Fractional-order Discrete-time Systems Research Articles

A three-dimensional discrete fractional-order HIV-1 model related to cancer cells, dynamical analysis and chaos control

In this paper, we study a three-dimensional discrete-time model to describe the behavior of cancer cells in the presence of healthy cells and HIV-infected cells. Based on the Caputo-like difference operator, we construct the fractional-order biological system. This study's significance lies in developing a new approach to presenting a biological dynamical system. Since the qualitative analysis related to existence, uniqueness, and stability is almost the same as can be found in numerous existing papers, and comparing this study to other research, constructing a biological discrete system using the Caputo difference operator can be particularly important. Using powerful tools of nonlinear theory such as phase plots, bifurcation diagrams, Lyapunov exponent spectrum, and the 0-1 test, we establish that the proposed system can exhibit different biological states, including stable, periodic, and chaotic behaviors. Here, the route leading to chaos is period-doubling bifurcation. Furthermore, the level of chaos in the system is quantified using $C_{0}$ complexity and approximate entropy algorithms. The stabilization or suppression of chaotic motions in the fractional-order system is presented, where an efficient controller is designed based on the stability theory of the discrete-time fractional-order systems. Numerical simulations are provided to validate the theoretical results derived in this research paper.

Global exponential convergence and synchronization for exponential numerical competitive neural networks with different time scales and fuzzy logic

For nonlocal fuzzy delayed competitive neural networks, a novel difference model is first developed in this article using the Mittag-Leffler Euler difference method. Secondly, the existence of a sole globally bounded solution and globally exponential stability to the discrete-time networks are addressed based on Banach contractive mapping principle and the constant variation formula for sequences. Besides, exponential synchronization of the discrete-time networks is investigated and a feedback controller is designed from the viewpoint of the theory of controls. It is the first time to consider nonlocal fuzzy delayed competitive neural networks by using Mittag-Leffler Euler differences. In contrast to nonlocal discrete approaches, such difference competitive neural networks could preferably figure the features of continuous networks. The difference method in this paper extends and improves the traditional difference approaches, for example, Adams-Bashforth Euler differences. Moreover, this article explores some avenue for the research of discrete-time fractional-order systems and develops a set of theories and methodologies for the future investigations.

A Finite-Dimensional Control Scheme for Fractional-Order Systems under Denial-of-Service Attacks

In this article, the security control problem of discrete-time fractional-order networked systems under denial-of-service (DoS) attacks is considered. A practically applicable finite-dimensional control strategy will be developed for fractional-order systems that possess nonlocal characteristics. By employing the Lyapunov method, it is theoretically proved that under the proposed controller, the obtained closed-loop fractional system is globally input-to-state stable (ISS), even in the presence of DoS attacks. Finally, the effectiveness of the designed control method is demonstrated by the numerical example.

On the combined estimation of the parameters and the states of fractional-order systems

In this article, we propose an approach for system identification for a class of discrete-time fractional-order Hammerstein systems using only input–output data. Using a combined state and parameter estimation approach, we develop an algorithm serving to estimate simultaneously, the system parameters, the system orders, and the system states. By minimizing the defining criterion, which is non-convex and nonlinear in the parameters, the model parameters are estimated using the recursive least squares and the Levenberg–Marquardt algorithms. Next, the system states are estimated using the Luenberger observer. Then, the convergence analysis of the proposed algorithm is proved. Finally, the Monte Carlo simulation analysis and an application to Ultracapacitor system are used to demonstrate the effectiveness of the suggested method.

Discretization of Fractional Order Operator in Delta Domain

The fractional order operator is the backbone of the fractional order system (FOS). The fractional order operator (FOO) is generally represented as s^(±μ) (0<μ<1). Discrete time FOS can be obtained through the discretization of the fractional order operator. The FOO is the general form of either fractional order differentiator (FOD) or integrator (FOI) depending upon the values of μ. Out of the two discretization methods, direct discretization outperforms the method of indirect discretization. The mapping between the continuous time and discrete time domain is done with the development of generating function. Continuous fraction expansion (CFE) is used expand the generating function for the rational approximation of the FOO. There is an inherent problem associated with the discretization of FOO in discrete z-domain particularly at very fast sampling rate. In the other hand, discretization using delta operator parameterization provides the continuous time and discrete time results in hand to hand, when the continuous time systems are sampled at very fast sampling rate and circumventing the problem with shift operator parameterization at fast sampling rate. In this work, a new generating function is proposed to discretize the FOO using the Gauss-Legendre 3-point quadrature rule and generating function is expanded using the CFE to form rational approximation of the FOO in delta domain. The benchmark fractional order systems are considered in this work for the simulation purpose and comparison of results are made to prove the efficacy of the proposed method using MATLAB.

Equilibrium point, exponential stability and synchronization of numerical fractional-order shunting inhibitory cellular neural networks with piecewise feature

In recent years, exponential Euler difference methods for first-order semi-linear differential equations have been developed rapidly, and various exponential Euler difference methods have become a very important research topic. For fractional-order shunting inhibitory cellular neural networks with time lags, this article first establishes a new difference method named Mittag-Leffler Euler difference, which includes exponential Euler difference. Second, the existence of unique global bounded solution and equilibrium point, exponential stability and synchronization of the derived difference model are investigated. Compared with the classical fractional-order Euler difference method, the Mittag-Leffler discrete shunting inhibitory cellular neural networks can better describe and maintain the dynamic properties of the corresponding continuous-time models. More importantly, it opens up a new way to study discrete-time fractional-order systems and establishes a set of theory and methods to study Mittag-Leffler discrete neural networks.

Inverse-Optimal Consensus Control of Fractional-Order Multiagent Systems

In this article, the problems of inverse-optimal consensus control (IOCC) for single-integral dynamics in Caputo continuous-time and Grunwald–Letnikov (G-L) discrete-time fractional-order multiagent systems (FOMASs) are investigated. For continuous-time FOMASs, by virtue of the variational method, the optimal solution under the constraint of the global performance index function is presented, and then the optimal state-feedback gain matrix (OSFGM) is derived through the Laplace transform and the inverse transform of fractional order. For discrete-time FOMASs, the OSFGM based on the algebraic Riccati equation (ARE) is computed, then it is shown that the OSFGM is a (asymmetric) Laplace matrix which has a zero eigenvalue and the corresponding network structure is a complete graph. Moreover, we rigorously prove that consensus can be achieved under the optimal state-feedback gain matrices. Finally, two simulation examples are shown to check the validity of the proposed theorems.

Stability analysis and synchronized control of fuzzy Mittag-Leffler discrete-time genetic regulatory networks with time delays

Exponential Euler differences for semi-linear differential equations of first order have got rapid development in the past few years and a variety of exponential Euler difference methods have become very significant researching topics. In allusion to fuzzy genetic regulatory networks of fractional order, this paper firstly establishes a novel difference method called Mittag-Leffler Euler difference, which includes the exponential Euler difference. In the second place, the existence of a unique global bounded solution and equilibrium point, global exponential stability and synchronization of the derived difference models are investigated. Compared with the classical fractional Euler differences, fuzzy Mittag-Leffler discrete-time genetic regulatory networks can better depict and retain the dynamic characteristics of the corresponding continuous-time models. What’s more important is that it starts a new avenue for studying discrete-time fractional-order systems and a set of theories and methods is constructed in studying Mittag-Leffler discrete models.

Infinite line of equilibriums in a novel fractional map with coexisting infinitely many attractors and initial offset boosting

Abstract The study of the chaotic dynamics in fractional-order discrete-time systems has received great attention in the past years. In this paper, we propose a new 2D fractional map with the simplest algebraic structure reported to date and with an infinite line of equilibrium. The conceived map possesses an interesting property not explored in literature so far, i.e., it is characterized, for various fractional-order values, by the coexistence of various kinds of periodic, chaotic and hyper-chaotic attractors. Bifurcation diagrams, computation of the maximum Lyapunov exponents, phase plots and 0–1 test are reported, with the aim to analyse the dynamics of the 2D fractional map as well as to highlight the coexistence of initial-boosting chaotic and hyperchaotic attractors in commensurate and incommensurate order. Results show that the 2D fractional map has an infinite number of coexistence symmetrical chaotic and hyper-chaotic attractors. Finally, the complexity of the fractional-order map is investigated in detail via approximate entropy.

On chaos and projective synchronization of a fractional difference map with no equilibria using a fuzzy-based state feedback control

In this paper, by considering the Caputo-like delta difference definition, a fractional difference order map with chaotic dynamics and with no equilibria is proposed. The complex dynamical behaviors associated with fractional difference order maps are analyzed employing the phase portraits, bifurcations diagrams, and Lyapunov exponents. The complexity of the sequence generated by the chaotic difference map is studied using the permutation entropy approach. Afterwards, projective synchronization of the systems is investigated. Fuzzy logic engines as intelligent schemes are strong tools for control of various systems. However, studies that apply fuzzy logic engines for control of fractional-order discrete-time systems are rare. Hence, in the current study, by taking advantages of fuzzy systems, a new controller is proposed for the fractional-order discrete-time map. The fuzzy logic engine is implemented in order to enhance the performance and agility of the proposed control technique. The stability of the closed-loop systems and asymptotic convergence of the projective synchronization error based on the proposed control scheme are proven. Finally, numerical simulations which clearly confirm that the offered control technique is able to push the states of the fractional-order discrete-time system to the desired value in a short period of time are presented.

LQ optimal control of fractional-order discrete-time uncertain systems

This paper primarily focuses on LQ optimal control problem of fractional-order discrete-time systems based on uncertainty theory that is a significant tool to model belief degree. First, an equivalent LQ problem, which is integer-order one, is presented by expanding the state variables with the unchanging control variables. This equivalent problem is then solved by dynamic programming approach. Accordingly, the solution of the proposed LQ optimal control problem reduces to solve a system of backward matrix difference equations. Finally, a numerical example is provided to show how to solve the proposed LQ optimal control problem. As an application, an LQ optimal control problem of macroeconomic system is discussed by the achieved results.

LMI-Based Robust Stability Analysis of Discrete-Time Fractional-Order Systems With Interval Uncertainties

Robust stability problem of discrete-time fractional-order systems (DTFOSs) with interval uncertainties is investigated in this paper. Firstly, a new theorem for matrix root-clustering in union-region is established. Based on this theorem, the stability regions of DTFOSs are described as the union-region of closed sub-regions, and sufficient conditions for stability of DTFOSs are presented. Then, new sufficient conditions for robust stability of DTFOSs with interval uncertainties are derived. All the results are obtained in terms of linear matrix inequalities (LMIs) which are more tractable than the existing ones. Finally, numerical examples are given to show that our results are valid and less conservative than the existing ones.

An Unprecedented 2-Dimensional Discrete-Time Fractional-Order System and Its Hidden Chaotic Attractors

Some endeavors have been recently dedicated to explore the dynamic properties of the fractional-order discrete-time chaotic systems. To date, attention has been mainly focused on fractional-order discrete-time systems with “self-excited attractors.” This paper makes a contribution to the topic of fractional-order discrete-time systems with “hidden attractors” by presenting a new 2-dimensional discrete-time system without equilibrium points. The conceived system possesses an interesting property not explored in the literature so far, i.e., it is characterized, for various fractional-order values, by the coexistence of various kinds of chaotic attractors. Bifurcation diagrams, computation of the largest Lyapunov exponents, phase plots, and the 0-1 test method are reported, with the aim to analyze the dynamics of the system, as well as to highlight the coexistence of chaotic attractors. Finally, an entropy algorithm is used to measure the complexity of the proposed system.

The effect of caputo fractional difference operator on a novel game theory model

It is well-known that fractional-order discrete-time systems have a major advantage over their integer-order counterparts, because they can better describe the memory characteristics and the historical dependence of the underlying physical phenomenon. This paper presents a novel fractional-order triopoly game with bounded rationality, where three firms producing differentiated products compete over a common market. The proposed game theory model consists of three fractional-order difference equations and is characterized by eight equilibria, including the Nash fixed point. When suitable values for the fractional order are considered, the stability of the Nash equilibrium is lost via a Neimark-Sacker bifurcation or via a flip bifurcation. As a consequence, a number of chaotic attractors appear in the system dynamics, indicating that the behaviour of the economic model becomes unpredictable, independently of the actions of the considered firm. The presence of chaos is confirmed via both the computation of the maximum Lyapunov exponent and the 0-1 test. Finally, an entropy algorithm is used to measure the complexity of the proposed game theory model.

The discrete fractional duffing system: Chaos, 0-1 test, C0 complexity, entropy, and control.

In this paper, we study the dynamics and control of a Caputo fractional difference form of the Duffing map. We use phase plots, bifurcation diagrams, and Lyapunov exponents to establish the existence of chaos over a wide range of fractional orders and examine the nature of the dynamics. Also, we present the 0-1 test to detect chaos and C0 complexity, which is an alternative nonlinear statistical measure that can quantify the regularity of a time series. In addition, we measure the approximate entropy to see the performance of our numerical results. Through phase plots and bifurcation diagrams, it is shown that the proposed fractional map exhibits a range of different dynamical behaviors including chaos and coexisting attractors. A one-dimensional feedback stabilization controller is proposed. The asymptotic convergence of the proposed controller is established by means of the stability theory of linear fractional order discrete-time systems. Simulation results have been carried out to illustrate the findings of the study.

Leader-following group consensus of discrete-time fractional-order double-integrator multi-agent systems

The current study concerns with the leader-following group consensus issue for discrete-time fractional-order multi-agent systems. The agents are assumed as fractional-order double-integrators. The interconnection among the agents is characterized by a bidirectional fixed graph. By considering some assumptions on the interaction graph, it is proved that leader-following group consensus will be realized if the controller parameters lie in a specific two-dimensional region. In the beginning, the proposed distributed control structure’s ability is evaluated for two groups with two leaders. Then, it is extended to an arbitrary number of groups with an arbitrary number of leaders. Simulation results indicate the efficiency of this distributed control structure.

Chaos and control of a three-dimensional fractional order discrete-time system with no equilibrium and its synchronization

Chaotic systems with no equilibrium are a very important topic in nonlinear dynamics. In this paper, a new fractional order discrete-time system with no equilibrium is proposed, and the complex dynamical behaviors of such a system are discussed numerically by means of a bifurcation diagram, the largest Lyapunov exponents, a phase portrait, and a 0–1 test. In addition, a one-dimensional controller is proposed. The asymptotic convergence of the proposed controller is established by means of the stability theory of linear fractional order discrete-time systems. Next, a synchronization control scheme for two different fractional order discrete-time systems with hidden attractors is reported, where the master system is a two-dimensional system that has been reported in the literature. Numerical results are presented to confirm the results.

Finite-dimensional control of linear discrete-time fractional-order systems

This paper addresses the design of finite-dimensional feedback control laws for linear discrete-time fractional-order systems with additive state disturbance. A set of sufficient conditions are provided to guarantee convergence of the state trajectories to an ultimate bound around the origin with size increasing with the magnitude of the disturbances. Performing a suitable change of coordinates, the latter result can be used to design a controller that is able to track reference trajectories that are solutions of the unperturbed fractional-order system. To overcome the challenges associated with the generation of such solutions, we address the practical case where the references to be tracked are generated as a solution of a specific finite-dimensional approximation of the original fractional-order system. In this case, the tracking error trajectory is driven to an asymptotic bound that is increasing with the magnitude of the disturbances and decreases with the increment in the accuracy of the approximation. The proposed controllers are finite-dimensional, in the sense that the computation of the control input only requires a finite number of previous state and input vectors of the system. Numerical simulations illustrate the proposed design methods in different scenarios.

Effect of initialization on optimal control problem of fractional order discrete-time system

Initialization effect on optimal control problem (OCP) of fractional order discrete-time system (FODS) is considered here. Formulation and numerical solution of OCP of initialized FODS is presented. Performance index (PI) considered is of general form and the system dynamics are given by fractional order differential equations (FDEs) in discrete-time domain. By using Hamiltonian approach, we obtain the necessary conditions. Optimal conditions are solved using a general numerical method. The effect of initialization is studied by changing length of the history period as well as nature of the history function. Numerical example is included to illustrate the initialization effect. It has been observed that initialization effect reduces with the increase in order of the fractional derivative.

A New Matrix Projective Synchronization of Fractional-Order Discrete-Time Systems and its Application in Secure Communication

In this study, it is proposed that a new matrix projective synchronization of fractional-order (FO) chaotic maps in discrete-time. A new synchronization error is introduced and a control law is constructed, which makes the synchronization error converge towards zero in sufficient time under the stability theory of linearization method of FO systems. Numerical simulation results are presented to illustrate the feasibility of the scheme. Finally, a secure communication scheme based on FO discrete-time (FODT) systems was proposed.