Adaptive control for fractional order induced chaotic fuzzy cellular neural networks and its application to image encryption

Abstract

The main concern of this paper is to address the synchronization problem of chaotic fractional-order fuzzy cellular neural networks (FOFCNNs) through designing the novel adaptive control scheme. The objective of the study is to explore the importance of considering fractional order derivatives (FODs) and time-varying delays. Even though numerous works have been reported in the literature regarding the derivation of sufficient conditions, there has been a lack of research on involving the dynamical analysis of FOFCNNs. Hence, this study focuses on the dynamical analysis of FOFCNNs. Particularly, both asymptotical and exponential synchronization of drive-response FOFCNN model is guaranteed via sufficient conditions that are derived by constructing the fractional Lyapunov functional candidate and solvable linear matrix inequalities (LMIs). Besides that, numerical simulations are performed to reveal the significance of the FODs. Also, an image encryption algorithm is designed based on the chaotic FOFCNNs solutions that result in better security measures. In summary, the overall contribution of the study is categorized into two: (1) sufficient conditions which ensure the global asymptotic and exponential stability are derived in a novel manner; (2) an image encryption algorithm is proposed by considering the FOFCNN as pseudo-random number generator (PRNG), which outperforms the existing encryption algorithms.

Introduction

Since the cellular neural networks (CNNs) have been proposed by the authors in [1], numerous research results have been published in the literature regarding the investigation of dynamical properties of CNNs that include stability, bifurcation, synchronization, and chaos analyses, [2], [3]. As a development, fuzzy cellular neural networks (FCNNs) have been introduced instead of CNNs in order to handle the vagueness and uncertainties while implementing the neural networks through the differential model, see [4], [5], [6]. FCNN is more popular than CNN because of its compatibility with various image processing applications such as edge detection and pattern recognition. Especially, in this paper, our analysis focuses on both theoretical controller design and practical implementation via image encryption/decryption algorithm. Further, time delays play a significant role in the implementation of neural networks due to its finite switching speed in the amplifier and have an ability to affect the stability of the dynamical systems. Till date, the investigations on qualitative issues of FCNNs by involving the time delays have not been fully dealt with, which is the main objective of the this study, [7], [8], [9].

Fractional order systems (FOSs) have received much attention among the researchers due to its infinite memory and heredity properties. The solid applications of FOSs include electronic circuits, engineering applications, and mechanics. In this regard, significant research reports have been reported for both integer and fractional order neural networks (FONNs) based on Laplace transformation method and linear stability theory. However, There is a limit to applying those methods to investigate nonlinear FONNs (see, [3], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19]). In order to relax these limitations, a fractional order Lyapunov direct method has been employed for analyzing the global stability via Mittag-Leffler sense [20]. In detail, authors in [20] derived the stability conditions by considering one-norm and the $\infty -$norm in the Lyapunov function. However, those results are applicable only to the matrix elements and not compatible with a complete matrix. With this point, two-norm Lyapunov function for FONNs was proposed to provide better results via linear matrix inequality (LMI) conditions [21]. On the other hand, authors in [22] formulated the differential system for CNNs and proved the dynamical characteristics of cell structure with a non-integer order, say, two-cell fractional order cellular neural network (FOCNN) which exhibits various chaotic behaviors with respect to fractional order. Authors in [23] studied the global stability of FONNs based on strict LMI conditions by employing two-norm in the Lyapunov function. Also, authors in [3] analyzed the finite-time stability and synchronization problem of memristor-based FOFCNN through providing existence and uniqueness of the Filippov solution by utilizing Banach fixed point theorem.

In general, the different types of synchronization criteria have been proposed for FONNs that include complete synchronization, lag synchronization, projective synchronization, and generalized projective synchronization, for more details, refer, [9], [24]). In order to attain the synchronous criteria, various types of controller schemes have been designed in the literature such as feedback controller, sliding mode controller, adaptive controller, and so on. Among those control schemes, the adaptive control scheme is considered to be more effective others as it has an ability to deal with the systems having unpredictable parameter deviations and uncertainties. Hence, in this study, the adaptive control scheme is employed which guarantees the synchronization of drive-response FOFCNNs model. For more information on the investigation of synchronization issues, we have listed some of the recent works [25], [26], [27] and references cited therein. Recently, authors in [28] have proposed the synchronization criteria for FOFCNNs with different coupling strengths and proposed the adaptive controller based on Lyapunov–Krasovskii functional (LKF) candidate, in that work, authors have derived the synchronous criterion for two fractional fuzzy coupled networks based on the traditional Lyapunov function without any integral terms. Also, authors in [29] have investigated the $D-$ stability and stabilization for FOSs via LMIs and claimed that the fractional systems can be analyzed in a similar manner as integer order systems. Besides that, recently, in [30], admissibility and robust stabilization problems of continuous-time linear singular FOSs were addressed, and a set of three different necessary and sufficient conditions were derived in terms of strict LMIs. Therefore, according to these works, it is clear that one can design the FOFCNNs and investigate its properties in a similar way as the traditional integer order FCNNs through Lyapunov stability theory and LMI approach.

Despite, this paper designs an image encryption algorithm by utilizing the FOFCNN model as PRNG, which proves the direct application of the proposed criteria. However, in order to utilize those FCNN models, it has to satisfy the following properties: the solutions must have a chaotic behavior in nature, and at the same time, the proposed controllers must be activated within the period of time. Based on these properties, we design an image encryption algorithm with the proposed FOFCNNs as crypto-system. We claim that the designed image encryption algorithm based on the synchronization of chaotic systems will ensure better security over transmissions of confidential information through the collection of pixels. Indeed, existing literatures contain numerous algorithms which were designed through utilizing PRNG sequence obtained from chaotic solutions of differential equations, but, the studies that combine the theoretical sufficient conditions and practical evaluations have been very limited. Here, we have listed a few works that discuss the LMI conditions and the image encryption algorithm [31], [32], [33], [34], [35], [36] and the common feature shared by these works is integer differential order in the neural network models. Comprehensively, authors in [34] designed an image encryption algorithm with chaotic FCNNs, and proved the effectiveness of their algorithm through security analyzes. In [35], authors investigated the synchronization problem of chaotic memristive multidirectional associative memory neural networks model based on Lyapunov stability theory, and illustrated their application via an image encryption/decryption process. Nonetheless, there have been no results reported regarding the investigation of synchronization issues of delayed FOFCNNs model based on Lyapunov stability theory with an application to image encryption.

Motivated by the aforementioned works, the overall contributions of the paper is listed below.

• The adaptive controller is designed with an updated law under the fractional domain which is distinct to the traditional control schemes.

• The sufficient conditions that ensure the global asymptotical and exponential stability of the proposed FOFCNNs are derived by constructing the novel LKF, and the conditions proposed are expressed in terms of solvable LMIs. In addition, if fractional order is chosen as $\alpha =1$ then the sufficient conditions are applicable to the integer order FCNNs.

• The numerical bifurcation analysis is performed, which reveals the effect of time-varying delays in FOFCNNs.

• An image encryption algorithm is designed by involving the proposed FOFCNNs as crypto-system, in which the pixels are scrambled with unpredictable solutions which provide high-level security on image transactions.

Notations 1

The notations used in the whole mathematical derivations are summarized as follows: ${\mathbb{R}}^n$ represents the domain as Euclidean n-dimensional space and ${\mathbb{R}}^{n\times n}$ denotes the set of all n × n real matrices. T to a power denotes the transposition operator. X ≥ Y (X > Y), denote symmetric matrices, with $X-Y$ is a positive semi-definite (positive definite), respectively. diag{⋅⋅⋅} denotes a block diagonal matrix. Notation * stands for the symmetric block in a symmetric matrix. ‖ · ‖ represents the Euclidean norm in ${\mathbb{R}}^n$.