Modified projective synchronization of distributive fractional order complex dynamic networks with model uncertainty via adaptive control

Abstract

Adaptive modified projective function synchronization of non-linear distributive fractional order complex dynamical networks (DFCDN) is addressed in this paper. Firstly, we have created a model for DFCDN with model uncertainty, external disturbances and uncertain parameters. Based on the Laplace and inverse Laplace transform property of distributive order fractional differential equations and Lyapunov stability theory, we realize the projective synchronization between the DFCDN system, according to the given scaling function by using the novel adaptive controller that we have constructed. Finally two numerical examples and simulations are given to show our proposed work is more realistic and more effective than the existing ones in the literature.

Introduction

More physical implications were successfully modelled using fractional order operators in some experiments, and more recently scientists have been focusing on fractional order dynamic systems and complex networks. This is because, fractional calculus enables the design of various processes with a high precision of long memory relative to classic calculus models. This benefit also extends to complex, distributed-order structures, modelled structures that occur in various natural phenomena. Caputo proposed the first notion of the distributed order differential equation in 1969 and implemented it in 1995. Since then, various applications of the distributed order calculus have appeared, including [1], [2], [3], [4], [5]. This differential distribution order incorporating di-electrical equations, diffusion equations, composite material rheological features and multidimensional random walking models were mainly developed. Distribution systems are actually known as generalizations of fractional order systems that use distributor systems to model physical mechanism behaviour. Some physical explanations for using differential equations in distributed order are already exists, for instance, in the literature, the distribution order fractional kinetics was discussed in [6]. The ultra-slow and lateral diffusion processes were studied in [7]. It is important to remember that integrator and fractional order systems are the special cases of distribution systems. Especially, distributed order operators become a more effective method for understanding and illustrating such particular physical manifestations, including network dynamics, non-linear, non-homogeneous, multi-scale and multi-spectral, etc.

Complex dynamical networks are now an interesting field of research for various disciplines such as mathematics, biology, and sociology because of their potential real-life applications in neural networks, social networks, ecological networks, communications networks, information technology, and smart power grids etc. These networks involve multiple nodes which communicate with each other on certain links. Synchronization is the most significant and critical of collaborative tasks in complex dynamical networks. Various types of synchronization namely, Mittag-Leffler synchronization [8], [9], projective synchronization [10], modified function projective synchronization [11], projective lag synchronization [12], [13], stochastic synchronization [14], finite time synchronization [15], [16], [17], [18], [19], adaptive impulsive synchronization [20], etc are developed as the most important dynamic behavior and many useful and significant results were obtained. The study of complex systems with distributed order is seen as a new field of research. However, recent significant works have shown that the concept of the distributed order of both simulation and tracking of real-world processes is prospective. Distributed order fractional calculus has drawn interest in recent years among researchers since it enables the creation of more suitable models for some physical and dynamical systems.

Recently, a novel synchronization about MFPS was discovered and deeply explored by many of the researchers around the world. The MFPS is a more general definition, for that many synchronization involving projective synchronization, lag synchronization, projective lag synchronization and so on can be regard as the special cases of MFPS when the so-called scaling function matrix takes different forms. As far as we know, there are no findings on problems of modified projective synchronization analysis in distributed dynamical systems in complex networks. In [21], the authors studied the projective synchronization of fractional order complex dynamical networks with unknown external disturbances and unknown parameters. The modified projective synchronization of CDN with model uncertainty with time varying delays by using adaptive control was studied by the researchers in [22]. Also the robust projective lag synchronization of complex networks with different dimensional nodes was developed in [23]. In [24], the researchers investigated a problem on modified lag projective synchronization criteria for complex networks with uncertain coupling strength. In [25], the distributive filtering of memristor networks with switching topology was discussed. The authors in [26] investigated the adaptive synchronization of memristor based networks with uncertain parameters. Most of the existing results of modified projective function synchronization of CDN considered in integer order dynamical systems. Only a few literature are available in fractional order although very less while considering the environmental disturbances and other impact factors in real world phenomena. But some of the results regarding the distributed order is reported, for instance, in [27], the stability analysis of nonlinear distributive order systems and in [28], the condition for stability of linear time invariant distributive order dynamic systems were investigated. As very recently, the asymptotic stability criteria for nonlinear distributive fractional order dynamical systems was investigated in [29]. However, our considered system in this manuscript are more general than the compared above references. Up to now, there has been no results in the literature that investigates the synchronization analysis for distributed order fractional complex dynamical networks.

As we know, in the real world systems, external disturbances and uncertainty are always exists, which cannot be ignored. In engineering it is impossible, for certain unpredictable variables, such as fluctuations in the atmosphere and data accidents, to acquire exact values of designed parameters. In this case, the consideration should be made with unknown parameters [30]. Moreover, the control structures play a vital part in our day to day life. The basic purpose of control is to develop the principles and techniques for planning engineering processes that naturally respond to ecological changes to protect attractive efficiency. It should be remembered that some networks cannot be explicitly coordinated. The powerful control schemes, such as sample data control [31], [32], impulsive control [24], [33], [34], [35], pinning control [36], adaptive control [21], hybrid control [19], [37], [38], [39], or bifurcation control [40], sliding mode control [41] have been programmed to synchronise complex network systems. Among these, it is well known that the advantage of adaptive control is that the control parameters can adjust themselves according to some suitable updating laws, which are designed under control purpose according to the characteristics of the considered system. It involves a variety of techniques that provide a methodical solution for ongoing self-adjustment of controllers.

Motivated by the above-mentioned studies, our main contribution of the paper are listed as follows:

• Firstly, the adaptive synchronization of nonlinear distributive fractional order complex dynamic networks with model uncertainty and couplings uncertain parameters with external disturbances is investigated for the first time.

• Considering the model uncertainty, in distributive order case is a new idea in the literature, which gives the model more realistic and complicated.

• The uncertain parameter with external disturbances are also considered along with the distributed order case, which has not been taken in any of the previous literature in CDN.

• Also, we investigated the couplings uncertainty in complex networks and in the distributive fractional order cases, which becomes more general realistic model when compared to any of the available sources in the literature in complex dynamical networks.

• By using the property of Laplace and inverse Laplace transform of distributive order, the Lyapunov stability theory and some matrix inequality theory are flexibly used in proving our main theoretical research to obtain the projective synchronization criterion easily.

• Finally, we present two numerical examples with their simulations for our new proposed model in order to demonstrate the efficiency of our main proofs.

This paper is further relaxed in the following way. The preliminaries, definition, lemmas and problem formulation are presented in Section 2 for the considered distributive fractional order modified projective function synchronization of complex dynamical networks (DFMPSCDNs). The analysis of projective synchronization of distributive fractional order complex networks with model uncertainty was examined in section 3.1. The DFCDN with uncertain inner couplings was built and its synchronization criteria was analyzed in section 3.2. Section 4 reveals the two examples with numerical simulations to demonstrate the performance regarding our proposed model. At the end, we have given conclusion in Section 5.

Notations: $D_t^{c\left(\beta \right)}$ represents the distributive order, where $c\left(\beta \right)>0$. $\otimes$ denotes the kronecker product, $\mathbb{R}$ be the set of real numbers. ${\mathbb{R}}^{n\times m}$ be the real matrices of the $n\times m$. ${\mathbb{R}}^n$ be the $n$ dimensional Euclidean space to the vector norm $\parallel ·\parallel$. I be the Identity matrix of the suitable dimension. $\Theta$ be the symmetric positive definite matrices of appropriate dimension. ${\psi }^T$ denotes the transpose of $\psi$. ${\lambda }_{\max }$ and ${\lambda }_{\min }$ be the largest and smallest eigen value of the corresponding matrix respectively.