# Finite-Time Synchronization and Identification between Two Nonlinear Coupled Multiweighted Markovian Switching Delayed Networks with Uncertain Models and External Disturbances

Authors: Xiaofei Chen; Haifeng Qiu; Chong Xu; Zhipeng Sun
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, two different nonlinear coupled Markovian switching delayed network models with multiple weights are established, and the effects of parameter and model uncertainties and external disturbances are fully considered. Firstly, based on the finite-time theory, several sufficient conditions for the finite-time synchronization of Markovian switching networks are obtained. Secondly, under the feedback controller with adaptive laws, when the synchronization of the networks is achieved in a finite time, the uncertain parameters of the networks can also be identified as true values. Finally, a representative numerical simulation is given to further illustrate the validity and practicability of the theoretical analysis and proof of this paper.


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## Body

## 1. Introduction

The complex dynamic network is a complex system with certain states and functions that is composed of many nodes and the topological connections formed by the edges between the nodes. In addition, as an important class of nonlinear systems, complex networks have received extensive attention from all walks of life. Researchers studying complex systems in the form of complex networks can not only deepen their understanding of the structure of complex systems but also gain a deeper understanding of complex systems in nature and society (including power grid, Internet, transportation network, and communication network) [1, 2]. In the research of complex networks, synchronization of complex networks has always been one of the hot topics. This is mainly because the synchronization phenomenon has a vital impact on real life. For example, the synchronization of the power grid ensures the safe and reliable operation of the power system, and the synchronization of the transportation network ensures the safety and smoothness of the road. Generally speaking, synchronization can be divided into inner synchronization and outer synchronization. In order to achieve global robust exponential synchronization and global robust asymptotical synchronization of drive response-based discontinuous Cohen–Grossberg neural networks, the pinning state-feedback and adaptive controllers were designed in [3]. In [4], a boundary controller relying to distributed measurement was designed to achieve nonlinear time-delayed complex spatiotemporal network (CSN). Compared with the inner synchronization, there are relatively few studies on out synchronization. Outer synchronization means that two or more systems with completely the same or similar properties gradually approach and reach the same state through the interaction of the systems under different initial conditions. Therefore, this paper mainly studies the outer synchronization between two different complex networks.It is well known that the network topology has an important influence on the network synchronization. This is because different topological structures have different dynamic states. At present, the research on fixed topology has made great achievements. However, the network topology is always affected by various factors in practice, such as the influence of network attacks, electromagnetic interference, and equipment damage, and the network topology will randomly change under the influence of these factors [5, 6]. This kind of random switching can depend on the Markovian process, so the research on the synchronization of Markovian switching networks is more practical [7–9]. In [10], the authors studied the finite-time synchronization of the Markovian jump complex dynamical framework with hybrid coupling. Cluster synchronization of Markovian switching complex networks with hybrid couplings and stochastic perturbations was studied in [11]. In addition, the Markovian switching complex networks considered above are all modeled with a single weight. But many real networks are complex systems with multiple weights, and each node of these systems is connected through multiple weights [12, 13]. Therefore, in order to make the network models closer to the real system, the Markovian switching network models with multiple weights should be considered.In the real world, the coupling relationship of Markovian switching networks cannot be all linear, and nonlinear coupling systems are more common. In some cases, the variablext cannot be directly obtained, but only hxt can be obtained, and the synchronization of the system can only be realized by hxt at this time. Therefore, the synchronization of nonlinear coupled systems also has great research value, and the results are more practical. In [14], the authors concerned with the problem of synchronization in nonlinear complex networks with multiple time-varying delays via adaptive aperiodic intermittent control. In [4], the authors dealt with the problem for synchronization of a nonlinear time-delayed complex spatiotemporal network (CSN), modeled by semilinear parabolic partial differential-difference equations. Furthermore, in the actual signal transmission process, time delays are often caused by factors, such as transmission speed limitations, network information congestion, and the influence of the transmission environment [15, 16]. Thus, considering the above reasons, time delays should be considered in the synchronization of Markovian switching complex networks.Most of the system models considered above are deterministic. It is well known that the parameters of the real system may be difficult to determine in advance, and the parameters of the system are generally unknown or uncertain [17]. In addition, under the influence of certain external environments and system structure changes, there may be some uncertain models and external disturbances in the dynamics of the system. The synchronization process of the system may be affected or even destroyed by these uncertainties. The robust finite-time state estimation problem of the uncertain Markovian jump neural networks with partly unknown transition probabilities was investigated in [18]. In [19], the effects of parameter and model uncertainties and external disturbances were fully taken into account, and a sliding model control approach was introduced to realize modified projective synchronization between two different chaotic systems in a finite time. Therefore, it is necessary to study the synchronization between Markovian switching complex networks with uncertainty.In actual engineering, people not only pay attention to its gradual stability characteristics in infinite time but also are more interested in realizing the synchronization state of the system in a finite time, such as aerospace systems, robot control systems, and power systems; they are often required that the synchronization be reached in a finite time. The finite-time control technology can realize this idea, and the finite-time control scheme can make the network have better robustness [20]. In [15], the author investigated finite-time synchronization of complexed-valued neural networks with multiple time-varying delays and infinite distributed delays. The multiweighted coupled neural network (MWCNN) models with and without coupling delays were investigated in [21]. Therefore, it is also valuable to realize the finite-time synchronization of Markovian switching complex networks in practice.Based on the above discussion, this paper mainly studied the finite-time synchronization and identification problems of nonlinear coupled Markovian switching delayed networks with multiple weights, in presence of uncertain parameters, uncertain model, and external disturbances on the networks. To the best of our knowledge, there are rarely studies on the finite-time synchronization and parameters identification between two nonlinear coupled Markovian switching delayed networks with multiple weights and uncertainties. Therefore, the research has significance in terms of theory and practice.The rest of this paper is arranged as follows. In Section2, some notations and necessary conditions will be given. The theoretical proof of network finite-time synchronization and identification is given in Section 3. In Section 4, in order to prove the validity of the above theoretical analysis, a representative numerical example is given. The conclusions are drawn in Section 5.


## 2. Preliminaries

In order to complete the theoretical proof and analysis of this paper, some important mathematical symbols and necessary conditions are given in this section. First of all, some important mathematical symbol definitions will be given. The matrixAT (or xT) is defined as the transposed matrix of A (or x). The symbol ⊗ is defined as the Kronecker product of matrixes. Define x2 as the 2-norm of x. Define In∈Rn×n as the n-dimensional identity matrix. E is defined as mathematical expectation. Secondly, some necessary conditions will be given below.Assumption 1.
Assume that a Lipschitz condition is satisfied by noise intensity functionσit,eit,rt, and there exists a constant ρi ≥ 0 such that(1)traceσiTt,eit,rtσit,eit,rt≤ρirteiTteit.Assumption 2.
For arbitraryxt,yt∈Rn, if there exists a constant L > 0, then following inequality holds:(2)fyt−fxt≤Lyt−xt.Assumption 3.
(see [22]). Let 0<γ<1 and κ>0; assume that there exists a continuous function g (g0>0) such that, for any 0≤u≤t,(3)gt−gu≤−κ∫utgsγds.Lemma 1.
(see [23]). Assume that a positive definite, continuous function Vt satisfies the differential inequality V˙t≤−αVεt,∀t≥t0,Vt0≥0, where α > 0, 0<ε<1. For any given t0, Vt satisfies the inequality V1−εt≤V1−εt0−α1−εt−t0,t0≤t≤t1, and Vt≡0,∀t≥t1; we can get t1:(4)t1=t0+V1−εt0α1−ε.Lemma 2.
(see [23]). For any vectors xt,yt∈Rn and a positive definite matrix K>0, the following inequality holds:(5)2xTy≤xTK−1x+yTKy.Lemma 3.
(see [24]). b1,b2,…,bn∈Rn are any vectors, and the following inequality holds:(6)b1c+b2c+⋯+bnc≥b12+b22+⋯+bn2c/2,i.e.,(7)∑i=1nbic≥∑i=1nbi2c/2,where c (0 < c < 2) is a real number.Definition 1.
(see [24]). Define f:Rn×R+×S⟶Rn and g:Rn×R+×S⟶Rn×m are differentiable functions satisfying the following n-dimensional functional stochastic differential equation:(8)dxt=fxt,t,rtdt+gxt,t,rtdωt.
According to Itô’s formula, ifVxt,t,r∈C2,1Rn×R+×S⟶R+, the operator ℒV is defined as(9)ℒVxt,t,p=Vtxt,t,p+Vxxt,t,pfxt,t,p+12tracegTxt,t,pVxxgxt,t,p+∑q=1mπpqVxt,t,q,where Vtxt,t,p=∂Vxt,t,p/∂t, Vxxt,t,p=∂xt,t,p/∂x1,…,∂xt,t,p/∂xn, and Vxxxt,t,p=∂2Vxt,t,p/∂xixjn×n.


## 3. Main Results

In this section, the nonlinear coupled Markovian switching delayed network models with multiple weights and uncertainties will be established, and the feedback controller with update laws is designed; then, the finite-time synchronization and identification of the network models is achieved by above controller and some inequality techniques.


### 3.1. Network Models

In this paper, then-dimensional drive and response networks with uncertain parameters, models, and external disturbances are given as follows:The drive network is as follows:(10)x˙it=fi1xit,xit−τ+fi2xit,xit−ταit+∑d=1l∑j=1NaijdrtΓdhxjt+Δfi1xit+Δi1,i=1,2,…,N,where xit=xi1t,xi2t,…,xintT∈Rn is the state vector of the i-th node; fi1:Rn⟶Rn and fi2:Rn⟶Rn×n are continuous vector function and matrix function, respectively; Δfi1:Rn⟶Rn is the model uncertainty; h⋅:Rn⟶Rn is the nonlinear coupling function; αit is an unknown parameter vector of the dynamic node; τ>0 is the time delays of node i; Δi1 is external disturbances of the network; and Γd∈Rn×n (d = 1, 2, …, l) is the inner coupling matrix of the d-th coupling form. Adrt=aijdrtN×N∈RN×N is the topological structure matrix of the d-th coupling form, which represents topological structure of the network at time t in mode rt, and Adrt can be defined as follows: if there exists a connection between node i and node jj≠i for the d-th coupling form, then aijd≠0; otherwise, aijd=0. The diagonal elements of Art are defined as(11)aiidrt=−∑j=1,j≠iNaijdrt,d=1,2,…,l;i=1,2,…,N.Letrt,t≥0 be a right-continuous Markovian process in probability space, which takes values in a finite space S=1,2,…,l with generator Π=πpqm×m. The transition probability from the p-th mode at time t to the q-th mode at time t+Δt is defined as follows:(12)Prt+Δt=q|rt=p=πpqΔt+oΔt,ifq≠p,1+πppΔt+oΔt,ifq=p,where Δt>0, limΔt⟶0oΔt/Δt=0, and transition rate πpqπpq>0 satisfies(13)πpp=−∑q=1,q≠pmπpq.The response network is as follows:(14)y˙it=gi1yit,yit−τ+gi2yit,yit−τβit+∑d=1l∑j=1NbijdrtΓdhyjt+Δi2+Δgi1yit+σit,eit,rtω˙t+uit,i=1,2,…,N,where yit=yi1t,yi2t,…,yintT∈Rn is the state vector of the i-th node; gi1:Rn⟶Rn and gi2:Rn⟶Rn×n are continuous vector function and matrix function, respectively; Δgi1:Rn⟶Rn is the model uncertainty; βit is an unknown parameter vector of the dynamic node with the same definition as αit; Δi2 is the external disturbances of the network; Bdrt=bijdrtN×N∈RN×N is the same as the definition of Adrt above; and Bdrt can be defined as follows: if there exists a connection between node i and node jj≠i for the d-th coupling form, then bijd≠0; otherwise, bijd=0. The diagonal elements of Brt is defined as(15)biidrt=−∑j=1,j≠iNbijdrt,d=1,2,…,l;i=1,2,…,N.Moreover,uit is a nonlinear controller; ωt=ω1t,ω2t,…,ωntT is a n-dimensional Weiner process.Define synchronization error of the system as follows:(16)eit=yit−xit,i=1,2,…,N,where eit=ei1t,ei2t,…,eintT represents the state error of the i-th node.Definition 2.
If there is a positive constantt and gives an arbitrary t1>t, such that(17)limt⟶t1Eeit=limt⟶t1Eyit−xit2=0,i=1,2,…,N,and eit2=yit−xit2≡0, then synchronization of Markovian switching complex networks (10) and (14) can be realized at a finite-time t1.Remark 1.
When the drive system (10) and the response system (14) gradually realize synchronization at a finite-time t1, the unknown parameter vectors αit and βit will be identified and expressed as(18)limt⟶t1∑i=1Nα^it−αi2=0,limt⟶t1∑i=1Nβ^it−βi2=0,where α^it and β^it are estimated values of αit and βit, respectively.


### 3.2. Main Results

In this section, finite-time synchronization between two different nonlinear coupled Markovian switching complex delayed networks with multiple weights and uncertainties will be demonstrated and analyzed.According to network equations (10) and (14), the synchronization error equation of system can be obtained:(19)e˙it=gi1yit,yit−τ+Δgi1yit+gi2yit,yit−τβit+Δi2−Δi1−fi1xit,xit−τ−fi2xit,xit−ταit+σit,eit,rtω˙t−Δfi1xit+∑d=1l∑j=1NbijdrtΓdhyjt−∑d=1l∑j=1NaijdrtΓdhxjt+uit.Obviously, when the synchronization error (19) converges to zero in a finite time, then finite-time synchronization and identification of the network models will also be achieved. Therefore, it is very important to design an effective controller so that the synchronization error converges to zero.Theorem 1.
Design an effective controller with update laws as follows:(20)uit=−ηirteit−gi1yit,yit−τ−gi2yit,yit−τβ^it+fi1xit,xit−τ+fi2xit,xit−τα^it−krtsigneiteitγ+∑d=1l∑j=1Naijdrt−bijdrtΓdhxjt−Δ^isigneit−v^isigneit,α^⋅it=−λirtfi2Txit,xit−τeit−ψirtsignα^itα^it+Mαγ,β^⋅it=μirtgi2Tyit,yit−τeit−ψirtsignβ^itβ^it+Mβγ,Δ^⋅it=φirteiTtsigneit−ψirtsignΔ^itΔ^it+M△γv^⋅it=ϕirteiTtsigneit−ψirtsignv^itv^it+Mvγ,where control parameters ηirt>0, kirt>0, λirt>0, μirt>0, φirt>0, ϕirt>0, and ψirt>0, r∈S; Mα,Mβ,MΔ, andMv are, respectively, known constants satisfying conditions Mα≥αi, Mβ≥βi, MΔ≥Δi, and Mv≥vi, if the following inequality holds:(21)∑q=1mπpqDq−Qp≤0,K˜+Ω+12Θrt+O˜−Ξrt⊗Im≤0,where Dq is a positive definite matrix of appropriate dimension, Qq is an arbitrary symmetric matrix, Θrt=diagρ1rt,ρ2rt,…,ρNrt;signeit=diagsignei1t,signei2t,…,signeint; Ω=1/2∑d=1lBd⊗ΓdKd−1Bd⊗ΓdT, ϖp=∑q=1mDq−Qp/2, Ο˜=1/2lL12KdIN;Κ˜=IN⊗ϖp, and Ξrt=diagη1rt,η2rt,…,ηNrt.Then, finite-time synchronization between two different nonlinear coupled multiweighted Markovian switching delayed networks (10) and (14) can be achieved under the controller (20) and inequality condition (21). According to Lemma 1, the finite-time t1 can be expressed as follows:(22)t1≤t0+Vt0,rt01−εξ2ε1−ε,where Vt0,rt0 is the function value of Vt,rt at the initial time and ξ=minmink,minψ,κ/21+γ/2.Proof.
Construct the Lyapunov function as follows:(23)Vt,et,p=12∑i=1NeiTteit+12∑i=1N1λipα˜iTtα˜it+12∑i=1N1μipβ˜iTtβ˜it12∑i=1N1φipΔ˜i2t+12∑i=1N1ϕipv˜i2t+∑i=1N∫t−τteiTθeiθdθ,where α˜it, β˜it, v˜it, and Δ˜it are the error between the true value and the estimated value.
According to Definition1, the differential operator ℒ can be given as(24)ℒVt,et,p=∑i=1NeiTte˙it+∑i=1N1λipα˜⋅iTtα˜it+∑i=1N1μipβ˜⋅iTtβ˜it+∑i=1N1φipΔ˜itΔ˜⋅it+∑i=1N1ϕipv˜itv˜⋅it+∑i=1NeiTteit−∑i=1NeiTt−τeit−τ+∑q=1mπpq∑i=1N∫t−τteiTθeiθdθ+12∑i=1NtraceσiTt,eit,pσit,eit,p+∑q=1mπpq12∑i=1NeiTteit.
Taking the controller (20) into (24), we can get(25)ℒVt,et,p=∑i=1NeiTt−gi2yit,yit−τβ˜it+fi2xit,xit−τα˜it−ηipeit+∑d=1l∑j=1NbijdpΓdhyjt−hxjt−Δ^isigneit−v^isigneit+Δi2−Δi1−kpsigneiteitγ+Δgi1yit−Δfi1xit+∑i=1NeiTteit+∑q=1mπpq12∑i=1NeiTtDqeit+12∑i=1NtraceσiTt,eit,pσit,eit,p−∑i=1NeiTtfi2xit,xit−τα˜it+∑i=1NeiTtgi2yit,yit−τβ˜it−ψp∑i=1N1λipsignα^itα^it+Mαγα˜it+∑i=1NΔ˜iteiTtsigneit−ψp∑i=1N1μipsignβ^itβ^it+Mβγβ˜it+∑i=1Nv˜iteiTtsigneit−ψp∑i=1N1φipsignΔ^itΔ^it+MΔγΔ˜it−∑i=1NeiTt−τeit−τ−ψp∑i=1N1ϕipsignv^itv^it+Mvγv˜it+∑q=1mπpq∑i=1N∫t−τteiTθDqeiθdθ.
In addition, if uncertaintiesΔf1xit, Δg1yit and disturbances Δi1 and Δi2 are all bounded, then there exist non-negative constants v and Δ, then(26)Δg1yit−Δf1xit≤v≤Mv,Δi2−Δi1≤Δ≤MΔ.
By Assumption1 and (26), equation (25) can be simplified as follows:(27)ℒVt,et,p=∑i=1NeiTt−ηipeit−kpsigneiteitγ+∑d=1l∑j=1NbijdpΓdHejt+∑q=1mπpq∑i=1N∫t−τteiTθDqeiθdθ+∑q=1mπpq12∑i=1NeiTtDqeit−ψp∑i=1N1λipsignα^it|α^it|+Mαγα˜it−∑i=1NeiTt−τeit−τ−ψp∑i=1N1μipsignβ^it|β^it|+Mβγβ˜it+∑i=1NeiTteit−ψp∑i=1N1ϕipsignv^it|v^it|+Mvγv˜it+12ρirteiTteit−ψp∑i=1N1φipsignΔ^it|Δ^it|+MΔγΔ˜it,where Hejt=hyjt−hxjt.
According to Lemma2 and Assumption 2, we can get(28)∑i=1NeiTt∑d=1l∑j=1NbijdpΓdHejt=∑d=1leTtBd⊗ΓdHet≤12∑d=1leTtBd⊗ΓdKd−1Bd⊗ΓdTet+12lHTetKdHet≤12∑d=1lcdeTtBd⊗ΓdKd−1Bd⊗ΓdTet+12lL12KdeTtet.
According to Assumption3, and bringing the inequality (28) into inequality (27), we can get(29)ℒVt,et,p≤12lL12Kd∑i=1NeiTteit−kp∑i=1NeiTtsigneiteitγ−κ∑i=1N∫t−τteiTθeiθ1+γ2dθ+∑q=1mπpq12∑i=1NeiTtDqeit−ψp∑i=1N1λipsignα^it|α^it|+Mαγα˜it+12ρirteiTteit+∑q=1mπpq∑i=1N∫t−τteiTθDqeiθdθ−ψp∑i=1N1μipsignβ^it|β^it|+Mβγβ˜it−ψp∑i=1N1ϕipsignv^it|v^it|+Mvγv˜it−ηip∑i=1NeiTteit−ψp∑i=1N1φipsignΔ^it|Δ^it|+MΔγΔ˜it.
Based on Lemma3, we have(30)−∑i=1NeiTtsigneiteitγ=−∑i=1N∑j=1Neijteijtγsigneijt=−∑i=1N∑j=1Neijt1+γ≤−∑i=1N∑j=1Neijt21+γ/2=−∑i=1NeiTteit1+γ/2.
From some inequality relationships, we can obtain(31)α˜it=α^it−αit≤α^it+αit≤α^it+Mα.
Hence,(32)−∑i=1N|α^it|+Mαγsignα^itα˜it=−∑i=1N|α^it|+Mαγα˜it≤−∑i=1Nα^it−αitγα˜it=−∑i=1Nα˜itγ+1≤−∑i=1Nα˜iTtα˜it1+γ/2,(33)−∑i=1N|β^it|+Mβγsignβ^itβ˜it=−∑i=1N|β^it|+Mβγβ˜it≤−∑i=1Nβ^it−βitγβ˜it=−∑i=1Nβ˜itγ+1≤−∑i=1Nβ˜iTtβ˜it1+γ/2,(34)−∑i=1N|Δ^it|+MΔγsignΔ^itΔ˜it=−∑i=1N|Δ^it|+MΔγΔ˜it≤−∑i=1NΔ^it−ΔitγΔ˜it=−∑i=1NΔ˜itγ+1≤−∑i=1NΔ˜i2t1+γ/2,(35)−∑i=1N|v^it|+Mvγsignv^itv˜it=−∑i=1N|v^it|+Mvγv˜it≤−∑i=1Nv^it−vitγv˜it=−∑i=1Nv˜itγ+1≤−∑i=1Nv˜i2t1+γ2.
According to inequalities (30) and (32)–(35), we can get(36)ℒVt,et,p≤−kp∑i=1NeiTteit1+γ/2−κ∑i=1N∫t−τteiTθeiθ1+γ/2−ψp∑i=1N1λipα˜iTtα˜it1+γ/2−ψp∑i=1N1μipβ˜iTtβ˜it1+γ/2−ψp∑i=1N1φipΔ˜i2t1+γ/2−ψp∑i=1N1ϕipv˜i2t1+γ/2−ηip∑i=1NeiTteit+12lL12Kd∑i=1NeiTteit+∑q=1mπpq12∑i=1NeiTtDq−Qpeit+12ρirteiTteit+∑q=1mπpq∑i=1N∫t−τteiTθDq−Qpeiθdθ.
From Lemma3 and inequality (21), we can get(37)ℒVt,et,p≤∑i=1NeiTtΚ˜+Ω+12Θrt+Ο˜−Ξrt⊗Ineit−ξ∑i=1NeiTteit+∑i=1N1λipα˜iTtα˜it+∑i=1N1μipβ˜iTtβ˜it+∑i=1N1φipΔ˜i2t+∑i=1N1ϕipv˜i2t+2∑i=1N∫t−τteiTθeiθdθ1+γ/2=∑i=1NeiTtPeit−ξ21+γ2V1+γ2t,et,p.
According to conditions (21), the above inequality (37) can be written as(38)ℒVt,et,p≤−ξ2εVεt,et,p,where ε=1+γ/2.
From Lemma1 and inequality (22), if t0=0, finite-time t1 can be estimated as(39)t1≤V0,r01−εθ2ε1−ε.Remark 2.
Based on an effective controller with update laws, when networks (10) and (14) achieve synchronization in a finite-time t1, the unknown parameter vectors αit and βit of the networks will be identified as true value.Remark 3.
In the finite-time synchronization and parameters identification process between two nonlinear coupled multiweighted Markovian switching delayed networks with uncertainties (10) and (14), inequality (21) is only a sufficient condition, not a necessary condition.Remark 4.
In the synchronization and parameters identification process between two Markovian switching networks, the speed of synchronization and unknown parameters identification will be affected by these control parametersηi,k,λi,μi,φ,ϕi,ψi,Δi,vi in controller (20).


## 3.1. Network Models

In this paper, then-dimensional drive and response networks with uncertain parameters, models, and external disturbances are given as follows:The drive network is as follows:(10)x˙it=fi1xit,xit−τ+fi2xit,xit−ταit+∑d=1l∑j=1NaijdrtΓdhxjt+Δfi1xit+Δi1,i=1,2,…,N,where xit=xi1t,xi2t,…,xintT∈Rn is the state vector of the i-th node; fi1:Rn⟶Rn and fi2:Rn⟶Rn×n are continuous vector function and matrix function, respectively; Δfi1:Rn⟶Rn is the model uncertainty; h⋅:Rn⟶Rn is the nonlinear coupling function; αit is an unknown parameter vector of the dynamic node; τ>0 is the time delays of node i; Δi1 is external disturbances of the network; and Γd∈Rn×n (d = 1, 2, …, l) is the inner coupling matrix of the d-th coupling form. Adrt=aijdrtN×N∈RN×N is the topological structure matrix of the d-th coupling form, which represents topological structure of the network at time t in mode rt, and Adrt can be defined as follows: if there exists a connection between node i and node jj≠i for the d-th coupling form, then aijd≠0; otherwise, aijd=0. The diagonal elements of Art are defined as(11)aiidrt=−∑j=1,j≠iNaijdrt,d=1,2,…,l;i=1,2,…,N.Letrt,t≥0 be a right-continuous Markovian process in probability space, which takes values in a finite space S=1,2,…,l with generator Π=πpqm×m. The transition probability from the p-th mode at time t to the q-th mode at time t+Δt is defined as follows:(12)Prt+Δt=q|rt=p=πpqΔt+oΔt,ifq≠p,1+πppΔt+oΔt,ifq=p,where Δt>0, limΔt⟶0oΔt/Δt=0, and transition rate πpqπpq>0 satisfies(13)πpp=−∑q=1,q≠pmπpq.The response network is as follows:(14)y˙it=gi1yit,yit−τ+gi2yit,yit−τβit+∑d=1l∑j=1NbijdrtΓdhyjt+Δi2+Δgi1yit+σit,eit,rtω˙t+uit,i=1,2,…,N,where yit=yi1t,yi2t,…,yintT∈Rn is the state vector of the i-th node; gi1:Rn⟶Rn and gi2:Rn⟶Rn×n are continuous vector function and matrix function, respectively; Δgi1:Rn⟶Rn is the model uncertainty; βit is an unknown parameter vector of the dynamic node with the same definition as αit; Δi2 is the external disturbances of the network; Bdrt=bijdrtN×N∈RN×N is the same as the definition of Adrt above; and Bdrt can be defined as follows: if there exists a connection between node i and node jj≠i for the d-th coupling form, then bijd≠0; otherwise, bijd=0. The diagonal elements of Brt is defined as(15)biidrt=−∑j=1,j≠iNbijdrt,d=1,2,…,l;i=1,2,…,N.Moreover,uit is a nonlinear controller; ωt=ω1t,ω2t,…,ωntT is a n-dimensional Weiner process.Define synchronization error of the system as follows:(16)eit=yit−xit,i=1,2,…,N,where eit=ei1t,ei2t,…,eintT represents the state error of the i-th node.Definition 2.
If there is a positive constantt and gives an arbitrary t1>t, such that(17)limt⟶t1Eeit=limt⟶t1Eyit−xit2=0,i=1,2,…,N,and eit2=yit−xit2≡0, then synchronization of Markovian switching complex networks (10) and (14) can be realized at a finite-time t1.Remark 1.
When the drive system (10) and the response system (14) gradually realize synchronization at a finite-time t1, the unknown parameter vectors αit and βit will be identified and expressed as(18)limt⟶t1∑i=1Nα^it−αi2=0,limt⟶t1∑i=1Nβ^it−βi2=0,where α^it and β^it are estimated values of αit and βit, respectively.


## 3.2. Main Results

In this section, finite-time synchronization between two different nonlinear coupled Markovian switching complex delayed networks with multiple weights and uncertainties will be demonstrated and analyzed.According to network equations (10) and (14), the synchronization error equation of system can be obtained:(19)e˙it=gi1yit,yit−τ+Δgi1yit+gi2yit,yit−τβit+Δi2−Δi1−fi1xit,xit−τ−fi2xit,xit−ταit+σit,eit,rtω˙t−Δfi1xit+∑d=1l∑j=1NbijdrtΓdhyjt−∑d=1l∑j=1NaijdrtΓdhxjt+uit.Obviously, when the synchronization error (19) converges to zero in a finite time, then finite-time synchronization and identification of the network models will also be achieved. Therefore, it is very important to design an effective controller so that the synchronization error converges to zero.Theorem 1.
Design an effective controller with update laws as follows:(20)uit=−ηirteit−gi1yit,yit−τ−gi2yit,yit−τβ^it+fi1xit,xit−τ+fi2xit,xit−τα^it−krtsigneiteitγ+∑d=1l∑j=1Naijdrt−bijdrtΓdhxjt−Δ^isigneit−v^isigneit,α^⋅it=−λirtfi2Txit,xit−τeit−ψirtsignα^itα^it+Mαγ,β^⋅it=μirtgi2Tyit,yit−τeit−ψirtsignβ^itβ^it+Mβγ,Δ^⋅it=φirteiTtsigneit−ψirtsignΔ^itΔ^it+M△γv^⋅it=ϕirteiTtsigneit−ψirtsignv^itv^it+Mvγ,where control parameters ηirt>0, kirt>0, λirt>0, μirt>0, φirt>0, ϕirt>0, and ψirt>0, r∈S; Mα,Mβ,MΔ, andMv are, respectively, known constants satisfying conditions Mα≥αi, Mβ≥βi, MΔ≥Δi, and Mv≥vi, if the following inequality holds:(21)∑q=1mπpqDq−Qp≤0,K˜+Ω+12Θrt+O˜−Ξrt⊗Im≤0,where Dq is a positive definite matrix of appropriate dimension, Qq is an arbitrary symmetric matrix, Θrt=diagρ1rt,ρ2rt,…,ρNrt;signeit=diagsignei1t,signei2t,…,signeint; Ω=1/2∑d=1lBd⊗ΓdKd−1Bd⊗ΓdT, ϖp=∑q=1mDq−Qp/2, Ο˜=1/2lL12KdIN;Κ˜=IN⊗ϖp, and Ξrt=diagη1rt,η2rt,…,ηNrt.Then, finite-time synchronization between two different nonlinear coupled multiweighted Markovian switching delayed networks (10) and (14) can be achieved under the controller (20) and inequality condition (21). According to Lemma 1, the finite-time t1 can be expressed as follows:(22)t1≤t0+Vt0,rt01−εξ2ε1−ε,where Vt0,rt0 is the function value of Vt,rt at the initial time and ξ=minmink,minψ,κ/21+γ/2.Proof.
Construct the Lyapunov function as follows:(23)Vt,et,p=12∑i=1NeiTteit+12∑i=1N1λipα˜iTtα˜it+12∑i=1N1μipβ˜iTtβ˜it12∑i=1N1φipΔ˜i2t+12∑i=1N1ϕipv˜i2t+∑i=1N∫t−τteiTθeiθdθ,where α˜it, β˜it, v˜it, and Δ˜it are the error between the true value and the estimated value.
According to Definition1, the differential operator ℒ can be given as(24)ℒVt,et,p=∑i=1NeiTte˙it+∑i=1N1λipα˜⋅iTtα˜it+∑i=1N1μipβ˜⋅iTtβ˜it+∑i=1N1φipΔ˜itΔ˜⋅it+∑i=1N1ϕipv˜itv˜⋅it+∑i=1NeiTteit−∑i=1NeiTt−τeit−τ+∑q=1mπpq∑i=1N∫t−τteiTθeiθdθ+12∑i=1NtraceσiTt,eit,pσit,eit,p+∑q=1mπpq12∑i=1NeiTteit.
Taking the controller (20) into (24), we can get(25)ℒVt,et,p=∑i=1NeiTt−gi2yit,yit−τβ˜it+fi2xit,xit−τα˜it−ηipeit+∑d=1l∑j=1NbijdpΓdhyjt−hxjt−Δ^isigneit−v^isigneit+Δi2−Δi1−kpsigneiteitγ+Δgi1yit−Δfi1xit+∑i=1NeiTteit+∑q=1mπpq12∑i=1NeiTtDqeit+12∑i=1NtraceσiTt,eit,pσit,eit,p−∑i=1NeiTtfi2xit,xit−τα˜it+∑i=1NeiTtgi2yit,yit−τβ˜it−ψp∑i=1N1λipsignα^itα^it+Mαγα˜it+∑i=1NΔ˜iteiTtsigneit−ψp∑i=1N1μipsignβ^itβ^it+Mβγβ˜it+∑i=1Nv˜iteiTtsigneit−ψp∑i=1N1φipsignΔ^itΔ^it+MΔγΔ˜it−∑i=1NeiTt−τeit−τ−ψp∑i=1N1ϕipsignv^itv^it+Mvγv˜it+∑q=1mπpq∑i=1N∫t−τteiTθDqeiθdθ.
In addition, if uncertaintiesΔf1xit, Δg1yit and disturbances Δi1 and Δi2 are all bounded, then there exist non-negative constants v and Δ, then(26)Δg1yit−Δf1xit≤v≤Mv,Δi2−Δi1≤Δ≤MΔ.
By Assumption1 and (26), equation (25) can be simplified as follows:(27)ℒVt,et,p=∑i=1NeiTt−ηipeit−kpsigneiteitγ+∑d=1l∑j=1NbijdpΓdHejt+∑q=1mπpq∑i=1N∫t−τteiTθDqeiθdθ+∑q=1mπpq12∑i=1NeiTtDqeit−ψp∑i=1N1λipsignα^it|α^it|+Mαγα˜it−∑i=1NeiTt−τeit−τ−ψp∑i=1N1μipsignβ^it|β^it|+Mβγβ˜it+∑i=1NeiTteit−ψp∑i=1N1ϕipsignv^it|v^it|+Mvγv˜it+12ρirteiTteit−ψp∑i=1N1φipsignΔ^it|Δ^it|+MΔγΔ˜it,where Hejt=hyjt−hxjt.
According to Lemma2 and Assumption 2, we can get(28)∑i=1NeiTt∑d=1l∑j=1NbijdpΓdHejt=∑d=1leTtBd⊗ΓdHet≤12∑d=1leTtBd⊗ΓdKd−1Bd⊗ΓdTet+12lHTetKdHet≤12∑d=1lcdeTtBd⊗ΓdKd−1Bd⊗ΓdTet+12lL12KdeTtet.
According to Assumption3, and bringing the inequality (28) into inequality (27), we can get(29)ℒVt,et,p≤12lL12Kd∑i=1NeiTteit−kp∑i=1NeiTtsigneiteitγ−κ∑i=1N∫t−τteiTθeiθ1+γ2dθ+∑q=1mπpq12∑i=1NeiTtDqeit−ψp∑i=1N1λipsignα^it|α^it|+Mαγα˜it+12ρirteiTteit+∑q=1mπpq∑i=1N∫t−τteiTθDqeiθdθ−ψp∑i=1N1μipsignβ^it|β^it|+Mβγβ˜it−ψp∑i=1N1ϕipsignv^it|v^it|+Mvγv˜it−ηip∑i=1NeiTteit−ψp∑i=1N1φipsignΔ^it|Δ^it|+MΔγΔ˜it.
Based on Lemma3, we have(30)−∑i=1NeiTtsigneiteitγ=−∑i=1N∑j=1Neijteijtγsigneijt=−∑i=1N∑j=1Neijt1+γ≤−∑i=1N∑j=1Neijt21+γ/2=−∑i=1NeiTteit1+γ/2.
From some inequality relationships, we can obtain(31)α˜it=α^it−αit≤α^it+αit≤α^it+Mα.
Hence,(32)−∑i=1N|α^it|+Mαγsignα^itα˜it=−∑i=1N|α^it|+Mαγα˜it≤−∑i=1Nα^it−αitγα˜it=−∑i=1Nα˜itγ+1≤−∑i=1Nα˜iTtα˜it1+γ/2,(33)−∑i=1N|β^it|+Mβγsignβ^itβ˜it=−∑i=1N|β^it|+Mβγβ˜it≤−∑i=1Nβ^it−βitγβ˜it=−∑i=1Nβ˜itγ+1≤−∑i=1Nβ˜iTtβ˜it1+γ/2,(34)−∑i=1N|Δ^it|+MΔγsignΔ^itΔ˜it=−∑i=1N|Δ^it|+MΔγΔ˜it≤−∑i=1NΔ^it−ΔitγΔ˜it=−∑i=1NΔ˜itγ+1≤−∑i=1NΔ˜i2t1+γ/2,(35)−∑i=1N|v^it|+Mvγsignv^itv˜it=−∑i=1N|v^it|+Mvγv˜it≤−∑i=1Nv^it−vitγv˜it=−∑i=1Nv˜itγ+1≤−∑i=1Nv˜i2t1+γ2.
According to inequalities (30) and (32)–(35), we can get(36)ℒVt,et,p≤−kp∑i=1NeiTteit1+γ/2−κ∑i=1N∫t−τteiTθeiθ1+γ/2−ψp∑i=1N1λipα˜iTtα˜it1+γ/2−ψp∑i=1N1μipβ˜iTtβ˜it1+γ/2−ψp∑i=1N1φipΔ˜i2t1+γ/2−ψp∑i=1N1ϕipv˜i2t1+γ/2−ηip∑i=1NeiTteit+12lL12Kd∑i=1NeiTteit+∑q=1mπpq12∑i=1NeiTtDq−Qpeit+12ρirteiTteit+∑q=1mπpq∑i=1N∫t−τteiTθDq−Qpeiθdθ.
From Lemma3 and inequality (21), we can get(37)ℒVt,et,p≤∑i=1NeiTtΚ˜+Ω+12Θrt+Ο˜−Ξrt⊗Ineit−ξ∑i=1NeiTteit+∑i=1N1λipα˜iTtα˜it+∑i=1N1μipβ˜iTtβ˜it+∑i=1N1φipΔ˜i2t+∑i=1N1ϕipv˜i2t+2∑i=1N∫t−τteiTθeiθdθ1+γ/2=∑i=1NeiTtPeit−ξ21+γ2V1+γ2t,et,p.
According to conditions (21), the above inequality (37) can be written as(38)ℒVt,et,p≤−ξ2εVεt,et,p,where ε=1+γ/2.
From Lemma1 and inequality (22), if t0=0, finite-time t1 can be estimated as(39)t1≤V0,r01−εθ2ε1−ε.Remark 2.
Based on an effective controller with update laws, when networks (10) and (14) achieve synchronization in a finite-time t1, the unknown parameter vectors αit and βit of the networks will be identified as true value.Remark 3.
In the finite-time synchronization and parameters identification process between two nonlinear coupled multiweighted Markovian switching delayed networks with uncertainties (10) and (14), inequality (21) is only a sufficient condition, not a necessary condition.Remark 4.
In the synchronization and parameters identification process between two Markovian switching networks, the speed of synchronization and unknown parameters identification will be affected by these control parametersηi,k,λi,μi,φ,ϕi,ψi,Δi,vi in controller (20).


## 4. Simulink Results

In this section, a representative mathematical simulation is given to prove the validity and feasibility of the above theoretical analysis. Firstly, the node dynamics equations of the driving network can be described by a chaotic system:(40)x˙i1t=αi1xi2t−xi1t,x˙i2t=αi3xi1t−xi1txi3t−xi2t,x˙i3t=xi1txi2t−βi2xi3t.The Lorenz system is taken as the node dynamic of driving complex network, and system (40) can be expressed as(41)x˙i1tx˙i2tx˙i3t=0−xi1txi3t−xi2txi1txi2t+xi2t−xi1t0000xi2t0−xi3t0×αi1αi2αi3,where the identification value of the unknown parameter vector αi is αi=αi1,αi2,αi3T=10,8/3,28T.Secondly, the node dynamics equations of the response network can be described by a chaotic system:(42)y˙i1t=βi1yi2t−yi1t,y˙i2t=βi3−βi1yi1t−yi1tyi3t+βi3yi2t,y˙i3t=yi1tyi2t−βi2yi3t.The Chen system is taken as the node dynamic of response complex network, and the system (42) can be expressed as(43)y˙i1ty˙i2ty˙i3t=0−yi1tyi3tyi1tyi2t+yi2t−yi1t00−yi1t0yi1t+yi2t0−yi3t0×βi1βi2βi3,where the identification value of the unknown parameter vector βi is βi=β1,β2,β3T=35,3,28T.The number of system nodes is 20 in numerical simulation. Moreover, Figure1 shows the system mode switching process. The other relevant parameters are as follows:Figure 1
Switching of system mode.l=2,hx=sinx, Γ1=100010001, Γ2=200020002, Δf1xit=0.1cosx1t0.2cosx2t0.3cosx3t, Δg1xit=0.1sinx1t0.2sinx2t0.3sinx3t, Δ1=cos0.1x1tcos0.2x2tcos0.3x3t, and Δ2=sin0.1x1tsin0.2x2tsin0.3x3t.In the follow, according to the controller (20) and condition (21), some control parameters of controller (20) are selected as η1=η2=10; k (1) = 12, k (2) = 15; λ1=λ2=15; μ1 = 24, μ2 = 18; φ1=φ2=5; ψ1 = 9, ψ2 = 4; ϕ1=ϕ2=15; γ = 0.6; Mα = 100, Mβ = 100, Mα = 100, and Mv = 100. In this simulation process, the start time of the simulation is selected at t0 = 0 and time delay τ = 0.01. In addition, through Lemma 1 and (39), we can obtain synchronization time t1 ≤ 9.3287.The time-varying curves of networks synchronization errorseit (i = 1, 2, 3) are shown in Figure 2. In Figure 2, synchronization error curves of networks gradually converge to 0. Moreover, Figure 3 is the identification process of the unknown parameter vectors αi (i = 1, 2, 3), and the unknown parameter vector is finally identified as αi = (10, 8/3, 28)T. Figure 4 is the identification process of the unknown parameter vectors βi (i = 1, 2, 3), and the unknown parameter vector is finally identified as βi = (35, 3, 28)T. From the curve change trend in Figures 2–4, it can be seen that the networks synchronization and parameter identification are achieved at about t = 7. By comparing calculated values with simulated values, the theoretical proof of the third part is also verified.Figure 2
Synchronization errorseit with system evolution.
(a)(b)(c)Figure 3
Identification of the uncertain parameterαi.
(a)(b)(c)Figure 4
Identification of the uncertain parameterβi.
(a)(b)(c)


## 5. Conclusion

This paper mainly solved the finite-time synchronization and parameter identification problem between two nonlinear coupled Markovian switching complex delayed networks with multiple weights uncertainties. Based on finite-time stability theory and some inequality technologies, an effective controller with update laws is designed to achieve finite-time synchronization of the networks; in the meantime, the unknown parameters are identified in a finite time. Finally, the results of numerical simulation have shown that the synchronization of the networks can be realized at aboutt≈5; at the same time, the unknown parameter vector of the networks can also be identified true value αi = (10, 8/3, 28)T and βi = (35, 3, 28)T at about t≈7. In addition, the feasibility and applicability of the theoretical proof in this paper has been further confirmed.



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