# Mixedℋ-Infinity and Passive Synchronization of Markovian Jumping Neutral-Type Complex Dynamical Networks with Randomly Occurring Distributed Coupling Time-Varying Delays and Actuator Faults

Authors: N. Boonsatit; R. Sugumar; D. Ajay; G. Rajchakit; C. P. Lim; P. Hammachukiattikul; M. Usha; P. Agarwal
Journal: Complexity
 (2021)

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## Abstract

This article examines mixedℋ-infinity and passivity synchronization of Markovian jumping neutral-type complex dynamical network (MJNTCDN) models with randomly occurring coupling delays and actuator faults. The randomly occurring coupling delays are considered to design the complex dynamical networks in practice. These delays complied with certain Bernoulli distributed white noise sequences. The relevant data including limits of actuator faults, bounds of the nonlinear terms, and external disturbances are available for designing the controller structure. Novel Lyapunov–Krasovskii functional (LKF) is constructed to verify the stability of the error model and performance level. Jensen’s inequality and a new integral inequality are applied to derive the outcomes. Sufficient conditions for the synchronization error system (SES) are given in terms of linear matrix inequalities (LMIs), which can be analyzed easily by utilizing general numerical programming. Numerical illustrations are given to exhibit the usefulness of the obtained results.


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## Body

## 1. Introduction

Complex dynamical network (CDN) models are firstly investigated by Watts and Strogatz [1]. These models are sets of large-scale coupled nodes of interconnected systems, e.g., chemical substrates, microprocessors, and computers [2, 3]. Generally, CDN models are epitomized in practice, e.g., the World Wide Web, food webs, cellular and metabolic networks, neural systems, aviation systems, transportation systems, and electricity distribution systems [4, 5]. A number of these systems exhibit complexities in their topological and dynamical properties. The application of these networks can be found in streamlining the structure as well as reducing the control costs [6].Particularly, synchronization remains normal along with the significant group of powerful performance as concerns CDNs, which has turned into an expanding concern furthermore, close inquire about the theme with its expressive ongoing years [7]. Synchronization is a regular and significant aggregate conduct of a complex network. As a key look into zone broad, endeavors are given to the comprehension of the synchronization system of such organization. As a central wonder in moving practices about systems, synchronization issue as concerns CDNs has appealed to an impressive consideration against different range, for example, environment, aerology, building, control matrices as well as banking sector [8, 9], because of its productive applications. Since inadequate data frequently show up in some practical systems, for example, just 5% of excitatory neurotransmitters can be received between two associated cortical regions of brain systems [10], not all the diverted data of clients in wireless systems are known to the transmitter [11]. Immediately, the synchronization issue of complex networks with partial coupling is tended to in [12]. For the model, a time-varying complex dynamical networks model and its synchronization criteria are investigated in [13]. Synchronization of CDNs with a weighted time-varying adjacency matrix was explored by means of distributed adaptive control in [14]. The coupled network models are regularly utilized as scientific instruments to demonstrate in break down systems. For asynchronous complex networks, it is specifically noteworthy to locate a lot of controllers to accomplish synchronization of dynamical practices of all nodes in systems.Markovian jump networks (MJNs) are a class of stochastic hybrid models. They are dynamic networks whose structures are subjected to random parameter that changes abrupt condition disturbances, changing subsystem interconnections that occur nonlinearly [15–21]. When CDN models get unexpected changes in their structure, we can represent them using Markovian jumping complex networks, which have been studied by many researchers [22,23].In practical systems, time delays are regularly experienced. Consequently, time delays in couplings and in dynamical nodes have received considerable attention [24, 25]. On the other hand, time delays are unavoidable as a result of constrained transmission speeds and traffic congestion. Many researchers have investigated the effects of time delays on stability and synchronization [26]. Indeed, the existence of time delays can be found in numerous systems, such as atomic reactors, population dynamic models, aircraft stabilization, natural frameworks, chemical engineering systems, and ship stabilization circuit theory [27, 28].It is natural that actuator faults occur frequently, which can lead to unstable system operation [29]. It is imperative to design controllers with the capability of ensuring the stability of closed-loop systems in the event of faults. In this regard, various control methodologies are available to realize the control objectives subject to faults. Fault-tolerant control (FTC) systems can be divided into two types: passive and active approaches. Passive FTC controllers usually have basic structures with restricted fault-tolerant capabilities. In contrast, active FTC controllers have structures with better self-organization. Fault identification or estimation generally requires dynamic fault-tolerant controllers that can be changed in accordance with data limitations (see [30, 31]).Motivated by the above discussion, we study mixedℋ-infinity and passivity-based synchronization of Markovian jumping neutral-type complex dynamical network (MJNTCDN) models with time-varying distributed coupling delays and actuator faults in this paper that are as follows:(1)
We analyze the mixedℋ-infinity and passivity synchronization of MJNTCDN models with distributed random coupling time-varying delays and actuator faults(2)
The randomly occurring coupling delays satisfy the Bernoulli random binary procedure(3)
Delay-dependent conditions are derived to guarantee the MJNTCDN models is mixedℋ-infinity and passive performance at level γ(4)
Improved Jensen’s inequalities and integral inequalities are utilized to infer the sufficient conditions in terms of LMIs(5)
Numerical results are provided to exhibit the effectiveness of the proposed methodNotations. The following notations are used throughout this paper. ℝn and ℝm×n denote the n-dimensional Euclidean space and the set of all m×n real matrices, respectively, while P1>0 means that matrix P1 is real symmetric and positive definite. Superscript “T” stands for the transpose; diag⋅ stands for a block diagonal matrix; A−1 denotes the matrix inverse of A; E⋅ stands for the mathematical expectation operator with respect to the given probability measure; ℒ20,∞ stands for the space of square-integrable vector functions over interval 0,∞; ⋅ denotes the Euclidean norm of a vector and its induced norm of a matrix. The symbol “⋆” is used to represent the term of a symmetric matrix which can be inferred by symmetry; symbol “⊗” stands for the Kronecker product. If not explicitly stated, all matrices are assumed to have compatible dimensions for algebraic operations. Given a complete probability space Ω,ℱ,ℱtt≥0,P with a natural filtration ℱtt≥0 fulfilling the typical conditions (i.e., the filtration contains all P-invalid sets and is right continuous), where Ω is the sample space, ℱ is the algebra of events, and P is the probability measure defined on ℱ. The process δt,t≤0 is a continuous homogeneous Markovian process with right continuous trajectories and takes values in the finite space S=1,2,…,N. In particular, δt is related to the change likelihood network Θ=θpqN×Np,q∈S given by the transition rates:(1)Pδt+h=q|δt=p=θpqh+oh,p≠q,1+θpph+oh,p=q,where h>0 is the visit time and limh⟶0oh/h=0. Here, θpq≥0 is the change rate from mode p at time t to mode q at time t+h if p≠q and θpp=−∑q=1,q≠pNθij.


## 2. Problem Statement and Preliminaries

We recognize a class of MJNTCDN models composed ofN indistinguishable coupled nodes, in which each node has n-dimensional randomly occurring distributed coupling delays and actuator faults and position of the elements of the zth hub is spoken to through the accompanying nonlinear dynamical subsystem:(2)r^˙zt=A1δtr^zt+B1δtft,r^zt+B2δtft,r^zt−ϑt+Cδtr^˙zt−κt+1−βt∑j=1Ngzj1Γ1δtr^jt+βt∑j=1Ngzj2Γ2δt∫t−ρttr^jsds+D1δtuzft+E1δtvzt,y^zt=A2δtr^zt,z=1,2,…,N,r^zt=Φzt,∀t∈−h,0,h=maxϑ2,κ2,where r^zt=r^z1t,r^z2t,…,r^zntT∈ℝn stands for the state vector zth node with respect to the model at time t, y^zt is the measured output corresponding to the zth node; Φzt is the continuous initial function of the zth node; uzft∈ℝm denotes the fault control input vector of the zth hub; wzt∈ℝp represents the constant external input vector which belongs to ℒ20,∞; f:ℝ×ℝn⟶ℝn is continuously differentiable smooth nonlinear vector-valued field, which satisfies sector-bounded conditions which will be characterized later; Gq=gzjl∈ℝN×Nl=1,2 are the outer-coupling configuration matrices representing the topological structure of the model, in which gzj is defined as follows: if there is a connection from node z to node jz≠j, then the coupling matrix gzj1=gjz1=1 and gzj2=gjz2=1; otherwise, gzj1=gjz1=0 and gzj2=gjz2=0. The diagonal elements are defined by(3)gzz1=−∑j=1,j≠zNgzj1,gzz2=−∑j=1,j≠zNgzj2,z=1,2,…,N.Γaδt=diagua1δt,ua2δt,…,uanδta=1,2 are constant diagonal inner-coupling matrices. In addition, A1δt, A2δt, B1δt, B2δt, C1δt, D1δt, and E1δt are constant matrices with appropriate dimensions. For the time delay signals, ϑt, κt, and ρt denote the retarded delays, distributed delays, and neutral time-varying delays, respectively, and are assumed to satisfy the following inequalities:(4)0≤ϑ1≤ϑt≤ϑ2,ϑ.t≤ς1<1,0≤κ1≤κt≤κ2,κ˙t≤ς2<1,0≤ρt≤ρ¯,ρt.≤μ3,where ϑ1, ϑ2, κ1, κ2, ς1, ς2, μ3, and ρ¯ are real constant scalars.The randomly occurring coupling delayβt∈ℝ denotes a stochastic variable which is in the form of a Bernoulli distributed sequence defined by(5)βt=1,if delayed information exchanges happen,0,if delayed information exchanges do not happen,The probability occurrence of stochastic variableβt is given by(6)Probβt=1=β¯,Probβt=0=1−β¯,where β¯∈0,1 is a known constant.Then, we can obtain(7)Eβt−β¯=0,Eβt−β¯2=β¯1−β¯,Assumption 1.
For nonlinear functionf⋅:ℝn⟶ℝn, there exist constant matrices G1 and G2 such that(8)ft,r^zt≤G1r^zt,ft,r^zt−ϑt≤G2r^zt−τt,for any r^zt∈ℝn.
To synchronize all theN indistinguishable nodes in model (2) to a common value, let us characterize the synchronization error vector e^zt=r^zt−s^t be the synchronization error, where s^t∈ℝn represents the state vector of the unforced isolate node and is chosen as(9)s^˙t=A1δts^t+B1δtft,s^t+B2δtft,s^t−ϑt+C1δts^˙t−κt,y^st=A2δts^t,where y^st is the unforced isolate output vector. We define the synchronization error dynamics as(10)e^˙zt=A1δte^zt+B1δth1t,e^zt+B2δth2t,e^zt−ϑt+C1δte^˙zt−κt+1−βt∑j=1Ngzj1Γ1δte^jt+βt∑j=1Ngzj2Γ2δt∫t−ρtte^jsds+D1δtuzft+E1δtvzt,y˜zt=A2δte^zt,z=1,2,…,N,e^zt=Φzt,∀t∈−h,0,h=maxϑ2,κ2,where h1t,e^zt=ft,r^zt−ft,s^t, h2t,e^zt−ϑt=ft,r^zt−ϑt−ft,s^t−ϑt, and y˜zt=y^zt−y^st.
The actuator fault model is used as [32, 33](11)uz,jft=1−λz,jtuz,jt,0≤λz,jt≤λ∗j<1,where z=1,2,…,N denotes the zth agent and j=1,2,…,m denotes the jth actuator, uz,j is the input signal of the actuator, uz,jf is the output signal from the actuator, λz,jt is an unknown and piecewise continuous bounded actuator failure factor which shows the degree of effectiveness of the actuator, and λ∗j is a known constant which represents the upper limits of λz,jt, ∀z. Note that, when λz,jt=0, there is no deficiency of the actuator, i.e., the jth actuator of the zth agent is sound or typical; when 0<λz,jt<1, the jth actuator is subject to fault.
We indicate(12)uzft=uz,1ft,uz,2ft,…,uz,mftT,λzt=diagλz,1t,λz,2t,…,λz,mt.
In this case, a uniform actuator fault model is represented as(13)uzft=ℐm−λztuzt.
Taking the actuator fault (13) into account, model (10) can be represented as follows:(14)e^˙zt=A1δte^zt+B1δth1t,e^zt+B2δth2t,e^zt−ϑt+C1δte^˙zt−κt+1−βt∑j=1Ngzj1Γ1δte^jt+βt∑j=1Ngzj2Γ2δt∫t−ρtte^jsds+D1δtℐm−λztuzt+E1δtvzt,y˜zt=A2δte^zt,z=1,2,…,N,e^zt=Φzt,∀t∈−h,0,h=maxϑ2,κ2,where λ¯=diagλ¯∗,1,λ¯∗,2,…,λ¯∗,m.
The following state feedback controller law is employed for the synchronization error model:(15)uzt=Kzδte^zt,z=1,2,…,N,where Kzδt is a real constant matrix representing the gain matrix of the feedback controller to be determined.
Therefore, the control input is substituted into the error model (14) leading to(16)e^˙zt=A1δte^zt+B1δth1t,e^zt+B2δth2t,e^zt−ϑt+C1δte^˙zt−κt+1−βt∑j=1Ngzj1Γ1δte^jt+βt∑j=1Ngzj2Γ2δt∫t−ρtte^jsds+D1δtℐm−λztKzδte^zt+E1δtvzt,y˜zt=A2δte^zt,z=1,2,…,N,e^zt=Φzt,∀t∈−h,0,h=maxϑ2,κ2.
For convenience, every conceivable estimation ofδt is meant by p, p∈S in the sequel. As such, we have(17)A1δt=A1p,A2δt=A2p,B1δt=B1p,B2δt=B2p,C1δt=C1p,D1δt=D1p,E1δt=E1p,Γ1δt=Γ1p,Γ2δt=Γ2p,where A1p, A2p, B1p, B2p, C1p, D1p, and E1p for any p∈S are known constant matrices of appropriate dimensions.
Then, error dynamical model (16) in virtue of the Kronecker product can be written in the following compact form:(18)e^˙t=ℐN⊗A1p+ℐN⊗D1pℐN⊗Kp−ℐN⊗D1pλtℐN⊗Kpe^t+ℐN⊗B1pℍ1t,e^t+ℐN⊗B2pℍ2t,e^t−ϑt+ℐN⊗C1pe^˙t−κt+1−βtG1⊗Γ1pe^t+βtG2⊗Γ2p∫t−ρtte^sds+ℐN⊗E1pvt,y˜t=ℐN⊗A2pe^t,where(19)e^t≔e^1Tte^2Tt,…,e^NTtT,ℍ1t,e^t≔h1Tt,e^1th1Tt,e^2t,…,h1Tt,e^NtT,ℍ2t,e^t−ϑt≔h2Tt,e^1t−ϑth2Tt,e^2t−ϑt,…,h2Tt,e^Nt−ϑtT,λt≔diagλ1tλ2t,…,λNt,vt≔v1Ttv2Tt,…,vNTtT,y˜t≔y˜1Tty˜2Tt,…,y˜NTtT,Kp≔diagK1pK2p,…,KNp.
In the following, we present the associated definitions and lemmas, which are required to derive the main results of this paper.Definition 1 (see [34]).
Model (18) is said to have mixed ℋ-infinity and passive performance γ, if there exists a constant γ>0 such that(20)∫0Tp−σy˜Tty˜t+21−σγy˜Ttvtdt≥−γ2∫0TpvTtvtdt,for any Tp≥0 and any nonzero vt∈ℒ20,∞, where σ∈0,1 represents the parameter that defines the trade-off between ℋ-infinity and passive performance.Definition 2.
The mixedℋ-infinity and passive synchronization of model (18) can achieve the mixed ℋ-infinity and passive performance at level γ.Definition 3 (see [35]).
For model (18), if there exists a scalar υ>0 such that the derivative of the Lyapunov function with respect to time t satisfies(21)V˙t≤−υet2,then model (18) with vt=0 is said to be quadratically stable.Remark 1.
This article considered the complex dynamic network system with the linear coupling strength on the mode at timet, while the rest of the coefficient is constant matrix, in the process of jump. This makes the complex dynamical network system more general.Remark 2.
Note that model (20) incorporates mixed ℋ-infinity and passivity performance level, which is a unique instance of dissipativity. For instance, if σ is estimated as 1, model (20) decreases to ℋ-infinity performance level; if σ is estimated as 0, model (20) decreases to the passivity performance; when σ takes 0,1, model (20) represents the mixed ℋ-infinity and passivity performance level.Lemma 1 (see [36]).
For any matrixN∈ℝn×n and N=NT>0, and given scalar λ>0, the vector function is Φ:0,λ⟶ℝn such that the following relation holds:(22)−λ∫0λΦTsNΦsds≤−∫0λΦsdsTN∫0λΦsds.Lemma 2 (see [36]).
For any matrixN∈ℝn×n and N=NT>0, and given a scalar function λ≔λt>0, the vector-valued function is Φ:−λ,0⟶ℝn such that the following relation holds:(23)−λ∫t−λtΦ˙TsNΦ˙sds≤ΦtΦt−λT−NN⋆−NΦtΦt−λ.Lemma 3 (see [37]).
For a given matricesA11, A12, A21, and A22 with appropriate dimensions, A11A12A21TA22<0 holds if and only if A22<0, A12−A12A22−1A12T<0.Lemma 4 (see [38]).
The Kronecker product has the following properties:(1)
αA⊗B=A⊗αB(2)
A+B⊗C=A⊗C+B⊗C(3)
A⊗BC⊗D=AC⊗BD(4)
A⊗BT=AT⊗BT


## 3. Main Results

In this section, we establish the LMI to determine a fault-tolerant controller which is mixedℋ-infinity and passive with synchronization error model (18).Theorem 1.
Given some constants such asϑ1, ϑ2, κ1, κ2, ς1, ς2, μ3, ρ¯, β1, β2, σ,β¯∈0,1, γ, λ¯∗ and diagonal matrices Γ1p, Γ2pp∈S, G1, G2, and G1, G2 are known constant matrices, error model (18) satisfies a mixed ℋ-infinity and passivity performance γ>0 in the sense of Definitions 1 and 2, if there exist symmetric positive definite matrices Xp>0p∈S, Pp>0, Um>0m=2,…,11, Z>0, and Yp and Mpp∈S are of appropriate dimension matrices such that the following successive LMIs hold:(24)Ψ¯=Ψ^p14×14χ^p⋆−F^p<0,where(25)Ψ^p11=θppℐN⊗Xp+ℐN⊗U^2+ℐN⊗U^4+ℐN⊗U^5+ℐN⊗U^7−1ϑ1ℐN⊗U^8−1ϑ2ℐN⊗U^9−1κ1ℐN⊗U^10−1κ2ℐN⊗U^11+ℐN⊗A1pℐN⊗Xp+ℐN⊗XpℐN⊗A1pT+ℐN⊗D1p×ℐN⊗Yp+ℐN⊗YpTℐN⊗D1pT−ℐN⊗D1pλ¯∗ℐN⊗Yp−ℐN⊗YpTλ¯∗TℐN⊗D1p+1−β¯G1⊗Γ1pℐN⊗Xp+1−β¯ℐN⊗XpTG1⊗Γ1pT+ρ¯2ℐN⊗Z^,Ψ^p12=ℐN⊗P^p−ℐN⊗Xp+β1ℐN⊗XpTℐN⊗A1pT+β1ℐN⊗YpTℐN⊗D1pT−β1ℐN⊗YpTλ¯∗TℐN⊗D1pT+1−β¯β1ℐN⊗XpG1⊗Γ1pT,Ψ^p13=1ϑ1ℐN⊗U^8,Ψ^p14=β2ℐN⊗XpTℐN⊗A1p+β1ℐN⊗YpTℐN⊗D1p−β2ℐN⊗YpTλ¯∗TℐN⊗D1pT+β21−β¯ℐN⊗XpG1⊗Γ1pT,Ψ^p15=1ϑ2ℐN⊗U^9,Ψ^p16=1κ1ℐN⊗U^10,Ψ^p17=1κ2ℐN⊗U^11,Ψ^p18=ℐN⊗C1pℐN⊗Xp,Ψ^p19=ℐN⊗B1p+β2ℐN⊗B1p,Ψ^p110=ℐN⊗B2p+β2ℐN⊗B2p,Ψ^p111=β¯G2⊗Γ2pℐN⊗Xp,Ψ^p112=−2ℐN⊗Xp×1−σγℐN⊗A2pT,Ψ^p113=ℐN⊗XpℐN⊗G1,Ψ^p114=σℐN⊗XpℐN⊗A2p,Ψ^p22=ℐN⊗U^6+ϑ1ℐN⊗U^8+ϑ2ℐN⊗U^9+κ1ℐN⊗U^10+κ2ℐN⊗U^11−ℐN⊗Xpβ1−β1ℐN⊗XpT,Ψ^p24=−β2ℐN⊗Xp,Ψ^p28=β1ℐN⊗C1pℐN⊗Xp,Ψ^p29=β1ℐN⊗B1p,Ψ^p210=β1ℐN⊗B2p,Ψ^p211=β1G2⊗Γ2pℐN⊗Xp+β1ℐN⊗Xp,Ψ^p212=β1ℐN⊗E1p,Ψ^p33=−ℐN⊗U^2−1ϑ1ℐN⊗U^8,Ψ^p44=−1−ς1ℐN⊗U^3,Ψ^p48=β2ℐN⊗C1pℐN⊗Xp,Ψ^p411=β2β¯G2⊗Γ2pℐN⊗Xp,Ψ^p412=β2ℐN⊗E1p,Ψ^p413=ℐN⊗XpℐN⊗G2,Ψ^p55=−ℐN⊗U^4−1ϑ2ℐN⊗U^9,Ψ^p66=−ℐN⊗U^5−1κ1ℐN⊗U10,Ψ^p77=−ℐN⊗U^7−1κ2ℐN⊗U11,Ψ^p88=−1−ς2ℐN⊗U^6,Ψ^p99=−ℐN⊗ℐN,Ψ^p1010=−ℐN⊗ℐN,Ψ^p1111=−1−ς1ℐN⊗Z^,Ψ^p1212=−γ2I,Ψ^p1313=−ℐN⊗ℐN,Ψ^p1414=−ℐN⊗ℐN,χ^p=θp1ℐN⊗XpT,…,θpp−1ℐN⊗XpT,θpp+1ℐN⊗XpT,…,θpNℐN⊗XpT,F^p=diagℐN⊗X1,…,ℐN⊗Xp−1,ℐN⊗Xp+1,…,ℐN⊗XN,and the remaining parameters are zero. Then, the feedback controller gain is given by Kp=YpXp−1 which stabilizes the error dynamics.Proof.
We consider the following LKF with integral terms for the synchronization error model (18)(26)Vt,e^t,p=∑r=14Vrt,e^t,p,where(27)V1t,e^t,p=e^TtℐN⊗Ppe^t,V2t,e^t,p=∫t−ϑ1te^TtℐN⊗U2e^sds+∫t−ϑtte^TtℐN⊗U3e^sds+∫t−ϑ2te^TtℐN⊗U4e^sds+∫t−κ1te^TtℐN⊗U5e^sds+∫t−κtte^˙TsℐN⊗U6e^˙sds+∫t−κ2te^TtℐN⊗U7e^sds,V3t,e^t,p=∫t−ϑ1t∫ute^˙TsℐN⊗U8e^˙sdsdu+∫t−ϑ2t∫ute^˙TsℐN⊗U9e^˙sdsdu+∫t−κ1t∫ute^˙TsℐN⊗U10e^˙sdsdu+∫t−κ2t∫ute^˙TsℐN⊗U11e^˙sdsdu,V4t,e^t,p=ρ¯∫−ρt0∫t+ute^sℐN⊗Ze^sdsdu.
We useℒVt,e^t,p to denote the weak infinitesimal operator of Vt,e^t,p [39], which is defined as(28)ℒVt,e^t,p=limΔ⟶0+1ΔEVt+Δ,e^t+Δ,δt+Δ|t,e^t,p−Vt,e^t,p,ℒVt,e^t,p=∑r=14ℒVrt,e^t,p.
The derivative and mathematical expectation of (26) along the trajectories of the error model (18) is(29)EℒV1t,e^t,p=E2e^TtℐN⊗Ppe^˙t+e^Tt∑q=1NℐN⊗Pqe^t,EℒV2t,e^t,p=Ee^TtℐN⊗U2+ℐN⊗U3+ℐN⊗U4+ℐN⊗U5+ℐN⊗U7e^t+e^˙TtℐN⊗U6e^˙t+e^Tt−ϑ1ℐN⊗U2e^t−ϑ1−1−ς1e^Tt−ϑtℐN⊗U3×e^t−ϑt−e^Tt−ϑ2ℐN⊗U4e^t−ϑ2−e^Tt−κ1ℐN⊗U5e^t−κ1−1−ς2e^˙Tt−κtℐN⊗U6e^˙t−κt−e^Tt−κ2ℐN⊗U7e^t−κ2,EℒV3t,e^t,p=Ee^˙TtℐN⊗U8+ϑ2ℐN⊗U9+κ1ℐN⊗U10+κ2ℐN⊗U11e^˙t−∫t−ϑ1te^˙TsℐN⊗U8e^˙sds−∫t−ϑ2te^˙TsℐN⊗U9e^˙sds−∫t−κ1te^˙TsℐN⊗U10e^˙sds−∫t−κ2te^˙TsℐN⊗U11e^˙sds,ΕℒV4t,e^t,p=ρ¯2e^TtℐN⊗Ze^t−1−μ3ρ¯∫t−ρtte^TsℐN⊗Ze^sds.
According to Lemmas1 and 2, the integral terms in (29) can be rewritten as(30)−∫t−ϑ1te^˙TsℐN⊗U8e^˙sds≤−1ϑ1∫t−ϑ1te^˙sdsTℐN⊗U8∫t−ϑ1te^˙sds,≤−1ϑ1e^t−e^t−ϑ1TℐN⊗U8e^t−e^t−ϑ1,−∫t−ϑ2te^˙TsℐN⊗U9e^˙sds≤−1ϑ2∫t−ϑ2te^˙sdsTℐN⊗U9∫t−ϑ2te^˙sds,≤−1ϑ2e^t−e^t−ϑ2TℐN⊗U9e^t−e^t−ϑ2,−∫t−κ1te^˙TsℐN⊗U10e^˙sds≤−1κ1∫t−κ1te^˙sdsTℐN⊗U10∫t−κ1te^˙sds,≤−1κ1e^t−e^t−κ1TℐN⊗U10e^t−e^t−κ1,−∫t−κ2te^˙TsℐN⊗U11e^˙sds≤−1κ2∫t−κ2te^˙sdsTℐN⊗U11∫t−κ2te^˙sds≤−1κ2e^t−e^t−κ2TℐN⊗U11e^t−e^t−κ2.
Applying Lemma1 in (29), it follows that(31)−ρ¯∫t−ρtte^TsℐN⊗Ze^sds≤−∫t−ρtte^sdsTℐN⊗Z∫t−ρtte^sds.
Besides, as per Assumption1, we can obtain the following inequality:(32)e^TtℐN⊗G1ℐN⊗G1Te^t−ℍ1Tt,e^tℍ1t,e^t>0,e^Tt−ϑtℐN⊗G2ℐN⊗G2Te^t−ϑt−ℍ1Tt,e^t−ϑtℍ1t,e^t−ϑt>0.
Besides that, for any matrixMp, p∈S with appropriate dimension and scalars β1, β2>0, and taking the mathematical expectation, the following condition holds:(33)0=E2e^Tt+β1e^˙Tt+β2e^Tt−ϑtℐN⊗MpℐN⊗A1p+ℐN⊗D1pℐN⊗Kp−ℐN⊗D1pλt×ℐN⊗Kpe^t+ℐN⊗B1pℍ1t,e^t+I⊗B2pℍ2t,e^t−ϑt+I⊗C1pe^˙t−κt+1−βtG1⊗Γ1p+β¯−βtG1⊗Γ1pe^t+β¯G2⊗Γ2p+βt−β¯G2⊗Γ2p×∫t−ρtte^sds+ℐN⊗E1pvt−e^˙t.
For the performance index, we can obtain(34)Jpt=σy˜Tty˜t−21−σγy˜Ttvt−γ2vTtvt,t≥0.
Substituting (32) and (33) into ℒVt,e^t,p, and from (29)–(31), it yields that(35)EℒVt,e^t,p+Jpt≤EζTtΨp12×12ζt,where(36)ζt=e^Tte^˙Tte^Tt−ϑ1e^Tt−ϑte^Tt−ϑ2e^Tt−κ1e^Tt−κ2e^˙Tt−κtℍ1Tt,e^tℍ2Tt,e^t−ϑt∫t−ρtte^sdsTvTtT,Ψp11=∑q=1NθpqℐN⊗Pq+ℐN⊗U2+ℐN⊗U4+ℐN⊗U5+ℐN⊗U7−1ϑ1ℐN⊗U8−1ϑ2ℐN⊗U9−1κ1ℐN⊗U10−1κ2ℐN⊗U11+ℐN⊗MpTℐN⊗A1p+ℐN⊗A1pT×ℐN⊗Mp+ℐN⊗MpTℐN⊗D1pℐN⊗Kp+ℐN⊗KpTℐN⊗D1pTℐN⊗Mp−ℐN⊗MpTℐN⊗D1pλ¯∗ℐN⊗Kp−ℐN⊗KpTλ¯∗TℐN⊗D1pTℐN⊗MpT+1−β¯ℐN⊗MpTG1⊗Γ1p+1−β¯G1⊗Γ1pTℐN⊗Mp+σℐN⊗A2pT×ℐN⊗A2p+ℐN⊗G1ℐN⊗G1T+ℐN⊗E1+ρ¯2ℐN⊗Z,Ψp12=ℐN⊗Pp+β1ℐN⊗MpℐN⊗A1pT+β1ℐN⊗KpTℐN⊗D1pTℐN⊗Mp−β1ℐN⊗KpTλ¯∗TℐN⊗D1pTℐN⊗Mp−ℐN⊗MpT+1−β¯β1G1⊗Γ1pTℐN⊗Mp,Ψp13=1ϑ1ℐN⊗U8,Ψp14=β2ℐN⊗A1pℐN⊗MpT+β2ℐN⊗KpTℐN⊗D1pT×ℐN⊗Mp−β1ℐN⊗KpTλ¯∗TℐN⊗D1pTℐN⊗Mp+β21−β¯ℐN⊗MpG1⊗Γ1p,Ψp15=1ϑ2ℐN⊗U9,Ψp16=1κ1ℐN⊗U10,Ψp17=1κ2ℐN⊗U11,Ψp18=ℐN⊗MpTℐN⊗C1p,Ψp19=ℐN⊗MpTℐN⊗B1p+β2ℐN⊗MpℐN⊗B1p,Ψp110=ℐN⊗MpTℐN⊗B2p+β2ℐN⊗MpℐN⊗B2p,Ψp111=β¯ℐN⊗MpTG2⊗Γ2p,Ψp112=−21−σγ×ℐN⊗A2pT+ℐN⊗MpTℐN⊗E1p,Ψp22=ℐN⊗U6+ϑ1ℐN⊗U8+ϑ2ℐN⊗U9+κ1ℐN⊗U10+κ2ℐN⊗U11−β1ℐN⊗MpT−ℐN⊗Mpβ1,Ψp24=−ℐN⊗MpTβ2,Ψp28=β1ℐN⊗MpTℐN⊗C1p,Ψp29=β1ℐN⊗MpTℐN⊗B1p,Ψp210=β1I⊗MpTℐN⊗B2p,Ψp211=β1β¯ℐN⊗MpTG2⊗Γ2p+β1ℐN⊗MpT,Ψp212=βℐN⊗MpTℐN⊗E1p,Ψp33=−ℐN⊗U2−1ϑ1ℐN⊗U8,Ψp44=−1−ς1ℐN⊗U3+ℐN⊗G2TℐN⊗G2,Ψp48=β1ℐN⊗MpℐN⊗C1p,Ψp411=β2ℐN⊗Mpβ¯G2⊗Γ2p,Ψp412=β2ℐN⊗E1p×ℐN⊗Mp,Ψp55=−ℐN⊗U4−1ϑ2ℐN⊗U9,Ψp66=−ℐN⊗U5−1ϑ2ℐN⊗U10,Ψp77=−ℐN⊗U7−1ϑ2ℐN⊗U11,Ψp88=−1−ς2ℐN⊗U6,Ψp99=−ℐN⊗ℐN,Ψp1010=−ℐN⊗ℐN,Ψp1111=−1−ς1ℐN⊗Z,Ψp1212=−γ2I,and the remaining elements of Ψp are zero. Let ℐN⊗Xp=ℐN⊗Mp−1, and pre- and postmultiply matrix Ψp by diagℐN⊗XpℐN⊗XpℐN⊗XpℐN⊗XpℐN⊗XpℐN⊗XpℐN⊗XpℐN⊗XpIIℐN⊗XpI. By using the Schur complement (Lemma 3), we can obtain matrix Ψ^p in (24). Now we introduce the following new variables:(37)ℐN⊗XpTℐN⊗UmℐN⊗Xp=ℐN⊗U^m,m=2,…,11,ℐN⊗XpTℐN⊗ZℐN⊗Xp=ℐN⊗Z^,Yp=KpXp.
These combinations are negative definite if the following conditions hold:(38)EℒVt,e^t,p+Jpt<EζTtΨ^pζt<0.
With zero initial condition, it can be inferred that for anyTp(39)E∫0Tpσy˜Tty˜t−21−σγy˜Ttvt−γ2vTtvtdt≤E∫0TpℒVt,e^t,p+σy˜Tty˜t−21−σγy˜Ttvt−γ2vTtvtdt<0,which indicates that(40)Jpt=E∫0Tpσy˜Tty˜t−21−σγy˜Ttvtdt≤E∫0Tpγ2vTtvtdtholds if Ψ¯<0, and it can be obtained by integrating both sides of (40) that Jpt≤0 holds, which means that the mixed ℋ-infinity and passive synchronization of the model is well achieved according to Definition 2. This completes the proof.Remark 3.
If there are no Markovian jumping parameters, then model (18) is simplified to(41)e^˙t=ℐN⊗A1+ℐN⊗D1ℐN⊗K−ℐN⊗D1λtℐN⊗Ke^t+ℐN⊗B1ℍ1t,e^t+ℐN⊗B2ℍ2t,e^t−ϑt+ℐN⊗C1e^˙t−κt+1−βtG1⊗Γ1e^t+βtG2⊗Γ2∫t−ρtte^sds+ℐN⊗E1vt,y˜t=ℐN⊗A2e^t,Corollary 1.
Given some constants such asϑ1, ϑ2, κ1, κ2, ς1, ς2, μ3, ρ¯, β1, β2, σ,β¯∈0,1, γ, λ¯∗ and diagonal matrices Γ1p, Γ2pp∈S, G1, G2, and G1, G2 are known constant matrices, error model (41) satisfies a mixed ℋ-infinity and passivity performance γ>0 in the sense of Definitions 1 and 2, if there exist symmetric positive definite matrices X1>0, P1>0, Um>0m=2,…,11, Z>0, and Y1 and M1 are of appropriate dimension matrices such that the following LMI holds:(42)Ψ^¯=Ψ^¯14×14<0,where(43)Ψ^¯11=ℐN⊗U^2+ℐN⊗U^4+ℐN⊗U^5+ℐN⊗U^7−1ϑ1ℐN⊗U^8−1ϑ2ℐN⊗U^9−1ϑ1ℐN⊗U^10−1κ2ℐN⊗U^11+ℐN⊗A1ℐN⊗X1+ℐN⊗X1ℐN⊗A1T+ℐN⊗D1ℐN⊗Y1+ℐN⊗Y1T×ℐN⊗D1pT−ℐN⊗D1λ¯∗ℐN⊗Y1−ℐN⊗Y1Tλ¯∗TℐN⊗D1+1−β¯G1⊗Γ1ℐN⊗X1+1−β¯ℐN⊗X1TG1⊗Γ1T+ρ¯2ℐN⊗Z^,Ψ^¯12=ℐN⊗P^1−ℐN⊗X1+β1ℐN⊗X1T×ℐN⊗A1T+β1ℐN⊗Y1TℐN⊗D1T−β1ℐN⊗Y1Tλ¯∗TℐN⊗D1T+1−β¯β1ℐN⊗X1×G1⊗Γ1T,Ψ^¯13=1ϑ1ℐN⊗U^8,Ψ^¯14=β2ℐN⊗X1TℐN⊗A1+β1ℐN⊗Y1TℐN⊗D1−β2ℐN⊗Y1Tλ¯∗TℐN⊗D1T+β21−β¯ℐN⊗X1G1⊗Γ1T,Ψ^¯15=1ϑ2ℐN⊗U^9,Ψ^¯16=1κ1ℐN⊗U^10,Ψ^¯17=1κ2ℐN⊗U^11,Ψ^¯18=ℐN⊗C1ℐN⊗X1,Ψ^¯p19=ℐN⊗B1+β2ℐN⊗B1,Ψ^¯110=ℐN⊗B2+β2ℐN⊗B2,Ψ^¯111=β¯G2⊗Γ2ℐN⊗X1,Ψ^¯112=−2ℐN⊗X11−σγℐN⊗A2T,Ψ^¯113=ℐN⊗X1ℐN⊗G1,Ψ^¯114=σℐN⊗XpℐN⊗A2p,Ψ^¯22=ℐN⊗U^6+ϑ1ℐN⊗U^8+ϑ2ℐN⊗U^9+κ1ℐN⊗U^10+κ2ℐN⊗U^11−ℐN⊗X1β1−β1ℐN⊗X1T,Ψ^¯24=−β2ℐN⊗X1,Ψ^¯28=β1ℐN⊗C1ℐN⊗X1,Ψ^¯29=β1ℐN⊗B1,Ψ^¯210=β1ℐN⊗B2,Ψ^¯211=β1G2⊗Γ2pℐN⊗X1+β1ℐN⊗X1,Ψ^¯212=β1ℐN⊗E1,Ψ^¯33=−ℐN⊗U^2−1ϑ1ℐN⊗U^8,Ψ^¯44=−1−ς1ℐN⊗U^3,Ψ^¯48=β2ℐN⊗C1ℐN⊗X1,Ψ^¯411=β2β¯G2⊗Γ2ℐN⊗X1,Ψ^¯412=β2ℐN⊗E1,Ψ^¯413=ℐN⊗X1ℐN⊗G2,Ψ^¯55=−ℐN⊗U^4−1ϑ2ℐN⊗U^9,Ψ^¯66=−ℐN⊗U^5−1κ1ℐN⊗U10,Ψ^¯77=−ℐN⊗U^7−1κ2ℐN⊗U11,Ψ^¯88=−1−ς2ℐN⊗U^6,Ψ^¯99=−ℐN⊗ℐN,Ψ^¯1010=−ℐN⊗ℐN,Ψ^¯1111=−1−ς1ℐN⊗Z^,Ψ^¯1212=−γ2I,Ψ^p1313=−ℐN⊗ℐN,Ψ^p1414=−ℐN⊗ℐN.
Then, the feedback controller gain is given byK=Y1X1−1.Proof.
The proof is similar to that of Theorem1 by choosing S=1, θpq=0. Hence, the proof is omitted.Remark 4.
For the case ofβ=0, i.e., without Markovian jumping, random coupling matrices, distributed coupling time-varying delays, and external disturbance, model (2) becomes(44)e^˙t=ℐN⊗A1+ℐN⊗D1ℐN⊗K−ℐN⊗D1λtℐN⊗Ke^t+ℐN⊗B1ℍ1t,e^t+ℐN⊗B2ℍ2t,e^t−ϑt+ℐN⊗C1e^˙t−κt.Corollary 2.
Given some constantsϑ1, ϑ2, κ1, κ2, ς1, ς2, β1, β2, λ¯∗ and diagonal matrices G1 and G2, error model (44) is quadratically stable, if there exist symmetric positive definite matrices X1>0, P^1>0, U^m>0m=2,…,11, and Y1 and M1 are of appropriate dimension matrices such that following LMI holds:(45)ϒ^=ϒ^11×11<0,where(46)ϒ^11=ℐN⊗U^2+ℐN⊗U^3+ℐN⊗U^4+ℐN⊗U^5+ℐN⊗U^7−1ϑ1ℐN⊗U^8−1ϑ2ℐN⊗U^9−1κ1ℐN⊗U^10−1κ2ℐN⊗U^11+ℐN⊗A1ℐN⊗X1+ℐN⊗D1ℐN⊗Y1+ℐN⊗Y1T×ℐN⊗D1T−ℐN⊗D1λ¯∗ℐN⊗Y1−ℐN⊗Y1Tλ¯∗TℐN⊗D1T,ϒ^12=ℐN⊗P^1−ℐN⊗X1,ϒ^13=1ϑ1ℐN⊗U^8,ϒ^14=β2ℐN⊗X1TℐN⊗A1T+β2ℐN⊗Y1TℐN⊗D1T−β2ℐN⊗D1T×λ¯∗TℐN⊗Y1T,ϒ^15=1ϑ2ℐN⊗U9,ϒ^16=1κ1ℐN⊗U^10,ϒ^18=ℐN⊗C1ℐN⊗X1,ϒ^111=−ℐN⊗X1ℐN⊗G1,ϒ^22=ℐN⊗U^6+ℐN⊗U^8+ϑ2ℐN⊗U^9+κ1ℐN⊗U^10+κ2ℐN⊗U^11−β1ℐN⊗X1,ϒ^24=−β2ℐN⊗X1,ϒ^28=β1ℐN⊗C1ℐN⊗X1,ϒ^29=β1ℐN⊗B1ℐN⊗X1,ϒ^210=β1ℐN⊗B2ℐN⊗X1,ϒ^33=ℐN⊗U^2−1ϑ1ℐN⊗U^8,ϒ^44=−1−ς1ℐN⊗U^3,ϒ^48=β2ℐN⊗C1ℐN⊗X1,ϒ^49=β2ℐN⊗B1ℐN⊗X1,ϒ^410=β2ℐN⊗B2ℐN⊗X1,ϒ^411=ℐN⊗X1ℐN⊗G2,ϒ^55=−ℐN⊗U^4−1ϑ2ℐN⊗U^9,ϒ^66=−ℐN⊗U^5−1κ1ℐN⊗U^10,ϒ^77=−ℐN⊗U^7−1κ2ℐN⊗U^11,ϒ^88=−1−ς2ℐN⊗U^6,ϒ^99=−ℐN⊗ℐN,ϒ^1010=−ℐN⊗ℐN,ϒ^1111=−ℐN⊗ℐN.
Then, the feedback controller gain is given byK=Y1X1−1.Proof.
Consider the following Lyapunov–Krasovskii functional candidate:(47)Vt,e^t=∑r1=13Vr1t,e^t,and V1t,e^t, V2t,e^t, and V3t,e^t are the same as defined in (26), and construct(48)ΠTt=e^Tte^˙Tte^Tt−ϑ1e^Tt−ϑte^Tt−ϑ2e^Tt−κ1e^Tt−κ2e^˙Tt−κtℍ1Tt,e^tℍ1Tt,e^t−ϑt.
The proof of Corollary2 is similar to that of Theorem 1. Therefore, it is omitted.Remark 5.
It is advantage specifying that a short time ago, many works regarding synchronization of CDNs have been reported in the literature, for instance, see [7, 40, 41]. Nevertheless, only very few papers have been focused on the issue of synchronization of NCDNs [42, 43]. It is noticed that all the abovementioned works on NCDNs have not considered the influence of sudden changes in parameters or environment, which can be represented using by Markovian jump parameters [44–48]. Moreover, so far in the literature, no work has been reported on ℋ-infinity and passivity synchronization problem of Markovian jump NCDNs with distributed random coupling delay. Thus, the main contribution of this paper is to fill such a gap through employing a fault-tolerant control law based on ℋ-infinity and passivity performance for achieving robust synchronization in Markovian jump NCDNs with random coupling delay against distributed time-varying actuator faults, which makes this work different from the existing works on Markovian jump NCDNs.Remark 6.
It should be mentioned that Theorem1 provides a set of sufficient conditions for mean-square asymptotic synchronization of the Markov jump NCDNs subject to time-varying actuator faults and random coupling delay. It is noted that the proof of Theorem 1 is mainly based on the Lyapunov–Krasovskii stability theory, wherein the number of decision variables in the obtained LMI constraints plays an important role. It is obvious that the computational complexity increases when the number of decision variables becomes larger. Moreover, in this paper, we construct the Lyapunov–Krasovskii functional consisting of double integral terms. So, the estimation of those integral terms brings out some difficulties. In order to overcome this, we employ Jensen’s integral inequality. It should be noted that the advantage of using Jensen’s integral inequality is that it can significantly reduce the number of decision variables in the derivation of the main results. Besides, fortunately, all the computations in the main results are off-line, and with the aid of the existing standard convex optimization software, the proposed LMI conditions can be easily solved.Remark 7.
Theorem1 develops mixed ℋ-infinity and passivity performance synchronization of MJNTCDNs with randomly occurring actuator faults. Theorem 1 makes full use of the information of the upper bounds of the discrete and distributed time-varying delays, which also brings us less conservativeness.


## 4. Numerical Examples

In this section, two numerical examples are presented to illustrate the effectiveness of the proposed method to analyze synchronization with respect to MJNTCDN models.Example 1.
Consider the following MJNTCDN model along with distributed time-varying delays with 3 nodes, modep=1,2 and N=3.(49)e^˙t=ℐN⊗A1p+ℐN⊗D1pℐN⊗Kp−ℐN⊗D1pλtℐN⊗Kpe^t+ℐN⊗B1pℍ1t,e^t+ℐN⊗B2pℍ2t,e^t−ϑt+ℐN⊗C1pe^˙t−κt+1−βtG1⊗Γ1pe^t+βtG2⊗Γ2p∫t−ρtte^sds+ℐN⊗E1pvt,y˜t=ℐN⊗A2pe^t,where e^t=e^1Tt,e^2TtT and the relevant parameters are given as follows:Mode 1:(50)A11=−3.90.260.31−2.8,A21=0.130.140.160.13,B11=0.250.750.350.25,B21=0.140.150.450.23,C11=0.0001000.0001,D11=0.50.60.50.6,E11=0.20.10.10.2.Mode 2:(51)A12=2.5002,A22=0.15−1.080.752.019,B12=−0.10.1−0.1−0.2,B22=−0.231.020.561.021,C12=−0.025−1.012.31.75,D12=−0.025−1.01−1.31.75,E12=3004,
In addition, the inner-coupling matrices of nondelayed and delayed terms are taken as follows:(52)Γ11=0.3000.3,Γ21=0.5000.5,Γ12=0.8000.8,Γ22=1001.
The outer-coupling matrices are assumed to begzj1=G1, gzj2=G2 with(53)G1=−1010−11−112,G2=−2200−2220−2.
The nonlinear functionft,r^zt is chosen as(54)ft,r^zt=0.2r^z1t−tanh0.1r^z1t0.1r^z2t.
It can be observed thatft,r^zt fulfills Assumption 1 with the following:(55)G1=0.1000.1,G2=0.1000.1.
The state and neutral, distributed time-varying delays satisfyϑt=0.5+0.5sint, and κt=0.25+0.05sint, ρt=0.5+0.5cost−1, respectively.
Here we chooseϑ1=0.001, ϑ2=1, γ=0.6, β1=0.05, β2=0.41, κ1=0.001, κ2=0.3, ρ¯=1, ς1=0.01, ς2=0.1, μ3=0.5, β¯=0.5, σ=0.3, and λ¯∗=0.9. In addition, the Markov chain δt=p,t≥0 takes value in finite state space N=1,2 and the transition probability matrix is given by(56)δt=0.50.50.20.2.
By using MATLAB LMI Toolbox, we solved LMIs (24) and obtained the feasible solutions as follows:(57)P1=65.1265−34.7055−34.705562.4038,P2=22.9921−2.4665−2.466516.6365,U2=41.7827−4.6555−4.655535.9923,U3=387.918685.208385.2083736.9597,U4=184.2395−7.9235−7.9235166.1189,U5=41.7827−4.6555−4.655535.9923,U6=31.7143−2.0245−2.024528.9686,U7=48.5381−5.3557−5.355743.9116,U8=0.09180.00240.00240.0915,U9=−69.79013.90913.9091−62.5765,U10=0.09180.00240.00240.0915,U11=18.49740.24750.247517.9481,X1=60.0845−35.8264−35.826455.3508,X2=5.05801.06931.06932.9108,Y1=103−1.7000−0.5398−0.5398−1.9253,Y2=1031.63313.30463.3046−0.4509,Z=96.6356−11.9607−11.960786.4221.
We can obtain the following state feedback controller gains:(58)K1=Y1X1−1=−55.5454−45.7051−48.4065−66.1147,K2=Y2X2−1=1030.08991.10230.7439−0.4282.
Thus, it can be concluded that model (18) is mixed ℋ-infinity and passive at performance level γ. The maximum allowable upper bounds of ϑ2 with different values ϑ1 are given in Table 1.Table 1
Maximum allowable bounds ofϑ2 with different values of lower bound ϑ1 for Example 1.
ϑ10.10.20.30.40.50.60.7ϑ22.12732.61242.61242.93453.02124.12135.2134Example 2.
Consider a class of model in the form of (44) consisting of two-dimensional nodes with the following coefficient matrices:(59)A1=−0.90.260.31−0.8,B1=0.250.750.350.25,B2=0.140.150.450.23,C1=−0.60.40.5−0.4,D1=1.21.40.80.3.
It can be found thatf satisfies Assumption 1 with(60)G1=−0.50.200.95,G2=−0.30.200.2,
Based on the above parameters, we use the MATLAB LMI Toolbox, to solve the LMI in Corollary2. We obtain the feasible solutions as follows. Therefore, the concerned model with time-varying delays is quadratically stable.(61)P1=−4.4613−0.4878−0.4878−3.8748,U2=19.53180.07750.077519.3311,U3=22.88510.15390.153922.6357,U4=73.00730.54190.541972.4158,U5=19.53180.07750.077519.3311,U6=10.23960.00420.004210.3032,U7=−99.7995−0.3808−0.3808−98.4768,U8=0.0032−0.0000−0.00000.0033,U9=−32.4779−0.1621−0.1621−32.3669,U10=0.0032−0.0000−0.00000.0033,U11=36.70910.13160.131636.2684,X1=−2.1495−0.5201−0.5201−1.4914,Y1=1.69941.38791.38790.4728.
The corresponding control gain matrix is given as(62)K=−0.6175−0.7152−0.6214−0.1003.


## 5. Conclusion

This article is concerned with mixedℋ-infinity and passivity synchronization of MJNTCDNs models with randomly occurring distributed coupling time-varying delays and actuator faults. We have designed the fault-tolerant state feedback controller that is modeled by the Bernoulli random variable. By utilizing the Lyapunov–Krasovskii functional approach, the sufficient conditions are ensuring the mixed ℋ-infinity passive performance for the MJNCDN models which have been established in terms of LMIs. Two numerical examples are presented to illustrate the effectiveness of the proposed method. In future, we would investigate the occurrences of discrete sampled data control be described using stochastic variables and probability density functions [49]. Also, it is important to extend our results to analyze stochastic synchronization of MJNTCDN models with multiple time-varying delays via impulsive control and pinning control.



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