Sliding mode control for networked systems with randomly varying nonlinearities and stochastic communication delays under uncertain occurrence probabilities

Communicated by Dr. Ma Lifeng Ma

Accepted Manuscript

Sliding mode control for networked systems with randomly varying nonlinearities and stochastic communication delays under uncertain occurrence probabilities Panpan Zhang, Jun Hu, Hongjian Liu, Changlu Zhang PII: DOI: Reference:

S0925-2312(18)30993-7 https://doi.org/10.1016/j.neucom.2018.08.043 NEUCOM 19891

To appear in:

Neurocomputing

Received date: Revised date: Accepted date:

28 May 2018 16 August 2018 20 August 2018

Please cite this article as: Panpan Zhang, Jun Hu, Hongjian Liu, Changlu Zhang, Sliding mode control for networked systems with randomly varying nonlinearities and stochastic communication delays under uncertain occurrence probabilities, Neurocomputing (2018), doi: https://doi.org/10.1016/j.neucom.2018.08.043

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ACCEPTED MANUSCRIPT

Sliding mode control for networked systems with randomly varying nonlinearities and stochastic communication delays under uncertain occurrence probabilities

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Panpan Zhang a , Jun Hu a,b,∗ , Hongjian Liu c , Changlu Zhang a

Department of Mathematics, Harbin University of Science and Technology, Harbin 150080, China b

School of Mathematics and Physics, Anhui Polytechnic University, Wuhu 241000, China

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c

School of Engineering, University of South Wales, Pontypridd CF37 1DL, UK

Abstract

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In this paper, we aim to propose the robust sliding mode control (SMC) scheme for discrete networked systems subject to randomly occurring uncertainty (ROU), randomly varying nonlinearities (RVNs) and multiple stochastic communication delays (MSCDs). Here, a series of mutually independent Bernoulli distributed random variables is introduced to model the phenomena of the ROU, RVNs and MSCDs, where the occurrence probabilities of above phenomena are allowed to be uncertain. For the addressed systems, an SMC strategy is given such that, for above network-induced phenomena, the stability of the resulted sliding motion can be guaranteed by presenting a new delay-dependent sufficient criterion via the delay-fractioning method. Moreover, the discrete sliding mode controller is synthesized such that the state trajectories of the system are driven onto a neighborhood of the specified sliding surface and remained thereafter, i.e., the reachability condition in discrete-time setting is verified. Finally, the usefulness of the proposed SMC method is illustrated by utilizing a numerical example.

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Key words: Sliding mode control; Networked systems; Randomly varying nonlinearities; Multiple stochastic communication delays; Uncertain occurrence probabilities.

Introduction

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Over the past several decades, as an effective control strategy, the sliding mode control (SMC) technique has been extensively applied in many practical domains due to its superiorities, such as the robustness against parametric uncertainties and unknown disturbances on the prescribed sliding surface. Accordingly, the application problems of SMC have gained certain research attention, such as in dynamical networks, robotics and spacecraft [1–4]. As such, many important SMC methods have been developed for various systems according to their

Preprint submitted to Neurocomputing

structural characteristics, see e.g. [5,6]. For example, the SMC problem has been studied in [5] for uncertain delayed multi-agent systems with external disturbances. It is worthy of noting that, since most control algorithms have been implemented digitally in modern industrial fields, the discrete-time SMC design of control systems has increasingly become prevalent [7–11]. For example, the quasi-sliding mode idea has been proposed in [9] and the SMC problems for complex systems based on the quasi-sliding mode have been extensively addressed. For instance, based on the method in [9], the reaching law

5 September 2018

ACCEPTED MANUSCRIPT the so-called randomly occurring nonlinearities (RONs) have been modeled in [28] to depict the stochastic disturbances and the randomly occurring uncertainty (ROU) has been characterized in [29]. In [10, 29], the robust SMC algorithms have been designed for networked systems with RONs based on the delay fractioning idea, where new stochastic stability criteria have been presented for resulted sliding motion. In addition, due to the unreliability of the networks or other reasons, the randomly varying nonlinearities (RVNs) have been modeled from a mathematical viewpoint. Accordingly, the effects of the RVNs have been discussed and some effective analysis methods have been given for various systems, such as the state estimation method [30], fault detection scheme and H∞ SMC approach [31, 32]. To the best of our knowledge, the robust SMC problem has not been thoroughly studied for discrete networked systems with ROU, RVNs and MSCDs, not to mention the case that the occurrence probabilities are allowed to be uncertain. Consequently, we decide to further investigate the SMC problem for networked systems with RVNs under uncertain occurrence probabilities, and propose a new robust SMC strategy by resorting to the delay-fractioning idea.

with the disturbance compensation scheme has been developed in [11] for a class of uncertain discrete-time systems.

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As it is well known, the time-delays are often encountered in a large number of engineering systems probably because of the limited bandwidth and speed of the information processing. In fact, the existence of time-delays inevitably deteriorates the system performance [12, 13]. Therefore, the problems of analysis and synthesis for delayed systems have been investigated with hope to attenuate the negative effects of time-delays. Regarding the mixed time-delays, an efficient SMC method has been presented in [14] for discrete Markovian jump systems by employing the delay-fractioning approach. Actually, it is more common that the stochastic communication delays occur when the signals are transmitted by communication channels [15, 16]. Recently, the robust control methods have been proposed in [17, 18] for nonlinear systems with MSCDs via the sliding mode conception. However, most of the existing results suppose that the occurrence probabilities of stochastic communication delays are deterministic, which would be a bit conservative in some cases, e.g., in the networked setting. Compared to the existing results, the MSCDs considered in this paper are allowed to have uncertain occurrence probabilities, which really reflects the case in practical applications. Besides, the delay-fractioning method is employed to handle such MSCDs with uncertain occurrence probabilities with hope to reduce the conservativeness. Hence, we aim to characterize the phenomenon of MSCDs with uncertain occurrence probabilities in a mathematical way firstly and then reveal the resulted effects onto the control systems. Hence, we aim to characterize the phenomena of MSCDs with uncertain occurrence probabilities in a mathematical way and reveal the resulted effects onto the control system design.

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On the other hand, it is well recognized that both additive uncertainties and nonlinearities may lead to deteriorate the desired system performance if not appropriately addressed [19–23]. As such, a great number of techniques have been proposed for uncertain nonlinear systems, see e.g. [24–26]. For example, a neural-networkbased SMC method has been developed in [27] for delayed systems with unknown nonlinearities. Recently,

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In this paper, the robust SMC problem is addressed for networked systems subject to ROU, RVNs and MSCDs under uncertain occurrence probabilities. Firstly, a linear switching surface is designed. Next, a delay-dependent sufficient criterion is established by the Lyapunov stability theorem to ensure the robustly asymptotical mean-square stability of the resulted sliding mode dynamics. Besides, the original non-convex problem is transformed into an optimal one by introducing a computational algorithm. The novelties of the paper lie in that: (i) we examine the effects from the network-induced phenomena of the ROU, RVNs as well as MSCDs in a same framework; (ii) we make the first attempt to discuss the SMC problem for networked systems with MSCDs under uncertain occurrence probabilities, and the delay-fractioning approach is employed to better address such type of MSCDs with uncertain occurrence probabilities; and (iii) the information of addressed network-induced phenomena has been fully taken into account in order to achieve desired performance, and then a new SMC algorithm has been obtained to examine the effects from the ROU, RVNs and

ACCEPTED MANUSCRIPT MSCDs under uncertain occurrence probabilities onto whole control system performance. At last, we propose numerical simulations to illustrate the usefulness of the presented control method.

(Ω, F, P) with

Notations. The following notations will be used throughout this paper. (Ω, F, P) denotes a probability space, where Ω is a sample space, F is the σ-algebra of subsets of Ω, and P is the probability measure on F. E{x} is the mathematical expectation of random variable x. The superscript “T” denotes the matrix transposition. Rn and Rn×m represent, respectively, the ndimensional Euclidean space and the set of all n×m real matrices. P > 0 (P ≥ 0) means that P is a symmetric and positive definite (positive semi-definite) matrix. I and 0 represent the identity matrix and the zero matrix with proper dimensions. diag{Y1 , Y2 , . . . , Yn } stands for a block diagonal matrix where the diagonal blocks are matrices Y1 , Y2 , . . . , Yn . col {. . . } represents a vector column with blocks given by the vectors in {. . . }. k · k represents the Euclidean norm of a vector and its induced norm of a matrix. The star (∗) in a matrix denotes the term that can be induced by the symmetry. Matrices, if not explicitly stated, are assumed to be compatible for algebraic operations.

The unknown matrix ∆A stands for the parameter uncertainty satisfying

E{ωk2 } = 1, E{ωk ωj } = 0 (k 6= j).

∆A = HF E, F T F ≤ I,

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where H and E are known matrices of appropriate dimensions, and F is an unknown matrix. The nonlinear function gi (x) : Rn −→ Rn (i = 1, 2) represents the mismatched nonlinearity satisfying the following sector-bounded condition: [gi (x) − F1i x]T [gi (x) − F2i x] ≤ 0, ∀x ∈ Rn ,

(2)

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where F1i and F2i are known matrices and Fi = F1i − F2i > 0.

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The mutually independent random variables αk , βk,i (i = 1, 2, . . . , q) and γk , which are employed to model the phenomena of ROU, RVNs and MSCDs, satisfy the Bernoulli distribution with

Problem Formulation and Preliminaries

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In this paper, we consider the following class of discrete uncertain delayed nonlinear systems:

Prob{γk = 1} = E{γk } = γ¯ + ∆γ,

Prob{γk = 0} = 1 − E{γk } = 1 − (¯ γ + ∆γ),

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+γk D1 g1 (xk ) + (1 − γk )D2 g2 (xk ) + E1 xk ωk , q X x ek = βk,i xk−τk,i , i=1

Prob{αk = 0} = 1 − E{αk } = 1 − (¯ α + ∆α), Prob{βk,i = 1} = E{βk,i } = β¯i + ∆βi ,

Prob{βk,i = 0} = 1 − E{βk,i } = 1 − (β¯i + ∆βi ),

xk+1 = (A + αk ∆A)xk + Aτ x ek + B(uk + f (xk )) xk = φk , k = [−τM , 0],

Prob{αk = 1} = E{αk } = α ¯ + ∆α,

(3)

where α ¯ + ∆α, β¯i + ∆βi and γ¯ + ∆γ ∈ [0, 1], α ¯ , β¯i and γ¯ are known scalars. Here, ∆α, ∆βi and ∆γ depict the uncertain occurrence probabilities satisfying |∆α| ≤ 1 , |∆βi | ≤ 2 and |∆γ| ≤ 3 with j (j = 1, 2, 3) being nonnegative scalars. Obviously, we have 0 ≤ 1 ≤ min{¯ α, 1− α ¯ }, 0 ≤ 2 ≤ min{β¯i , 1− β¯i } and 0 ≤ 3 ≤ min{¯ γ , 1− γ¯ }.

(1)

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where xk ∈ Rn is the system state, uk ∈ Rm stands for the control input, τk,i ∈ [τm , τM ] (i = 1, 2, . . . , q) are time-varying delays, where τM and τm are known bounds, respectively. Besides, the lower upper τm can be expressed by τm = τ p with τ and p being integers. f (xk ) is a bounded nonlinearity in Euclidean norm. A, Aτ , B, D1 , D2 and E1 are known matrices, among which the matrix B is assumed to have full column rank. ωk ∈ R1 is a zero-mean Wiener process on a probability space

Remark 1 In (3), the Bernoulli distributed random variables αk , βk,i (i = 1, 2, . . . , q) and γk are introduced to respectively characterize the so-called ROU, MSCDs and RVNs. It should be mentioned that MSCDs are employed to describe the possible existence of different delays during the signals transmission via the various communication channels. Here, we make the attempt to

3

ACCEPTED MANUSCRIPT that Q + εHH T + ε−1 E T E < 0 or,

characterize the case when the system in reality is unavoidably influenced by various types of network-induced phenomena and the occurrence probabilities could be uncertain due to some reasons [33, 34]. To be specific, the scalars ∆α, ∆βi (i = 1, 2, . . . , q) and ∆γ are used to depict the uncertain occurrence probabilities due probably to case that the exact probability information may not be easily obtained. In reality, by the repeated statistical tests, it is not technically difficult to obtain the corresponding information of RVNs with uncertain occurrence probabilities (i.e., the occurrence probabilities and the related upper bounds). Subsequently, the major effort is made to propose new robust SMC scheme for addressed networked systems against the ROU, RVNs and MSCDs under uncertain occurrence probabilities.



εH E T

Q



     εH T −εI 0  < 0.     E 0 −εI Lemma 3 (Schur complement) Given constant matrices U1 , U2 , U3 where U1 = U1T and U2 = U2T > 0, then U1 + U3T U2−1 U3 < 0 if and only if 



 −U U 2 3    < 0. ∗ U1

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 T U U  1 3   <0 ∗ −U2

or

Lemma 4 [35] Let M ∈ Rn×n be a positive semidefinite matrix. For xi ∈ Rn and ai ≥ 0 (i = 1, 2, . . . ), if the series concerned are convergent, then

Remark 2 It is worthwhile to notice that the considered q different communication delays τk,i (i = 1, 2, . . . , q) in (1) are time-varying and our aim is to propose new control method by attenuating the effects from the communication delays. In order to simplify the design idea of SMC, we let the upper/lower bounds of communication delays be same with the upper bound τM and lower bound τm . In fact, if we consider more general situations (e.g. τk,i ∈ [τm,i , τM,i ]), one possible common upper/lower bounds could be obtained easily by setting τm = min {τm,i }, τM = max {τM,i }.

T

X +∞

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X +∞ i=1

3

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1≤i≤q

1≤i≤q

ai xi

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The objective of the paper is to propose a robust SMC strategy such that, for all ROU, MSCDs and RVNs, the robust mean-square asymptotical stability of the resulted sliding motion is ensured by proposing a delaydependent sufficient criterion.

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To proceed, we introduce the following necessary lemmas.

ai xi

i=1



≤

X +∞ i=1

ai

X +∞

ai xTi M xi .

i=1

Main Results

In this section, we aim to discuss the robust SMC synthesis problem for addressed networked systems with MSCDs, RVNs and ROU. Firstly, a linear switching surface is introduced. subsequently, the robust stability analysis problem of the corresponding sliding motion is discussed and a delay-dependent sufficient condition is given in terms of the delay fractioning method. In what follows, an LMIs-based minimization problem is given by introducing a computational algorithm. Moreover, the discrete-time reachability condition is tested by proposing a robust sliding mode controller. 3.1

Lemma 1 For a, b and P > 0 of compatible dimensions, we have

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Switching surface

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In the paper, choose the following linear sliding surface: sk = Gxk − GAxk−1 ,

aT b + bT a ≤ aT P a + bT P −1 b.

(4)

where real matrix G ∈ Rm×n is to be determined b = 0 with such that GB is non-singular and GD  b := D D D E . In what follows, we select

Lemma 2 (S-procedure) Let Q = QT , H and E be real matrices of compatible dimensions, and uncertain matrix F satisfies F T F ≤ I. Then Q + HF E + E T F T H T < 0 holds, if and only if there exists a positive constant ε such

1

T

2

1

G = B P with P > 0 to guarantee the non-singularity of GB.

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sk+1 = sk = 0.

(5)

Then, it follows from (1), (4) and (5) that the equivalent controller can be given by:  q X −1 ueq = −(GB) G (¯ α + ∆α)∆Axk + Aτ (β¯i + ∆βi ) i=1



×xk−τk,i − f (xk ).

where

(6)

0 0 0 0 −P ∗



0    0    0   <0, (9)  0    PH    −εI

b B T P D=0, (10)

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bT b √ eT  Λ + εE E τM X dZj ΞA P B   ∗ −ρP 0 0     ∗ ∗ −ρP 0     ∗ ∗ ∗ −B T P B    ∗ ∗ ∗ ∗   ∗ ∗ ∗ ∗

It is worthy of noting that the ideal quasi-sliding mode satisfies the following condition

Substituting (6) into (1) yields the following sliding mode dynamics equation:

˜ TA P Ξ ˜A + Ξ ˜T P Ξ ˜ E + ΞT [(2ρ~ − 1)P + Q]ΞI Λ = 8Ξ E I T˜ T +Ξ Φτ Ξτ + W RWR + W T SWS

xk+1 = Ak − B(GB)−1 GAk + B(GB)−1 GAxk q X +Aτ [βk,i − (β¯i + ∆βi )]xk−τk,i

Q = (τM − τm + 1)

+[αk − (¯ α + ∆α)]∆Axk + γk D1 g1 (xk )

+(1 − γk )D2 g2 (xk ) + E1 xk wk ,

(7)

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where q X

R S T T T +γ1 Ξg1 D1 P D1 Ξg1 + γ2 Ξg2 D2T P D2 Ξg2 +ΠTg1 F1 Πg1 + ΠTg2 F2 Πg2 , q X

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i=1

(β¯i + ∆βi )xk−τk,i .

i=1

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Ak = [A + (¯ α + ∆α)∆A]xk + Aτ

τ

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Now, we analyze the robust asymptotic stability in mean square of the resulted closed-loop system (7) via the Lyapunov stability theorem.

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Theorem 1 Given a constant ρ ∈ (0, 1), if there exist matrices P > 0, Qj > 0, R > 0, S > 0, real matrices X , Yj and Zj (j = 1, 2, . . . , q), and scalar ε > 0 satisfying

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eT bT b √  Λ + εE E τM X dYj ΞA P B   ∗ −ρP 0 0     ∗ ∗ −ρP 0     ∗ ∗ ∗ −B T P B    ∗ ∗ ∗ ∗   ∗ ∗ ∗ ∗

0 0 0 0 −P ∗

 0    0    0   <0, (8)  0    PH    −εI 5

j=1

Qj ,

  ΞI = In×n 0n×(m+q+3)n ,   ˜ A = hA ˆ n×n 0n×(m+q+3)n , Ξ   ˜ ˆ 1 )n×n 0n×(m+q+3)n , ΞE = h(E   Ξτ = 0qn×(m+1)n Iqn×qn 0qn×3n ,   I 0 n×n n×(m+q+3)n   WS =  , 0n×(m+q+1)n In×n 0n×2n   Imn×mn 0mn×(q+4)n   WR =  , 0mn×n Imn×mn 0mn×(q+3)n   Ξg1 = 0n×(m+q+2)n In×n 0n×n ,   Ξg2 = 0n×(m+q+3)n In×n , R = diag {R, −R} ,   In×n 0n×(m+q+3)n   Πg1 =  , S =diag {S, −S} , 0n×(m+q+2)n In×n 0n×n   " q # X I 0  n×n n×(m+q+3)n  b Πg 2 =  (β¯i + 2 ) , ,β = 4 i=1 0n×(m+q+3)n In×n n o ˜ τ = diag Φ ˜ 1τ , Φ ˜ 2τ , . . . , Φ ˜ qτ , Φ ˜ jτ = βj AT P Aτ − Qj , Φ τ

ACCEPTED MANUSCRIPT 

bT =  E 

α1 E

T

0(m+q+3)n×nF





 ˜ ¯   −Fi Fi   , Fi =   (i = 1, 2), ∗ −2I

+[1 − (¯ γ + ∆γ)]g2T (xk )D2T P D2 g2 (xk ) − xTk P xk +xTk E1T P E1 xk

i=1

T T T T F˜i = F1i F2i + F2i F1i , F¯i = F1i + F2i , p α1 = (2ρ~ + 1)(α ¯ + 1 )(7¯ α + 91 + 1), ¯ b βj = (2ρ~ + 1)(βj + 2 )(β + 1 − β¯j + 2 ),

×ATτ P Aτ

γ1 = 2(2ρ~ + 1)(¯ γ + 3 ), γ2 =2(2ρ~ + 1)(1 − γ¯ + 3 ), p √ ˆ h = 2ρ~ + 1, ~ = 2τM − τm , d = τM − τm , (11)

≤(¯ γ +∆γ)ATk P Ak +(¯ γ +3 )g1T (xk )D1T P D1 g1 (xk ), (14) 2[1 − (¯ γ + ∆γ)]ATk P D2 g2 (xk )

≤ [1 − (¯ γ + ∆γ)]ATk P Ak + [1 − (¯ γ + ∆γ)]g2T (xk )D2T

(12)

×P D2 g2 (xk )

0 X

k−1 X

ηlT P ηl + ρ

j=ˆ τM l=k−1+j

−τ Xp

k−1 X

j=ˆ τM l=k−1+j

ηl = xl+1 − xl , τˆM = −τM + 1

ATk P Ak ≤ 2xTk [A + " q X +2

i=1

ηlT P ηl ,

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 Γl = col xl , xl−τ , . . . , xl−(m−1)τ ,

×D2 g2 (xk ),

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V1k = xTk P xk , k−1 k−1 X X V2k = ΓTl RΓl , V3k = xTl Sxl , l=k−τ l=k−τM q q −τ k−1 k−1 m X X X X X V4k = xTi Qj xi + xTi Qj xi , j=1 i=k−τk,j j=1 s=ˆ τM i=k+s

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≤ [1 − (¯ γ + ∆γ)]ATk P Ak + (1 − γ¯ + 3 )g2T (xk )D2T P

where

E {∆V1k } (

=E

ATk P Ak

−

ATk GT (GB)−1 GAk

i=1

≤ 4xTk AT P Axk + 4(¯ α + ∆α)2 xTk ∆AT P ∆Axk " q # q X X +2 (β¯i + ∆βi ) (β¯i + ∆βi )xT AT P Aτ τ

i=1

≤ 4xTk AT P Axk + 4(¯ α + 1 )2 xTk ∆AT P ∆Axk q βb X ¯ (βi + 2 )xTk−τk,i ATτ P Aτ xk−τk,i . + 2 i=1

(16)

Notice that

E{∆Vik },

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where

τ

×xk−τk,i

PT

i=1

×xk−τk,i

k,i

#

k−τk,i

CE

5 X

(15)

(¯ α + ∆α)∆A]T P [A + (¯ α + ∆α)∆A]xk #T " q X T ¯ (βi + ∆βi )xk−τ A P Aτ (β¯i + ∆βi )

i=1

with P > 0, Qj > 0 (j = 1, 2, . . . , q), R > 0 and S > 0 being matrices to be designed. Then, let’s calculate the difference of Vk along (7) E{∆Vk } =

(13)

≤(¯ γ +∆γ)ATk P Ak +(¯ γ +∆γ)g1T (xk )D1T P D1 g1 (xk )

i=1

V5k = ρ

i=1

!)

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Vik ,

q X [βk,i − (β¯i + ∆βi )]xk−τk,i

!T

2(¯ γ +∆γ)ATk P D1 g1 (xk )

Proof : Firstly, we choose the following LyapunovKrasovskii functional: 5 X

[βk,i − (β¯i + ∆βi )]xk−τk,i

with Ak being defined below (7). By using Lemmas 1 and 4, it follows that

then the sliding mode dynamics (7) is robustly asymptotically stable in mean square sense.

Vk =

+

q X

 X q T E [βk,i − (β¯i + ∆βi )]xk−τk,i ATτ P Aτ i=1

× + 2(¯ γ + ∆γ)

q X i=1

[βk,i − (β¯i + ∆βi )]xk−τk,i



q nX o =E [βk,i − (β¯i + ∆βi )]2 xTk−τk,i ATτ P Aτ xk−τk,i

×ATk P D1 g1 (xk ) + 2[1 − (¯ γ + ∆γ)]ATk P D2 g2 (xk )

i=1

q nX o ≤E (β¯i +2 )(1−β¯i +2 )xTk−τk,i ATτ P Aτ xk−τk,i ,(17)

+xTk AT GT (GB)−1 GAxk + (α ¯ + ∆α)[1 − (¯ α + ∆α)]

×xTk ∆AT P ∆Axk + (¯ γ + ∆γ)g1T (xk )D1T P D1 g1 (xk )

i=1

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ACCEPTED MANUSCRIPT Similarly, it can be derived that

then it follows from (13)-(17) that

E {∆V2k } ( k X =E

E {∆V1k }  ≤ E 8xTk AT P Axk + 8(α ¯ + 1 )2 xTk ∆AT P ∆Axk

l=k+1−τ  T = E Γk RΓk −

(β¯i + 2 )xTk−τk,i ATτ P Aτ xk−τk,i + xTk AT GT −1

1 )xTk ∆AT P

×(GB)

GAxk + (α ¯ + 1 )(1 − α ¯+ q X ×∆Axk + (β¯i + 2 )(1 − β¯i + 2 )xTk−τk,i ATτ P Aτ

l=k+1−τ

×xk−τk,i + 2(¯ γ + 3 )g1T (xk )D1T P D1 g1 (xk )

E {∆V4k } X q k X =E

+2(1 − γ¯ + 3 )g2T (xk )D2T P D2 g2 (xk ) + xTk E1T P E1  ×xk − xTk P xk n = E 8xTk AT P Axk + xTk AT GT (GB)−1 GAxk + xTk E1T i=1

βei

j=1 i=k+1−τk+1,j

+

+

−

CE xTk−τk,1

xTk−τk,q

xTk−τM

q X

l=k−τ

(20)

q X

k−1 X

xTi Qj xi

j=1 i=k−τk,j

  xTk Qj xk − xTk+s Qj xk+s

k−τ Xm

q X

k−τ Xm

xTi Qj xi + (τM − τm )

xTi Qj xi

j=1 i=k+1−τM



q X

xTk Qj xk

j=1

 q q X X = E (τM − τm + 1) xTk Qj xk − xTk−τk,j Qj


j=1



×xk−τk,j ,   E ∆V5k

 =E ρ +ρ

j=1

(21)

0 X

k X

j=−τM +1 l=k+j −τ Xp

k X

ηlT P ηl − ρ

ηlT P ηl

j=−τM +1 l=k+j

−ρ

0 X

k−1 X

ηlT P ηl

j=−τM +1 l=k−1+j −τ Xp

k−1 X

j=−τM +1 l=k−1+j

ηlT P ηl



 k−1 X = E ρτM ηkT P ηk − ρ ηlT P ηl + ρ(τM − τm )ηkT P ηk

T

, ... n o Φτ = diag βe1 ATτ P Aτ , βe2 ATτ P Aτ , . . . , βeq ATτ P Aτ , p ¯ + 1 )2 + (α ¯ + 1 )(1 − α ¯ + 1 ), α e = 8(α e ¯ b ¯ βi = (βi + 2 )(β + 1 − βi + 2 ), γ e1 = 2(¯ γ + 3 ), xTk−τm

AC

ξk,τ =



−τ m X

j=1 i=k+1−τM

PT

T ξk = ΓTk ξk,τ , g1T (xk ) g2T (xk )   ΞA = An×n 0n×(m+q+3)n ,   ΞE = (E1 )n×n 0n×(m+q+3)n ,   ΞδA = α e∆An×n 0n×(m+q+3)n ,

xTl Sxl

)

j=1

M

T

q X

j=1 s=−τM +1

ED



k−1 X

X  q  T T ≤E xk Qj xk − xk−τk,j Qj xk−τk,j

×xTk−τk,i ATτ P Aτ xk−τk,i + γ e1 g1T (xk )D1T P D1 g1 (xk ) o +e γ2 g2T (xk )D2T P D2 g2 (xk ) n  = E ξkT 8ΞTA P ΞA + ΞTA GT (GB)−1 GΞA + ΞTE P ΞE − ΞTI  ×P ΞI + ΞTδA P ΞδA + ΞTτ Φτ Ξτ ξk + γ e1 g T (xk )D1T P D1 o 1 ×g1 (xk ) + γ e2 g2T (xk )D2T P D2 g2 (xk ) , (18)

where

(19)

xTi Qj xi −

AN US

×P E1 xk − xTk P xk + α e2 xTk ∆AT P ∆Axk +

xTl Sxl −

)

M  M = E xTk Sxk − xTk−τM Sxk−τM ,

i=1

q X

ΓTl RΓl

l=k−τ T Γk−τ RΓk−τ ,

E {∆V3k } ( k X =E

i=1

−

k−1 X

CR IP T

+βb

q X

ΓTl RΓl

l=k−τM

−

k−τ m −1 X l=k−τM

(

=E

γ e2 = 2(1 − γ¯ + 3 ),

1 − q

and Ξτ , ΞI as well as βb are defined in (11). 7

ηlT ρP ηl

ηkT ρ~P ηk

−

k−τ m −1 X

l=k−τk,1



k−1 X

ηlT ρP ηl

l=k−τM

ηlT ρP ηl +

k−τk,1 −1

X

l=k−τM

ηlT ρP ηl

!

ACCEPTED MANUSCRIPT

ηlT ρP ηl

+

l=k−τk,2 k−τ m −1 X

1 − q

k−τk,2 −1

X

ηlT ρP ηl

!

ηlT ρP ηl

!)

l=k−τM

ηlT ρP ηl

+

l=k−τk,q

k−τk,q −1

X

l=k−τM

where   ˜ δ = α ∆A Ξ A 1 n×n 0n×(m+q+3)n , (22)

and other matrices as well as parameters can be found in (11). By considering (2), it is not difficult to obtain that

with ~ = 2τM − τm . Then, it follows from (19)-(22) that 5 X i=2

(



  ˜ ¯  xk   −Fi Fi   xk       ≥ 0 (i = 1, 2), (25) gi (xk ) ∗ −2I gi (xk )

E {∆Vik }

T T T T . Further+ F2i F1i and F¯i = F1i F2i + F2i where F˜i = F1i more, for any matrices X , Yj , Zj (j = 1, 2, . . . , q) with appropriate dimensions, we arrive at

≤ E 2ρ~xTk+1 P xk+1 + 2ρ~xTk P xk + ΓTk RΓk − ΓTk−τ R ×Γk−τ + xTk Sxk − xTk−τM Sxk−τM + (τM − τm + 1)

1 q

1 − q ... 1 − q

k−τ m −1 X

q X j=1

xTk−τk,j Qj xk−τk,j−

ηlT ρP ηl +

X

ηlT ρP ηl

!

X

ηlT ρP ηl

!

ηlT ρP ηl

!)

k−τk,1 −1 l=k−τM

k−τ m −1 X

k−τk,2 −1

ηlT ρP ηl +

l=k−τk,2

l=k−τM

ηlT ρP ηl

"

ηlT ρP ηl

0 = 2ξkT X xk − xk−τM −

l=k−τM

l=k−τk,1

k−τ m −1 X

k−1 X

+

l=k−τk,q

k−τk,q −1

X

l=k−τM

In view of (18) and (23), we have

.

(23)

PT

E {∆Vk } (  ˜ TA P Ξ ˜A + Ξ ˜ TE P Ξ ˜ E + (2ρ~ − 1)ΞTI P ΞI ≤ E ξkT 8Ξ

AC

CE

˜ TA GT (GB)−1 GΞ ˜A + Ξ ˜ Tδ P Ξ ˜δ +ΞTI QΞI + Ξ A A T˜ T T T +Ξτ Φτ Ξτ + WR RWR + WS SWS + γ1 Ξg1 D1T P  k−1 X T T ×D1 Ξg1 + γ2 Ξg2 D2 P D2 Ξg2 ξk − ηlT ρP ηl 1 − q

1 − q ... 1 − q

k−τ m −1 X

ηlT ρP ηl

+

ηlT ρP ηl

!

X

ηlT ρP ηl

!

ηlT ρP ηl

!)

l=k−τk,1

l=k−τM

k−τ m −1 X

k−τk,2 −1

ηlT ρP ηl

+

l=k−τk,2

l=k−τM

k−τ m −1 X

k−τk,q −1

l=k−τk,q

ηlT ρP ηl +

X

l=k−τM

l=k−τM

#

ηl ,

l=k−τM

Then, substituting (25) and (26) into (24) yields

l=k−τM

X

k−τk,1 −1

k−1 X

  k−τ m −1 X 1 0 = × 2ξkT Y1 xk−τm − xk−τk,1 − ηl  , q l=k−τk,1   k−τk,1 −1 X 1 0 = × 2ξkT Z1 xk−τk,1 − xk−τM − ηl  , q l=k−τM   k−τ m −1 X 1 0 = × 2ξkT Y2 xk−τm − xk−τk,2 − ηl  , q l=k−τk,2   k−τk,2 −1 X 1 ηl  , 0 = × 2ξkT Z2 xk−τk,2 − xk−τM − q l=k−τM ...   k−τ m −1 X 1 0 = × 2ξkT Yq xk−τm − xk−τk,q − ηl  , q l=k−τk,q   k−τk,q −1 X 1 0 = × 2ξkT Zq xk−τk,q − xk−τM − ηl  . (26) q

AN US

j=1

xTk Qj xk −

M

−

q X

ED

×

T 

CR IP T

k−τ m −1 X

1 − q ...

E {∆Vk } (  ˜T P Ξ ˜A + Ξ ˜T P Ξ ˜ E + (2ρ~ − 1)ΞT P ΞI + ΞT ≤E ξkT 8Ξ A E I I ˜ TA GT (GB)−1 GΞ ˜A + Ξ ˜ Tδ P Ξ ˜ δ + ΞTτ Φ ˜ τ Ξτ ×QΞI + Ξ A A

,

+WRT RWR + WST SWS + γ1 ΞTg1 D1T P D1 Ξg1 + γ2

×ΞTg2 D2T P D2 Ξg2 + ΠTg1 F1 Πg1 + ΠTg2 F2 Πg2 + Π1  +ΠT1 + τM X (ρP )−1 X T ξk

(24)

8

ACCEPTED MANUSCRIPT

  −1 T Π + τM X (ρP ) X ξk

1 + (τk,1 − τm )ξkT Y1 (ρP )−1 Y1T ξk q 1 + (τM − τk,1 )ξkT Z1 (ρP )−1 Z1T ξk q 1 + (τk,2 − τm )ξkT Y2 (ρP )−1 Y2T ξk q 1 + (τM − τk,2 )ξkT Z2 (ρP )−1 Z2T ξk q ... 1 + (τk,q − τm )ξkT Yq (ρP )−1 YqT ξk q

˜ TA P Ξ ˜A + Ξ ˜T P Ξ ˜ E + ΞT [(2ρ~ − 1)P + Q]ΞI Π = 8Ξ E I ˜ δ + ΞT Φ ˜ τ Ξτ ˜ T GT (GB)−1 GΞ ˜A + Ξ ˜T P Ξ +Ξ A

A

τ

+γ2 ΞTg2 D2T P D2 Ξg2 + ΠTg1 F1 Πg1 + ΠTg2 F2 Πg2

+Π1 + ΠT1 ,   1 Π1 = X Y 1 Z1 Y 2 Z2 . . . Y q Z q q   qI 0 − qI 0 n×n n×n n×2n n×(m+q)n       0n×mn In×n − In×n 0n×(q+2)n        0n×(m+1)n In×n 0n×(q−1)n − In×n 0n×2n         0n×mn In×n 0n×n − In×n 0n×(q+1)n  , ×    0n×(m+2)n In×n 0n×(q−2)n − In×n 0n×2n          ...      0  n×mn In×n 0n×(q−1)n − In×n 0n×3n     0n×(m+q)n In×n − In×n 0n×2n k−1 X

Σ0 =

l=k−τM

)

δA

+WRT RWR + WST SWS + γ1 ΞTg1 D1T P D1 Ξg1

AN US

≤E

ξkT

M

(

where

CR IP T

1 + (τk,1 − τm )ξkT Y1 (ρP )−1 Y1T ξk q 1 + (τM − τk,1 )ξkT Z1 (ρP )−1 Z1T ξk q 1 + (τk,2 − τm )ξkT Y2 (ρP )−1 Y2T ξk q 1 + (τM − τk,2 )ξkT Z2 (ρP )−1 Z2T ξk q ... 1 + (τk,q − τm )ξkT Yq (ρP )−1 YqT ξk q 1 + (τM − τk,q )ξkT Zq (ρP )−1 ZqT ξk q  ) − Σ0 + Σ11 + Σ12 + Σ21 + Σ22 + · · · + Σq,1 + Σq,2

1 + (τM − τk,q )ξkT Zq (ρP )−1 ZqT ξk q  h 1 τk,1 − τm  =E ξkT × Π + τM X (ρP )−1 X T q τM − τm  1 τ −τ M k,1 +(τM − τm )Y1 (ρP )−1 Y1T + × q τM − τm   × Π + τM X (ρP )−1 X T + (τM − τm )Z1 (ρP )−1 Z1T 1 τk,2 − τm  + × Π + τM X × (ρP )−1 X T + (τM − τm ) q τM − τm   1 τ −τ M k,2 ×Y2 (ρP )−1 Y2T + × × Π + τM X (ρP )−1 q τM − τm  T ×X + (τM − τm )Z2 (ρP )−1 Z2T

Σ11 =

ED

Σ12

1 = q

Σ21 =

PT

1 q

Σ22 =

1 q 1 q

CE

... Σq,1 =

AC

... 1 τk,q − τm  + × Π + τM X (ρP )−1 X T + (τM − τm ) q τM − τm  1 τ −τ  M k,q ×Yq (ρP )−1 YqT + × Π + τM X (ρP )−1 q τM − τm i  ×X T + (τM − τm )Zq (ρP )−1 ZqT ξk ,

Σq,2 =

1 q 1 q

(ρP ηl + X T ξk )T (ρP )−1 (ρP ηl + X T ξk ),

k−τ m −1 X

(ρP ηl + Y1T ξk )T (ρP )−1 (ρP ηl + Y1T ξk ),

X

(ρP ηl + Z1T ξk )T (ρP )−1 (ρP ηl + Z1T ξk ),

l=k−τk,1

k−τk,1 −1 l=k−τM k−τ m −1 X

(ρP ηl + Y2T ξk )T (ρP )−1 (ρP ηl + Y2T ξk ),

X

(ρP ηl + Z2T ξk )T (ρP )−1 (ρP ηl + Z2T ξk ),

k−τ m −1 X

(ρP ηl + YqT ξk )T (ρP )−1 (ρP ηl + YqT ξk ),

X

(ρP ηl + ZqT ξk )T (ρP )−1 (ρP ηl + ZqT ξk ),

l=k−τk,2

k−τk,2 −1 l=k−τM

l=k−τk,q

k−τk,q −1 l=k−τM

and Πg1 , Πg2 , Fi (i = 1, 2) are defined in (11). Finally, by utilizing the Lemmas 2-3, we can conclude that E {∆Vk } < 0 can be guaranteed by matrix inequalities (8) and (9). Hence, the sliding mode dynamics (7) is robustly asymptotically mean-square stable.

9

ACCEPTED MANUSCRIPT Remark 3 During the proof of Theorem 1, we can see that more matrices should be found due to the utilization of the delay-fractioning approach. Accompanying with the rapid developments of the computation techniques, the resulted issue of the computation burden is not serious. Actually, there indeed exists a tradeoff between the computation burden and the conservatism. For this issue, we decide to adopt a compromise in an acceptable range, i.e., reduce the conservatism but increase certain computation burden.

3.3

Remark 4 In Theorem 1, a delay-dependent stability condition is presented for the sliding mode dynamics. It should be pointed out that, due to the existence of the equality constraint (10), the feasibility of newly proposed sufficient criterion cannot be easily checked. By utilizing the subsequent computational algorithm, the desired performance requirement can be ensured by solving a minimization problem.

where U = diag{µ1 , µ2 , . . . , µm } ∈ Rm×m , V = diag{ν1 , ν2 , . . . , νm } ∈ Rm×m , κ represents the sampling period, µi > 0 and νi > 0 (i = 1, 2, . . . , m) are properly chosen constants satisfying 0 < 1 − κνi < 1. To proceed, set

In this subsection, the reachability of the sliding surface (4) is discussed by designing the SMC law. As in [9], we need to check the following inequalities with hope to achieve the desired performance:

Computational Algorithm

M

ED

(27)

i

T  i i ˆi = δ a + δ a , b a = δˆ1 δˆ2 . . . δˆm ∆ , δ a a a a 2 i n o δ a − δ ia 1 ˜2 m i ˜ ˜ ˜ e ∆a = diag δa , δa , . . . , δa , δa = , 2  T i i ˆi = δ τ + δ τ , b τ = δˆ1 δˆ2 . . . δˆm ∆ , δ τ τ τ τ 2 i i n o e τ = diag δ˜1 , δ˜2 , . . . , δ˜m , δ˜i = δ τ − δ τ , ∆ τ τ τ τ 2 i  T i δ + δ f ˆi = f b f = δˆ1 δˆ2 . . . δˆm ∆ , δ , f f f f 2 i n o δ − δ if e f = diag δ˜1 , δ˜2 , . . . , δ˜m , δ˜i = f ∆ , f f f f 2

CE

AC

subject to (8), (9) and (27).

i

i

δ ia≤δai (k)≤δ a , δ iτ ≤δτi (k) ≤ δ τ , δ if ≤ δfi (k)≤δ f , where δai (k), δτi (k) and δfi (k) (i = 1, 2, . . . , m) are the ith elements of ∆a (k), ∆τ (k) and ∆f (k), respectively. In the subsequent, set

Then, it is clear that the original non-convex problem can be converted into the minimization problem as follows: min µ

i=1

Then, suppose that ∆a (k), ∆τ (k) and ∆f (k) are bounded in Euclidean norm. This means that there exist i i i known bounds δ ia , δ a , δ iτ , δ τ δ if and δ f (i = 1, 2, . . . , m) satisfying

PT

 bT P B −µI D    ≤ 0.  T b B P D −I

∆sk =sk+1 − sk ≥−κU sgn[sk ] − κV sk , if sk <0,

∆a (k) := αk G∆Axk , ∆f (k) := GBf (xk ), q X ∆τ (k) := GAτ x ek = GAτ βk,i xk−τk,i .

It should be pointed out that the sufficient condition in Theorem 1 is non-convex due to the equality constraint (10). Based on the technique developed in [36], the condition (10) can be equivalently expressed by b T B T P D] b = 0. Next, by introducing the matr[(B T P D) T b T BT P D b ≤ µI with µ > 0 and trix inequality (B P D) utilizing the Lemma 3, it is easy to obtain 

∆sk =sk+1 − sk ≤−κU sgn[sk ] − κV sk , if sk >0,

CR IP T

(

AN US

3.2

Design of sliding mode controller

(28)

Remark 5 It is worthwhile to mention that the minimization problem in (28) is an LMI-based one and its feasibility can be easily checked via the standard numerical software, then the stability problem of the resulted sliding motion is solved.

then we will design the desired robust sliding mode controller, which can ensure the reachability analysis.

10

ACCEPTED MANUSCRIPT the scalars τm , τM , β¯i and 2 reflect the time-varying MSCDs, and the matrices H, E and scalars α ¯ , 1 refer to the ROU. Therefore, the related information of the above mentioned factors has been clearly reflected in main results.

Theorem 2 For the sliding surface (4) with G = B T P , if the minimization problem (28) is solvable and P is the solution to (28), then the sliding mode controller given by

Remark 7 Note that the addressed networked systems are influenced simultaneously by several stochastic phenomena, we have made great efforts to propose new robust control method. More specifically, the stability analysis problem of the sliding motion has been studied by proposing a new sufficient condition in terms of the delay fractioning method. In addition, the desired sliding mode controller has been designed, which can drive the system state trajectories onto the pre-defined sliding surface.

(29)

can guarantee the discrete-time reaching condition. Proof : Note that the proof of this theorem can be easily obtained and hence the proof is omitted for brevity. Remark 6 In this paper, we make great efforts to address robust SMC problem of addressed systems with network-induced phenomena, and several types of network-induced phenomena have been well discussed. The main difficulties during the derivation of new results can be summarized as follows: (i) how to fully consider the phenomenon of MSCDs and reflect the related information in main results? (ii) how to address the related sum terms induced by the MSCDs via the proper method? and (iii) how to integrate the SMC method with the delay-fractioning idea to attenuate the effects from MSCDs, ROU and RVNs subject to uncertain occurrence probabilities? To reply the above three issues, 1) we make an attempt to divide the sum term Pk−τm −1 T “− l=k−τ ηl ρP ηl ” in (22) as M

An illustrative example

AN US

4

CR IP T

 uk = −(GB)−1 κU sgn[sk ] + (κV − I)sk ba + ∆ e a sgn[sk ]) + (∆ bτ + ∆ e τ sgn[sk ]) +(∆  bf + ∆ e f sgn[sk ]) +(∆

In this section, we provide numerical simulations to illustrate the usefulness of the obtained SMC method. The system parameters in (1) are given by:

k−τ m −1 X

ηlT ρP ηl

l=k−τM

1 − q ...

ηlT ρP ηl

l=k−τk,1 k−τ m −1 X

k−τk,1 −1

X

+

ηlT ρP ηl +

l=k−τk,2 k−τ m −1 X

l=k−τk,q

ηlT ρP ηl

!

ηlT ρP ηl

!

ηlT ρP ηl

!

l=k−τM

k−τk,2 −1

X

l=k−τM

ηlT ρP ηl

AC

1 − q

k−τ m −1 X

PT

1 =− q

CE

−

ED

M



+

k−τk,q −1

X

l=k−τM

;







0.15 −0.25 0 0.1817 0.4286            A= , B =  0 0.13 0.01   0.1597 0.793  ,     0.03 0 −0.05 0.1138 0.0581     0.025 0.1 0 −0.03 0 −0.01            , D = Aτ =   0.02 0.03 0  1  0 −0.03 0  ,     0.04 0.035 −0.01 0.04 0.05 −0.01     0.015 0 −0.01 −0.05 0.037 −0.36            D2 = 0 0.03 0  , E1 =  0.01 0.015 0  ,     0.04 0.035 0.01 0.02 0.025 −0.01     T H = 0.14 0.2 0.17 , E = 0.2 0.1 0 , Fk =sin(0.6k). Let the nonlinearities be given by:

2) we use the free-weighting matrix technique to deal with the above sum terms induced by the MSCDs; and 3) we introduce the term V5k below (12) with a parameter ρ to enhance the feasibility of the developed method. Moreover, the matrices F1i , F2i and scalars γ¯ , 3 are there for RVNs subject to uncertain occurrence probabilities,

 T f (xk ) = 0.49 sin(x1,k x3,k ) 0.13 sin(x2,k ) ,

g1 (xk ) = 0.5(F11 + F21 )xk + 0.5(F21 − F11 ) sin(xk )xk ,

g2 (xk ) = 0.5(F12 + F22 )xk + 0.5(F22 − F12 ) cos(xk )xk ,

11

ACCEPTED MANUSCRIPT where

10

F11 = F12 = diag{0.4, 0.5, 0.8},

8

F21 = F22 = diag{0.3, 0.2, 0.6},

6

2 0

4

sin(xk ) := diag{sin(x1,k ), sin(x2,k ), sin(x3,k )},

x1,k x2,k x3,k

−2 30

35

40

2

cos(xk ) := diag{cos(x1,k ), cos(x2,k ), cos(x3,k )},

0

and xi,k (i = 1, 2, 3) is the i-th element of xk . Let α ¯= ¯ ¯ 0.7, 1 = 0.1, β1 = 0.6, β2 = 0.5, 2 = 0.15, γ¯ = 0.82, 3 = 0.07. The bounds of the time-delay τk,j (j = 1, 2) are set as τm = 3 and τM = 6. Choose the lower and upper bounds for δai (k), δτi (k) and δfi (k) (i = 1, 2, . . . , m) as

−2 −4

−8 −10

10

i

δ ia = − k GH kk Exk k, δ a =k GH kk Exk k,

10

i

δ τ = k GAτ kk xk−τk,1 k + k GAτ kk xk−τk,2 k,

8

70

80

90

100

x1,k x2,k x3,k

0

4

−2 30

35

40

2 0



−2 −4

ED

M

−6 −8

−10

10

20

30

40 50 60 No. of samples. K

70

80

90

100

Fig. 2. The trajectory of state xk (κ=0.6)

Remark 8 It should be noted that the proposed stability conditions in Theorem 1 are delay-dependent, which are clearly affected by the delay interval. Accordingly, the actual discontinuous control will increase the gain and the chattering effects should be considered. From the simulations, we can see that smaller sampling period can reduce the chattering effects, which has been clearly shown in Figs. 1-6. Hence, during the implementation, we can increase the sampling frequency and then alleviate the chattering effects effectively.

PT

ε = 0.1952, µ = 0.0106.

40 50 60 No. of samples. K

2

6

Then, for prescribed scalars p = 1 and ρ = 1.2 × 10−4 , solving the minimization problem (28) yields 1.4008 −0.7145 0.2304      P =  −0.7145 3.1500 −0.4516  ,   0.2304 −0.4516 2.9226   0.1666 0.3218 0.3023   G= , 0.0471 2.1655 −0.0896

30

AN US

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Fig. 1. The trajectory of state xk (κ=0.06)

δ iτ = −(k GAτ kk xk−τk,1 k + k GAτ kk xk−τk,2 k), δ if = − k GB kk f (xk ) k, δ f =k GB kk f (xk ) k .

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In the simulation, let κ = 0.06 and µi = νi = 0.1 (i = 1, 2). By applying newly synthesized sliding mode controller (29), the simulation results are obtained. Among them, Fig. 1 clearly shows that the system state trajectories x1,k , x2,k and x3,k converge to a small neighborhood quickly in finite time. The control signal uk is plotted in Fig. 5 and the communication delay τk,i (i = 1, 2) is shown in Figs. 7-8. Besides, the sliding surface sk and signal ∆sk are plotted respectively in Figs. 3 and 9. Besides, for the comparison purpose, we provide the related simulations by setting κ = 0.6 in order to illustrate the chattering effects, which have been shown in Figs. 2, 4 and 6. Overall, from the above simulation results, we can conclude that the proposed SMC method performs well.

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Conclusion

In this paper, we have addressed the robust SMC problem for discrete networked systems subject to the ROU, RVNs and MSCDs under uncertain occurrence

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40

s1,k s2,k

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u1,k u2,k

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6 20 4 10

2 0

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−10

−4 −20 −30

−8 −10

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Fig. 3. The trajectory of sliding mode function sk (κ=0.06) 10

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Fig. 5. The control signal uk (κ=0.06) 40

s1,k s2,k

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u1,k u2,k

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Fig. 4. The trajectory of sliding mode function sk (κ=0.6)

−30 −40

10

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40 50 60 No. of samples. K

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Fig. 6. The control signal uk (κ=0.6)

ple has been employed to show the feasibility of the new control method. It should be noted that it is of important significance to handle the problems of analysis and synthesis for networked systems with abruptly changed structures and incomplete measurements. Hence, further research topics include the extensions of the proposed SMC method to deal with the SMC problems for networked systems with Markovian jumping parameters and communication protocols.

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probabilities. The network-induced phenomena of ROU, RVNs and MSCDs have been characterized by a set of Bernoulli distributed random variables with uncertain occurrence probabilities. Firstly, we have introduced a linear switching surface. Secondly, a sufficient criterion based on the delay fractioning idea has been established to ensure the robust asymptotic stability of the sliding motion in the mean square. In the sequel, a minimization algorithm has been provided for convenience of examining the feasibility of the proposed method and an SMC law has been designed properly to guarantee the reachability analysis. It is worthwhile to note that the all parameter matrices in the sliding surface and SMC law can be easily obtained by solving an optimal problem with certain LMI constraints. Finally, a numerical exam-

Acknowledgments This work was supported in part by the National Natural Science Foundation of China under Grants 61673141, 11301118 and 61503001, the Fok Ying Tung Education Foundation of China under Grant 151004,

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10

8 ∆s1,k ∆s2,k

τk,1 9

6

8

4

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2 30

6 4 35

40

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6 5

0

4

−2

3 −4 −6

1 0

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Fig. 7. The communication delay τk,1 (κ=0.06)

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Fig. 9. The signal ∆sk (κ=0.06)

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controller design for synchronization of complex dynamical

τk,2 6

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4

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networks, Nonlinear Dynamics, 88 (4) (2017) 2637–2649.

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[3] H. Li, J. Yu, C. Hilton, and H. Liu, Adaptive slidingmode control for nonlinear active suspension vehicle systems

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using T-S fuzzy approach, IEEE Transactions on Industrial

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Electronics, 60 (8) (2013) 3328–3338.

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[4] A. T. Boum, G. B. D. Keubeng, and L. Bitjoka, Sliding

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mode control of a three-phase parallel active filter based on

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a two-level voltage converter, Systems Science and Control

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[5] J. Zhang, M. Lyu, T. Shen, L. Liu, and Y. Bo, Sliding mode control for a class of nonlinear multi-agent system with timedelay and uncertainties, IEEE Transactions on Industrial Electronics, 65 (1) (2018) 865–875. [6] L. Wu, P. Shi, and H. Gao, State estimation and sliding-

Fig. 8. The communication delay τk,2 (κ=0.06)

mode control of markovian jump singular systems, IEEE

the Outstanding Youth Science Foundation of Heilongjiang Province of China under grant JC2018001, the University Nursing Program for Young Scholars with Creative Talents in Heilongjiang Province of China under grant UNPYSCT-2016029, the Science Funds for the Young Innovative Talents of HUST of China under grant 201508, and the Alexander von Humboldt Foundation of Germany.

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ACCEPTED MANUSCRIPT Hongjian Liu received his B.Sc. degree in applied mathematics in 2003 from Anhui University, Hefei, China and the M.Sc. degree in detection technology and automation equipments in 2009 from Anhui Polytechnic University, Wuhu, China, and the Ph.D. degree in control theory and control engineering in 2018 from Donghua University, Shanghai, China. He is currently an Associate Professor in the School of Mathematics and Physics, Anhui Polytechnic University, Wuhu, China. His current research interests include filtering theory, memristive neural networks and network communication systems. He is a very active reviewer for many international journals.

probabilities, International Journal of General Systems, 47 (5) (2018) 422–437. [35] Y. Liu, Z. Wang, J. Liang, and X. Liu, Synchronization and state estimation for discrete-time complex networks with distributed delays, IEEE Transactions on Systems, Man, and Cybernetics-Part B: Cybernetics, 38 (5) (2008) 1314–1325. [36] Y. Niu, D. W. C. Ho, and J. Lam, Robust integral sliding mode control for uncertain stochastic systems with timevarying delay, Automatica, 41 (5) (2005) 873–880.

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Panpan Zhang received the B.Sc. degree in Mathematics and Applied Mathematics from Qufu Normal University, Qufu, China, in 2016. She is currently working toward the M.Sc. degree in Mathematics from Harbin University of Science and Technology, Harbin, China. Her current research interests include time-delay systems, sliding mode control for networked systems. She is an active reviewer for some international journals.

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Changlu Zhang is now working toward his B.Sc. degree in Information and Computation Science at Harbin University of Science and Technology, Harbin, China. Changlu Zhang’s research interests include nonlinear control and optimal estimation over the networked transmissions.

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Jun Hu received the B.Sc. degree in Information and Computation Science and M.Sc. degree in Applied Mathematics from Harbin University of Science and Technology, Harbin, China, in 2006 and 2009, respectively, and the Ph.D. degree in Control Science and Engineering from Harbin Institute of Technology, Harbin, China, in 2013. From September 2010 to September 2012, he was a Visiting Ph.D. Student in the Department of Information Systems and Computing, Brunel University, U.K. From May 2014 to April 2016, he was an Alexander von Humboldt research fellow at the University of Kaiserslautern, Kaiserslautern, Germany. His research interests include nonlinear control, filtering and fault estimation, time-varying systems and complex networks. He has published more than 30 papers in refereed international journals. He serves as a reviewer for Mathematical Reviews, as an editor for IEEE Access, Neurocomputing, Journal of Intelligent and Fuzzy Systems, Neural Processing Letters, Systems Science and Control Engineering, and as a guest editor for International Journal of General Systems and Information Fusion.

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