Fractional-order Discrete-time Research Articles

A Tuning Method via Borges Derivative of a Neural Network-Based Discrete-Time Fractional-Order PID Controller with Hausdorff Difference and Hausdorff Sum

In this paper, the fractal derivative is introduced into a neural network-based discrete-time fractional-order PID controller in two areas, namely, in the controller’s structure and in the parameter optimization algorithm. The first use of the fractal derivative is to reconstruct the fractional-order PID controller by using the Hausdorff difference and Hausdorff sum derived from the Hausdorff derivative and Hausdorff integral. It can avoid the derivation of the Gamma function for the order updating to realize the parameter and order tuning based on neural networks. The other use is the optimization of order and parameters by using Borges derivative. Borges derivative is a kind of fractal derivative as a local fractional-order derivative. The chain rule of composite function is consistent with the integral-order derivative. It is suitable for updating the parameters and the order of the fractional-order PID controller based on neural networks. This paper improves the neural network-based PID controller in two aspects, which accelerates the response speed and improves the control accuracy. Two illustrative examples are given to verify the effectiveness of the proposed neural network-based discrete-time fractional-order PID control scheme with fractal derivatives.

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.

Multidimensional scaling analysis of generalized mean discrete-time fractional order controllers

The dynamics of discrete-time fractional order control systems depends on the method used for implementing the fractional derivatives and integrals. A reliable numerical approach adopts the generalized mean of the continuous to discrete conversion. The extra freedom provided by the proposed method must be carefully optimized by the user. This paper investigates the use of multidimensional scaling for evaluating and visualizing the performance of generalized mean discrete-time fractional controllers. Two alternative performance indices are adopted for comparing the time and frequency responses of the controlled system when adopting different combinations of the parameters. Numerical experiments with a fractional PID and two linear plants demonstrate the feasibility of the method for comparing and visualizing multiple test cases.

Implementation of Discrete-Time Fractional-Order Derivative Controller for a Class of Double Integrating System

Implementation of Discrete-Time Fractional-Order Derivative Controller for a Class of Double Integrating System

Dynamical Analysis of a New Chaotic Fractional Discrete-Time System and Its Control.

This article proposes a new fractional-order discrete-time chaotic system, without equilibria, included two quadratic nonlinearities terms. The dynamics of this system were experimentally investigated via bifurcation diagrams and largest Lyapunov exponent. Besides, some chaotic tests such as the 0–1 test and approximate entropy (ApEn) were included to detect the performance of our numerical results. Furthermore, a valid control method of stabilization is introduced to regulate the proposed system in such a way as to force all its states to adaptively tend toward the equilibrium point at zero. All theoretical findings in this work have been verified numerically using MATLAB software package.

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.

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.

Commensurate and Non-Commensurate Fractional-Order Discrete Models of an Electric Individual-Wheel Drive on an Autonomous Platform.

This paper presents integer and linear time-invariant fractional order (FO) models of a closed-loop electric individual-wheel drive implemented on an autonomous platform. Two discrete-time FO models are tested: non-commensurate and commensurate. A classical model described by the second-order linear difference equation is used as the reference. According to the sum of the squared error criterion (SSE), we compare a two-parameter integer order model with four-parameter non-commensurate and three-parameter commensurate FO descriptions. The computer simulation results are compared with the measured velocity of a real autonomous platform powered by a closed-loop electric individual-wheel drive.

Discrete-Time Fractional Order SIR Epidemic Model with Saturated Treatment Function

Abstract In this paper, a discrete-time fractional-order SIR epidemic model with saturated treatment function is investigated. The local asymptotic stability of the equilibrium points is analyzed and the threshold condition basic reproduction number is derived. Backward bifurcation is shown when the model possesses a stable disease-free equilibrium point and a stable endemic point coexisting together when the basic reproduction number is less than unity. It is also shown that when the treatment is partially effective, a transcritical bifurcation occurs at ℜ 0 = 1 $\Re_{0}=1$ and reappears again when the effect of delayed treatment is getting stronger at ℜ 0 < 1 $\Re_{0}<1$ . The analysis of backward and forward bifurcations associated with the transcritical, saddle-node, period-doubling and Neimark–Sacker bifurcations are discussed. Numerical simulations are carried out to illustrate the complex dynamical behaviors of the model. By carrying out bifurcation analysis, it is shown that the delayed treatment parameter ε should be less than two critical values ε 1 and ε 2 so as to avoid ℜ 0 $\Re_{0}$ belonging to the dangerous range ℜ 0 , 1 $\left[ \Re_{0},1\right]$ . The results of the numerical simulations support the theoretical analysis.

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.

A Separation Principle for Discrete-Time Fractional-Order Dynamical Systems and Its Implications to Closed-Loop Neurotechnology

Closed-loop neurotechnology requires the capability to predict the state evolution and its regulation under (possibly) partial measurements. There is evidence that neurophysiological dynamics can be modeled by fractional-order dynamical systems (FODS). Therefore, we propose to establish a separation principle for discrete-time FODS, which are inherently nonlinear and are able to capture spatiotemporal relations that exhibit non-Markovian properties. The separation principle states that the problems of controller and state estimator design can be done independently of each other while ensuring proper estimation and control in closed-loop setups. Lastly, we illustrate, as proof-of-concept, the application of the separation principle when designing controllers and estimators for these classes of systems in the context of neurophysiological data. In particular, we rely on real data to derive the models used to assess and regulate the evolution of closed-loop neurotechnologies based on electroencephalographic data.

Discrete-Time Fractional Order Integral Sliding Mode Control of an Antagonistic Actuator Driven by Pneumatic Artificial Muscles

Recently, pneumatic artificial muscles (PAMs), a lightweight and high-compliant actuator, have been increasingly used in assistive rehabilitation robots. PAM-based applications must overcome two inherent drawbacks. The first is the nonlinearity due to the compressibility of the air, and the second is the hysteresis due to its geometric construction. Because of these drawbacks, it is difficult to construct not only an accurate mathematical model but also a high-performance control scheme. In this paper, the discrete-time fractional order integral sliding mode control approach is investigated to deal with the drawbacks of PAMs. First, a discrete-time second order plus dead time mathematical model is chosen to approximate the characteristics of PAMs in the antagonistic configuration. Then, the fractional order integral sliding mode control approach is employed together with a disturbance observer to improve the trajectory tracking performance. The effectiveness of the proposed control method is verified in multi-scenario experiments using a physical actuator.

New Conditions and Numerical Checking Method for the Practical Stability of Fractional Order Positive Discrete-Time Linear Systems

Abstract Practical stability of a fractional order discrete-time linear state-space systems was put up in recent years. It is usually checked by the eigenvalues of the state matrix, some methods have been given during these years. But if the size of the state matrix is large, the computations of eigenvalues can be very onerous. In this paper, some new conditions on practical stability for positive fractional discrete-time linear system are presented. Numerically checking method of practical stability is presented based on the new conditions given in this paper. It is illustrated by the numerical examples that our checking method is effective and true. Compared to the now existing methods, numerically checking method is an attractive method because it’s easily implemented.

Synchronization of fractional–order discrete–time chaotic systems by an exact delayed state reconstructor: Application to secure communication

Abstract This paper deals with the synchronization of fractional-order chaotic discrete-time systems. First, some new concepts regarding the output-memory observability of non-linear fractional-order discrete-time systems are developed. A rank criterion for output-memory observability is derived. Second, a dead-beat observer which recovers exactly the true state system from the knowledge of a finite number of delayed inputs and delayed outputs is proposed. The case of the presence of an unknown input is also studied. Third, secure data communication based on a generalized fractional-order Hénon map is proposed. Numerical simulations and application to secure speech communication are presented to show the efficiency of the proposed approach.

Fractional order chaotic cryptography in colored noise environment by using fractional order interpolatory cubature Kalman filter

In this paper, a novel estimation algorithm based on fractional order interpolatory cubature Kalman filter is introduced to achieve the state estimation in nonlinear discrete-time fractional order ...

Optimal Setting and Control Strategy for Industrial Process Based on Discrete-Time Fractional-Order PI$^{\lambda}$ D$^{\mu}$

Generally, the main task of the optimal operation in the industrial process is to satisfy the process technical requirements, and the PID controller is a well-known controller in industrial control applications. Nevertheless, due to the complicated mechanisms and the dynamic characteristics of a complex industrial process, the conventional PID controller fails to provide effective control to such systems. In this paper, we develop a novel discrete-time fractional order PID (DFOPID) control strategy to achieve the technical requirements of the complex industrial process. The proposed work is conducted through a combination of three novel interdependent efforts. First, on the basis of the widely used Tustin operator and its Taylor series, a digital structure of the DFOPID is proposed. Second, in order to solve the stability problem of the complex industrial process, an optimal setting of the approximation function's order (N) and five parameters (λ, μ, K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> , K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> , K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sub> ) is necessary. Hence, an integral time absolute error (ITAE) criterion is applied to convert the optimal setting problem to a nonconvex optimization problem. Finally, a novel intelligent optimization search algorithm called state transition algorithm is employed to carry out the aforementioned design procedure. Furthermore, the performance of the DFOPID control strategy in some practical industrial control systems, including the copper removal process and the electrochemical process of zinc are also investigated.

Identification of dynamic errors-in-variables bilinear systems of fractional order

An approach for the identification of dynamic single-input single-output bilinear discrete-time fractional order system models within the errors-in-variables framework for the case of white input and output noise sequences is presented. A criterion is obtained that allows obtaining highly consistent estimates of the parameters of the system. The propose algorithm was realized in Matlab. The simulation results show the high efficiency of the propose algorithm.

The Co-existence of Different Synchronization Types in Fractional-order Discrete-time Chaotic Systems with Non-identical Dimensions and Orders.

This paper is concerned with the co-existence of different synchronization types for fractional-order discrete-time chaotic systems with different dimensions. In particular, we show that through appropriate nonlinear control, projective synchronization (PS), full state hybrid projective synchronization (FSHPS), and generalized synchronization (GS) can be achieved simultaneously. A second nonlinear control scheme is developed whereby inverse full state hybrid projective synchronization (IFSHPS) and inverse generalized synchronization (IGS) are shown to co-exist. Numerical examples are presented to confirm the findings.