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Implementing Discrete-time Fractional-order Controllers.
... In recent decades, the application of the fractional diffusion equation (FDE) in many fields has become more and more extensive [1][2][3]. FDE has become an indispensable tool for describing many of phenomena in mechanics and physics [4][5][6]. It has also attracted a lot of attention in biology [7], finance [8], image processing [9] and many other fields. ...
For solving a block lower triangular Toeplitz linear system arising from the time-space fractional diffusion equations more effectively, a single-parameter two-step split iterative method (TSS) is introduced, its convergence theory is established and the corresponding preconditioner is also presented. Theoretical analysis shows that the original coefficient matrix after preconditioned can be expressed as the sum of the identity matrix, a low-rank matrix, and a small norm matrix. Numerical experiments show that the preconditioner improve the calculation efficiency of the Krylov subspace iteration method.
... The need of many scientific areas for the use of fractional partial differential equations (FPDEs) to describe their processes has been widely recognized. Nowadays, the interest of scientists with FPDEs in fields of finance [1], engineering [2], viscoelasticity [3], control systems [4], diffusion procedures [5] and many other scientific areas has no limit. Many anomalous diffusion processes which existed in some physical and biological areas can be modeled by the time fractional reaction diffusion wave equation [6,7]. ...
In this paper, we consider a numerical scheme for a class of non-linear time delay fractional diffusion equations with distributed order in time. This study covers the unique solvability, convergence and stability of the resulted numerical solution by means of the discrete energy method. The derivation of a linearized difference scheme with convergence order O(τ+(δα)4+h4) in L∞-norm is the main purpose of this study. Numerical experiments are carried out to support the obtained theoretical results.
... It is a well-known fact that the integer order differential operator is a local operator whereas the fractional order differential operator is nonlocal in the sense that the next state of the system depends not only upon its current state but also upon all of its proceeding states. In the last decade, many authors have made notable contribution to both theory and application of fractional differential equations in areas as diverse as finance [10], physics [11, 12], control theory [13], and hydrology [14, 15]. Several papers have been written [16, 17] to show the equivalence between the transport equations using fractional order derivatives and some heavytailed motions, thus extending the predictive capability of models built on the stochastic process of Brownian motion, which is basis for the classical ADE. ...
We propose optimal variational asymptotic method to solve time fractional nonlinear partial differential equations. In the proposed method, an arbitrary number of auxiliary parameters γ0,γ1,γ2,… and auxiliary functions H0(x),H1(x),H2(x),… are introduced in the correction functional of the standard variational iteration method. The optimal values of these parameters are obtained by minimizing the square residual error. To test the method, we apply it to solve two important classes of nonlinear partial differential equations: (1) the fractional advection-diffusion equation with nonlinear source term and (2) the fractional Swift-Hohenberg equation. Only few iterations are required to achieve fairly accurate solutions of both the first and second problems.
... Joints 1 and 3 show similar velocity spectra. In the author's best knowledge the foh are aspects of fractional dynamics [26][27][28], but a final and assertive conclusion about a physical interpretation is a matter still to be explored. ...
Fractional Calculus (FC) goes back to the beginning of the theory of differential calculus. Nevertheless, the application of FC just emerged in the last two decades, due to the progress in the area of chaos that revealed subtle relationships with the FC concepts. In the field of dynamical systems theory some work has been carried out but the proposed models and algorithms are still in a preliminary stage of establishment. Having these ideas in mind, the paper discusses a FC perspective in the study of the dynamics and control of several systems. This article illustrates several applications of fractional calculus in science and engineering. It has been recognized the advantageous use of this mathematical tool in the modeling and control of many dynamical systems. In this perspective, this paper investigates the use of FC in the fields of controller tuning, electrical systems, digital circuit synthesis, evolutionary computing, redundant robots, legged robots, robotic manipulators, nonlinear friction and financial modeling.
This paper present a method for computation of a minimal realisation of a given proper transfer function of continuous-time fractional linear systems in the electrical circuit. For the proposed method, a digraph-based algorithm was constructed. We have also shown how after using the constant phase element method we can realise such a system. The proposed method was discussed and illustrated with some examples.
The aim of this paper is to employ fractional order proportional integral derivative (FO-PID) controller and integer order PID controller to control the position of the levitated object in a magnetic levitation system (MLS), which is inherently nonlinear and unstable system. The proposal is to deploy discrete optimal pole-zero approximation method for realization of digital fractional order controller. An approach of phase shaping by slope cancellation of asymptotic phase plots for zeros and poles within given bandwidth is explored. The controller parameters are tuned using dynamic particle swarm optimization (dPSO) technique. Effectiveness of the proposed control scheme is verified by simulation and experimental results. The performance of realized digital FO-PID controller has been compared with that of the integer order PID controllers. It is observed that effort required in fractional order control is smaller as compared with its integer counterpart for obtaining the same system performance.
The aim of paper is to employ digital fractional order proportional integral derivative (FO-PID) controller for speed control of buck converter fed DC motor. Optimal pole-zero approximation method in discrete form is proposed for realization of digital fractional order controller. The stand-alone controller is implemented on embedded platform using digital signal processor TMS320F28027. The five tuning parameters of controller enhance the performance of control scheme. For tuning of the controller parameters, dynamic particle swarm optimization technique is employed. The proposed control scheme is simulated on MATLAB and verified by experimental results. Performance comparison shows better speed control of separately excited DC motor with the realized digital FO-PID controller than that of the integer order PID controller.
The aim of the present paper is to investigate the performance of a fractional-order sigma–delta modulator wherein the integer-order integrator is replaced by a fractional integrator of order \( \alpha \,(1 <\alpha < 2)\). A generalized approach to both linear frequency domain and non-linear time domain modeling and characterization of fractional-order sigma–delta modulator has been discussed. The performance of such modulator has been studied and compared with the corresponding integer-order modulators through simulation.
The paper deals with an aspect of the analysis of nonlinear feedback control systems with a fractional order transfer function. A review of the classical describing function (DF) method is given and its application to a control system with a fractional order plant is demonstrated. Unlike the DF method the frequency domain approach of Tsypkin is known to give exact results for limit cycles in relay systems and it is shown that this approach extends to systems with fractional order transfer functions. The formulation is done in terms of A loci which are related to and more general than the Tsypkin loci. Programs have been developed in MATLAB to compute the limit cycle frequency and also to show the results graphically. Examples are provided to illustrate the approach for a relay with no dead zone.
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