Commensurate Fractional Order Research Articles

Representations of Solutions of Time-Fractional Multi-Order Systems of Differential-Operator Equations

This paper is devoted to the general theory of systems of linear time-fractional differential-operator equations. The representation formulas for solutions of systems of ordinary differential equations with single (commensurate) fractional order is known through the matrix-valued Mittag-Leffler function. Multi-order (incommensurate) systems with rational components can be reduced to single-order systems, and, hence, representation formulas are also known. However, for arbitrary fractional multi-order (not necessarily with rational components) systems of differential equations, the representation formulas are still unknown, even in the case of fractional-order ordinary differential equations. In this paper, we obtain representation formulas for the solutions of arbitrary fractional multi-order systems of differential-operator equations. The existence and uniqueness theorems in appropriate topological vector spaces are also provided. Moreover, we introduce vector-indexed Mittag-Leffler functions and prove some of their properties.

Dynamics of chaotic and hyperchaotic modified nonlinear Schrödinger equations and their compound synchronization

We present in this paper four versions of chaotic and hyperchaotic modified nonlinear Schrödinger equations (MNSEs). These versions are hyperchaotic integer order, hyperchaotic commensurate fractional order, chaotic non-commensurate fractional order, and chaotic distributed order MNSEs. These models are regarded as extensions of previous models found in literature. We also studied their dynamics which include symmetry, stability, chaotic and hyperchaotic solutions. The sufficient condition is stated as a theorem to study the existence and uniqueness of the solutions of hyperchaotic integer order MNSE. We state and prove another theorem to test the dependence of the solution of hyperchaotic integer order MNSE on initial conditions. By similar way, we can introduce the previous two theorems for the other versions of MNSEs. The Runge-Kutta of the order 4, the Predictor-Corrector and the modified spectral numerical methods are used to evaluate the numerical solutions for integer, fractional and distributed orders MNSEs, respectively. We calculate numerically using the Lyapunov exponents the intervals of parameters of the purposed models at which hyperchaotic, chaotic and stable solutions are exist. The MNSEs have an important role in many fields of science and technology, such as nonlinear optics, electromagnetic theory, superconductivity, chemical and biological dynamics, lasers and plasmas. The compound synchronization for these chaotic and hyperchaotic models is investigated. We state its scheme using the tracking control technique among three integer commensurate and non-commensurate orders as the derive models and one distributed order as a slave model. We presented and proved a theorem that provides us with the analytical formula for the control functions which are required to achieve compound synchronization. The analytical results are supported by numerical calculations and agreement is found.

A new approximate solution of the fractional trigonometric functions of commensurate order to a regular linear system

This paper introduces a novel approach to the approximate solution of linear differential equations associated with principal fractional trigonometry and the R function. This method proposes a solution that is expressed by adding appropriate fractional linear fundamental functions. Laplace transforms of these functions are irrational. Therefore, we rounded these functions to obtain rational functions in the form of damped cosine, damped sine, cosine, sine and exponential functions. This transformation was achieved by utilizing the concept of fractional commensurate order and, as a result, has direct practical relevance to real-world physics. The precision and effectiveness of the approach are demonstrated through illustrative examples of solving fractional linear systems.

Bi-fractional-order reference model parameters-based control system design

It is known that analytical manipulation of the time-domain representation of fractional order transfer function is, hitherto, a challenging task. In this study, a controller design methodology based on the direct synthesis design method is proposed using bi-fractional order transfer function as a reference model and certain time-domain criteria. First, an analysis examines the effects of bi-fractional order transfer function parameters, that is, commensurate fractional order, damping ratio and natural frequency, on the system time-domain criteria. This examination has allowed us to set up a relation between damping ratio and commensurate fractional order. Next, a polynomial function fitting is established in order to express the damping ratio in terms of this commensurate order. Bi-fractional-order reference model is regenerated using the above-mentioned relationship. Furthermore, a control design algorithm that utilizes the newly derived bi-fractional order reference model is developed by considering control signal limitations. The proposed fractional order control design method is then compared with globally optimized fractional order proportional-integral-derivative (PID) controllers under the same circumstances and performance index. Finally, the implementation of the proposed controller design algorithm is done on a real-time active suspension system. The results are very satisfactory and incoherent with the simulations.

Complexity and Chaos Analysis for Two-Dimensional Discrete-Time Predator–Prey Leslie–Gower Model with Fractional Orders

The paper introduces a novel two-dimensional fractional discrete-time predator–prey Leslie–Gower model with an Allee effect on the predator population. The model’s nonlinear dynamics are explored using various numerical techniques, including phase portraits, bifurcations and maximum Lyapunov exponent, with consideration given to both commensurate and incommensurate fractional orders. These techniques reveal that the fractional-order predator–prey Leslie–Gower model exhibits intricate and diverse dynamical characteristics, including stable trajectories, periodic motion, and chaotic attractors, which are affected by the variance of the system parameters, the commensurate fractional order, and the incommensurate fractional order. Finally, we employ the 0–1 method, the approximate entropy test and the C0 algorithm to measure complexity and confirm chaos in the proposed system.

Predictive Generalized Active Disturbance Rejection Control for Fractional Order Systems with Time Delay

Abstract In this paper, two different active disturbance rejection control techniques based on notion of Smith Predictor are proposed for control of fractional order stable systems with time delay. In the proposed Smith Predictor based Fractional order linearised active disturbance rejection control (SP-FO-LADRC) technique, fractional order dynamics are treated as a part of generalized disturbance and concept of Smith Predictor is used to obtain delay less output, which is fed to an extended state observer (ESO) and an integer order control strategy is developed. To further ameliorate control performance of this technique, complete plant information of commensurate fractional order system is used to formulate another new technique, namely Smith Predictor based fractional order generalized active disturbance rejection control (SP-FO-GADRC), which incorporates fractional ESO and leads to formulation of fractional order control law. An extensive simulation study and robustness analysis conducted on three different examples is testimony to efficacy of proposed techniques.

A chaos control strategy for the fractional 3D Lotka–Volterra like attractor

In this paper, we have considered a three-dimensional Lotka–Volterra attractor in the frame of the Caputo fractional derivative to examine its dynamics. The theoretical concepts like existence and uniqueness and boundedness of the solution are analyzed. To regulate the chaos in this fractional-order system, we have developed a sliding mode controller and conditions for global stability of the controlled system with and without uncertainties and outside disruptions are derived. The ability of the designed controller is examined in terms of both commensurate and non-commensurate fractional order derivatives for all the aspects. The Lyapunov exponent is the novelty of this paper which is used to illustrate the behavior of the chaos and demonstrate the dissipativeness of the considered chaotic system. We have examined the effect of fractional order derivatives in this system. With the help of numerical simulations, the theoretical claims regarding the impact of the controller on the system are established.

A fractional PI observer for incommensurate fractional order systems under parametric uncertainties

The problem of state observation in incommensurate fractional order systems has been poorly studied. Currently some observers that have been proposed are based on a copy of the system, which causes them to be highly dependent on the system parameters, additionally they are redundant (estimate variables that are available). So this paper proposes a novel fractional observer against parametric uncertainties for a certain type of incommensurate fractional order systems. The fractional observer design is based on a property concerning observability in incommensurate fractional order systems which allows us to construct the observer only considering the available output and its fractional derivatives. On the other hand, the convergence analysis of the observation error is carried out using a particular approach of fractional order systems related to the Global Mittag-Leffler boundedness. We prove that there is a compact set GMLA (Globally Mittag-Leffler Attractive, according to Definition 4) where the system that represents the observation error dynamics is attractive and we also prove that the observation error is uniformly bounded. Additionally, the fractional observer is model-free i.e., a system copy is not required, this gives robustness in spite of parametric uncertainties and it is also reduced order therefore one observer must be designed for each variable that we want to estimate consequently the observer is non-redundant (no estimation of variables that are already available). Moreover, our proposed fractional observer can be designed for commensurate fractional order systems and we also show that if we consider integer derivative order, the proposed fractional observer presents certain properties. Finally, in order to show the effectiveness of the proposed fractional observer, an incommensurate fractional order Rössler hyperchaotic system is considered as a numerical example and an incommensurate fractional model of the COVID-19 pandemic as a real-world application.

Fractional System Identification using Binary Measurements

This paper focuses on the identification of commensurate fractional order systems from binary measurements on the output. The proposed algorithm allows the estimation of both the parameters of the model and the commensurate order. The algorithm is based on a specific design of the output signal as a sum of sinusoids. It allows the parametrization of the output signal on a basis of sinusoids and the estimation of the high resolution output. Numerical simulation results are provided to show the efficiency of the method.

Mikhailov stability criterion for fractional commensurate order systems with delays

In this paper, we present the extension of the Mikhailov stability criterion to linear fractional commensurate order systems with delays of the retarded type. The extension is obtained by generalizing the Mikhailov stability criterion of fractional commensurate order and integer order delay systems. The validity of the results is illustrated by means of several examples.

On the Stability of Commensurate Fractional-Order Lorenz System

On the Stability of Commensurate Fractional-Order Lorenz System

Dynamical distributed controller for the synchronization problem of integer and fractional order partial differential equation systems

In this work we present a methodology to design dynamical distributed controllers for the synchronization problem of systems governed by partial differential equations of integer and fractional order. We consider fractional systems whose space derivatives are of integer order and time derivatives are of fractional commensurate order. The methodology is based on finding canonical forms by means of a change of variable, such that a distributed controller can be designed in a natural way in the form of a chain of integrators. To study the stability of the integer order closed loop system, we propose to use the spectral and semi-group theory for infinite dimensional Hilbert spaces. On the other hand, the stability criterion previously established by Matignon is used for the stability analysis of the fractional order case. Additionally, we tackle the synchronization problem of multiple systems, which is reduced to a problem of generalized multi-synchronization. To illustrate the effectiveness of the proposed methodology, examples and numerical results are given.

Stability and Robustness Analysis of Commensurate Fractional-Order Networks

Motivated by biochemical reaction networks, a generalization of the classical secant condition for the stability analysis of cyclic interconnected commensurate fractional-order systems is provided. The main result presents a sufficient condition for the stability of networks of cyclic interconnection of fractional-order systems when the digraph describing the network conforms to a single circuit. The condition becomes necessary under a special situation where coupling weights are uniform. We then investigate the robustness of fractional-order linear networks. Robustness performance of a fractional-order linear network is quantified using the $\mathcal H_2$ -norm of the dynamical system. The theoretical results are confirmed via some numerical illustrations.

On the Stability of Linear Incommensurate Fractional-Order Difference Systems

To follow up on the progress made on exploring the stability investigation of linear commensurate Fractional-order Difference Systems (FoDSs), such topic of its extended version that appears with incommensurate orders is discussed and examined in this work. Some simple applicable conditions for judging the stability of these systems are reported as novel results. These results are formulated by converting the linear incommensurate FoDS into another equivalent system consists of fractional-order difference equations of Volterra convolution-type as well as by using some properties of the Z-transform method. All results of this work are verified numerically by illustrating some examples that deal with the stability of solutions of such systems.

Direct fractional order MRAC adaptive control design for a class of fractional order commensurate linear systems

In this paper, we introduce a direct fractional order adaptive control design based on model reference adaptive control (MRAC) structure for a class of commensurate fractional order linear systems with an arbitrary relative degree, and whose parameters are unknown. By generalising the application of standard direct MRAC strategy to plants described by fractional order models, we develop a fractional adaptive control scheme (FOMRAC) based on the output feedback. We also define an adaptation control law ensuring the stability of the closed-loop system and the good tracking of the reference trajectory. The asymptotic stability of the fractional order control system is proven using an extension of the Lyapunov theorem. Simulation results show the effectiveness of the proposed control method even for plants with model parametric variations and additive noises.

Robust fractional Active Disturbance Rejection Control: A unified approach

This paper presents an adaptation of the Active Disturbance Rejection Control (ADRC) method within the fractional calculus context. This ADRC version is characterized by a single active cancellation of the lumped effects of all the unmodeled dynamics, external disturbances and parameter uncertainty associated with the linear model of commensurate fractional order. This methodology proposes suitable simplifications to fit the resulting system to one of appropriate commensurate fractional order. The design is reduced to a linear observer-based control of fractional order that estimates the unified disturbance. A numerical stability analysis is presented to quantify the tracking and estimate errors bounds and provide guidelines in the parameters of observer-based controller configuration. The proposed control strategy is experimentally validated in linear and nonlinear cases of fractional order by means of approximate analog implementations.

Frequency domain subspace identification of fractional order systems using time domain data with outliers

AbstractThis paper focuses on the identification of the multiple‐input multiple‐output commensurate fractional order systems. Different from the assumption of known frequency domain data and prior information about noise in the general frequency domain identification algorithm, the reliable frequency domain data is measured in this paper by designing special excitation signals. Then, in order to suppress the impact of noise, the data used in algorithm are truncated from the low frequency band. In addition, this paper considers the case where the sampling data are disturbed by outliers, and a matrix decomposition method with threshold value is developed to eliminate the influence of outliers. After that, the parameters of the system to be identified are estimated by the frequency domain subspace identification method. The validity of the proposed method is demonstrated by an illustrative numerical example.

Lyapunov exponents spectrum estimation of fractional order nonlinear systems using Cloned Dynamics

This work presents the determination of Lyapunov spectrum of fractional order dynamical systems. The method considers the Cloned Dynamic technique, which is based on time evaluation of exact copies of dynamic equations with small initial conditions disturbances. As its calculations are simple, this method can be applied to various dynamic systems types, in particular fractional order, nonlinear and time varying systems, and non linear systems where Jacobian are impossible to evaluate. The predictor-corrector Adams-Bashforth-Moulton algorithm is used for fractional order system numerical evaluation. All Lyapunov exponents are estimated based on the convergence or divergence of the small disturbed clones with respect to the reference (fiducial) trajectory and application of Gram-Schmidt Reorthonormalization after each convenient periodic time interval. To evaluate the application of this method, three third order dynamical systems are explored: Jerk, Financial, and a Four Wing systems. Integer order, commensurate fractional order and incommensurate fractional order are explored. To confirm chaotic or non-chaotic behavior, 0-1 Test is used and Jerk system is studied with a classical method for Lyapunov exponents determination also. The advantages and contribution of the proposed method the possibility to study incommensurate orders fractional cases and explore non linear systems without need of Jacobian determination, as is proved with applications shown. The Lyapunov Exponents and the phase portraits were simulated together to validate this method and the obtained results are in good agreement.

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.

Application of modified least squares method for order reduction of commensurate higher order fractional systems

The paper related to the reduction and investigation of family of commensurate fractional order systems. The fractional order system is first transformed to integer order and then a hybrid method is applied as a model reduction scheme. In this scheme the reduced denominator is acquired by least square method and the numerator is achieved by time moment matching. This formulated reduced integer model is reconverted to reduced fractional model. Three examples are conferred to authenticate and emphasize the efficacy of the proposed approach. The proposed method is verified by MATLAB simulation, and shows better performance in the estimation process.