Linear Fractional Order Research Articles

A NEW ALGORITHM FOR GENERATING POWER SERIES SOLUTIONS FOR A BROAD CLASS OF FRACTIONAL PDES: APPLICATIONS TO INTERESTING PROBLEMS

This research offers a precise analytical solution for initial value problems of both linear and nonlinear fractional order partial differential equations. A new approach called the limit residual function method is developed and utilized to generate a series solution for the equations. The key tools of this method are the concepts of power series, residual function, and the limit at zero. The new method is characterized by the speed of determining the series solution coefficients and the limited mathematical operations used. The study examines five significant applications of fractional partial differential equations using this new method. To validate the accuracy of the results, they are compared with exact solutions in the classical case for the discussed applications. We conclude that the proposed approach is straightforward, effective, and successful in solving linear and nonlinear fractional differential equations.

Deviation analysis in transient response of fractional-order systems: An elementary function-based lower bound

In this paper, transient response of commensurate fractional-order systems with non-zero initial conditions is investigated in the viewpoint of the existence of deviation from initial condition. In particular, firstly peak effect for a class of linear fractional order systems is inspected by presenting a lower bound for their responses. Such a lower bound is described by an elementary function, where the commensurate order of the system is considered as a rational number. Furthermore, it has been demonstrated that under particular circumstances the derived lower bound can be extended to apply for deviation analysis in response of a class of fractional-order nonlinear systems. Moreover, included are various illustrative examples intended to assess the applicability of the obtained lower bound in prediction of the deviation value and the time instance at which the peak effect occurs.

Analytical Methods for the Solution of Linear Fractional Order Systems

Analytical Methods for the Solution of Linear Fractional Order Systems

System Identification Based on Experimental Technique Using Stability Boundary Locus Method for Linear Fractional Order Systems

Fractional calculus is an important mathematical tool that is widely used in control systems. It is established in the literature that fractional order models are more accurate and more effective in system modelling. In this study, an alternative and novel technique is proposed to identify the fractional order time-delayed model of an unknown system. The method is based on obtaining the approximate stability boundary locus (SBL) curve of the unknown system by applying three different experimental tests. Three points on the SBL curve are determined by the experimental tests and then the parameters of the fractional order time-delayed model are computed by solving the nonlinear systems of equation. The system model with double fractional order element plus a time delay is obtained using the proposed method. The proposed method is explained through simulations on a twin rotor system. The proposed method is also used in model order reduction calculation of the higher order transfer functions.

Domination in linear fractional-order distributed systems

This paper investigates the notion of domination in linear fractional-order distributed systems in a finite-dimensional state. The objective is to compare or classify the input operators with respect to the output ones, and we present the characterization and property results of this concept. Then, we examine the relationship between controllability and the notion of domination. Finally, we provide a numerical example to illustrate our results.

Novel applications of oustaloup recursive approximation and modified oustaloup recursive approximation methods in linear fractional order control design

Novel applications of oustaloup recursive approximation and modified oustaloup recursive approximation methods in linear fractional order control design

Modeling and Initialization of Nonlinear and Chaotic Fractional Order Systems Based on the Infinite State Representation

Based on the infinite state representation, any linear or nonlinear fractional order differential system can be modelized by a finite-dimension set of integer order differential equations. Consequently, the recurrent issue of the Caputo derivative initialization disappears since the initial conditions of the fractional order system are those of its distributed integer order differential system, as proven by the numerical simulations presented in the paper. Moreover, this technique applies directly to fractional-order chaotic systems, like the Chen system. The true interest of the fractional order approach is to multiply the number of equations to increase the complexity of the chaotic original system, which is essential for the confidentiality of coded communications. Moreover, the sensitivity to initial conditions of this augmented system generalizes the Lorenz approach. Determining the Lyapunov exponents by an experimental technique and with the G.S. spectrum algorithm provides proof of the validity of the infinite state representation approach.

Robust fractional order singular Kalman filter

AbstractIn this article, the state estimation problem of linear fractional order singular (FOS) systems subject to matrix uncertainties is investigated where a recursive robust algorithm is derived. Considering an uncertain discrete‐time linear FOS system with added process and measurement noises, we aim to design a robust Kalman‐type state estimation algorithm based on an optimal data fitting approach with a given sequence of observations. As a substitute for the stochastic formulation, this general filter is obtained by minimizing a completely deterministic regularized residual norm in its worst‐possible form at each step over admissible uncertainties. Analysis of the algorithm shows that not only does the proposed robust filter cover the traditional robust Kalman filters (KFs), but it also represents an extension of the nominal fractional singular KF (FSKF) when the system is not subject to uncertainties. Furthermore, besides giving a sufficient condition for the existence of the robust filter, we derive conditions for the asymptotic properties of the filter, where we demonstrate that the filter and the Riccati equation are stable and converge when an equivalent system is detectable and stabilizable. A numerical example is included to demonstrate the performance of the introduced filter.

LMI based stability condition for delta fractional order system with sector approximation

This paper concerns the stability of linear time invariant delay delta fractional order systems. Due to the complexity of the stable region, it is difficult or even impossible to develop a sufficient and necessary linear matrix inequality (LMI) condition. After analyzing the feature of the suggested stable region, a sector region is constructed to be the proper subset of true stable region. To control the angle of the sector region, a variable θ is introduced which improves the degree of freedom. Afterwards, the equivalent LMI condition corresponding to the sector region is developed. Notably, both the cases of α∈(0,1) and α∈(1,2) are considered. Besides the stability analysis, the developed method also be applied in controller design. Finally, the validity and efficacy of the elaborated method are illustrated by simulation study.

Linear and nonlinear fractional order diffusion equations with initial and bowndary conditions

The usual differentiation and integration are expanded to any non-integer order in fractional calculus. The topic predates the development of differential calculus by Leibnitz and Newton and is therefore as old as classical theory. The concept of fractional calculus has generated interest not only among mathematicians but also among physicists and engineers. This concept is calculated in the same way as the classical methods of differential and integral calculus, and also dates back to the time when Leibniz and Newton invented differential calculus. The idea of calculating fractional order is of interest not only among individual mathematicians, but also among physicists and engineers. The method of upper and lower solutions has been extended to FDEs using these minimum-maximum principles, and various existence results have been established. In this paper, a one-dimensional subdiffusion equation is investigated using the principle of the maximum of the Riemann-Liouville derivative of fractional order. It is proved that for fractional order diffusion equations in linear and nonlinear time there is a unique classical solution to the initial boundary value problem and that the solution continuously depends on the initial and boundary conditions.

A Comparative Study on Solution Methods for Fractional order Delay Differential Equations and its Applications

Fractional calculus is one of the evolving fields in applied sciences. Delay differential equation of non-integer order plays a vital role in epidemiology, population growth, physiology economy, medicine, chemistry, control, and electrodynamics and many mathematical modeling problems Fractional Delay differential equations usually lacks analytic solutions and some of these equations can only be solved by some numerical methods. In this review article we present a comparative study on some standard numerical methods applied to solve linear fractional order differential equations with time delay. Fractional finite difference method (FFDM), Predictor-corrector method (PCM) with new and extended versions has been discussed in this article. All above mentioned methods use the Caputo type fractional differential operator to define fractional derivatives. Solution of a real-life problem formulated by FDDEs has been discussed under these methods. Results have been presented in tabular and graphical form to analyze the efficiency and scarcity of mentioned methods. These graphical and numerical comparisons are provided to illustrate and corroborate the similarity and differences between these methods.

Numerical approximation of Atangana-Baleanu Caputo derivative for space-time fractional diffusion equations

<abstract><p>In this study, we attempt to obtain the approximate solution for the time-space fractional linear and nonlinear diffusion equations. A finite difference approach is given for the solution of both linear and nonlinear fractional order diffusion problems. The Riesz fractional derivative in space is specifically approximated using the centered difference scheme. A system of Atangana-Baleanu Caputo equations that have been converted through spatial discretization is solved using a newly developed modified Simpson's 1/3 formula. A study of the proposed scheme is done to ascertain its stability and convergence. It has been shown that for mesh size h and time steps $ \delta t $ the recommended method converges at a rate of $ O(\delta t^2 + h^2) $. Based on graphic results and numerical examples, the application of the model is also examined.</p></abstract>

PI Control of Loudspeakers Based on Linear Fractional Order Model

This paper aims at the proportional-integral (PI) control of the cone vibration of the electrodynamic loudspeakers system recently described using a linear fractional order model. After introducing the fractional order model of the circuit of these loudspeakers, firstly, a new method is developed to design a fractional order PI controller to place the poles of the system in a desired area of the complex plane which is called D-stabilizing. The design parameters of the method depend directly on the speed of the system output response, the cone vibration. Moreover, the offered fractional order controller avoids any non-minimum phase zero, which causes undesired undershoots in the output, for the closed-loop control system. Secondly, considering uncertainties in the coefficients of the model, a methodology is presented to determine up to how much the uncertainties can increase such that the controller is still able to maintain both D-stability and the absence of non-minimum phase zeros for the control system. Finally, the merit of the presented results and the superiority of the designed fractional order controller over its conventional integer order counterpart are illustrated through numerical simulations.

Rigid plate submerged in a Newtonian fluid and fractional differential equation problems via Caputo fractional derivative

The study of fractional variational problems with derivatives in the sense of Caputo is a recent subject in Newtonian fluids. In this article, the homotopy perturbation method (HPM), the variational iteration method (VIM), and the Akbari–Ganji method (AGM) were used to solve linear fractional differential equations order in fluid mechanics and fractional optimal control problems in the sense of Caputo. Achieving an exact solution to these equations has been one of the researchers’ goals; according to the research results, the proposed methods provide an exact solution. Comparing the results obtained by the proposed methods with those obtained by the Shehu method reveals that the present methods are very effective and convenient in applied fields. Also, the VIM and AGM methods are more proper than HPM in solving these equations. The solutions obtained by the proposed methods agree with the solutions available in the literature.

Fractional Fourier Transform and Ulam Stability of Fractional Differential Equation with Fractional Caputo-Type Derivative

In this paper, we study the Ulam-Hyers-Mittag-Leffler stability for a linear fractional order differential equation with a fractional Caputo-type derivative using the fractional Fourier transform. Finally, we provide an enumeration of the chemical reactions of the differential equation.

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.

Controller Design for the Pitch Control of an Autonomous Underwater Vehicle

In recent years, the Autonomous Underwater Vehicle (AUV) has found its application in a large number of areas, especially in the ocean environment. But due to its highly non-linear nature with six degrees of freedom and the presence of hydrodynamic forces, the equations for AUV control become complex and difficult to design. Hence, in order to overcome this complexity and non-linearity, a reduced-order subsystem is derived for controlling the pitch. Linear Quadratic Regulator (LQR) and Fractional Order PID (FOPID) techniques have been applied for determining the controller for better performance of pitch control in the presence of disturbance.

Finite-time stability in measure for nabla uncertain discrete linear fractional order systems

Finite-time stability in measure for nabla uncertain discrete linear fractional order systems

On the robust stability of commensurate fractional-order systems

Based on a recent generalised version of the Mikhailov stability criterion, this paper presents a Kharitonov–like test for a class of linear fractional–order systems described by transfer functions whose coefficients are subject to interval uncertainties. To this purpose, first the transfer function is associated with an integer-order complex polynomial function of the generalised frequency (i.e. the current coordinate along the boundary radii of the instability sector) whose coefficients are uncertain. Then the geometrical form of the value set of this characteristic polynomial is determined from the direct examination of its monomial terms. To show how the test operates, it is finally applied to two fractional–order transfer functions whose coefficients belong to given intervals.

Stability analysis of fixed point of fractional-order coupled map lattices

We study the stability of synchronized fixed-point state for linear fractional-order coupled map lattice (CML). We observe that the eigenvalues of the connectivity matrix determine the stability of this system as in integer-order CML. These eigenvalues can be determined exactly in certain cases. We find exact bounds in a one-dimensional lattice with translationally invariant coupling using the theory of circulant matrices. This can be extended to any finite dimension. Similar analysis can be carried out for the synchronized fixed point of nonlinear coupled fractional maps where eigenvalues of the Jacobian matrix play the same role. The analysis is generic and demonstrates that the eigenvalues of connectivity matrix play a pivotal role in the stability analysis of synchronized fixed point even in coupled fractional maps.