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

In this paper, two different nonlinear coupledMarkovian switching delayed networkmodels withmultiple weights are established, and the effects of parameter and model uncertainties and external disturbances are fully considered. Firstly, based on the finitetime 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.


Introduction
e 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.
is 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. erefore, 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. is 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].
is kind of random switching can depend on the Markovian process, so the research on the synchronization of Markovian switching networks is more practical [7][8][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]. erefore, 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 variable x(t) cannot be directly obtained, but only h(x(t)) can be obtained, and the synchronization of the system can only be realized by h(x(t)) at this time.
erefore, 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 differentialdifference 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]. us, 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. e synchronization process of the system may be affected or even destroyed by these uncertainties. e 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.
erefore, 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. e 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. e multiweighted coupled neural network (MWCNN) models with and without coupling delays were investigated in [21]. erefore, 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. erefore, the research has significance in terms of theory and practice. e rest of this paper is arranged as follows. In Section 2, some notations and necessary conditions will be given. e 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. e conclusions are drawn in Section 5.

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. e matrix A T (or x T ) is defined as the transposed matrix of A (or x). e symbol ⊗ is defined as the Kronecker product of matrixes. Define ‖x‖ 2 as the 2-norm of x. Define I n ∈ R n×n as the n-dimensional identity matrix. E is defined as mathematical expectation. Secondly, some necessary conditions will be given below.
Definition 1 (see [24]). Define f: R n × R + × S ⟶ R n and g: R n × R + × S ⟶ R n×m are differentiable functions satisfying the following n-dimensional functional stochastic differential equation:

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.

Network Models.
In this paper, the n-dimensional drive and response networks with uncertain parameters, models, and external disturbances are given as follows: e drive network is as follows: where . , x in (t) ) T ∈ R n is the state vector of the i-th node; f i1 : R n ⟶ R n and f i2 : R n ⟶ R n×n are continuous vector function and matrix function, respectively; Δf i1 : R n ⟶ R n is the model uncertainty; h(·): R n ⟶ R n is the nonlinear coupling function; α i (t) is an unknown parameter vector of the dynamic node; τ > 0 is the time delays of node i; Δ i1 is external disturbances of the network; and Γ d ∈ R n×n (d � 1, 2, . . ., l) is the inner coupling matrix of the d-th coupling form. A d (r(t)) � (a d ij (r(t))) N×N ∈ R N×N is the topological structure matrix of the d-th coupling form, which represents topological structure of the network at time t in mode r(t), and A d (r(t)) can be defined as follows: if there exists a connection between node i and node j(j ≠ i) for the d-th coupling form, then a d ij ≠ 0; otherwise, a d ij � 0. e diagonal elements of A(r(t)) are defined as Let r(t), 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 � (π pq ) m×m . e transition probability from the p-th mode at time t to the q-th mode at time t + Δt is defined as follows: where Δt > 0, lim Δt⟶0 o(Δt)/Δt � 0, and transition rate π pq (π pq > 0) satisfies e response network is as follows: where y i (t) � ( y i1 (t), y i2 (t), . . . , y in (t) ) T ∈ R n is the state vector of the i-th node; g i1 : R n ⟶ R n and g i2 : R n ⟶ R n×n are continuous vector function and matrix function, respectively; Δg i1 : R n ⟶ R n is the model uncertainty; β i (t) is an unknown parameter vector of the dynamic node with the same definition as α i (t); Δ i2 is the external disturbances of the network; B d (r(t)) � (b d ij (r(t))) N×N ∈ R N×N is the same as the definition of A d (r(t)) above; and B d (r(t)) can be defined as follows: if there exists a connection between node i and node j(j ≠ i) for the d-th coupling form, Moreover, u i (t) is a nonlinear controller; ω(t) � ( ω 1 (t), ω 2 (t), . . . , ω n (t) ) T is a n-dimensional Weiner process.
Define synchronization error of the system as follows: where e i (t) � ( e i1 (t), e i2 (t), . . . , e in (t) ) T represents the state error of the i-th node.

Definition 2.
If there is a positive constant t and gives an arbitrary t 1 > t, such that and ‖e i (t)‖ 2 � ‖y i (t) − x i (t)‖ 2 ≡ 0, then synchronization of Markovian switching complex networks (10) and (14) can be realized at a finite-time t 1 .
Remark 1. When the drive system (10) and the response system (14) gradually realize synchronization at a finite-time t 1 , the unknown parameter vectors α i (t) and β i (t) will be identified and expressed as where α i (t) and β i (t) are estimated values of α i (t) and β i (t), respectively.

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: 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.
erefore, it is very important to design an effective controller so that the synchronization error converges to zero.
en, finite-time synchronization between two different nonlinear coupled multiweighted Markovian switching Mathematical Problems in Engineering delayed networks (10) and (14) can be achieved under the controller (20) and inequality condition (21). According to Lemma 1, the finite-time t 1 can be expressed as follows: where V(t 0 , r(t 0 )) is the function value of V(t, r(t)) at the initial time and ξ � min min(k), min(ψ), κ/2 (1+c)/2 .
Proof. Construct the Lyapunov function as follows: where α i (t), β i (t), v i (t), and Δ i (t) are the error between the true value and the estimated value.
According to Definition 1, the differential operator L can be given as Taking the controller (20) into (24), we can get In addition, if uncertainties Δf 1 ( x i (t) ), Δg 1 ( y i (t) ) and disturbances Δ i1 and Δ i2 are all bounded, then there exist non-negative constants v and Δ, then By Assumption 1 and (26), equation (25) can be simplified as follows: where H(e j (t)) � h(y j (t)) − h(x j (t)). According to Lemma 2 and Assumption 2, we can get

Mathematical Problems in Engineering
According to Assumption 3, and bringing the inequality (28) into inequality (27), we can get Based on Lemma 3, we have From some inequality relationships, we can obtain Hence,

Mathematical Problems in Engineering
Remark 2. Based on an effective controller with update laws, when networks (10) and (14) achieve synchronization in a finite-time t 1 , the unknown parameter vectors α i (t) and β i (t) 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 (20).

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: e Lorenz system is taken as the node dynamic of driving complex network, and system (40) can be expressed as where the identification value of the unknown parameter vector Secondly, the node dynamics equations of the response network can be described by a chaotic system: e Chen system is taken as the node dynamic of response complex network, and the system (42) can be expressed as where the identification value of the unknown parameter vector β i is β i � (β 1 , β 2 , β 3 ) T � (35, 3, 28) T . e number of system nodes is 20 in numerical simulation. Moreover, Figure 1 shows the system mode switching process. e other relevant parameters are as follows: ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ , and Δ 2 � sin(0.1x 1 (t)) sin(0.2x 2 (t)) sin(0.3x 3 (t)) ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ .

Conclusion
is 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 about t ≈ 5; at the same time, the unknown parameter vector of the networks can also be identified true value α i � (10, 8/3, 28) T and β i � (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.

Data Availability
e date used to support the findings of this study are included within the article.

Disclosure
is work was not supported by any fund.