Finite-time synchronization of delayed complex dynamical networks via sampled-data controller

1.   Introduction

A complex dynamical networks(CDNs) generally comprises a large-scale arrangement of interconnected nodes, where all nodes resemble a dynamical network. The dynamical networks are often represented by a differential equation that includes a coupling term between the various nodes involved in the dynamics. However, the network structure enables and limits dynamic behaviors within the network. In real life, complex networks can be modeled in many areas, such as biological systems, complex circuits, social networks, food webs, power grids, the World Wide Web, electricity distribution networks, and epidemiology, and it is attractive in both industry and academic areas ^[1,2]. These applications make it attractive to study CDNs. In addition, time-delays occur in CDNs owing to component capabilities, and congestion often exists in complex networks during acquisition and signal transmission ^[3,4]. Thus, it's crucial to consider the effects of time-varying delay on complex networks. Hence, coupled differential equations can be used to describe a wide range of interesting collective phenomena in CDNs, including spatiotemporal chaos, synchronization, and self-organization ^[5,6]. The dynamical nodes of complex networks exhibit a fascinating and essential property, namely, synchronization, which has been extensively studied in ^[7]. Synchronization is a significant nonlinear phenomenon that has played a vital role in the dynamic features of complex network fields in recent years ^[8,9]. Subsequently, the investigation focused on the synchronization issue between all dynamic nodes in a complex network, which denotes that every node in a complex network over time aligns with every other node. This phenomenon is significant because it may be observed in daily life, including the synchronous transfer of signals (analog or digital) in communication networks and engineering applications, which require synchronization for rapid convergence.

Typically, synchronization in complex networks does not occur spontaneously and requires external controls. Therefore, a complex network synchronization control is critical. To attain synchronization, numerous efficient control strategies have been put forward, including observer ^[10], adaptive ^[11], feedback ^[12], pinning ^[13] controls, and so on. Controlling all nodes simultaneously is very difficult because it causes problems such as high costs and difficulty in increasing the control. As a result, controlling a complex network through controllers is typically challenging, and a natural technique to maintain a complicated network is pinning a few of the nodes ^[14]. Therefore, specific nodes within a complex network need to be controlled to reduce problems and enhance feasibility, called pinning control. This strategy only requires some key nodes to control the complex network comprehensively (see ^[15,16]). Pinned control and dynamic feedback quantizers have recently been employed to achieve finite-horizon $H_{\infty}$ synchronization for CDNs ^[17]. Using a pinning control technique, the authors in ^[18,19] examined a complex network synchronization with or without time-varying delay couplings.

Furthermore, sampled-data control (SDC) is gaining popularity in stabilizing the system with appropriate and good control performance. In ^[20], the sampled-data synchronization control issue was studied for general CDNs with time-varying delays. A sufficient condition was derived to ensure asymptotic stability in the error system, apply the widely used Jensen inequality, and create the planned sampled-data feedback controller. However, the sampling period is essential for improving the control performance. In particular, a long sampling period can result in limited communication bandwidth and capacity. Hence, researchers focused on improving the large sampling period using different approaches, including continuous Lyapunov functionals (LFs) ^[21] and discontinuous LFs ^[22]. The looped functions have received much attention among researchers due to their fundamentally positive definite only in sampling instants, resulting in less conservative stability criteria in terms of the sampling period. A Lyapunov function $V(t)$ is said to be a looped LF if it satisfies

The key advantage of looped LFs is a relaxed constraint on the positivity of $V_i(t)$ at the sample instants $t_q$ and $t_{q+1}$. Additionally, it utilizes the whole information of $x(t_q)$ to $x(t)$ and $x(t)$ to $x(t_{q+1})$. Therefore, looped LFs have gained interest in SDC stability analysis ^[23,24]. To date, the synchronization of CDNs has been defined over infinite time intervals. Finite-time synchronization (FTS) optimizes the convergence or settling time. To this aim, FTS techniques have been derived for time-varying delayed complex networks ^[25,26,27] to achieve stability or synchronization in finite-time. FTS in CDNs has attracted more attention in specific fields, such as ^[28,29], where FTS occurs when the error system reaches the origin within finite-time. The works mentioned on FTS are that the derived conditions do not include information on two-sided looped functionals in FTS. To our knowledge, finite-time CDNs with two-sided looped LFs have yet to be considered for synchronization of delayed CDNs.

Motivated by the analysis mentioned above, designing two-sided looped functionals in the delayed CDNs for FTS is the primary goal of this study. Based on the above explanation, the main implications of this study are as follows:

● The synchronization issue between the isolated node CDNs and the $i^{th}$ node CDNs with time-varying delay is investigated under SDC-based pinning control, and improved looped LFs are constructed to check the stability of the proposed approach in terms of linear matrix inequalities (LMIs).

● The information of $x(t_q)$ to $x(t)$ and $x(t)$ to $x(t_{q+1})$ and the upper bound of the time-varying delay are involved in the proposed looped LFs.

● The derived sufficient conditions guarantee the FTS for the error states of the CDNs with time-varying delay. The numerical example and its simulation results prove the effectiveness of the proposed criteria.

Notations: The $n$-dimensional Euclidean space is denoted by $\mathbb{R}^n$, the set of all $n\times n$ real matrices is $\mathbb{R}^{n\times n}$, positive real numbers are denoted by $\mathbb{R}^{+}$, and an identity matrix is denoted by $I_n$. The maximum and minimum eigenvalues of a real symmetric matrix Q are denoted by $\lambda_{\mathrm{max}}(Q)$ and $\lambda_{\mathrm{min}}(Q)$, respectively. $Q > 0$ or $Q < 0$ indicates the positive definite or negative definite matrix of $Q$. The symmetric terms in the symmetric matrix are shown by "$*$". The transpose of the matrix $X$ is represented by $X^T$. $||\cdot||$ represents the norm of a Euclidean vector. Kronecker product is represented as "$\otimes$".

2.   Preliminaries and problem formulations

This paper considers CDNs with identical nodes "$N$". The tracking target is denoted by node index $0$, and other nodes $1, 2, ..., N$ are denoted by usual nodes within the network. The representation of the tracking target as an isolated node is explained by

where target node's state vector is denoted by

and

denotes the nonlinear functions and expresses the dynamics of the nodes. $A\in\mathbb{R}^{n\times n}$ is a constant real matrix. $\tau(t)$ represents the time-varying delay,

where

and

are known constants. The $\mathfrak{i}^{th}$ node of dynamics is represented below as:

where $\mathfrak{i} = 1, 2, ..., N$; the state vector of the $\mathfrak{i}^{th}$ node is

The coupling strength of the whole network is $\mathfrak{C}$. To show the topology of the network, the coupling matrix is

$b_{\mathfrak{ij}} > 0$ if node $\mathfrak{i}$ is able to receive information from node $\mathfrak{j}$; otherwise,

Then,

denoted as diagonal elements. The internal coupling matrix,

is semi-positive definite and signifies the precise connection relationship between two nodes. If two nodes communicate at $\mathfrak{i}^{th}$ state, then

otherwise,

Achieving synchronization in large-scale networks by controlling all nodes is more challenging. Thus, pinning control selects specific network nodes for $m$ nodes that are pinned $(1 \leq m \leq N)$; the set of all nodes are

and the set of pinned nodes are

The CDNs with pinning controller are given below as:

Let us consider the SDC as follows:

where $\mathfrak{i} = 1, 2, ..., N$ and $K \in \mathbb{R}^{n\times n}$ is a control gain matrix. The $h_q$ is a sampling interval that satisfies

and is denoted by

where $q \in \mathbb{N}$. Then, substitute (2.3) in (2.2), and we get

Define the synchronization error of CDNs as

Then, we get that the error dynamics of the complex network becomes:

where,

For simplicity, the error system can be rewritten as follows:

If we let

Assumption 2.1. The nonlinear function satisfies the following condition as similar in ^[30],

where $x, y \in \mathbb{R}^n$ and $\mathrm{U}, \mathrm{V}$ are known real-valued matrices.

Lemma 2.1. ^[31] For any positive symmetric constant matrix

scalars $\mathtt{r}_1, \mathtt{r}_2$ satisfy

a vector function

such that the integrations concerned are well-defined, then

Lemma 2.2. ^[32] Let $r(t)$ be a differentiable function,

For any positive matrix $W \in \mathbb{R}^n$, any matrices $S_1$ and $S_2 \in \mathbb{R}^{k \times n}$, and any vector $\xi(t)$, the following inequality holds:

where

Definition 2.1. Given that $Q$ is a positive matrix, $c_1, c_2, T$ is the positive constants with $c_1 < c_2$, and the error system is FTS with respect to $(c_1, c_2, T)$, if

3.   Main results

In this section, the looped LFs, assumption and lemmas are used to construct the stability criteria for the error system (2.6) and were summarized as follows:

Theorem 3.1. For given constants

scalar $\alpha$, and unknown control gain matrix K, the CDNs (2.6) can achieve FTS with respect to $(c_1, c_2, T)$ with $c_1 < c_2$, if there exist symmetric matrices

and a scalars $\lambda > 0, \; \omega > 0$, such that the following LMIs hold:

and

For simplicity,

Here,

Then,

is a control gain matrix.

Proof. Choose the LFs as follows:

where

Here,

The derivative of V(t) along state trajectories can be calculated as

By using Lemma 2.1 in (3.7) to process the integral part, we get

By applying 2.2 in $\dot{V}_6(t)$, the integral terms are estimated as:

where,

Now the following inequality is obtained by 2.1, where $\mathrm{U}, \mathrm{V}$ are diagonal matrices, and for any $\lambda > 0$,

For any non-singular matrix $J$ with appropriate dimensions and scalar $\alpha$, the Eq (2.6) is given as follows:

Let,

Thus, by adding (3.5) to (3.17) inequalities, we get

where,

with

and

Then, it can be clearly verified that

are convex in $t \in [t_q, t_{q+1})$ if, and only if,

Therefore, by applying the Schur complement in Eqs (3.19) and (3.20), one can get LMIs (3.1) and (3.2), and it can be shown that

Thus, the error system is asymptotically stable.

Notice that,

According to (3.1) and (3.2), one has

i.e.,

We have

Integrating from $0$ to $T$ in (3.22), with $t \in [0, T]$, it follows that

Then, by the Definition 2.1 of $V_{\mathfrak{i}}(t)\ (\mathfrak{i} = 1, 2, 3, 4),$ we can obtain that

By following this,

becomes zero, when $t = 0$, i.e.,

Combining the (3.23) and (3.24) inequalities, we get

Therefore,

In Theorem 3.1, under (3.3), we can obtain

for all $t \in [0, T]$. Subsequently, the system achieves FTS concerning $c_1, c_2, T$. Hence, the theorem completes. □

Remark 3.1. In ^[24], for sampling information of the system, they considered one-sided looped LFs. In Eq (3.4), we proposed two-sided looped LFs in the stability analysis of CDNs with time-varying delay. Therefore, utilizing sampling information from the interval $t_q$ to $t$ and $t$ to $t_{q+1}$, it helps to reduce the convergence of the system.

Remark 3.2. The conditions (3.1)–(3.3) can be turned into standard LMIs by fixing the $\omega$ value. By the definition of FTS, the synchronization error stays within the time interval of which we fixed. Specifically, the condition (3.3) is undertaken by the following LMI conditions,

for some positive numbers $\theta_1$–$\theta_5.$

Remark 3.3. Here, $c_1$, $c_2$, and $T$ values are all predetermined in the numerical problem. Definition 2.1 says that CDNs are FTS when constrained by an initial state error. The synchronization error remains bounded within the specified time limit. Additionally, scalar value $\omega$ is fixed, and $T$ value increases with $c_1$ decreases or $c_2$ increases.

Remark 3.4. The evaluation techniques presented in the researches ^{[33,34,35,36]} are entirely distinct from the methods introduced in Theorem 3.1. A two-sided LF is designed for the FTS of the delayed CDNs. First, we investigated the delayed CDNs under SDC with looped LFs for stability analysis. Second, we introduced the upper bound of time-varying delay in the looped LF. Third, the sufficient conditions are derived for FTS. Therefore, the results presented in this paper provide a new approach for studying FTS, which leads a less conservatism.

Remark 3.5. Based on Theorem 3.1, we have calculated the number of decision variables as $20n^2 + 6n +1$ for the derived conditions. Here, the computational complexity depends on the order of $n^2$, i.e., $O(n^2)$. The computational complexity is increased if $n$ becomes a large value.

4.   Numerical example

This section illustrates the numerical example to demonstrate the effectiveness of the network synchronization criteria. Consider three nodes in the CDNs. The pinning nodes are considered as the first two nodes in the following parameters:

and

The Assumption 2.1 is satisfied with

Let us consider

A comparison with the existing method for determining the maximum sampling period $h_M$ is shown in Table 1. By taking $\mathfrak{C} = 0.1$, Table 1 shows a comparison between the existing result, and it shows that our approach yields a less conservative outcome than the existing work in ^[24].

From Table 1, the proposed method is shown to be superior than the existing method in ^[24]. Using $h_M$ and Theorem 3.1, we can arrive at the following feasible solutions:

and

to verify Theorem 3.1 to get feasible for the value

with respect to $(c_1, c_2, T)$ and using Matlab LMI toolbox to obtained the control gain matrix

as

Let's consider the initial conditions for the three nodes as

Under the proposed CDNs with time-varying delay in Theorem 3.1, Figure 1 shows the behavior of the proposed SDC. Figure 2 shows the error system (2.6) with $\mathfrak{j}\ (\mathfrak{j} = 1, 2)$ with the gain matrix converging to origin at finite-time $T$.

Additionally, we obtained the state response (2.4) with $\mathfrak{j}\ (\mathfrak{j} = 1, 2)$ of the nodes in Figures 3 and 4, respectively. Hence, one can see from these figures that the desired performance of the CDNs was achieved through synchronization using a designed controller.

5.   Conclusions

This article presented the design of a FTS problem for the CDNs with time-varying delay using an SDC. Synchronization criteria were obtained using the appropriate looped LF method with integral terms. The SDC is designed to achieve FTS, and the gain matrix is obtained by solving the corresponding LMIs. Numerical examples are given to demonstrate the effectiveness of the synchronization criteria in finite time. In ^[37], stochastic switching is used to study switched Takagi-Sugeno fuzzy systems, where they considered mode-dependent event-triggered control for the switched systems. Therefore, the future direction of the work will focus on applying the switching method to study the proposed CDNs and determining the synchronization criteria with the stochastic switching signal under the mode-dependent event-triggered control scheme.

Use of Generative-AI tools declaration

The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported by Science and Engineering Research Board, DST, Govt. of India, SURE Project under File No. SUR/2022/001625.

Conflict of interest

The authors declare that there are no conflicts of interest.