen as wd (t) = 0 and H(t , x(t)) = 5x1 (t) sin(t) respectively. Then, the corresponding simulation results are provided in Figs. 7–9. It is clear from Figs. 7 and 8 that the control effort and steady state tracking error are highly reduced while applying IEID-based robust repetitive controller instead of conventional MRC. The Fig. 9 displays the estimation of nonlinear uncertainty H(t , x(t)) and the corresponding Markov jump modes. Thus, the IEID estimator provides an efficient compensation of the external unknown nonlinear uncertainties. Moreover, to compare the advantage of the proposed technique with the existing result, we borrow the values

[

KD2 =

{

0.0000 −0.3631

0.0000 0.9665

]

w ˙ (t) = W w(t) + Md3 (t), wd (t) = V¯ w(t)[V ],

S. Mohanapriya, R. Sakthivel, O.M. Kwon et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 130–142

Fig. 4. Control performance of the closed-loop system Eq. (17) with and without IEID.

Fig. 5. Estimated disturbance using IEID and tracking error e(t) with and without IEID.

Fig. 6. State xr (t) and output yr (t) of modified repetitive controller with and without IEID.

Fig. 7. Tracking performance of the closed-loop system (17) with and without IEID.

139

140

S. Mohanapriya, R. Sakthivel, O.M. Kwon et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 130–142

Fig. 8. Control signals of the closed-loop system Eq. (17) and tracking error e(t) with and without IEID.

Fig. 9. Estimated nonlinear H(t , x(t)) using IEID and the corresponding Markov jump mode.

Fig. 10. Estimated disturbance using IEID & DO and the corresponding Markov jump mode.

[ ] 0 1 0.1 25 sin(π t), 5 ≤ t ≤ 45 , V¯ = 1 0 , M = , and d3 (t) = , H(t , x(t)) = x1 (t) sin t. −1 0 0.3 0, elsewhere. as in [45] in addition to the parameters (28) and F = [1 0]T , H = [0 0]T . By letting λ = 1, γ = 1, then by solving the LMIs in Theorem 1 of [45], [ ] the gain of the observer and controller [ can be obtained as K]1 = [−4.0000 − 2.9564] , L1 = 0.1373 −0.3809 0.1369 −0.3821 , K2 = [−2.5000 − 2.9508] , L2 = . To be specific, Fig. 10 shows the −0.4650 −0.3750 −0.4655 −0.3743 [

]

[

]

{

where W =

exact lumped disturbance and its estimation by the designed IEID-based controller and the disturbance observer (DO)based method [45]. From Fig. 10, it is confirmed that the proposed IEID-based controller method delivers better disturbance estimation performance than that of the DO-based method. Further, to describe the advantage of IEID approach with the EID based approach we can consider KD1 = KD2 = diag {1, 1}. For given scalars α = 1, β = 10, γ = 0.01, ϵ1 = 0.001, ϵ2 = 3.2, ϵ3 = 0.1, ϵ4 = 0.00009, ϵ5 = 0.001, ϵ6 = 0.01, ϵ7 = 1.999, ϵ8 = 0.00006, µ = 0.7, ρ = 0.00001 and

h = 0, by solving the LMIs in Theorem a set the following [ 3.2, we can obtain ] [ of feasible solutions with ] [ ]gain matrices KY1 = 140.0589 −2.0001 3.6635 1 0 −121.1574, KR1 = 903.2333, L1 = , Av1 = , KD 1 = , KY2 = −91.7233, −105.0996 1.0006 4.341 0 1

S. Mohanapriya, R. Sakthivel, O.M. Kwon et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 130–142

141

Fig. 11. Error signal e(t) using IEID & EID.

KR2 = 918.5622, L2 =

[

162.3194 −67.9272

[

] , Av2 =

−1.6999 1.0006

2.1413 4.2285

] , KD 2 =

[

1 0

0 1

] . Also, Fig. 11 displays that the

error obtained by using IEID approach is below 0.5 which is better than 33% of the error obtain by EID based approach. It is concluded from the simulation that the IEID-based robust MRC technique not only rejects the external disturbances but also rejects the effect of input nonlinear uncertainties and tracks the periodic reference without steady-state error in the control systems. The results reveal that an IEID-based robust modified repetitive controller produces high accuracy of estimation, active compensation for the total disturbance and also improves the tracking performance for the DC motor driving model. 5. Conclusion In this study, we have discussed the periodic output tracking performance of SMJSs with time-varying delay and nonlinear uncertainties by employing the IEID-based robust modified repetitive controller. By integrating a novel Lyapunov–Krasovskii functional together with LMI approach, a new set of sufficient conditions has been developed for ensuring the meansquare asymptotic tracking performance of the considered SMJSs and for obtaining the control design parameters. Finally, a numerical example and its simulation results have been provided to exhibit the advantages of the proposed control design technique. The results reveal that the proposed IEID-based robust modified repetitive controller is conceptually simple and impressive due to its accurate tracking performance. Moreover, in the future work we plan to use the proposed technique to the UDP-like systems [48,49], which are a special type of Markov jump systems with unknown jump mode and multiple delays. Acknowledgements This research work of S. Mohanapriya is financially supported by DST Women Scientist Scheme [Grant No: SR/WOS-A/PM91/2016(G)], Government of India, New Delhi. References [1] S. Marir, M. Chadli, D. Bouagada, A novel approach of admissibility for singular linear continuous-time fractional-order systems, Int. J. Control Autom. Syst. 15 (2) (2017) 959–964. [2] M. Chadli, H.R. Karimi, P. Shi, On stability and stabilization of singular uncertain Takagi–Sugeno fuzzy systems, J. Franklin Inst. 351 (3) (2014) 1453–1463. [3] Y. Liu, Y. Kao, H.R. Karimi, Z. Gao, Input-to-state stability for discrete-time nonlinear switched singular systems, Inform. Sci. 359 (2016) 18–28. [4] S. Santra, R. Sakthivel, K. Mathiyalagan, S. Marshal Anthoni, Exponential passivity results for singular networked cascade control systems via sampleddata control, J. Dyn. Sys. Meas. Control 139 (3) (2017) 031001. [5] Z.G. Wu, H. Su, P. Shi, J. Chu, Analysis and Synthesis of Singular Systems with Time-Delays, Springer, Berlin, 2013. [6] J. Li, H. Su, Z. Wu, J. Chu, Robust stabilization for discrete-time nonlinear singular systems with mixed time delays, Asian J. Control 14 (5) (2012) 1411–1421. [7] M. Fang, Ju H. Park, Non-fragile synchronization of neural networks with time-varying delay and randomly occurring controller gain fluctuation, Appl. Math. Comput. 219 (15) (2013) 8009–8017. [8] C. Aouiti, Neutral impulsive shunting inhibitory cellular neural networks with time-varying coefficients and leakage delays, Cogn Neurodyn. 10 (6) (2016) 573–591. [9] C. Aouiti, Oscillation of impulsive neutral delay generalized high-order Hopfield neural networks, Neural Comput. Appl. 29 (9) (2018) 477–495. [10] C. Aouiti, I.B. Gharbia, J. Cao, M.S. M’hamdi, A. Alsaedi, Existence and global exponential stability of pseudo almost periodic solution for neutral delay BAM neural networks with time-varying delay in leakage terms, Chaos Solitons Fractals 107 (2018) 111–127. [11] H. Que, M. Fang, Z.G. Wu, H. Su, T. Huang, D. Zhang, Exponential synchronization via aperiodic sampling of complex delayed networks, IEEE Trans. Syst. Man Cybern. A http://dx.doi.org/10.1109/TSMC.2018.2858247. [12] C. Aouiti, E.A. Assali, J. Cao, A. Alsaedi, Global exponential convergence of neutral-type competitive neural networks with multi-proportional delays distributed delays and time-varying delay in leakage delays, Internat. J. Systems Sci. 49 (10) (2018) 2202–2214.

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