Finite-time synchronization of quaternion-valued neural networks with delays: A switching control method without decomposition

Highlights

• The two-step analysis is suitable for models that are not easily decomposed.

• A new switching controller is designed to deal with delays.

• Under 1-norm and 2-norm, the finite-time synchronization of a class of QVNNs is analyzed.

• A simulation of QVNNs dealing with high-dimensional data is presented.

Abstract

For a class of quaternion-valued neural networks (QVNNs) with discrete and distributed time delays, its finite-time synchronization (FTSYN) is addressed in this paper. Instead of decomposition, a direct analytical method named two-step analysis is proposed. That method can always be used to study FTSYN, under either 1-norm or 2-norm of quaternion. Compared with the decomposing method, the two-step method is also suitable for models that are not easily decomposed. Furthermore, a switching controller based on the two-step method is proposed. In addition, two criteria are given to realize the FTSYN of QVNNs. At last, three numerical examples illustrate the feasibility, effectiveness and practicability of our method.

Introduction

In recent years, finite-time synchronization (FTSYN) of neural networks is an important application of finite-time stability, and it has been investigated abundantly (Hu, Yu, Chen, Jiang, & Huang, 2017). The phenomenon of synchronization is often encountered in real life. However, how to improve synchronization efficiency is a concern when considering time cost. Different from exponential and asymptotic synchronization, FTSYN requires a synchronized time which is bounded and relevant to initial values. Moreover, for the complex dynamical characteristics of some practical systems, such as secure communications (Bowong, Kakmeni, & Koina, 2006), it is a challenge to realize the FTSYN. Fortunately, up to now, many excellent results have emerged so that we have the basis to study more results of FTSYN (Aghababa et al., 2011, Li and Tian, 2003, Liu and Chen, 2018, Yang, 2014, Yang and Cao, 2010). In Aghababa et al. (2011), the FTSYN analysis of two chaotic systems with completely unknown parameters was carried out by using sliding mode control method. In Yang (2014), Yang proposed a Lyapunov analytical method to achieve the FTSYN of neural networks with arbitrary delays. In Li and Tian (2003), Li et al. designed a continuous state feedback controller to achieve the FTSYN of two chaotic systems, and verified the FTSYN characteristics of Duffing systems and Lorenz systems by their method. In Yang and Cao (2010), the finite-time stochastic synchronization of system with stochastic disturbance was studied. In Liu and Chen (2018), under a pining controller, Liu et al. proposed several simple rules to achieve the finite/fixed-time cluster synchronization.

As mentioned above, the FTSYN of real-valued neural networks (RVNNs) has been extensively discussed. Nevertheless, the RVNNs do not perform well when it encounters 2-/3-dimensional affine geometric transformations (Arena, Fortuna, Muscato, & Xibilia, 1998). We think of complex-valued neural networks (CVNNs), which can solve these issues better (Peng, Qiu, Lu, Tu, & Cao, 2021), as an extension of RVNNs. And they are good at representing signals with phase and amplitude, especially for electrical and electronic engineering problems (Hirose, 2003). Recently, many published papers investigated the FTSYN of CVNNs (Liu and Li, 2020, Xiong et al., 2020, Zhang et al., 2018, Zhang et al., 2021, Zhou et al., 2017). Furthermore, the good ability of learning 3-D affine transformations of QVNNs was demonstrated in Kusakabe, Isokawa, Kouda, and Matsui (2002). Moreover, QVNNs have been used in many practical applications, such as aircraft, satellite TV, aerospace, 3-D wind processing, color night vision, image processing, polarized waves and space rotation (Syed Ali, Narayanan, Nahavandi, Wang, & Cao, 2021). Therefore, in order to solve some practical problems with neural network better, we can extend some research topics of CVNNs and RVNNs to QVNNs, such as stability, dissipativity and synchronization. For a quaternion has four real and imaginary parts and its multiplication is not commutative, it is difficult to study the related problems of QVNNs. However, with the emergence of various research ideas and the richness of results of RVNNs and CVNNs over FTSYN, it is possible to develop the relevant theoretical research of FTSYN of QVNNs.

Up to now, only a few results have paid attention to the FTSYN of QVNNs, and their analytical techniques are only decomposition, which doubles the actual computation. For instance, in Wei and Cao (2019), Fixed-time synchronization of a class of QVNNs was achieved by differential inclusion theory and Lyapunov analytical method. In Deng and Bao (2019), Deng et al. studied the fixed-time synchronization of QVNNs firstly by using new nonlinear feedback controllers. These decomposing methods of Wei and Cao (2019) and Deng and Bao (2019) can only be applied to QVNNs with the following activation functions: $f\left(a\left(\cdot \right)\right)=f_{\left(0\right)}\left(a_{\left(0\right)}\left(\cdot \right)\right)+f_{\left(1\right)}\left(a_{\left(1\right)}\left(\cdot \right)\right)\mathbb{i}+f_{\left(2\right)}\left(a_{\left(2\right)}\left(\cdot \right)\right)+\mathbb{k}{f}_{\left(3\right)}\left(a_{\left(3\right)}\left(\cdot \right)\right)$, where $a\left(\cdot \right)=a_{\left(0\right)}\left(\cdot \right)+\mathbb{i}{a}_{\left(1\right)}\left(\cdot \right)+\mathbb{j}{a}_{\left(2\right)}\left(\cdot \right)+\mathbb{k}{a}_{\left(3\right)}\left(\cdot \right)$ is a quaternion, and $\mathbb{i},\mathbb{j},\mathbb{k}$ are three imaginary units; While $f\left(a\left(\cdot \right)\right)=f_{\left(0\right)}(a_{\left(0\right)}\left(\cdot \right)$, $a_{\left(1\right)}\left(\cdot \right)$, $a_{\left(2\right)}\left(\cdot \right)$, $a_{\left(3\right)}\left(\cdot \right))+\mathbb{i}{f}_{\left(1\right)}(a_{\left(0\right)}\left(\cdot \right)$, $a_{\left(1\right)}\left(\cdot \right)$, $a_{\left(2\right)}\left(\cdot \right)$, $a_{\left(3\right)}\left(\cdot \right))+\mathbb{j}{f}_{\left(2\right)}(a_{\left(0\right)}\left(\cdot \right)$, $a_{\left(1\right)}\left(\cdot \right)$, $a_{\left(2\right)}\left(\cdot \right)$, $a_{\left(3\right)}\left(\cdot \right))+\mathbb{k}{f}_{\left(3\right)}(a_{\left(0\right)}\left(\cdot \right)$, $a_{\left(1\right)}\left(\cdot \right)$, $a_{\left(2\right)}\left(\cdot \right)$, $a_{\left(3\right)}\left(\cdot \right))$ is the most common in some practical problems. Therefore, we think that the decomposing method is conservative. Of course, there are still a few results that have done some research by non-decomposition method. For instance, in Li and Liu (2020), via a non-decomposition method, for a class of coupled QVNNs, Li et al. analyzed its finite time anti-synchronization. But overall, too little research has been done on this topic. Moreover, we think that the unbounded and non-decreasing function $\mu \left(t\right)$ of Li and Liu (2020) is difficult to find. Therefore, combined widely with the analysis methods of some existing literature on synchronization and QVNNs (Aouiti and Bessifi, 2021c, Chen and Song, 2019, Cheng et al., 2021a, Cheng et al., 2021b, Lin et al., 2020, Song and Chen, 2018, Wu et al., 2020, Zhang et al., 2021a, Zhang et al., 2021b), an interesting and difficult topic will be studied in this paper, that is, to find an easier and non-decomposition method to realize the FTSYN of more general QVNNs.

In many practical systems, time-delay phenomena are common (Jiang, Lu, & Liu, 2020), for influence of signal transmission distance and reaction speed of electronic components (Ji, He, Wu, & Zhang, 2015). Moreover, time delay is one of the adverse factors giving rise to networks’ instability (Lu, Jiang, & Zheng, 2021). Therefore, how to cope with the various time delays, such as discrete time delay, distributed time delay, constant time delay and time-varying delay, is an essential problem if we want to realize the FTSYN of delayed QVNNs smoothly. It is worth noting that the existing conclusion is not well suited to dealing with the FTSYN of the delayed system. Even so, as far as we know, some scholars have put forward some methods to neutralize the influence of time delay on Lyapunov analysis process of FTSYN. For example, some special Lyapunov functions were designed (Peng et al., 2021, Syed Ali et al., 2017b, Syed Ali et al., 2019, Yang, 2014, Zhou et al., 2017); A controller containing time delay was designed (Arslan et al., 2020, Feng et al., 2020, Zhang et al., 2018); The activation function was bounded (Chen, Zhu, Wang, Yang, & Zeng, 2020); Some new methods were used, such as two-phase-method (Li and Liu, 2020, Wang and Chen, 2019); LMI technologies (Syed Ali et al., 2017a, Syed Ali et al., 2015) were employed. Although, some results of FTSYN of the delayed neural networks have been obtained. But in these results, the RVNNs are the majority, namely, the literature of QVNNs is less. Hence, to achieve the FTSYN of delayed QVNNs, it is worth improving their methods of dealing with time delays.

Motivated by the previous discussions, i.e., to find an easier and non-decomposition method to realize the FTSYN of more general QVNNs, and to study a less conservative method to handling the time delays of finite/fixed-time synchronization of time-delay QVNNs, a class of QVNNs with time delays is investigated, and main contributions are listed as follows:

(1) To achieve the FTSYN of QVNNs with mixed delays smoothly, some new 1 or 2- norm inequalities of quaternions are deduced, and they are essential prerequisites for non- decomposing methods.

(2) Based on the above inequalities, to realize FTSYN of QVNNs, a non-decomposing two-step method is proposed: firstly, the synchronization error norm of state variables is reduced to 1 in a finite time if their initial values are more than 1; Secondly, the synchronization error norm is reduced to 0 from 1 in a fixed time. Different from the two-phase method in Chen et al., 2020, Li and Liu, 2020, Wang and Chen, 2019, the first step is inspired by Yang (2014), and the second step mainly roots in considering that when the synchronization error of state variables is very small, such as less than 1, it is easy to deal with the time delay terms.

(3) Based on the two-step method, a switching controller is proposed. The first step of the two-step method is fulfilled by a controller such that synchronization error decreases from initial values (¿1) to 1 finite-timely, and then that controller is switched to another one for the second step such that synchronization error decreases from 1 to 0 fixed-timely.

(4) To show the effectiveness of that method, 1-norm and 2-norm are all considered by that method. According to some scholars, FTSYN can only be realized by 1-norm, and other norm is not easy to realize (Yang, 2014, Zhou et al., 2017). However, in this paper, one tries to realize the FTSYN of QVNNs under 2-norm. Moreover, a simulation application of the process of FTSYN of image recovery is presented; it also shows the good effect of QVNNs for high-dimensional data processing.

The rest of this article is structured as follows. A synchronization error model is presented, and several inequalities of quaternion to be used later are derived in Section 2. A switching controller is designed, and under 1-norm and 2-norm the FTSYN of QVNNs is achieved in Section 3. The effectiveness of sufficient conditions is checked by three simulation examples in Section 4. We summarize the method of FTSYN of QVNNS in this paper in Section 5.

Notations

$\mathbb{R}$ and ${\mathbb{R}}_{+}$ represent all real and positive numbers, respectively. $\mathbb{Q}$ denotes all quaternions. $M,N\in {\mathbb{R}}_{+}$, ${\mathbb{R}}^{M\times N}$ and ${\mathbb{Q}}^{M\times N}$ represent all $M\times N$ real matrices and quaternion matrices, respectively. ${\mathbb{Q}}^M$ denotes all $M$-dimensional quaternion. The set of all continuous quaternion functions defined on $[x-a,x]$ is represented by $C([x-a,x],{\mathbb{Q}}^M)$. The $l$-norm ($l=1,2$) of real vector $\eta \in {\mathbb{R}}^M$ and quaternion $\kappa ={\kappa }_{\left(0\right)}+{\kappa }_{\left(1\right)}\mathbb{i}+{\kappa }_{\left(2\right)}\mathbb{j}+{\kappa }_{\left(3\right)}\mathbb{k}\in {\mathbb{Q}}^M$ (${\kappa }_{\left(\mu \right)}\in {\mathbb{R}}^M,\mu =0,1,2,3$) are represented by ${\Vert \eta \Vert }_l={\left({\sum }_{j=1}^M{\left|{\eta }_i\right|}^l\right)}^{\frac{1}{l}}$ and ${\Vert \kappa \Vert }_l={\left({\sum }_{\mu =0,1,2,3}{\Vert {\kappa }_{\left(\mu \right)}\Vert }_l^l\right)}^{\frac{1}{l}}$, respectively, and the norm of matrix $A=\left(a_{ij}\right)\in Q^{M\times N}$ is defined by ${\Vert A\Vert }_l={\left({\sum }_{i=1}^M{\sum }_{j=1}^N{\left|a_{ij}\right|}^l\right)}^{\frac{1}{l}}$, where for convenience of unified expression, F-norm of matrix $A$ is denoted by ${\Vert A\Vert }_2$. The real part of $\kappa$ is denoted by $Re\left(\kappa \right)$. ${\kappa }^T$ is the transpose of vector $\kappa$. And $\overline{\kappa }={\kappa }_{\left(0\right)}-\mathbb{i}{\kappa }_{\left(1\right)}-\mathbb{j}{\kappa }_{\left(2\right)}-\mathbb{k}{\kappa }_{\left(3\right)}$ and ${\kappa }^{\ast }=({\kappa }_{\left(0\right)}-{\kappa }_{\left(1\right)}\mathbb{i}-{\kappa }_{\left(2\right)}\mathbb{j}{-{\kappa }_{\left(3\right)}\mathbb{k})}^T$ denote the conjugate and the conjugate transpose of $\kappa$, respectively. For $n=(n_1,\dots ,{n_M)}^T\in {\mathbb{Q}}^M$ and $q>0$, denote sign function as $sign\left(n\right)=(sign\left(n_1\right),\dots ,{sign\left(n_M\right))}^T$, and ${\left[n\right]}_l^q$ = $({\left[n_1\right]}_l^q$, $\dots ,{{\left[n_M\right]}_l^q)}^T=(sign\left(n_1\right){\Vert n_1\Vert }_l^q$, $\dots ,{sign\left(n_M\right){\Vert n_M\Vert }_l^q)}^T$, $l=1,2$.