# Finite-Time Nonfragile Dissipative Control for Discrete-Time Neural Networks with Markovian Jumps and Mixed Time-Delays

Authors: Ling Hou; Dongyan Chen; Chan He
Journal: Complexity
 (2019)

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## Abstract

This paper considers the stochastic finite-time dissipative (SFTD) control problem based on nonfragile controller for discrete-time neural networks (NNS) with Markovian jumps and mixed delays, in which the mode switching phenomenon, is described as Markov chain, and the mixed delays are composed of discrete time-varying delay and distributed delays. First, by selecting an appropriate Lyapunov-Krasovskii functional and applying stochastic analysis methods, some parameters-dependent sufficient conditions for solvability of stochastic finite-time boundedness are derived. Then, the main results are extended to SFTD control. Furthermore, existence condition of nonfragile controller is derived based on solution of linear matrix inequalities (LMIs). Finally, two numerical examples are employed to show the effectiveness of the obtained methods.


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## Body

## 1. Introduction

NNS are a complex network system formed by a large number of simple processing units connected with each other. In the last decades, NNS have been widely used in many real-world problems, such as pattern recognition, smart antenna arrays, signal and image processing, target tracking, and combinatorial optimization; see [1–4]. As we all know, these practical applications rely heavily on the dynamic behavior of NNS. Thus, dynamic property of various types of NNS has become a hot research topic. The corresponding results can be found in [5–8] and the reference therein. In addition, in the electronic implementation of many biological and artificial NNS, time delays often occur in signals transmission between difference neurons, which may cause oscillation and instability of dynamic network behaviors. In the last decades, some important works on NNS with time delays have been presented, such as [9–13]. On the other hand, in NNS, information latching often occurs, which means that at different times the NNS may be limited to switching from one mode to another. Thus, in this case, the NNS can be handled as a Markov jump NNS. Up to now, there have been some results for this kind of system, such as [14–18]. In [14], slow state variables feedback stabilization problem for Markov jump system was investigated. In [15], the passivity problem of Markov jump NNS was discussed by using linear matrix inequality (LMI) technique. In [16], the exponential stability of the NNS with Markov jump parameters was studied. In [17, 18], the stability and synthesis problem of Markov jump NNS with model-dependent mixed delays were discussed.Dissipativity is a more general criterion compared with Lyapunov stability,H∞ performance, and passive performance. Its essence is that there is a nonnegative energy function so that the energy loss inside for the system is always less than the supply rate of external energy. Since Willems put forward the concept of dissipativity in 1972 [19], dissipative theory has played an important role in systems, networks, control theory, and control engineering. During the past several decades, many results about dissipative theory have been obtained. For example, the dissipative problems of linear systems were studied in [20–22]. In [23], Shen et al. discussed the dissipative performance of sampled-data Markov jump system.It is worth noting that all above mentioned results are mainly concerned with the dynamic behavior in an infinite interval. However, in pratice, we may be concern with the dynamic performance in a specified time interval, that is, finite-time stability (FTS). The concept of FTS was first introduced by Dorato [24]. And in recent years, the FTS attracts more and more consideration. For example, Lv et al. studied the FTS for a class of nonlinear system in [25]. Yang et al. dealt with the FTS for singular system [26]. Subsequently, FTS was extended to finite-time boundedness (FTB), the finite-time H∞ control, finite-time passive (FTP) control, and so on [27–31]. And recently, Ma et al. studied the finite-time dissipative (FTD) problem of singular discrete-time Markov jump systems [32]. In [33–35], Song, Mathiyalagan, and Chen studied the FTD problem for general network systems, stochastic interval systems, and network series control systems, respectively. However, in the abovementioned research, the designed controllers are all accurate. But in many industrial applications, controllers such as actuator damage and digital switching errors are inevitably inaccurate. Therefore, it is the more ideal method to design a controller that is insensitive to uncertainty coefficients and can provide sufficient tuning range. This leads to the study of nonfragile control. Many results about nonfragile state feedback controller design problem can be found in [36–38]. Recently, Ma et al. discussed the FTD problem based on nonfragile controller for uncertain T-S fuzzy systems in [39]. Furthermore, Gao and Xia studied the finite-time extended dissipative control for uncertain switched NNS based on nonfragile controller in [40, 41]. Summarizing the discussions made so far, stochastic finite-time nonfragile dissipative control problem for discrete-time NNS contain Markovian jumps and mixed delays has not yet been investigated, which remains a good challenge. Moreover, this kind of system model can be used to solve water pollution problems; see [32].The objective of this paper is to study the problem of SFTD analysis based on nonfragile controller for a class of discrete time NNS with Markovian jumps and mixed delays. Different from the existing works, our objective is to design a finite-time nonfragile dissipative controller by fully taking into account the Markovian jumps and mixed-delays. Through such a nonfragile controller, both stochastic finite-time boundedness and desired dissipative performance of a class of discrete-time NNS can be ensured. The main contributions of this paper can be highlighted.(1) A more comprehensive model of discrete-time NNS is proposed that contains Markovian jumps and mixed delays than existing literatures [30, 31], since the Markovian jumps and mixed delays were not considered in [30, 31]. (2) By applying stochastic analysis method and LMI tool, sufficient conditions for the stochastic finite-time boundedness and stochastic finite-time dissipativity of the discrete-time NNS with Markovian jumps and mixed delays are given. Our model and conditions are more general than those given in [27–30]. (3) A more general controller is designed, that is, the nonfragile state feedback controller, which can greatly reduce conservation of the system.Notation.Throughout this paper, Rn and Rm×n represent the n-dimension real Euclidean space and the set of m×n real matrices, respectively. E{·} expresses the mathematical expectation of matric X, and Prob{X} is used to denote the occurrence probability of the event. The notations X≥Y and X>Y mean that X-Y is positive semidefinite or positive definite, respectively, in which X and Y are symmetric matrices. I and 0 represent the identity and zero matrices with compatible dimensions respectively. Diag{·} denotes the block diagonal matrix. ∗ denotes symmetry terms in a symmetric matrix, and MT denotes the transposition of matrix M.


## 2. Problem Formulation and Preliminaries

The discrete-time NNS considered in this paper is with Markovian jumps and mixed-delays as follows:(1)xk+1=Aθkxk+Adθkxk-dk+Amθk∑m=1+∞μmxk-m,+Bθkuk+Cθkfxk+Dθkωk,zk=Fθkxk,xj=φj,-∞<j≤0.In system (1), x(k)=[x1(k),⋯,xn(k)]T∈Rn is the state vector of the NNS associated with n neurons. u(k)∈Rp denotes the control input, z(k)∈Rt denotes the controlled output, and ω(k) is external disturbance input. f(x(·))=[f(x1(·)),⋯,f(xn(·))]T is the neuron activation function, and φ(j) denotes the vector-value initial condition. ∑m=1+∞μmx(k-m) is distributed delay, where the constants μm satisfy the following convergence conditions:(2)μ¯=∑m=1+∞μm≤∑m=1+∞mμm<+∞,A(θk), Ad(θk), Am(θk), B(θk), C(θk), D(θk), and F(θk) are known constant matrices with compatible dimensions, and θk is discrete-time Markov stochastic process and takes discrete values in a finite set Λ={1,2,⋯,s} with transition matrix Π=(πij)s×s, that is, (3)πij=Prθk+1=j∣θk=i, where 0≤πij≤1 and ∑j=1sπij=1.To obtain the main results, the following assumptions are needed.Assumption 1.
d ( k ) is time-varying delay satisfying(4)dm≤dk≤dM,dm,dM>0.Assumption 2.
The neuron activation functionf(x(·)) is continuous and satisfies the following two conditions:(A1)
f(0)=0,(A2)
[f(x)-f(y)-U1(x-y)]T[f(x)-f(y)-U2(x-y)]≤0, ∀x, y∈Rn,
whereU1,U2∈Rn×n are real matrices of appropriate dimensions.
It should be pointed out that condition (A2) is more general than the Lipschitz condition.Assumption 3.
The external disturbance inputω(k) satisfies the constraint(5)∑k=0NωTkωk≤d,d≥0.
whered is a given positive number.
For simplifying the notation of this paper, we denote a matrixG(θk)=Gi for each possible θk=i(i∈Λ). For example, A(θk) and B(θk) will be presented by Ai and Bi, and so forth. Moreover, we denote x(k-d(k))=xd and x(k-m)=xm.
This paper adopts the following nonfragile controller for the discrete-time NNS (1):(6)uk=K¯xk, where K¯=Ki+ΔKi, Ki is the controller gain that will be determined later, and ΔKi is an uncertain matrix with(7)ΔKi=MiΔkNi, in which Mi and Ni are known real constant matrices with compatible dimensions and Δ(k) is an unknown perturbed matrix satisfying ΔT(k)Δ(k)≤I.Remark 4.
FTS means that the system state trajectories do not exceed a certain threshold during the specified finite-time interval under a given the initial condition. Up to now, most of the existing results concerning nonfragile controller design for NNS are considered in an infinite time interval, for example [37, 38], which cannot reflect the transient behavior of the system. In many practical applications, one may be interested in not only Lyapunov asymptotic stability on an infinite time interval but also bound of system trajectories over a fixed short period. Thus, it is necessary to study nonfragile controller design problem in a finite-time interval. Different from [37, 38], in this paper, nonfragile control problem for NNS is considered in a specified finite-time interval.
Then the resultant closed-loop system can be written as follows:(8)xk+1=A¯ixk+Adixd+Ami∑m=1+∞μmxm+Cifxk+Diωk,zk=Fixk,xj=φj,-∞<j≤0The aim of this article is to study nonfragile state feedback controller design problem which guarantees resultant closed-loop system is stochastic finite-time bounded (SFTB) and SFTD in spite of Markovian jumps and mixed delays. To study the stochastic finite-time boundedness and stochastic finite-time dissipativity of system (8), the some necessary definitions and lemmas are given below.Definition 5.
For given symmetric matricesRi>0, the system (8) is said to be SFTB with respect to (c1,c2,d,N,Ri), where 0<c1<c2, d>0, and N>0, if(9)sup-∞<l≤0ExlTRixl≤c1⇒ExkTRixk≤c2,k∈1,⋯,N, for all ω(k) satisfying Assumption 3.Definition 6.
For given symmetric matricesRi>0, the system (8) is said to be stochastic finite-time (W,S,R)-γ- dissipative with respect to (c1,c2,d,N,Ri) if the system (8) is SFTB with respect to (c1,c2,d,N,Ri) and, under the zero-initial condition, the output z(k) satisfies(10)∑k=0NEzTkWzk+2zTkSωk+ωTkRωk≥γ∑k=0NωTkωk, for any nonzero ω(k) satisfying Assumption 3, W, S, and R are specified real matrices with W and R symmetric. Also, we assume that W≤0 and denote -W=(W-)TW- for some W-.Remark 7.
It is worth noting that the concept of stochastic finite-time dissipativity is different from the conventional dissipative results [20–23]. Conventional dissipative theory discussed the asymptotic behavior of the system over an infinite-time interval, while the proposed dissipative methods of this paper provide an input-output energy-related characterization to the analysis in a specified finite-time interval. On the other hand, the dissipative control considered in this paper is more general than most of existing finite-time stabilization, finite-time H∞ control, and FTP control [25–31], which contains H∞ performance and passive performance as two special cases by choosing appropriate parameters W, S, and R in inequality (10). When setting W=-I, S=0, and R=(γ+γ2)I, the performance prescribed in Definition 6 becomes stochastic finite-time H∞ performance [27–30]; if W=0, S=I, and R=2I, then it corresponds to FTP performance [31].Lemma 8 ([40] (schur complement)).
Given constant matricesS1, S2, and S3, where S1=S1T and S2=S2T>0. Then S1+S3TS2-1S3<0 if and only if(11)S1S3T∗-S2<0or-S2S3∗S1<0.Lemma 9 (see [15]).
LetM∈Rn×n be a positive semidefinite matrix, xi∈Rn, and αi(i=1,2,⋯) be a constant. If the series αi≥1 is convergent, then(12)∑i=1+∞αixiTM∑i=1+∞αixi≤∑i=1+∞αi∑i=1+∞αixiTMxi.Lemma 10 ([40] (S-procedure)).
LetL=LT, M, and N be real matrices of appropriate dimensions with F satisfying FTF≤I, then L+MFN+NTFTMT<0 if and only if there exists a positive scalar ε such that L+ε-1MMT+εNTN<0, or, equivalently,(13)LMεNTMT-εI0εN0-εI<0.


## 3. Main Result

This section will establish the criteria for stochastic finite-time boundedness and stochastic finite-time dissipativity. On this basis, the design method of nonfragile controller is provided.In this subsection, we focus on the problem for stochastic finite-time boundedness of the system (8).Theorem 11.
For given a scalarδ≥1, symmetric matrices Ri>0, the system (8) is SFTB with respect to (c1,c2,d,N,Ri), if there exist symmetric matrices Pi>0, Q>0, and Z>0 and positive scalars εi and λj(j=1,2,3,4) such that the following conditions hold for all i∈Λ:(14)Πi=Πi11Πi12∗Πi22<0,(15)λ1Ri<Pi<λ2Ri,0<Q<λ3Ri,0<Z<λ4Ri,(16)γd-c2δλ1c1δN+1λ2ρ1λ3ρ2λ4∗-c1δN+1λ200∗∗-ρ1λ30∗∗∗-ρ2λ4<0,where(17)Πi11=Δ1i00-εiH2T0∗-δdmQ000∗∗-1μ¯Z00∗∗∗-εiI0∗∗∗∗-γδNI,Πi12=AiAdiAmiCiDiTπi1Iπi2I⋯πisI,Πi22=diag-P1-1,⋯,-Ps-1,H1=U1TU2+U2TU12,H2=-U1T+U2T2,Δ1i=-δPi+τQ-μ¯Z-εiH1,ρ1=δN+1dMδdm-1+τ1-1τ2δdm-2,ρ2=δN+1σ,τ1=dM-dm+1,τ2=dM+dm-1,σ=μ1+1+δμ2+⋯+1+δ+⋯+δdM-1μdMProof.
Choose Lyapunov-Krasovskii functional as follows:(18)Vθk,xk=∑i=13Vik,where(19)V1k=xTkPixk,V2k=∑s=k-dkk-1δk-1-sxTsQxs+∑s=-dM+2-dm+1∑t=k+s-1k-1δk-1-txTtQxt,V3k=∑m=1+∞μm∑l=k-mk-1δk-1-lxTlZxl.
Now, we assume that(20)ExlTRixl≤c1,-∞<l≤0.
Our purpose below is to prove thatEx(k)TRix(k)≤c2,k∈{1,⋯,N}. First, by taking the difference variation of V(k) along the trajectory of system (8), we can obtain that(21)EV1θk+1,xk+1-δV1θk,xk=ExTk+1Pixk+1-δxTkPixk=xTkA¯iTP^jA¯i-δPixk+2xTkA¯iTP^jAdixd+Ami∑m=1+∞μmxm+Cifxk+Diωk+xdTAdiTP^jAdixd+2xdTAdiTP^jAmi∑m=1+∞μmxm+Cifxk+Diωk+∑m=1+∞μmxmTAmiTP^jAmi∑m=1+∞μmxm+2∑m=1+∞μmxmTAmiTP^jCifxk+Diωk+fTxkCiTP^jCifxk+2fTxkCiTP^jDiωk+ωTkDiTP^jDiωk, where P^j=∑i=1sπijPi.
Similarly, we have(22)
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In addition, from Assumption2, we can obtain the following inequality:(29)xkfxkTH1H2∗Ixkfxk≤0. Let εi>0, it follows from (27) and (29) that(30)EVθk+1,xk+1-δVθk,xk≤ξTkΣ¯iξk+γδNωTkωk,where(31)Σ¯i=Δ2iA¯iTP^jAdiA¯iTP^jAmiA¯iTP^jCi-εiH2TA¯iTP^jD1i∗AdiTP^jAdi-δdmQAdiTP^jAmiAdiTP^jCiAdiTP^jDi∗∗AmiTP^jAmi-1μ¯ZAmiTP^jCiAmiTP^jDi∗∗∗CiTP^jCi-εiICiTP^jDi∗∗∗∗DiTP^jDi-γδNI,Δ2i=A¯iTP^jA¯i-δPi+τQ-μ¯Z-εiH1.
By Lemma8 and inequality (14), we have Σ¯i<0. Thus(32)EVθk+1,xk+1-δVθk,xk<γδNωTkωk,∀k∈Z+. It follows from (5) and (32) that(33)EVθk,xk<δkEV0+γδN∑s=0k-1δk-1-sωTsωs<δNEV0+γδNδN-1∑s=0N-1ωTsωs<δNEV0+γδd.Noticing that condition (15), it yields that(34)EV0=ExT0Pix0+∑s=-d0-1δ-1-sxTsQxs+∑s=-dM+2-dm+1∑t=s-1-1δ-1-txTtQxt+∑m=1+∞μm∑l=-m-1δ-1-lxTlZxl<λ2ExT0Rix0+λ3δdM-1∑s=-dM-1xTsRixs+λ3δdM-2∑s=-dM+2-dm+1∑t=-1+s-1xTtRixt+λ4∑m=1+∞μm∑l=-m-1δ-1-lxTlRixl. Owing to(35)∑m=1+∞μm∑l=-m-1δ-1-lxTlRixl=μ1xT-1Rix-1+μ2δxT-2Rix-2+xT-1Rix-1+μ3δ2xT-3Rix-3+δxT-2Rix-2+xT-1Rix-1+⋯+μmδm-1xT-mRix-m+δm-2xT-m+1Rix-m+1+⋯+xT-1Rix-1+⋯=μ1xT-1Rix-1+μ2δxT-2Rix-2+xT-1Rix-1+μ3δ2xT-3Rix-3+δxT-2Rix-2+xT-1Rix-1+⋯+μdMδdM-1xT-dMRix-dM+δdM-2xT-dM+1Rix-dM+1+⋯+xT-1Rix-1≜c1σ, we have(36)EV0<λ2+λ3dMδdM-1+λ3τ1-1τ2δdM-2+λ4σc1≜ς1. Taking (33) and (36) into account, we have(37)EVθk,xk<δNς1+γδd,∀k∈Z+. On the other hand(38)EVθk,xk⩾ExTkPixk>λ1ExTkRixk∀k∈Z+. Consequently, we get from (37) and (38) that(39)ExTkRixk<δN+1ς1+γdδλ1. Note that(40)δN+1ς1+γd-c2δλ1<0⇔γd-c2δλ1+c1δN+1λ2+c1ρ1λ3+c1ρ2λ4<0⇔γd-c2δλ1+c1δN+1λ2ρ1λ3ρ2λ4Φ1c1δN+1λ2ρ1λ3ρ2λ4T<0,where (41)Φ1=c1δN+1λ2000ρ1λ3000ρ2λ4-1.By Lemma 8, inequality (40) is equivalent to LMI (14). Then we have(42)ExTkRixk<c2. The proof is completed.


### 3.1. Stochastic Finite-Time(W,S,R)-γ Dissipative

In this subsection, we give the following SFTD conditions for system (8) based on Theorem 11.Theorem 12.
For given a scalarδ≥1, symmetric matrices Ri>0, and matrices W, S, and R satisfying that W and R are symmetric and W≤0, the system (8) is stochastic finite-time (W,S,R)-γ dissipative if there exist symmetric matrices Pi>0, Q>0, and Z>0 and positive scalars εi and λj(j=1,⋯,7) such that the following conditions hold for all i∈Λ:(43)Π^i=Π^i11Π^i12Π^i13∗Π^i220∗∗-I<0,(44)λ1Ri<Pi<λ2Ri,0<Q<λ3Ri,0<Z<λ4Ri,0<FiTFi<λ5Ri,0<STS<λ6I,0<R<λ7I,(45)-c2λ1c1δN+1λ2ρ1λ3ρ2λ4ρ3λ5ρ4λ6ρ5λ7∗-c1δN+1λ200000∗∗-ρ1λ30000∗∗∗-ρ2λ4000∗∗∗∗-ρ3λ500∗∗∗∗∗-ρ4λ60∗∗∗∗∗∗-ρ5λ7<0,where(46)Π^i11=Δ1i00-εiH2T-FiTS∗-δdmQ000∗∗-1μ¯Z00∗∗∗-εiI0∗∗∗∗γI-R,Π^i12=Πi12,Π^i13=W-Fi0000T,Π^i22=Πi22,ρ3=δN-1N,ρ4=ρ5=δN-1d.
Other parameters are shown in Theorem11.Proof.
Firstly, we will show that inequality (10) holds. Define the following function:(47)Jk=zTkWzk+2zTkSωk+ωTkR-γIωk Then by choosing the same Lyapunov-Krasovskii functional and using similar handling methods as Theorem 11, we can obtain that(48)EVθk+1,xk+1-δVθk,xk-Jk<ξTkΘiξk, where(49)Θi=Δ1i-FiTWFiA¯iTP^jAdiA¯iTP^jAmiA¯iTP^jCi-εiH2TA¯iTP^jDi-FiTS∗AdiTP^jAdi-δdmQAdiTP^jAmiAdiTP^jCiAdiTP^jDi∗∗AmiTP^jAmi-1μ¯ZAmiTP^jCiAmiTP^jDi∗∗∗CiTP^jCi-εiICiTP^jDi∗∗∗∗DiTP^jDi-R+γI.
By Lemma8, we know that the condition (43) ensures that Θi<0. This leads to(50)EVθk+1,xk+1<EδVθk,xk+Jk, and, by iteration, we get(51)EVθk,xk<EδkV0+∑s=0k-1δk-1-sJs. Notice that E{V(θk,x(k))}>0 and under the zero-initial condition, we can obtain that(52)∑s=0NδN-sEzTsWzs+2zTsSωs+ωTsRωk-γωTsωs>0. Owing to δ≥1, (55) immediately yields that(53)∑s=0NEzTsWzs+2zTsSωs+ωTsRωs-γωTsωs>0, that is,(54)∑s=0NEzTsWzs+2zTsSωs+ωTsRωs>γ∑s=0NωTsωs. Thus inequality (10) holds.
In the following, we prove that the system (8) is SFTB. From the above proof and Theorem 11, we can obtain that(55)EVθk,xk<δkς1+∑s=0k-1δk-1-sEJs<δNς1+δN-1∑s=0N-1EJs.Note that W<0, we derive that(56)∑s=0N-1EJs≤∑s=0N-12zTsSωs+ωTsRωs,EVθk,xk<δNς1+∑s=0N-1E2zTsSωs+ωTsRωs. Thus we have(57)ExTkRixk<EVrk,xkλ1<δNς1+∑s=0N-1E2zTsSωs+ωTsRωsλ1. Since(58)∑s=0N-12zTsSωs+ωTsRωs=∑s=0N-12xTsFiTSωs+ωTsRωs, by nature of the vector, we have(59)2xTsFiTSws≤xTsFiTFixs+ωTsSTSωs. It follows that(60)ExTkRixk<δN-1ζ2+λ6+λ7dλ1, where ζ2=δλ2+λ3[dMδdM+(τ1-1)τ2δdM-1+δλ4σ+Nλ5.
Therefore, it can be easily verified from inequality (60) that(61)ExTkRixk<c2. Then, we conclude that the system (8) is stochastic finite-time (W,S,R)-γ- dissipative with respect to (c1,c2,d,N,Ri) from Definition 6. The proof is now completed.


### 3.2. Nonfragile Feedback Controller Design

Now we will provide the design scheme of nonfragile controller for NNS (1).Theorem 13.
For given scalarδ≥1 and symmetric matrices Ri>0 and matrices W, S, and R satisfying that W and R are symmetric and W≤0, the system (1) is stochastic finite-time (W,S-,R¯)-γ dissipative if there exist symmetric matrices Ti>0, Xi>0, Q¯i>0, Z¯i>0, and Wji>0(j=1,⋯,7), a nonsingular matrix Li, and positive scalars ρl(l=1,2,3,4,5) and αi such that the following conditions hold for all i∈Λ:(62)Πˇi=Πˇi11M~iαiN~iT∗-αiI0∗∗-αiI<0,(63)Ti<W2i,Q¯i<W3i,Z¯i<W4i,F¯iTF¯i<W5i,S¯TS¯<W6i,R¯<W7i,(64)-W1ic1δNW2iρ1W3iρ2W4iρ3W5iρ4W6iρ5W7i∗-c1δNW2i00000∗∗-ρ1W3i0000∗∗∗-ρ2W4i000∗∗∗∗-ρ3W5i00∗∗∗∗∗-ρ4W6i0∗∗∗∗∗∗-ρ5W7i<0, The desired state feedback controller gain matrix is given by(65)Ki=LiTi-1,where(66)Πˇi11=Δ3i00-εiH2T-FiTSπ1iTiAiT+LiTBiT⋯πsiTiAiT+LiTBiTFiTW-∗-δdmQi000π1iAdiT⋯πsiAdiT0∗∗-1μ¯Zi00π1iAmiT⋯πsiAmiT0∗∗∗-εiI0π1iCiT⋯πsiCiT0∗∗∗∗γI-Rπ1iDiT⋯πsiDiT0∗∗∗∗∗-T1⋯00⋮⋮⋮⋮⋮⋮⋱⋮0∗∗∗∗∗∗∗-Ts0∗∗∗∗∗∗∗∗-I,Δ3i=-δUi+τQi-μ¯Zi-εiXi,M~i=0,0,0,0,0,π1iBiMi,⋯,0T,N~i=Ni,0,0,0,0,0,⋯,0.Proof.
From Theorem12, it is derived that(67)Π^i<0. Noting that(68)ΔKi=MiΔkNi, we have(69)Π^i=Ξi+M~iΔkN~i+N~iTΔkM~iT<0.Moreover, inequality (69) is equivalent to(70)Ξi+αi-1M~iM~iT+αi-1αiN~iTΔkαiN~i<0. By using Lemma 10, the following inequality holds:(71)ΞiM~iαiN~iT∗-αiI0∗∗-αiI<0,where(72)Ξi=Ξi11Ξi12Ξi13∗Ξi220∗∗Ξi33,Ξi11=Δ1i00-εiH2T-FiTS∗-δdmQi000∗∗-1μ¯Zi00∗∗∗-εiI0∗∗∗∗γI-R,Ξi12=π1iAi+BiKiT⋯πsiAi+BiKiTπ1iAdiT⋯πsiAdiTπ1iAmiT⋯πsiAmiTπ1iCiT⋯πsiCiTπ1iDiT⋯πsiDiT,Ξi13=W-Fi0000T,Ξi22=diag-U1,⋯,-Us,Ξi33=-I.
LettingJ=diag{Pi-1,Pi-1,Pi-1,I,I,I,I,I,⋯,I} and then pre- and postmultiplying inequality (71) with J and letting Pi-1Ki=Li, Ti=Pi-1, Qi=Pi-1QPi-1, Zi=Pi-1ZPi-1, and Fi¯=FiTFi(i∈Λ), it is easy to see that inequality (71) is equivalent to inequality (62).
For getting LMI (64), we note that the inequality (45) can be regarded as(73)-c2λ1Ic1δN+1λ2Iρ1λ3Iρ2λ4Iρ3λ5Iρ4λ6Iρ5λ7I∗-c1δN+1λ2I00000∗∗-ρ1λ3I0000∗∗∗-ρ2λ4I000∗∗∗∗-ρ3λ5I00∗∗∗∗∗-ρ4λ6I0∗∗∗∗∗∗-ρ5λ7I<0. Postmultiplying inequality (73) by the matrix diag{Ri,Ri,Ri,Ri,Ri,I,I}, we can get that(74)-c2λ1Ric1δN+1λ2Riρ1λ3Riρ2λ4Riρ3λ5Riρ4λ6Iρ5λ7I∗-c1δN+1λ2Ri00000∗∗-ρ1λ3Ri0000∗∗∗-ρ2λ4Ri000∗∗∗∗-ρ3λ5Ri00∗∗∗∗∗-ρ4λ6I0∗∗∗∗∗∗-ρ5λ7I<0. Denoting J1=diag{Pi-1,Pi-1,Pi-1,Pi-1,Pi-1,Pi-1,Pi-1}. By pre- and postmultiplying (74) by J1 and letting W1i=c2δPi-1(λ1Ri)Pi-1, W2i=Pi-1(λ2Ri)Pi-1, W3i=Pi-1(λ3Ri)Pi-1, W4i=Pi-1(λ4Ri)Pi-1,W5i=Pi-1(λ5Ri)Pi-1, W6i=Pi-1(λ6I)Pi-1, and W7i=Pi-1(λ7I)Pi-1, we can obtain inequality (65). This completes the proof.Remark 14.
In Theorem13, sufficient conditions for guaranteeing stochastic finite-time dissipativity are proposed. It is easy to see that the obtained results of this paper are influenced by all system parameters, such as the bounds on the time-delay and the jumping transition probabilities. In addition, the design problem of the nonfragile state gain can be transformed into the following optimization problem:(75)minc2+γs.t.64,65and69.


## 3.1. Stochastic Finite-Time(W,S,R)-γ Dissipative

In this subsection, we give the following SFTD conditions for system (8) based on Theorem 11.Theorem 12.
For given a scalarδ≥1, symmetric matrices Ri>0, and matrices W, S, and R satisfying that W and R are symmetric and W≤0, the system (8) is stochastic finite-time (W,S,R)-γ dissipative if there exist symmetric matrices Pi>0, Q>0, and Z>0 and positive scalars εi and λj(j=1,⋯,7) such that the following conditions hold for all i∈Λ:(43)Π^i=Π^i11Π^i12Π^i13∗Π^i220∗∗-I<0,(44)λ1Ri<Pi<λ2Ri,0<Q<λ3Ri,0<Z<λ4Ri,0<FiTFi<λ5Ri,0<STS<λ6I,0<R<λ7I,(45)-c2λ1c1δN+1λ2ρ1λ3ρ2λ4ρ3λ5ρ4λ6ρ5λ7∗-c1δN+1λ200000∗∗-ρ1λ30000∗∗∗-ρ2λ4000∗∗∗∗-ρ3λ500∗∗∗∗∗-ρ4λ60∗∗∗∗∗∗-ρ5λ7<0,where(46)Π^i11=Δ1i00-εiH2T-FiTS∗-δdmQ000∗∗-1μ¯Z00∗∗∗-εiI0∗∗∗∗γI-R,Π^i12=Πi12,Π^i13=W-Fi0000T,Π^i22=Πi22,ρ3=δN-1N,ρ4=ρ5=δN-1d.
Other parameters are shown in Theorem11.Proof.
Firstly, we will show that inequality (10) holds. Define the following function:(47)Jk=zTkWzk+2zTkSωk+ωTkR-γIωk Then by choosing the same Lyapunov-Krasovskii functional and using similar handling methods as Theorem 11, we can obtain that(48)EVθk+1,xk+1-δVθk,xk-Jk<ξTkΘiξk, where(49)Θi=Δ1i-FiTWFiA¯iTP^jAdiA¯iTP^jAmiA¯iTP^jCi-εiH2TA¯iTP^jDi-FiTS∗AdiTP^jAdi-δdmQAdiTP^jAmiAdiTP^jCiAdiTP^jDi∗∗AmiTP^jAmi-1μ¯ZAmiTP^jCiAmiTP^jDi∗∗∗CiTP^jCi-εiICiTP^jDi∗∗∗∗DiTP^jDi-R+γI.
By Lemma8, we know that the condition (43) ensures that Θi<0. This leads to(50)EVθk+1,xk+1<EδVθk,xk+Jk, and, by iteration, we get(51)EVθk,xk<EδkV0+∑s=0k-1δk-1-sJs. Notice that E{V(θk,x(k))}>0 and under the zero-initial condition, we can obtain that(52)∑s=0NδN-sEzTsWzs+2zTsSωs+ωTsRωk-γωTsωs>0. Owing to δ≥1, (55) immediately yields that(53)∑s=0NEzTsWzs+2zTsSωs+ωTsRωs-γωTsωs>0, that is,(54)∑s=0NEzTsWzs+2zTsSωs+ωTsRωs>γ∑s=0NωTsωs. Thus inequality (10) holds.
In the following, we prove that the system (8) is SFTB. From the above proof and Theorem 11, we can obtain that(55)EVθk,xk<δkς1+∑s=0k-1δk-1-sEJs<δNς1+δN-1∑s=0N-1EJs.Note that W<0, we derive that(56)∑s=0N-1EJs≤∑s=0N-12zTsSωs+ωTsRωs,EVθk,xk<δNς1+∑s=0N-1E2zTsSωs+ωTsRωs. Thus we have(57)ExTkRixk<EVrk,xkλ1<δNς1+∑s=0N-1E2zTsSωs+ωTsRωsλ1. Since(58)∑s=0N-12zTsSωs+ωTsRωs=∑s=0N-12xTsFiTSωs+ωTsRωs, by nature of the vector, we have(59)2xTsFiTSws≤xTsFiTFixs+ωTsSTSωs. It follows that(60)ExTkRixk<δN-1ζ2+λ6+λ7dλ1, where ζ2=δλ2+λ3[dMδdM+(τ1-1)τ2δdM-1+δλ4σ+Nλ5.
Therefore, it can be easily verified from inequality (60) that(61)ExTkRixk<c2. Then, we conclude that the system (8) is stochastic finite-time (W,S,R)-γ- dissipative with respect to (c1,c2,d,N,Ri) from Definition 6. The proof is now completed.


## 3.2. Nonfragile Feedback Controller Design

Now we will provide the design scheme of nonfragile controller for NNS (1).Theorem 13.
For given scalarδ≥1 and symmetric matrices Ri>0 and matrices W, S, and R satisfying that W and R are symmetric and W≤0, the system (1) is stochastic finite-time (W,S-,R¯)-γ dissipative if there exist symmetric matrices Ti>0, Xi>0, Q¯i>0, Z¯i>0, and Wji>0(j=1,⋯,7), a nonsingular matrix Li, and positive scalars ρl(l=1,2,3,4,5) and αi such that the following conditions hold for all i∈Λ:(62)Πˇi=Πˇi11M~iαiN~iT∗-αiI0∗∗-αiI<0,(63)Ti<W2i,Q¯i<W3i,Z¯i<W4i,F¯iTF¯i<W5i,S¯TS¯<W6i,R¯<W7i,(64)-W1ic1δNW2iρ1W3iρ2W4iρ3W5iρ4W6iρ5W7i∗-c1δNW2i00000∗∗-ρ1W3i0000∗∗∗-ρ2W4i000∗∗∗∗-ρ3W5i00∗∗∗∗∗-ρ4W6i0∗∗∗∗∗∗-ρ5W7i<0, The desired state feedback controller gain matrix is given by(65)Ki=LiTi-1,where(66)Πˇi11=Δ3i00-εiH2T-FiTSπ1iTiAiT+LiTBiT⋯πsiTiAiT+LiTBiTFiTW-∗-δdmQi000π1iAdiT⋯πsiAdiT0∗∗-1μ¯Zi00π1iAmiT⋯πsiAmiT0∗∗∗-εiI0π1iCiT⋯πsiCiT0∗∗∗∗γI-Rπ1iDiT⋯πsiDiT0∗∗∗∗∗-T1⋯00⋮⋮⋮⋮⋮⋮⋱⋮0∗∗∗∗∗∗∗-Ts0∗∗∗∗∗∗∗∗-I,Δ3i=-δUi+τQi-μ¯Zi-εiXi,M~i=0,0,0,0,0,π1iBiMi,⋯,0T,N~i=Ni,0,0,0,0,0,⋯,0.Proof.
From Theorem12, it is derived that(67)Π^i<0. Noting that(68)ΔKi=MiΔkNi, we have(69)Π^i=Ξi+M~iΔkN~i+N~iTΔkM~iT<0.Moreover, inequality (69) is equivalent to(70)Ξi+αi-1M~iM~iT+αi-1αiN~iTΔkαiN~i<0. By using Lemma 10, the following inequality holds:(71)ΞiM~iαiN~iT∗-αiI0∗∗-αiI<0,where(72)Ξi=Ξi11Ξi12Ξi13∗Ξi220∗∗Ξi33,Ξi11=Δ1i00-εiH2T-FiTS∗-δdmQi000∗∗-1μ¯Zi00∗∗∗-εiI0∗∗∗∗γI-R,Ξi12=π1iAi+BiKiT⋯πsiAi+BiKiTπ1iAdiT⋯πsiAdiTπ1iAmiT⋯πsiAmiTπ1iCiT⋯πsiCiTπ1iDiT⋯πsiDiT,Ξi13=W-Fi0000T,Ξi22=diag-U1,⋯,-Us,Ξi33=-I.
LettingJ=diag{Pi-1,Pi-1,Pi-1,I,I,I,I,I,⋯,I} and then pre- and postmultiplying inequality (71) with J and letting Pi-1Ki=Li, Ti=Pi-1, Qi=Pi-1QPi-1, Zi=Pi-1ZPi-1, and Fi¯=FiTFi(i∈Λ), it is easy to see that inequality (71) is equivalent to inequality (62).
For getting LMI (64), we note that the inequality (45) can be regarded as(73)-c2λ1Ic1δN+1λ2Iρ1λ3Iρ2λ4Iρ3λ5Iρ4λ6Iρ5λ7I∗-c1δN+1λ2I00000∗∗-ρ1λ3I0000∗∗∗-ρ2λ4I000∗∗∗∗-ρ3λ5I00∗∗∗∗∗-ρ4λ6I0∗∗∗∗∗∗-ρ5λ7I<0. Postmultiplying inequality (73) by the matrix diag{Ri,Ri,Ri,Ri,Ri,I,I}, we can get that(74)-c2λ1Ric1δN+1λ2Riρ1λ3Riρ2λ4Riρ3λ5Riρ4λ6Iρ5λ7I∗-c1δN+1λ2Ri00000∗∗-ρ1λ3Ri0000∗∗∗-ρ2λ4Ri000∗∗∗∗-ρ3λ5Ri00∗∗∗∗∗-ρ4λ6I0∗∗∗∗∗∗-ρ5λ7I<0. Denoting J1=diag{Pi-1,Pi-1,Pi-1,Pi-1,Pi-1,Pi-1,Pi-1}. By pre- and postmultiplying (74) by J1 and letting W1i=c2δPi-1(λ1Ri)Pi-1, W2i=Pi-1(λ2Ri)Pi-1, W3i=Pi-1(λ3Ri)Pi-1, W4i=Pi-1(λ4Ri)Pi-1,W5i=Pi-1(λ5Ri)Pi-1, W6i=Pi-1(λ6I)Pi-1, and W7i=Pi-1(λ7I)Pi-1, we can obtain inequality (65). This completes the proof.Remark 14.
In Theorem13, sufficient conditions for guaranteeing stochastic finite-time dissipativity are proposed. It is easy to see that the obtained results of this paper are influenced by all system parameters, such as the bounds on the time-delay and the jumping transition probabilities. In addition, the design problem of the nonfragile state gain can be transformed into the following optimization problem:(75)minc2+γs.t.64,65and69.


## 4. Numerical Examples

In this section, two numerical examples are provided to illustrate the effectiveness of the approach addressed.Example 1.
We consider the same example in [28]; the system parameters are as follows:(76)A1=-1.51-1.03-5.2,Ad1=0.020-0.010.5,Am1=0.040.020.050.04,A2=-0.560-1,Ad2=00.1200.2,Am2=-0.080.060-0.1,B1=-2.510-5,B2=-0.2000.5,C1=0.0400.05-0.14,C2=-0.18000.02,D1=0.020-0.040.1,D2=0.10.020.010.3,F1=-0.12-0.02,F2=0.040.5,
Assume thatN=15, c1=0.9, δ=1.5. We choose W=0, S=I, R=2I, and R1=R2=I; the time-varying delay d(k)=1+0.03sin2(0.5kπ) satisfies 0.97<d(k)<4. The transition matrix Π=0.70.30.60.4. Moreover, we can get ωT(k)ω(k)≤0.7=d when the external disturbance ω(k) is chosen as ωk=0.21/(1+k2),0.21/(1+k2)T.
In addition, we take the following activation function:(77)fxk=0,0T and we can conclude that(78)U1=V1=diag0,0,U2=V2=diag0,0 According to the methods in [28], we can get the optimal values γmin=5.9474 and c2min=11.5133, respectively. While using methods of this paper, we can obtain the optimal values γmin=4.3640 and c2min=9.2560, respectively, and the corresponding controller gain matrices are given as follows:(79)K1=-0.5959-0.0143-0.2016-1.0353,K2=-0.27703.7916-0.10202.2540. Since the smaller γ and c2, the better system performance is. Thus, our results are better than that in [28]. In addition, we can get that the corresponding system is SFTB and satisfies the desired finite-time H∞ performance by simulation results in Figures 1 and 2. This shows that the results of this paper can be applied to the stochastic finite-time H∞ control problem, which further verifies that the proposed results in this paper are more general than that in [28].Figure 1
The trajectory of statex(k).Figure 2
The curve ofE{xT(k)Rix(k)}.Example 2.
Consider the discrete time NNS (1). Its parameters are shown below:(80)A1=0.02-0.05-0.120.030.06-0.10.060.08-0.08,Ad1=-0.020.060.040.040.01-0.030.010.020.05,Am1=0.040.0200.050.0400.020.060.03,A2=0.02-0.060.070.060.04-0.040.010.020.014,Ad2=0.10.080.100.05-0.06-0.030.04-0.05,Am2=-0.080.0600-0.10.040.020.040.02,B1=-1.00.230.50.050.40.3,B2=0.8-0.171.00.40.30.26,C1=100010001,C2=100010001,D1=00.0010010.001000.001,D2=12112.62332,F1=00.0010.0100.010000.001,F2=1.21.30.122.52123,
Assume thatN=40, c1=1, and δ=1.0027 and M1, M2, N1, and N2 and Δ(k) are selected as follows:(81)M1=0.20.4-0.10.20.2-0.3T,M2=0.20.4-0.10.20.4-0.6T,N1=-0.10000.1000-0.3,N2=-0.10000.1000-0.15,Δk=sin0.2k000cos0.2k000sin0.2k. In addition, in this example, we take dissipativity matrices W=-0.3, S=120001200012T, R=10.10.10.110.10.10.11, R1=R2=30001.30001, and transition matrix Π=0.60.40.70.3. The constant sequence μm=3-(3+m). It can be checked that μ¯=∑m=1+∞μm=1/54<∑m=1+∞mμm=1/36<+∞, which satisfies the convergence condition (2). Moreover, we can get ωT(k)ω(k)≤0.62=d when the external disturbance ω(k) is chosen as ω(k)=0.21/(1+k2),0.21/(1+k2),0.21/(1+k2)T.
Activation function is given as(82)fxk=tanh⁡-1.2x1k,tanh⁡0.8x1k,tanh⁡x1kTand we can conclude that(83)U1=V1=diag0,0,0,U2=V2=diag1.2,0.8,1Other parameters are the same as Example 1. By solving the LMIs (65), (69), (70), and (71), we can obtain the optimal value γmin=1.340e-12, c2min=6.4624, and state feedback gain matrices as follows:(84)K1=-0.8625-0.4365-0.00900.2306-0.3940-0.0240,K2=0.20160.17250.0041-1.2945-0.5934-0.0324The effectiveness of the proposed methods will be shown in Figures3 and 4, in which Figure 3 shows the state trajectories of the closed-loop system and Figure 4 denotes the time history of E{x(k)Rix(k)}. From Figures 3 and 4, we can obtain that the system (8) is SFTD; this shows that designed nonfragile state feedback controller is effective. Furthermore, in order to show that the dissipative results in this paper are more general, we choose matrices on Remark 14 for each case. By using LMI toolbox in Matlab, the optimal value and the corresponding controller gain matrices can be obtained for each case in Tables 1 and 2.Table 1
Optimized variable for each case.
Case c 2 m i n γmin H ∞ control 7.2134 3.4302e-12 Passivity control 8.1320 1.8893e-12Table 2
Controller gain for each case.
Case K 1 K2 H ∞ control - 0.8688 - 0.4377 - 0.0090 0.2273 - 0.3513 - 0.0205 0.2038 0.1707 0.0038 - 1.2731 - 0.5455 - 0.0295  Passivity control - 0.8684 - 0.4398 - 0.0089 0.2325 - 0.4040 - 0.0241 0.20230.17400.0041-1.3098-0.6022-0.0323Figure 3
The trajectory of statex(k).Figure 4
The curve ofE{xT(k)Rix(k)}.The simulation results are shown in Figures5–8. Figures 5 and 7 show the state responses of the closed-loop system under two difference cases, respectively. Figures 6 and 8 show the time history of E{x(k)Rix(k)} under two difference cases, respectively. It is not difficult to find that the system is SFTB and satisfies corresponding performance under two difference cases from Figures 5–8. This shows that the dissipative control method obtained in this paper can also solve the H∞ control and passive control problem, which proves that dissipative performance is a more general result than H∞ performance and passive performance. These results show that different control goals can be achieved in the process of dissipative control. Therefore, in the actual system dissipative control can save costs and time.Figure 5
The trajectory of statex(k).Figure 6
The curve ofE{xT(k)Rix(k)}.Figure 7
The trajectory of statex(k).Figure 8
The curve ofE{xT(k)Rix(k)}.


## 5. Conclusions

In this paper, stochastic finite-time nonfragile dissipative control issue for a class of discrete-time NNS in the presence of Markovian jumps and mixed delays has been presented. By employing stochastic analysis method and LMI technique, some sufficient conditions of stochastic finite-time dissipativity for the system (8) are obtained. Through the feasible solution of LMIs, the analytic expression of the expected nonfragile controller gain is provided. Two numerical examples have been used to verify the effectiveness of the proposed theoretical results. It is interesting for us to extended derived results to SFTD filtering problems. On the other hand, the proposed approach in this paper can also further extended to more complicated discrete-time NNS such as actuator saturation [32] and fading measurements [33].



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