Impulsive synchronization of coupled delayed neural networks with actuator saturation and its application to image encryption

Abstract

The actuator of any physical control systems is constrained by amplitude and energy, which causes the control systems to be inevitably affected by actuator saturation. In this paper, impulsive synchronization of coupled delayed neural networks with actuator saturation is presented. A new controller is designed to introduce actuator saturation term into impulsive controller. Based on sector nonlinearity model approach, impulsive controls with actuator saturation and with partial actuator saturation are studied, respectively, and some effective sufficient conditions are obtained. Numerical simulation is presented to verify the validity of the theoretical analysis results. Finally, the impulsive synchronization is applied to image encryption. The experimental results show that the proposed image encryption system has high security properties.

Introduction

In recent years, the research about neural networks has become one of the important hot spots in the field of science. Neural network models have high nonlinearity, non-convexity, adaptability and self-learning ability, associative memory function and good tolerance to faults (Ding and Tao, 2018, Huang et al., 2020, Huang, Yang, and Liu, 2019, Li et al., 2014, Liu et al., 2014, Liu et al., 2016, Liu et al., 2020, Lv et al., 2018, Wan et al., 2017, Zhang, Tao, and Zhou, 2019). Therefore, it has been widely used in the fields of intelligent control and prediction (Lv et al., 2018, Wan et al., 2017), pattern recognition (Ding and Tao, 2018, Zhang, Tao, and Zhou, 2019), combinatorial optimization (Liu et al., 2014) and robot control (Li et al., 2014) in practical scientific engineering. A coupled neural network is a special neural network. Its complexity originates from not only the dynamic behavior of each node itself but also the complex structure produced by them interlaced together. In order to have some theoretical guidance in many engineering applications of neural networks, it is necessary to study the dynamic behavior of coupled neural networks. Synchronization has important applications in secure communication (Wen, Zeng, Huang, Meng, & Yao, 2015), system identification (Sorrentino & Ott, 2009), image processing (Grichuk, Kuzmina, & Manykin, 2009) and so on. The purpose of synchronization of coupled neural networks is to find a way to achieve state consistency for all nodes in the corresponding networks. The significance of synchronization is to study and make full use of its positive role to avoid negative effects. Synchronization of coupled neural networks has a broad research background, involving electronic communication, automatic control and social science, and its theoretical research and practical applications are of profound significance to the exploration of nature.

In the actual control system, time delay may be produced due to some factors such as system measurement, sampling process, speed limit of information transmission, energy conversion process and aging or wear of equipment component (Chen, Wu, Cao, and Liu, 2015, Chen et al., 2004, Hu et al., 2015, Huang et al., 2014). Time delay can often affect the performance of system to a certain extent, which may cause system state to have an oscillation or instability (Hu et al., 2010, Huang et al., 2018, Huang, Zhang, Cao, and Hu, 2019, Zhang et al., 2010). Time delay is an uncontrollable factor, and thus time delay cannot be reduced or eliminated by the input of its own control. Due to the important applications of delayed neural networks in the fields of pattern recognition, associative memory and combinatorial optimization, the dynamics of delayed neural networks (Chen, Wu, Cao, and Liu, 2015, Chen et al., 2004, Hu et al., 2015) have attracted wide attention in recent years. Chen et al. (2004) obtained global synchronization of coupled delayed neural networks. Additionally, synchronization of coupled inertial delayed neural networks was studied in Hu et al. (2015). Moreover, delayed neural networks were extended to fractional-order, and stability and synchronization based on memristor were studied in Chen, Wu, Cao, and Liu (2015).

At present, many control methods are applied to synchronization of delayed neural networks, such as impulsive control (Hu et al., 2010, Zhang et al., 2010), intermittent control (Zhang, Li, Huang, & Xiao, 2015), adaptive control (Wang & Wu, 2014), sliding mode control (Gan, Xu, & Yang, 2012), sampled-data control (Liu et al., 2019, Xiao et al., 2018), event-triggered control (Guo, Gong, Wen, & Huang, 2019) and so on. Impulsive control refers to the amount of impulsive control injected a certain intensity at the discrete point in time, which makes the state value of the system change. There are many practical systems in nature that require impulsive control, and continuous control methods cannot be applied, such as the adjustment of bank interest rates. Impulsive control can make system stable or change stability state of system, and has strong robustness. Compared with continuous control, the response speed of impulsive control is fast, and system has good convergence performance. In the application of encrypted communication based on synchronization, if impulsive control strategy is adopted, the response system could realize synchronization only by obtaining the communication information of the drive system at impulsive time, which greatly reduces the possibility of information being deciphered and improves the security. In recent years, impulsive control (Hu et al., 2010, Lv et al., 2018, Zhang et al., 2014, Zhu et al., 2018) has developed rapidly and has become one of the hot spots in the field of engineering control. Zhang et al. (2014) got some conditions for stochastic dynamical networks with time delay to ensure synchronization under impulsive control. Zhu et al. (2018) gave conditions to ensure stability for the continuous-time dynamic systems under event-based impulsive control and applied them to obtain synchronization conditions of memristive neural networks. In actual control systems, the output signal of the controller is generally transmitted to the controlled system through actuator, and then the controlled object is driven to complete the established task. Due to physical limitations, the size of the control signal that the actuator can transmit is within a specific range. When the control signal is too large, the output signal from the actuator to the controlled system will be distorted. Actuator saturation often results in a sharp decrease in the performance of the closed-loop system. If the saturation limit is not taken into account in the design of the control system, it will make the system performance worse, such as causing lag, increasing overshoot, increasing the oscillation, and even will lead to instability. Therefore, it is of not only theoretical value but also practical significance to design an effective controller for a control system with actuator saturation. Recently, many scholars have made some achievements in the analysis and design of control systems with actuator saturation (Chen, Tao, and Jiang, 2015, Ding et al., 2018, Selvaraj et al., 2018, Wang et al., 2019, Zeng et al., 2017). By designing dynamic surface control, Chen et al. obtained stability condition of uncertain nonlinear system with input saturation in Chen, Tao, and Jiang (2015). Selvaraj et al. (2018) and Zeng et al. (2017) discussed synchronization of chaotic neural networks with actuator saturation and stochastic coupled neural networks with input saturation, respectively. Ding et al. (2018) discussed event-triggered stabilization of neural networks with input saturation. Furthermore, Wang et al. (2019) expanded to discrete-time switched memristive neural networks and analyzed exponential dissipativity. As can be seen from the previous discussion, the research study of dynamic systems with actuator saturation under impulsive control is very few at present. Because of the complexity of the problems, it is difficult to study, and it still belongs to the open problem in control field. Impulsive control is affected by saturation nonlinearity, and therefore, the original analysis method of dynamic systems under impulsive control cannot be directly applied to dynamic systems with actuator saturation under impulsive control. At the same time, due to the existence of impulses, the existing saturation control theory for continuous systems cannot be directly applied to dynamic systems with actuator saturation under impulsive control. Therefore, the research on dynamic systems with actuator saturation via impulsive control is an innovative and challenging work.

A preliminary version of this paper appeared in Ouyang, Huang, Li, Chen, and Li (2018) as a poster paper, in which the synchronization of coupled neural networks with actuator saturation via impulsive control was studied. In this paper, in order to ensure synchronization of coupled delayed neural networks, two kinds of impulsive control strategies are studied by using sector nonlinearity model approach. Based on such approach, impulsive controls with actuator saturation and with partial actuator saturation are studied respectively, and some effective sufficient conditions are obtained. In addition, we build an image encryption system using synchronization conditions that have been identified. Theoretical analysis and experimental results verify that the new encryption strategy has high security, so it can be applied to the actual image encryption application. Compared with the existing works, this paper mainly has the following contributions.

• Compared with the existing results for coupled neural networks under impulsive control (Lv et al., 2018, Tang et al., 2018, Zhang and Sun, 2009), this paper considers impulsive synchronization of coupled delayed neural networks with actuator saturation. In impulsive control, if the effect of state saturation is ignored, it may make the controller fail to achieve the desired results in real life.

• Current methods of processing impulsive system are comparison approach (Liu, 2008), convex combination (Chen, Lu, & Zheng, 2015), impulse time window (Wang, Li, Huang, & Pan, 2014), B-equivalence method (Li, Zhou, Wang, & Huang, 2017) and so on, but these traditional methods cannot be directly applied to impulsive synchronization of coupled delayed neural networks with actuator saturation. According to sector nonlinearity model approach, impulsive synchronization of coupled delayed neural networks with actuator saturation is studied.

• Based on impulsive control to realize synchronization, the response system only needs to obtain the information of impulsive time of the drive system, which greatly improves the confidentiality of the information. The works of Lakshmanan et al., 2020, Prakash et al., 2016, Sheng et al., 2018 and Wen et al. (2015), use synchronization for image encryption, but they are not based on impulsive synchronization. Compared with the previous work (Chen, Luo, & Zheng, 2016), although image encryption based on impulsive synchronization is used, the designed image encryption algorithm cannot effectively resist differential attack. Based on the synchronization theory of this paper, a new image encryption algorithm based on substitution–diffusion structure is proposed, which can effectively resist all kinds of known attacks, such as exhaustive attack, statistical attack and differential attack.

The rest of this paper is organized as follows. In Section 2, a class of coupled delayed neural networks and some useful lemmas are given. In Section 3, actuator saturation is considered in impulsive controller, and some theorems and corollaries are established using sector nonlinearity model approach. In Section 4, an example to show the validity of the theoretical analysis results are given. Based on synchronization theorem of the paper, an image encryption system based on impulsive synchronization is proposed in Section 5, and a detailed encryption analysis is carried out. In Section 6, the conclusion and future prospects are presented.

Notations

Let $\mathbf{M}=\{1,2,\dots ,n\}$, $\mathbf{N}=\{1,2,\dots ,N\}$, $\aleph =\{1,2,\dots ,2^n\}$, $\mathbf{R}$, ${\mathbf{R}}_{+}$ and ${\mathbf{N}}_{+}$ denote the sets of real numbers, nonnegative real numbers and positive integer respectively. Moreover, ${\mathbf{R}}^n$ denotes the $n$-dimensional real column vector space endowed with the norm $\Vert \cdot \Vert$. For matrix $\mathcal{D}={\left(d_{ıȷ}\right)}_{n\times n}\in {\mathbf{R}}^{n\times n}$, ${\mathcal{D}}^{\mathbf{T}}$ and ${\mathcal{D}}^{-1}$ represent the transpose of and the inverse of matrix $\mathcal{D}$, and $\mathcal{D}>0$ means positive definite matrix. $\text{co}\left\{s\right\}$ represents the convex hull defined by the vertices $s$. $I_n$ is an identity matrix of $n\times n$ dimensions. Let $\pounds =\{{\pounds }_{ı}:ı\in \aleph \}$ be the set of $n\times n$ diagonal matrices whose diagonal element takes the value 1 or 0. ${\pounds }_{ı}^{-}=I_n-{\pounds }_i$ ($ı\in \aleph$).