# Dissipativity Analysis for a Class of Discrete-Time Neutral Stochastic Nonlinear Systems with Time Delay

Authors: Shengchun Yu; Yanzhen Pang; Guici Chen; Xin Zhou
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

This paper focuses on the problem of dissipativity analysis for a class of discrete-time neutral stochastic nonlinear systems (DTNSNSs) with time delay and parameter uncertainties. Different from the existing results on this topic of neutral system, a kind of discretizing the neutral system is considered. Firstly, a sufficient condition of the dissipativity, which is dependent on the solution of the Lyapunov–Krasovskii technique and linear matrix inequalities (LMIs), is established. Moreover, the state-feedback controller is designed to guarantee the dissipative performance of the closed-loop system. The effectiveness of the theoretical results is finally demonstrated by a numerical example.


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## Body

## 1. Introduction

Time delay is an ubiquitous phenomenon in nature, which is the main reason of oscillation, instability, and bad system performance of the control system. As a result, the problem of time delay is inevitably involved in many control systems [1–5]. At the same time, it is not only the time delay that has an impact on the system; random factors also exist objectively in the actual production, and the influence of random errors can hardly be ignored [1, 6–12]. Accordingly, the work of a stochastic time-delay system is particularly important.In the research of stochastic time-delay systems, neutral stochastic time-delay systems are a special and crucial kind of control systems with time delay, which are widely used in natural science and engineering technology, such as economics, automatic control theory, medicine, biology, and chemistry. Neutral time-delay systems are those in which both the state and state derivative are delayed. Furthermore, the delay in general system state is called discrete delay, and the delay in system state derivative is called neutral delay, which can reflect the change law of things more profoundly and accurately and reveal the essence of things. Therefore, it is necessary to study the neutral stochastic delay systems, which have captured the attention of a large number of scholars [1, 8–10].As we all know, most phenomena in nature are described by nonlinear systems [3, 13–16]. Although the research results of stochastic nonlinear systems are not as much as those of linear stochastic systems, there are also rich achievements [2, 17–22]. Based on fixed point theory, Wu et al. [17] investigated the asymptotic mean square stability of the nonlinear neutral stochastic differential equations with time-varying delays. Without assuming linear growth conditions, Shen et al. [19] studied the boundedness and exponential stability of the exact solution, whose results can be applied to practical applications.The notion of dissipative in control systems has caused many scholars’ concern since it was introduced by Willems firstly [23], due to its wide applications in control theory and practical systems such as robotic systems, electromechanical systems, power systems, internal combustion engine engineering, and chemical processes [3, 11, 24–26]. Dissipativity is a very important concept in a control system, and as special cases, the concepts of H∞ performance and passivity exist widely in the fields of physics, applied mathematics, and mechanics, (see, for example, [26, 27]. By constructing an appropriate Lyapunov–Krasovskii Functional (LKF), Cao et al. [3] focused on the problem of the global asymptotic stability and dissipativity problem for a class of neutral-type stochastic Markovian Jump Static Neural Networks (NTSMJSNNs) with time-varying delays. Yang [25] probed the robust control problem of nonlinear time-delay systems based on dissipative analysis. The problem of extended dissipativity analysis for delayed uncertain discrete-time singular neural networks (DTSNNs) with Markovian jump parameters and stochastic behavior was studied in [11]. By utilizing the linearization technique, Hanmei Wang and Jun Zhao have investigated passivity and H∞ control of switched discrete-time nonlinear systems [26].In a discrete system, namely, the discrete time system, the variables of all or key components of the system exist in the form of discrete signals, and the state of the system changes at discrete time points. Discrete systems need to be described by difference equations, which are widely used in social, economic, and engineering systems, such as automata, impulse control, sampling regulation, and digital control [12, 27–30]. In practical control systems, some discrete dynamical systems described by difference equations are often encountered. The stability research of them in the sense of Lyapunov–Krasovskii has been poured enough attention into [31, 32]. However, due to the existence of the neutral items xs+1−τ, there are few results of the work on discrete-time neutral systems [32].In this paper, we discuss a dissipative analysis problem for discrete-time neutral stochastic nonlinear systems (DTNSNSs) with time delay. The problem under consideration is to design a state-feedback controller for DTNSNSs with time delay such that the resulting closed-loop system is not only asymptotically stable in the mean square but also satisfies the strictlyX,Y,Z-dissipative condition. Using linear matrix inequalities, a sufficient condition for the solvability of the problem is established. When the condition is satisfied, the desired feedback controller is also obtained for DTNSNSs with time delay and parameter uncertainties.The rest of this paper is structured as follows. In Section2, a model description and preliminaries are stated. Our main results are provided in Section 3. In Section 4, simulation results are presented to explain the correctness of the theoretical methods. Finally, conclusions are drawn in Section 5.


## 2. Model Description and Preliminaries

We consider the following discrete-time neutral stochastic nonlinear systems (DTNSNSs) with time delay and parameter uncertainties(1)xs+1=A+ΔAsxs+Ad+ΔAdsxs−d+Bus+Gvs+f^s,xs+f^ds,xs−d+μxs+1−τ+H+ΔHsxs+Hd+ΔHdsxs−dws,zs=C+ΔCsxs+Cd+ΔCds×xs−d,xs=φs,∀s∈−maxd,τ,0,where d is a time delay, τ denotes a neutral delay, xs∈Rn is the state variable, vs∈Rq is the exogenous disturbance input or a reference signal, us∈Rn is the control input, zs∈Rp is the control output vector, φs is a continuous vector-valued initial function, f^s,xs,f^ds,xs−d∈Rn×Rn represent unknown nonlinear functions which describe the structure uncertainty of the system, and ws is a scalar Brownian motion on the complete probability space Ω,Ϝ,P with Ews=0, Ew2s=1. Here, A,Ad,B,G,H,Hd,C,Cd are known real matrices. ΔAs,ΔAds,ΔHs,ΔHds,ΔCs,ΔCds are unknown time-varying matrices, which are the parameter uncertainties and satisfy the following condition:(2)ΔAsΔAdsΔHsΔHdsΔCsΔCds=ℳ1ℳ2ℳ3ΔsN1N2,in which ℳ1,ℳ2,ℳ3,N1,N2 are determinate constant matrices with appropriate dimensions and Δs is an unknown real time-varying matrix function with Lebesgue norm measurable elements satisfying(3)ΔTsΔs≤I,∀s≥0.The parameter uncertainties are said to be admissible if both (2) and (3) hold. When Δs=0, the system is referred to as a nominal system.Remark 1.
Generally speaking, the structure of the parameter uncertainties with the form (2) and (3) has been widely considered in practical problems of control systems including both continuous and discrete time [1, 9, 18, 30]. Compared with the novel framework of discrete-time neutral systems [32], the stochastic and nonlinear perturbations are taken into this paper, which makes the model more practical.
For systems (1), we design the following state feedback controller:(4)us=Kxs,in which the matrix K is the gain matrix, which is to be designed. Therefore, the closed-loop DTNSNSs with time delay and parameter uncertainties can be rewritten as(5)xs+1=A⌣xs+A⌣dxs−d+Gvs+f^s,xs+f^ds,xs−d+μxs+1−τ+H⌣xs+H⌣dxs−dws,zs=C⌣xs+C⌣dxs−d,xs=φs,∀s∈−maxd,τ,0,where A⌣=A+BK+ΔAs, A⌣d=Ad+ΔAds, H⌣=H+ΔHs, H⌣d=Hd+ΔHds, C⌣=C+ΔCs, and C⌣d=Cd+ΔCds.
In the first place, we make the following assumptions.Assumption 1.
The unknown nonlinear perturbationsf^s,xs,f^ds,xs−d satisfy the following condition:(6)f^Ts,xsℜf^s,xs≤α2xTsℜxs,(7)f^dTs,xs−dℜf^ds,xs−d≤β2xTs−dℜxs−d,where α, β are the given positive real constants and ℜ is a matrix with appropriate dimensions.
As a matter of fact, Assumption1 is more general and has significant implications for dealing with the nonlinear function of the system [25].Assumption 2.
Without loss of generity, for the neutral termμxs+1−τ, it satisfies the following condition:(8)μxs+1−τ≤ηxs+1−τ,η∈0,1.
In fact, the abovementioned inequality is equivalent to(9)η2xTs+1−τxs+1−τ−μTxs+1−τμxs+1−τ≥0.
Before presenting the main results of this paper, two definitions for the DTNSNSs (1) with time delay are introduced, which will be necessary for the derivations of following theorems.Definition 1.
(see [17]). us=0 and vs=0 of systems (1) are called stochastic mean square stable if for any ε>0, there exists constant δε>0 such that(10)Exs2<ε,s>0,when sup−h≤κ≤0Exs2<δε, h=maxd,τ.
If, additionally,lims⟶∞Exs2=0 for any initial conditions, then systems (1) with us=0 and vs=0 are said to be stochastic mean square asymptotically stable.
For systems (1), the energy supply function is defined as(11)Φv,z,N=z,XzN+2v,YzN+v,ZvN,∀N≥0,where the matrices XT=X∈Xq×q≤0, YT=Y∈Yq×q>0, and ZT=Z∈Zq×q≥0, and(12)ℏ,ℑN≜∑s=0NℏTsℑs.Definition 2.
(see [4]). Given supply rates function Φv,z,N, the DTNSNSs (1) with time delay are called X,Y,Z-dissipative if under zero initial condition, the energy supply function satisfies(13)EΦv,z,N≥0,∀N≥0.
Furthermore, if there exists a small enough scalarϱ>0,(14)EΦv,z,N≥ϱEv,vN,∀N≥0.
The DTNSNSs (1) with time delay are called strictly X,Y,Z-dissipative.Remark 2.
The abovementioned performance of strictX,Y,Z-dissipativity includes the H∞ performance and passivity as special cases, which are listed as follows:(1)
WhenX=−I, Y=0, and Z=γ2I, strictly X,Y,Z-dissipative, (13) reduces to an H∞ performance requirement(2)
WhenX=0, Y=I, and Z=0, (1) corresponds to a strict passive problem
In addition, the following lemmas are used for proving our desired results.Lemma 1.
(see [2]). Let a∈Rn, b∈Rn, and ε>0; then, we have(15)aTb+bTa≤εaaT+ε−1bbT.Lemma 2.
(see [14]). For the given symmetric matrix Σ and matrices D and ℒ with appropriate dimensions, Σ+Dℱsℒ+ℒTℱTsDT<0 with ℱs satisfying ℱTsℱs≤I holds if and only if Σ+ɛDDT+ɛ−1ℒℒT≤0 holds for any ϵ>0.Lemma 3 (see [21]) (Schur Complement).
Given one positive definite matrixG2>0 and constant matrices G1 and G3, where G1=G1T, G1+G3TG2−1G3<0 if and only if(16)G1G3TG3−G2<0or−G2G3G3TG1<0.
The aim is to study the stability and dissipativity analysis for DTNSNSs with time delay and parameter uncertainties.


## 3. Main Results and Proofs

### 3.1. Dissipativity Analysis for DTNSNSs with Time Delay

Theorem 1.
We considerus=0 of the DTNSNSs (5) with time delay and parameter uncertainties. The nonlinear functions f^s,xs and f^ds,xs−d satisfy (6) and (7). If there exist matrices P>0, Q‡>0‡=1,2, Y, Z≥0, X≤0, and positive scalars λ††=1,2,3,4,5 such that(17)E=E11E12E13E14E15∗E22000∗∗E3300∗∗∗E440∗∗∗0E55<0,is strictly negative definite, where(18)E11=Q1−P0ηA⌣TQ2−C⌣TYP∗−Q1ηA⌣dTQ2−C⌣dTY0∗∗−Q2ηQ2G0∗∗∗♯10∗∗∗∗−P,E12=A⌣TH⌣TA⌣TH⌣TC⌣TA⌣dTH⌣dTA⌣dTH⌣dTC⌣dT00000GT0GT0000000,E13=αIαI00000βIβI00000ηI0000000000,E14=A⌣T0αIαI000A⌣dT00000000ηIηI000000000000,♯1=λ5GTG+ϱI−Z,E15=Φ1,Φ2,Φ3,Φ4,Φ1=αP,0,0,0,0T,Φ2=αQ2,0,0,0,0T,Φ3=0,βP,0,0,0T,Φ4=0,βQ2,0,0,0T,E22=diag−P−1,−P−1,−Q2−1,−Q2−1,X−1,E33=diag−P−1,−Q2−1,−P−1,−Q2−1,−Q2−1,E44=diag−λ5−1I,−λ5−1I,−14λ3−1I,−15λ4−1I,−14λ2−1I,−15λ4−1I,E55=diag−λ1I,−λ2I,−λ3I,−λ4I,then the DTNSNSs (5) with time delay are asymptotically stochastically stable in the mean square and strictly X,Y,Z-dissipative.Proof.
We construct the following Lyapunov–Krasovskii functional as follows:(19)Vs=∑j=13Vjs,with(20)V1s=xs−μxs−τTPxs−μxs−τ,V2s=∑i=s−ds−1xTiQ1xi,V3s=∑κ=s+1−τsxTκQ2xκ.
Calculating the forward difference ofVjs as ΔVjs=Vjs+1−Vjs and taking the mathematical expectation, we obtain(21)EΔV1s=xTs+1Pxs+1−xTs+1Pμxs+1−τ−μT×xs+1−τPxs+1+μTxs+1−τPμxs+1−τ−μTxs−τP×μxs−τ+xTsPμxs−τ+μTxs−τPxs−xTsPxs,EΔV2s=xTsQ1xs−xTs−dQ1xs−d,EΔV3s=xTs+1Q2xs+1−xTs+1−τQ2xs+1−τ.
Accordingly,(22)EΔVs=A⌣xs+A⌣dxs−d+f^s,xs+f^ds,xs−d+GvsTP+Q2A⌣xs+A⌣dxs−d+f^s,xs+f^ds,xs−d+Gvs+H⌣xs+H⌣dxs−dTP+Q2×H⌣xs+H⌣dxs−d+μTxs+1−τ×Q2A⌣xs+A⌣dxs−d+f^s,xs+f^ds,xs−d+Gvs+A⌣xs+A⌣dxs−d+f^s,xs+f^ds,xs−d+GvsTQ2×μxs+1−τ+μTxs+1−τQ2μxs+1−τ−xTsPxs+xTsPμxs−τ+μTxs−τPxs−μTxs−τPμxs−τ+xTsQ1xs−xTs−dQ1xs−d−xTs+1−τQ2xs+1−τ.
By virtue of Lemma1 and Assumption 1, we proceed to express(23)xTsA⌣TPf^s,xs+f^Ts,xsPA⌣xs≤ε1xTsA⌣TA⌣xs+ε1−1α2xTsPTPxs,xTsA⌣TPf^ds,xs−d+f^dTs,xs−dPA⌣xs≤ε2xTsA⌣TA⌣xs+ε2−1β2xTs−dPTPxs−d,xTs−dA⌣dTPf^s,xs+f^Ts,xsPA⌣dxs−d≤ε3xTs−dA⌣dTA⌣dxs−d+ε3−1α2xTsPTPxs,xTs−dA⌣dTPf^ds,xk−d+f^dTs,xs−dPA⌣dxs−d≤ε4xTs−dA⌣dTA⌣dxs−d+ε4−1β2xTs−dPTPxs−d,vTsGTPf^s,xs+f^Ts,xsPGvs≤ε5vTsGTGvs+ε5−1α2xTsPTPxs,vTsGTPf^ds,xs+f^dTs,xsPGvs≤ε6vTsGTGvs+ε6−1β2xTs−dPTPxs−d,f^Ts,xsPf^ds,xs−d+f^dTs,xs−dPf^s,xs≤ε7α2xTsxs+ε7−1β2xTs−dPTPxs−d,(24)f^Ts,xsPf^s,xs≤α2xTsPxs,f^dTs,xs−dPf^ds,xs−d≤β2xTs−dPxs−d.
Similar to the derivation given above, we denote(25)xTsA⌣TQ2f^s,xs+f^Ts,xsQ2A⌣xs≤ε8xTsA⌣TA⌣xs+ε8−1α2xTsQ2TQ2xs,xTsA⌣TQ2f^ds,xs−d+f^dTs,xs−dQ2A⌣xs≤ε9xTsA⌣TA⌣xs+ε9−1β2xTs−dQ2TQ2xs−d,xTs−dA⌣dTQ2f^s,xs+f^Ts,xsQ2A⌣dxs−d≤ε10xTs−dA⌣dTA⌣dxs−d+ε10−1α2xTsQ2TQ2xs,xTs−dA⌣dTQ2f^ds,xs−d+f^dTs,xs−dQ2A⌣dxs−d≤ε11xTs−dA⌣dTA⌣dxs−d+ε11−1β2xTs−dQ2TQ2xs−d,vTsGTQ2f^s,xs+f^Ts,xsQ2Gvs≤ε12vTsGTGvs+ε12−1α2xTsQ2TQ2xs,vTsGTQ2f^ds,xs+f^dTs,xsQ2Gvs≤ε13vTsGTGvs+ε13−1β2xTs−dQ2TQ2xs−d,f^Ts,xsQ2f^ds,xs−d+f^dTs,xs−dQ2f^s,xs≤ε14α2xTsxs+ε14−1β2xTs−d×Q2TQ2xs−d,(26)f^Ts,xsQ2f^s,xs≤α2xTsQ2xs,f^dTs,xs−dQ2f^ds,xs−d≤β2xTs−dQ2xs−d.
Together with Lemma1 and Assumptions 1 and 2, we get(27)f^Ts,xsQ2μxs+1−τ+μTxs+1−τQ2f^s,xs≤ε15−1α2xTsQ2TQ2xs+ε15η2xTs+1−τxs+1−τ,f^dTs,xs−dQ2μxs+1−τ+μTxs+1−τQ2f^ds,xs−d≤ε16−1β2xTs−dQ2TQ2xs−d+ε16η2xTs+1−τxs+1−τ,xTsA⌣TQ2μxs+1−τ+μTxs+1−τQ2A⌣xs≤ηxTsA⌣TQ2xs+1−τ+ηxTs+1−τQ2A⌣xs,xTs−dA⌣dTQ2μxs+1−τ+μTxs+1−τQ2A⌣dxs−d≤ηxTs−dA⌣dTQ2xs+1−τ+ηxTs+1−τQ2A⌣dxs−d,vTsGTQ2μxs+1−τ+μTxs+1−τQ2Gvs≤ηvTsGTQ2xs+1−τ+ηxTs+1−τQ2Gvs,μTxs+1−τQ2μxs+1−τ≤η2xTs+1−τQ2xs+1−τ.
Substituting (23)–(27) into (22), it can be derived that(28)EΔVs≤ζTsΛsζs,where(29)ζTs=xTs,xTs−d,xTs+1−τ,vTs,μTxs−τ,Λs=ℷ110ηA⌣TQ20P∗ℷ22ηA⌣dTQ200∗∗ℷ33ηQ2G0∗∗∗ℷ440∗∗∗∗−P+A⌣TA⌣dT0GT0PA⌣TA⌣dT0GT0T+H⌣TH⌣dT000PH⌣TH⌣dT000T+A⌣TA⌣dT0GT0Q2A⌣TA⌣dT0GT0T+H⌣TH⌣dT000Q2H⌣TH⌣dT000T,ℷ11=ε1+ε2+ε8+ε9A⌣TA⌣+ε1−1+ε3−1+ε5−1α2PTP+ε8−1+ε10−1+ε12−1+ε15−1α2Q2TQ2+ε7+ε14α2I+α2P+α2Q2+Q1−P,ℷ22=ε3+ε4+ε10+ε11A⌣dTA⌣d+ε2−1+ε4−1+ε6−1+ε7−1β2PTP+ε9−1+ε11−1+ε13−1+ε14−1+ε16−1β2Q2TQ2+β2P+β2Q2−Q1,ℷ33=ε15+ε16η2I+η2Q2−Q2,ℷ44=ε5+ε6+ε12+ε13GTG.
For simplicity, let(30)ε1=ε3=ε5=3λ1,ε8=ε10=ε12=ε15=4λ2,ε2=ε4=ε6=ε7=4λ3,ε9=ε11=ε13=ε14=ε16=5λ4,3λ1+4λ2+4λ3+5λ4=λ5.
Therefore,ℷ11, ℷ22, ℷ33, and ℷ44 can be rewritten as follows:(31)ℷ11=λ5A⌣TA⌣+λ1−1α2PTP+λ2−1α2Q2TQ2+4λ3+5λ4α2I+α2P+α2Q2+Q1−P,ℷ22=λ5A⌣dTA⌣d+λ3−1β2PTP+λ4−1β2Q2TQ2+β2P+β2Q2−Q1,ℷ33=4λ2+5λ4η2I+η2Q2−Q2,ℷ44=λ5GTG.
First of all, we aim to prove that the DTNSNSs (5) with time delay are stochastic mean square asymptotic stability.
Letvs=0, which results in(32)EΔVs≤ηTsΞsηs,where(33)ηTs=xTs,xTs−d,xTs+1−τ,μTxs−τ,Ξs=ℷ110ηA⌣TQ2P∗ℷ22ηA⌣dTQ20∗∗ℷ330∗∗∗−P+A⌣TA⌣dT00PA⌣TA⌣dT00T+H⌣TH⌣dT00PH⌣TH⌣dT00T+A⌣TA⌣dT00Q2A⌣TA⌣dT00T+H⌣TH⌣dT00Q2H⌣TH⌣dT00T.
Using Schur complement lemma, it can be written that(34)ψ11ψ12ψ13ψ14ψ15∗ψ22000∗∗ψ3300∗∗∗ψ440∗∗∗∗ψ55,where(35)ψ11=Q1−P0ηA⌣TQ2P∗−Q1ηA⌣dTQ20∗∗−Q20∗∗∗−P,ψ12=A⌣TH⌣TA⌣TH⌣TA⌣dTH⌣dTA⌣dTH⌣dT00000000,ψ13=αIαI00000βIβI00000ηI00000,ψ14=A⌣T0αIαI000A⌣dT00000000ηIηI000000,ψ15=π1,π2,π3,π4,π1=αP,0,0,0T,π2=αQ2,0,0,0T,π3=0,βP,0,0,π4=0,βQ2,0,0,ψ22=diag−P−1,−P−1,−Q2−1,−Q2−1,ψ33=diag−P−1,−Q2−1,−P−1,−Q2−1,−Q2−1,ψ44=diag−λ5−1I,−λ5−1I,−14λ3−1I,−15λ4−1I,−14λ2−1I,−15λ4−1I,ψ55=diag−λ1I,−λ2I,−λ3I,−λ4I.
Combined with (17), we find that Ξs<0⇒EΔVs<0. Therefore, there must exist a positive scalar λ>0 such that(36)ΔVs≤−λxs2,∀xs≠0.
From Definition1, the DTNSNSs (5) with time delay and vs=0 are asymptotically stochastically stable in the mean square.
Next, we will prove the strictlyX,Y,Z-dissipativeness of systems (5). When xs0=0, let vs≠0, and combined with (26), we can get(37)EΔVs−EzTsXzs+2vTsYzs+vTsZvs+ϱEvTsvs≤ζTsΛsζs−C⌣xs+C⌣dxs−dTXC⌣xs+C⌣dxs−d−2C⌣xs+C⌣d×xs−dTYvs+vTsϱI−Zvs=ζTsΛ˜sζs,where(38)Λ˜s=ℷ110ηA⌣TQ2−C⌣TYP∗ℷ22ηA⌣dTQ2−C⌣dTY0∗∗ℷ33ηQ2G0∗∗∗ℷ˜440∗∗∗∗−P+A⌣TA⌣dT0GT0PA⌣TA⌣dT0GT0T+H⌣TH⌣dT000PH⌣TH⌣dT000T+A⌣TA⌣dT0GT0Q2A⌣TA⌣dT0GT0T+H⌣TH⌣dT000Q2H⌣TH⌣dT000T+−C⌣T−C⌣dT000X−C⌣T−C⌣dT000T,ℷ˜44=λ5GTG+ϱI−Z.
By the Schur complement, (17) gives rise to(39)Λ˜s<0.
By means of (37), we have(40)EzTsXzs+2vTsYzs+vTsZvs−ϱEvTsvs−EVs+1+EVs>0.
As a consequence, for any integerN>0, summing up both sides of (40), from s=0 to s=N−1, and taking the mathematical expectation, it follows that(41)EV0+EΦv,z,N>ϱEv,vN+EVN.
On account of the zero initial condition, (41) means that(42)EΦv,z,N>ϱEv,vN.
Then, by Definition2, the DTNSNSs (5) with time delay are strictly X,Y,Z-dissipative. Thus, the proof is completed.Remark 3.
In the proof of Theorem1, we consider the existence of nonlinear perturbations f^s,xs and f^ds,xs−d and the neutral term μxs+1−τ of DTNSNSs (5) with time delay and parameter uncertainties, which are more complicated and laboured than that of general time-delay systems [12, 30, 32], and the inequality techniques are used. In order to solve them by LMI technology, we make the following: (1) the items containing matrix P+Q2 in (22) are separated into the items containing matrix P and matrix Q2, respectively, processed separately, and then, added, so that they can be converted into linear matrices; (2) by analyzing the relationship between εj−1j=1,2,…,16, λ††=1,2,3,4,5 are introduced to reduce the dimensions of the matrices (17).


### 3.2. State-Feedback Controller Design for DTNSNSs with Time Delay

In this part, the state-feedback controller for DTNSNSs (5) with time delay and parameter uncertainties is designed, and the following theorem is given in terms of LMI.Theorem 2.
Given a negative semidefinite matrixX≤0, a real matrix Y, and the positive matrix Z≥0, if there exist matrices X1>0, X2>0, Y, and Z‡‡=1,2>0 and positive scalars λ††=1,2,3,4,5 satisfying the following Θ which is strictly negative definite, such that(43)Θ=J11J12J13J14J15J16∗J22000J26∗∗J33000∗∗∗J440J46∗∗∗∗J550∗∗∗∗∗J66<0,where(44)J11=Z1−X10⋄1−X1CTYX1∗−Z1ηX1AdT−X1CdTY0∗∗−X2ηG0∗∗∗⋄20∗∗∗∗−X1,J12=⋄3X1HT⋄4X1HTX1CTX1AdTX1HdTX1AdTX1HdTX1CdT00000GT0GT0000000,J13=αX1αX100000βX1βX100000ηX20000000000,J14=⋄50αλ3X1αλ4X1000λ5X1AdT00000000ηλ2X2ηλ4X2000000000000,J15=αIαZ20000βIβZ200000000,J16=0X1N1T0X1N1T000X1N2T000X1N2Tɛ1ηℳ100000−ɛ1Yℳ300000000000,J22=diag−X1,−X1,−X2,−X2,X−1,J26=ϑ1,0,0,0,0,0,J33=diag−X1,−X2,−X1,−X2,−X2,J44=diag−λ5I,−λ5I,−14λ3I,−15λ4I,−14λ2I,−15λ4I,J46=ϑ2,ϑ3,0,0,0,0T,J55=diag−λ1I,−λ2I,−λ3I,−λ4I,J66=diag−ɛ1I,−ɛ1I−ɛ2I,−ɛ2I,−ɛ3I,−ɛ3I,ϑ1=ɛ1ℳ1T,ɛ1ℳ2T,ɛ1ℳ1T,ɛ1ℳ2T,ɛ1ℳ3TT,ϑ2T=0,0,ɛ2λ5ℳ1,0,0,0,ϑ3T=0,0,0,0,ɛ3λ5ℳ1,0,⋄1=ηX1AT+ηYTBT,⋄2=λ5GTG+ϱI−Z,⋄3=⋄4=X1AT+YTBT,⋄5=λ5X1AT+λ5YTBT,then the systems (5) are asymptotically stable in the mean square and strictly X,Y,Z-dissipative, and the state feedback gain is K=YX1−1.Proof.
Applying Lemma2, the matrix inequality (17) can be written as(45)£˜+ϒ1TΔTsϒ2+ϒ2TΔsϒ1+ϒ3TΔTsϒ4+ϒ4TΔsϒ3+ϒ5TΔTsϒ6+ϒ6TΔsϒ5<0,where(46)£˜=£˜11£˜12£˜13£˜14£˜15∗£˜22000∗∗£˜3300∗∗∗£˜440∗∗∗∗£˜55,£˜11=Q1−P0ηA+BKTQ2−CTYP∗−Q1ηAdTQ2−CdTY0∗∗−Q2ηQ2G0∗∗∗♯˜10∗∗∗∗−P,£˜12=A+BKTHTA+BKTHTCTAdTHdTAdTHdTCdT00000GT0GT0000000,£˜13=αIαI00000βIβI00000ηI0000000000,£˜14=A+BKT0αIαI000AdT00000000ηIηI000000000000,♯˜1=λ5GTG+ϱI−Z,£˜15=Φ˜1,Φ˜2,Φ˜3,Φ˜4,Φ˜1=αP,0,0,0,0T,Φ˜2=αQ2,0,0,0,0T,Φ˜3=0,βP,0,0,0T,Φ˜4=0,βQ2,0,0,0T,£˜22=diag−P−1,−P−1,−Q2−1,−Q2−1,X−1,£˜33=diag−P−1,−Q2−1,−P−1,−Q2−1,−Q2−1,£˜44=diag−λ5−1I,−λ5−1I,−14λ3−1I,−15λ4−1I,−14λ2−1I,−15λ4−1I,£˜55=diag−λ1I,−λ2I,−λ3I,−λ4I,ϒ1=N1,N2,01×23,ϒ3=N1,01×24,ϒ2=0,0,ηℳ1TQ2,−ℳ3TY,0,ℳ1T,ℳ2T,ℳ1T,ℳ2T,ℳ3T,01×15,ϒ4=01×15,ℳ1T,01×9,ϒ5=0,N2,01×23,ϒ6=01×16,ℳ1T,01×8.
Combined with Schur complement lemma and contragradient transformation, we defineZ1=X1Q1X1, Z2=X1Q2T, P−1=X1, Q2−1=X2, and KX1=Y, and we obtain that (43) implies (17). This completes the proof of Theorem 2.


## 3.1. Dissipativity Analysis for DTNSNSs with Time Delay

Theorem 1.
We considerus=0 of the DTNSNSs (5) with time delay and parameter uncertainties. The nonlinear functions f^s,xs and f^ds,xs−d satisfy (6) and (7). If there exist matrices P>0, Q‡>0‡=1,2, Y, Z≥0, X≤0, and positive scalars λ††=1,2,3,4,5 such that(17)E=E11E12E13E14E15∗E22000∗∗E3300∗∗∗E440∗∗∗0E55<0,is strictly negative definite, where(18)E11=Q1−P0ηA⌣TQ2−C⌣TYP∗−Q1ηA⌣dTQ2−C⌣dTY0∗∗−Q2ηQ2G0∗∗∗♯10∗∗∗∗−P,E12=A⌣TH⌣TA⌣TH⌣TC⌣TA⌣dTH⌣dTA⌣dTH⌣dTC⌣dT00000GT0GT0000000,E13=αIαI00000βIβI00000ηI0000000000,E14=A⌣T0αIαI000A⌣dT00000000ηIηI000000000000,♯1=λ5GTG+ϱI−Z,E15=Φ1,Φ2,Φ3,Φ4,Φ1=αP,0,0,0,0T,Φ2=αQ2,0,0,0,0T,Φ3=0,βP,0,0,0T,Φ4=0,βQ2,0,0,0T,E22=diag−P−1,−P−1,−Q2−1,−Q2−1,X−1,E33=diag−P−1,−Q2−1,−P−1,−Q2−1,−Q2−1,E44=diag−λ5−1I,−λ5−1I,−14λ3−1I,−15λ4−1I,−14λ2−1I,−15λ4−1I,E55=diag−λ1I,−λ2I,−λ3I,−λ4I,then the DTNSNSs (5) with time delay are asymptotically stochastically stable in the mean square and strictly X,Y,Z-dissipative.Proof.
We construct the following Lyapunov–Krasovskii functional as follows:(19)Vs=∑j=13Vjs,with(20)V1s=xs−μxs−τTPxs−μxs−τ,V2s=∑i=s−ds−1xTiQ1xi,V3s=∑κ=s+1−τsxTκQ2xκ.
Calculating the forward difference ofVjs as ΔVjs=Vjs+1−Vjs and taking the mathematical expectation, we obtain(21)EΔV1s=xTs+1Pxs+1−xTs+1Pμxs+1−τ−μT×xs+1−τPxs+1+μTxs+1−τPμxs+1−τ−μTxs−τP×μxs−τ+xTsPμxs−τ+μTxs−τPxs−xTsPxs,EΔV2s=xTsQ1xs−xTs−dQ1xs−d,EΔV3s=xTs+1Q2xs+1−xTs+1−τQ2xs+1−τ.
Accordingly,(22)EΔVs=A⌣xs+A⌣dxs−d+f^s,xs+f^ds,xs−d+GvsTP+Q2A⌣xs+A⌣dxs−d+f^s,xs+f^ds,xs−d+Gvs+H⌣xs+H⌣dxs−dTP+Q2×H⌣xs+H⌣dxs−d+μTxs+1−τ×Q2A⌣xs+A⌣dxs−d+f^s,xs+f^ds,xs−d+Gvs+A⌣xs+A⌣dxs−d+f^s,xs+f^ds,xs−d+GvsTQ2×μxs+1−τ+μTxs+1−τQ2μxs+1−τ−xTsPxs+xTsPμxs−τ+μTxs−τPxs−μTxs−τPμxs−τ+xTsQ1xs−xTs−dQ1xs−d−xTs+1−τQ2xs+1−τ.
By virtue of Lemma1 and Assumption 1, we proceed to express(23)xTsA⌣TPf^s,xs+f^Ts,xsPA⌣xs≤ε1xTsA⌣TA⌣xs+ε1−1α2xTsPTPxs,xTsA⌣TPf^ds,xs−d+f^dTs,xs−dPA⌣xs≤ε2xTsA⌣TA⌣xs+ε2−1β2xTs−dPTPxs−d,xTs−dA⌣dTPf^s,xs+f^Ts,xsPA⌣dxs−d≤ε3xTs−dA⌣dTA⌣dxs−d+ε3−1α2xTsPTPxs,xTs−dA⌣dTPf^ds,xk−d+f^dTs,xs−dPA⌣dxs−d≤ε4xTs−dA⌣dTA⌣dxs−d+ε4−1β2xTs−dPTPxs−d,vTsGTPf^s,xs+f^Ts,xsPGvs≤ε5vTsGTGvs+ε5−1α2xTsPTPxs,vTsGTPf^ds,xs+f^dTs,xsPGvs≤ε6vTsGTGvs+ε6−1β2xTs−dPTPxs−d,f^Ts,xsPf^ds,xs−d+f^dTs,xs−dPf^s,xs≤ε7α2xTsxs+ε7−1β2xTs−dPTPxs−d,(24)f^Ts,xsPf^s,xs≤α2xTsPxs,f^dTs,xs−dPf^ds,xs−d≤β2xTs−dPxs−d.
Similar to the derivation given above, we denote(25)xTsA⌣TQ2f^s,xs+f^Ts,xsQ2A⌣xs≤ε8xTsA⌣TA⌣xs+ε8−1α2xTsQ2TQ2xs,xTsA⌣TQ2f^ds,xs−d+f^dTs,xs−dQ2A⌣xs≤ε9xTsA⌣TA⌣xs+ε9−1β2xTs−dQ2TQ2xs−d,xTs−dA⌣dTQ2f^s,xs+f^Ts,xsQ2A⌣dxs−d≤ε10xTs−dA⌣dTA⌣dxs−d+ε10−1α2xTsQ2TQ2xs,xTs−dA⌣dTQ2f^ds,xs−d+f^dTs,xs−dQ2A⌣dxs−d≤ε11xTs−dA⌣dTA⌣dxs−d+ε11−1β2xTs−dQ2TQ2xs−d,vTsGTQ2f^s,xs+f^Ts,xsQ2Gvs≤ε12vTsGTGvs+ε12−1α2xTsQ2TQ2xs,vTsGTQ2f^ds,xs+f^dTs,xsQ2Gvs≤ε13vTsGTGvs+ε13−1β2xTs−dQ2TQ2xs−d,f^Ts,xsQ2f^ds,xs−d+f^dTs,xs−dQ2f^s,xs≤ε14α2xTsxs+ε14−1β2xTs−d×Q2TQ2xs−d,(26)f^Ts,xsQ2f^s,xs≤α2xTsQ2xs,f^dTs,xs−dQ2f^ds,xs−d≤β2xTs−dQ2xs−d.
Together with Lemma1 and Assumptions 1 and 2, we get(27)f^Ts,xsQ2μxs+1−τ+μTxs+1−τQ2f^s,xs≤ε15−1α2xTsQ2TQ2xs+ε15η2xTs+1−τxs+1−τ,f^dTs,xs−dQ2μxs+1−τ+μTxs+1−τQ2f^ds,xs−d≤ε16−1β2xTs−dQ2TQ2xs−d+ε16η2xTs+1−τxs+1−τ,xTsA⌣TQ2μxs+1−τ+μTxs+1−τQ2A⌣xs≤ηxTsA⌣TQ2xs+1−τ+ηxTs+1−τQ2A⌣xs,xTs−dA⌣dTQ2μxs+1−τ+μTxs+1−τQ2A⌣dxs−d≤ηxTs−dA⌣dTQ2xs+1−τ+ηxTs+1−τQ2A⌣dxs−d,vTsGTQ2μxs+1−τ+μTxs+1−τQ2Gvs≤ηvTsGTQ2xs+1−τ+ηxTs+1−τQ2Gvs,μTxs+1−τQ2μxs+1−τ≤η2xTs+1−τQ2xs+1−τ.
Substituting (23)–(27) into (22), it can be derived that(28)EΔVs≤ζTsΛsζs,where(29)ζTs=xTs,xTs−d,xTs+1−τ,vTs,μTxs−τ,Λs=ℷ110ηA⌣TQ20P∗ℷ22ηA⌣dTQ200∗∗ℷ33ηQ2G0∗∗∗ℷ440∗∗∗∗−P+A⌣TA⌣dT0GT0PA⌣TA⌣dT0GT0T+H⌣TH⌣dT000PH⌣TH⌣dT000T+A⌣TA⌣dT0GT0Q2A⌣TA⌣dT0GT0T+H⌣TH⌣dT000Q2H⌣TH⌣dT000T,ℷ11=ε1+ε2+ε8+ε9A⌣TA⌣+ε1−1+ε3−1+ε5−1α2PTP+ε8−1+ε10−1+ε12−1+ε15−1α2Q2TQ2+ε7+ε14α2I+α2P+α2Q2+Q1−P,ℷ22=ε3+ε4+ε10+ε11A⌣dTA⌣d+ε2−1+ε4−1+ε6−1+ε7−1β2PTP+ε9−1+ε11−1+ε13−1+ε14−1+ε16−1β2Q2TQ2+β2P+β2Q2−Q1,ℷ33=ε15+ε16η2I+η2Q2−Q2,ℷ44=ε5+ε6+ε12+ε13GTG.
For simplicity, let(30)ε1=ε3=ε5=3λ1,ε8=ε10=ε12=ε15=4λ2,ε2=ε4=ε6=ε7=4λ3,ε9=ε11=ε13=ε14=ε16=5λ4,3λ1+4λ2+4λ3+5λ4=λ5.
Therefore,ℷ11, ℷ22, ℷ33, and ℷ44 can be rewritten as follows:(31)ℷ11=λ5A⌣TA⌣+λ1−1α2PTP+λ2−1α2Q2TQ2+4λ3+5λ4α2I+α2P+α2Q2+Q1−P,ℷ22=λ5A⌣dTA⌣d+λ3−1β2PTP+λ4−1β2Q2TQ2+β2P+β2Q2−Q1,ℷ33=4λ2+5λ4η2I+η2Q2−Q2,ℷ44=λ5GTG.
First of all, we aim to prove that the DTNSNSs (5) with time delay are stochastic mean square asymptotic stability.
Letvs=0, which results in(32)EΔVs≤ηTsΞsηs,where(33)ηTs=xTs,xTs−d,xTs+1−τ,μTxs−τ,Ξs=ℷ110ηA⌣TQ2P∗ℷ22ηA⌣dTQ20∗∗ℷ330∗∗∗−P+A⌣TA⌣dT00PA⌣TA⌣dT00T+H⌣TH⌣dT00PH⌣TH⌣dT00T+A⌣TA⌣dT00Q2A⌣TA⌣dT00T+H⌣TH⌣dT00Q2H⌣TH⌣dT00T.
Using Schur complement lemma, it can be written that(34)ψ11ψ12ψ13ψ14ψ15∗ψ22000∗∗ψ3300∗∗∗ψ440∗∗∗∗ψ55,where(35)ψ11=Q1−P0ηA⌣TQ2P∗−Q1ηA⌣dTQ20∗∗−Q20∗∗∗−P,ψ12=A⌣TH⌣TA⌣TH⌣TA⌣dTH⌣dTA⌣dTH⌣dT00000000,ψ13=αIαI00000βIβI00000ηI00000,ψ14=A⌣T0αIαI000A⌣dT00000000ηIηI000000,ψ15=π1,π2,π3,π4,π1=αP,0,0,0T,π2=αQ2,0,0,0T,π3=0,βP,0,0,π4=0,βQ2,0,0,ψ22=diag−P−1,−P−1,−Q2−1,−Q2−1,ψ33=diag−P−1,−Q2−1,−P−1,−Q2−1,−Q2−1,ψ44=diag−λ5−1I,−λ5−1I,−14λ3−1I,−15λ4−1I,−14λ2−1I,−15λ4−1I,ψ55=diag−λ1I,−λ2I,−λ3I,−λ4I.
Combined with (17), we find that Ξs<0⇒EΔVs<0. Therefore, there must exist a positive scalar λ>0 such that(36)ΔVs≤−λxs2,∀xs≠0.
From Definition1, the DTNSNSs (5) with time delay and vs=0 are asymptotically stochastically stable in the mean square.
Next, we will prove the strictlyX,Y,Z-dissipativeness of systems (5). When xs0=0, let vs≠0, and combined with (26), we can get(37)EΔVs−EzTsXzs+2vTsYzs+vTsZvs+ϱEvTsvs≤ζTsΛsζs−C⌣xs+C⌣dxs−dTXC⌣xs+C⌣dxs−d−2C⌣xs+C⌣d×xs−dTYvs+vTsϱI−Zvs=ζTsΛ˜sζs,where(38)Λ˜s=ℷ110ηA⌣TQ2−C⌣TYP∗ℷ22ηA⌣dTQ2−C⌣dTY0∗∗ℷ33ηQ2G0∗∗∗ℷ˜440∗∗∗∗−P+A⌣TA⌣dT0GT0PA⌣TA⌣dT0GT0T+H⌣TH⌣dT000PH⌣TH⌣dT000T+A⌣TA⌣dT0GT0Q2A⌣TA⌣dT0GT0T+H⌣TH⌣dT000Q2H⌣TH⌣dT000T+−C⌣T−C⌣dT000X−C⌣T−C⌣dT000T,ℷ˜44=λ5GTG+ϱI−Z.
By the Schur complement, (17) gives rise to(39)Λ˜s<0.
By means of (37), we have(40)EzTsXzs+2vTsYzs+vTsZvs−ϱEvTsvs−EVs+1+EVs>0.
As a consequence, for any integerN>0, summing up both sides of (40), from s=0 to s=N−1, and taking the mathematical expectation, it follows that(41)EV0+EΦv,z,N>ϱEv,vN+EVN.
On account of the zero initial condition, (41) means that(42)EΦv,z,N>ϱEv,vN.
Then, by Definition2, the DTNSNSs (5) with time delay are strictly X,Y,Z-dissipative. Thus, the proof is completed.Remark 3.
In the proof of Theorem1, we consider the existence of nonlinear perturbations f^s,xs and f^ds,xs−d and the neutral term μxs+1−τ of DTNSNSs (5) with time delay and parameter uncertainties, which are more complicated and laboured than that of general time-delay systems [12, 30, 32], and the inequality techniques are used. In order to solve them by LMI technology, we make the following: (1) the items containing matrix P+Q2 in (22) are separated into the items containing matrix P and matrix Q2, respectively, processed separately, and then, added, so that they can be converted into linear matrices; (2) by analyzing the relationship between εj−1j=1,2,…,16, λ††=1,2,3,4,5 are introduced to reduce the dimensions of the matrices (17).


## 3.2. State-Feedback Controller Design for DTNSNSs with Time Delay

In this part, the state-feedback controller for DTNSNSs (5) with time delay and parameter uncertainties is designed, and the following theorem is given in terms of LMI.Theorem 2.
Given a negative semidefinite matrixX≤0, a real matrix Y, and the positive matrix Z≥0, if there exist matrices X1>0, X2>0, Y, and Z‡‡=1,2>0 and positive scalars λ††=1,2,3,4,5 satisfying the following Θ which is strictly negative definite, such that(43)Θ=J11J12J13J14J15J16∗J22000J26∗∗J33000∗∗∗J440J46∗∗∗∗J550∗∗∗∗∗J66<0,where(44)J11=Z1−X10⋄1−X1CTYX1∗−Z1ηX1AdT−X1CdTY0∗∗−X2ηG0∗∗∗⋄20∗∗∗∗−X1,J12=⋄3X1HT⋄4X1HTX1CTX1AdTX1HdTX1AdTX1HdTX1CdT00000GT0GT0000000,J13=αX1αX100000βX1βX100000ηX20000000000,J14=⋄50αλ3X1αλ4X1000λ5X1AdT00000000ηλ2X2ηλ4X2000000000000,J15=αIαZ20000βIβZ200000000,J16=0X1N1T0X1N1T000X1N2T000X1N2Tɛ1ηℳ100000−ɛ1Yℳ300000000000,J22=diag−X1,−X1,−X2,−X2,X−1,J26=ϑ1,0,0,0,0,0,J33=diag−X1,−X2,−X1,−X2,−X2,J44=diag−λ5I,−λ5I,−14λ3I,−15λ4I,−14λ2I,−15λ4I,J46=ϑ2,ϑ3,0,0,0,0T,J55=diag−λ1I,−λ2I,−λ3I,−λ4I,J66=diag−ɛ1I,−ɛ1I−ɛ2I,−ɛ2I,−ɛ3I,−ɛ3I,ϑ1=ɛ1ℳ1T,ɛ1ℳ2T,ɛ1ℳ1T,ɛ1ℳ2T,ɛ1ℳ3TT,ϑ2T=0,0,ɛ2λ5ℳ1,0,0,0,ϑ3T=0,0,0,0,ɛ3λ5ℳ1,0,⋄1=ηX1AT+ηYTBT,⋄2=λ5GTG+ϱI−Z,⋄3=⋄4=X1AT+YTBT,⋄5=λ5X1AT+λ5YTBT,then the systems (5) are asymptotically stable in the mean square and strictly X,Y,Z-dissipative, and the state feedback gain is K=YX1−1.Proof.
Applying Lemma2, the matrix inequality (17) can be written as(45)£˜+ϒ1TΔTsϒ2+ϒ2TΔsϒ1+ϒ3TΔTsϒ4+ϒ4TΔsϒ3+ϒ5TΔTsϒ6+ϒ6TΔsϒ5<0,where(46)£˜=£˜11£˜12£˜13£˜14£˜15∗£˜22000∗∗£˜3300∗∗∗£˜440∗∗∗∗£˜55,£˜11=Q1−P0ηA+BKTQ2−CTYP∗−Q1ηAdTQ2−CdTY0∗∗−Q2ηQ2G0∗∗∗♯˜10∗∗∗∗−P,£˜12=A+BKTHTA+BKTHTCTAdTHdTAdTHdTCdT00000GT0GT0000000,£˜13=αIαI00000βIβI00000ηI0000000000,£˜14=A+BKT0αIαI000AdT00000000ηIηI000000000000,♯˜1=λ5GTG+ϱI−Z,£˜15=Φ˜1,Φ˜2,Φ˜3,Φ˜4,Φ˜1=αP,0,0,0,0T,Φ˜2=αQ2,0,0,0,0T,Φ˜3=0,βP,0,0,0T,Φ˜4=0,βQ2,0,0,0T,£˜22=diag−P−1,−P−1,−Q2−1,−Q2−1,X−1,£˜33=diag−P−1,−Q2−1,−P−1,−Q2−1,−Q2−1,£˜44=diag−λ5−1I,−λ5−1I,−14λ3−1I,−15λ4−1I,−14λ2−1I,−15λ4−1I,£˜55=diag−λ1I,−λ2I,−λ3I,−λ4I,ϒ1=N1,N2,01×23,ϒ3=N1,01×24,ϒ2=0,0,ηℳ1TQ2,−ℳ3TY,0,ℳ1T,ℳ2T,ℳ1T,ℳ2T,ℳ3T,01×15,ϒ4=01×15,ℳ1T,01×9,ϒ5=0,N2,01×23,ϒ6=01×16,ℳ1T,01×8.
Combined with Schur complement lemma and contragradient transformation, we defineZ1=X1Q1X1, Z2=X1Q2T, P−1=X1, Q2−1=X2, and KX1=Y, and we obtain that (43) implies (17). This completes the proof of Theorem 2.


## 4. Illustrative Examples

In the following part, we use the previous conclusions to investigate a numerical simulation example, through which we indicate the usefulness of the provided controller and the correctness of the conclusions.For DTNSNSs (5) with time delay and parameter uncertainties, the corresponding parameters are listed as follows:(47)A=−0.3100.4,Ad=−0.50−0.5−0.3,B=−0.10.20.4−0.2,G=−0.100−0.1,H=−0.5−0.1−0.1−0.8,Hd=−0.60.7−1−0.1,C=−0.10.20.1−0.1,Cd=−0.2−0.1−0.1−0.1,ℳ1=0.2000.2,ℳ2=0.2000.2,ℳ3=0.2000.2,N1=0.2000.2,N2=0.2000.2,X=−300−3,Y=15000150,Z=1.45001.45.Givend=5,τ=6,α=0.003,β=0.001,η=0.8,ϱ=0.03,λ1=1/3,λ2=1/4,λ3=1/4,λ4=1/5,λ5=4, the exogenous disturbance input is(48)vs=e−ssinse−0.8scos2s.Without consideration of the delay-feedback control, the simulation of state trajectories for the open-loop system is shown in Figure1, and we find that the open DTNSNSs (5) with time delay are not stable or not dissipative. Therefore, we need to design the state-feedback controller as (4) for the DTNSNSs (5) with time delay such that dissipativity is satisfied. According to (43), the feasible solutions are obtained as follows:(49)X1=10−4×0.36820.14290.14290.5446,X2=0.2439−0.0015−0.00150.2437,Y=10−3×−0.0935−0.1159−0.1159−0.2293,Z1=10−4×0.11970.08690.08690.0081,Z2=0.3624−0.0392−0.03920.4074,ɛ1=1.8336×10−4,ɛ2=4.1671,ɛ3=4.1959.Figure 1
The state trajectories of open-loop DTNSNSs (5) with time delay.Therefore, the controller gain can be designed as follows:(50)K=−1.9071−1.6270−1.6847−3.7679.Hence, the simulation of the closed-loop DTNSNSs (5) with time delay is given in Figure 2. Obviously, systems (5) are asymptotically stochastically stable in the mean square and dissipative. From Figure 3, we can see more clearly that the closed-loop DTNSNSs (5) with time delay are strictly X,Y,Z-dissipative, and Figure 4 shows the variation of ϱ in EΦv,z,N≥ϱEv,vN,∀N≥0.Figure 2
The state trajectories of closed-loop DTNSNSs (5) with time delay and delay-feedback control.Figure 3
The trajectories of performance levelEΦv,z,N and Ev,vN of closed-loop DTNSNSs (5) with time delay.Figure 4
The variation curves ofϱ in EΦv,z,N≥ϱEv,vN of closed-loop DTNSNSs (5) with time delay.


## 5. Conclusions

The problem of dissipativity analysis for discrete-time neutral stochastic nonlinear systems (DTNSNSs) with time delay and parameter uncertainties has been investigated in this paper. A sufficient condition has been proposed by constructing a discrete-time Lyapunov–Krasovskii functional (LKF), which guarantees the DTNSNSs to be asymptotically stochastically stable in the mean square and strictlyX,Y,Z-dissipative. In light of this condition, a feedback controller of DTNSNSs with timedelay has been designed to ensure the stability and strictly dissipative performance of the resulting closed-loop system. At last, a numerical example has been demonstrated to validate the proposed results.



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*Source: 9932134-2021-06-24.xml*