BLDC Speed Control Based on Fractional PID Controller

Discrete fractional order system vibrations

Parameter identification of fractional order linear system based on Haar wavelet operational matrix

We argue that fractional order differential equations are more suitable to model complex adaptive systems. Hence they are applied in replicator equations for non-cooperative games. Rock-scissors-paper game is discussed. It is known that its integer order model does not have a stable equilibrium. Its fractional order model is shown to have a locally asymptotically stable internal solution. A fractional order asymmetric game is shown to have a locally asymptotically stable internal solution. This is not the case for its integer order counterpart.

This paper presents a numerical scheme for the solutions of Fractional Differential Equations (FDEs) of order α, 1 < α < 2 which have been expressed in terms of Caputo Fractional Derivative (FD). In this scheme, the properties of the Caputo derivative are used to reduce an FDE into a Volterra type integral equation. The entire domain is divided into several small domains, and the distribution of the unknown function over the domain is expressed in terms of the function values and its slopes at the node points. These approximations are then substituted into the Volterra type integral equation to reduce it to algebraic equations. Since the method enforces the continuity of variables at the node points, it provides a solution that is continuous and with a slope that is also continuous over the entire domain. The method is used to solve a simple FDE using two different types of polynomials, namely cubic order and fractional order. Results obtained using both types of polynomials agree well with the analytical results. However, the fractional order polynomials give more accurate results than the cubic order polynomials do. This suggests that for the numerical solutions of FDEs fractional order polynomials may be more suitable than the integer order polynomials. A series of numerical studies suggests that the algorithm is stable.

This paper presents a numerical scheme for the solutions of Fractional Differential Equations (FDEs) of order α, 1<α<2 which have been expressed in terms of Caputo Fractional Derivative (FD). In this scheme, the properties of the Caputo derivative are used to reduce an FDE into a Volterra-type integral equation. The entire domain is divided into several small domains, and the distribution of the unknown function over the domain is expressed in terms of the function values and its slopes at the node points. These approximations are then substituted into the Volterra-type integral equation to reduce it to algebraic equations. Since the method enforces the continuity of variables at the node points, it provides a solution that is continuous and with a slope that is also continuous over the entire domain. The method is used to solve two problems, linear and nonlinear, using two different types of polynomials, cubic order and fractional order. Results obtained using both types of polynomials agree well with the analytical results for problem 1 and the numerical results obtained using another scheme for problem 2. However, the fractional order polynomials give more accurate results than the cubic order polynomials do. This suggests that for the numerical solutions of FDEs fractional order polynomials may be more suitable than the integer order polynomials. A series of numerical studies suggests that the algorithm is stable.

The problem of analyzing the parameters of a chaotic system of fractional order is considered. As an example, a chaotic system based on a Chua circuit of an integer order was chosen. To obtain a Chua circuit of fractional order, it is proposed to replace the capacitive elements in the original circuit with elements with fractal impedance. The purpose of the work is to create a mathematical model of a fractional order Chua circuit with real values of the circuit elements, on the basis of which to evaluate existing software products in the Matlab environment suitable for modeling chaotic fractional order systems, as well as to search for software products suitable for calculating the spectrum of Lyapunov exponents. An analysis of ready-made solutions in the Matlab environment, from the literature known to us, showed that the most suitable option for modeling a system of differential equations of a fractional order is the «fde_pi12_pc» program. The selected program allows you to set all the system parameters of interest and has a relatively high accuracy due to the use of the predictor-corrector algorithm. Based on this program, a domain for determining fractional indicators of the Chua system was constructed in which the presence of a characteristic attractor is visually observed. Existing software products in the Matlab environment are analyzed, allowing the calculation of Lyapunov exponents for chaotic systems described by fractional order differential equations. Of the solutions considered, the best, in our opinion, is the «FO_NC_Lyapunov» program. Using the selected program, a spatial graph of the spectrum of Lyapunov exponents was constructed when changing the fractional exponents of the Chua system. Based on the spatial graph of the spectrum of Lyapunov exponents and the criteria for the chaotic behavior of the system based on the values of Lyapunov exponents, a definition area was constructed with visualization of zones of chaotic behavior of the Chua system. The domains of definition obtained by visual analysis of the presence of an attractor and calculating the spectrum of Lyapunov exponents for the same mathematical model did not coincide; therefore, there are currently no software products for reliably assessing the chaos of fractional order systems by calculating the spectrum of Lyapunov exponents.

In this paper, we have investigated that initial time difference boundedness criteria and Lagrange stability for fractional order differential equation in Caputo's sense are unified with Lyapunov-like functions to establish comparison result. The qualitative behavior of a perturbed fractional order differential equation with Caputo's derivative that differs in initial position and initial time with respect to the unperturbed fractional order differential equation with Caputo's derivative has been investigated. We present a comparison result that again gives the null solution a central role in the comparison fractional order differential equation when establishing initial time difference boundedness criteria and Lagrange stability of the perturbed fractional order differential equation with respect to the unperturbed fractional order differential equation in Caputo's sense.

Fractional order differential equations are an efficient tool to model various processes arising in science and engineering. Fractional models adequately reflect subtle internal properties, such as memory or hereditary properties, of complex processes that the classical integer order models neglect. In this chapter we will discuss the theoretical background of fractional modeling, that is the fractional calculus, including recent developments - distributed and variable fractional order differential operators.

In this paper, mathematical models for transmission dynamics of Potato leaf roll virus using integer and fractional order differential equations are developed. The models considered Potato as well as Vector populations. The potato population is sub-divided as susceptible ( and infected ( The vector population also sub-divided as susceptible ( and infected (). Firstly, we proposed the integer order of potato leaf roll virus (PLRV) model and then we extend into fractional order due to the reason that fractional order possesses memory and other benefit in modeling of real-life phenomena. Secondly, the qualitative behavior of the model including invariant region, positivity of future solution and equilibrium points are analyzed in both approaches. Finally, numerical simulation is done to investigate the effect of each parameter on the control of the disease for both integer and fractional orders. The results obtained from numerical simulation indicate that increasing elimination rate of infected potato and contact rate have a great contribution in combating potato leaf roll virus in the specified period of time.

This paper presents the identification, control and robustness analysis for a robotic system of 1 degree of freedom using fractional and integer PID controllers. Initially, an ADAMS-MATLAB co-simulation model is built to simulate the dynamic model of robotic system. The identification of the dynamic model parameters of the system employs the recursive least-squares method. The controller design uses a linearized model of the robotic system obtained by the method of input-output linearization, and then, the fractional PID and integer PID controllers are designed based on it. The robustness of the PID controllers is analyzed in the frequency and the time domain. The robustness analysis in the frequency domain uses The Gang of Four (sensibility, complementary sensibility, sensibility of the plant and sensibility of the controller effort). The time domain analysis uses the ADAMS-MATLAB co-simulation model, considering external disturbances, noise in the feedback loop, parametric uncertainty and set-point variation, using as performance criteria ISE, average value and standard deviation of the control action. The tests are performed for positioning and tracking tasks. The obtained results show that the fractional PID controller has a better behavior in presence of external disturbances and lower energy consumption.

Fractional order differential equations can be utilized to model the physical world systems more efficiently than their integer order differential equation counterparts. Most of the problems in science and engineering are represented in the form of non-integer order differential equation. The computation of fractional order systems’ step and impulse responses still pose an uphill task for the researchers. Thus, there is a need for an integer order approximation table that eases the process of fractional order simulation and realization of the real-world systems. In this paper, we have obtained approximated integer order coefficients for fractional order systems using teacher–learner-based optimization technique (TLBO). The TLBO technique performs better than the other optimization techniques as it is not bound by any algorithmic parameters. The obtained integer order approximation table using TLBO method is then analyzed and compared with step responses obtained by FOMCON toolbox and least square curve fitting method.

In this article, a technique called Haar wavelet-Picard technique is proposed to get the numerical solutions of nonlinear differential equations of fractional order. Picard iteration is used to linearize the nonlinear fractional order differential equations and then Haar wavelet method is applied to linearized fractional ordinary differential equations. In each iteration of Picard iteration, solution is updated by the Haar wavelet method. The results are compared with the exact solution.   Key words: Fractional differential equations, Wavelet analysis, Caputo derivative, Haar wavelets, Picard iteration.

This survey paper is devoted to the description of the results by M.M. Djrbashian related to the modern theory of Fractional Calculus. M.M. Djrbashian (1918-1994) is a well-known expert in complex analysis, harmonic analysis and approximation theory. Anyway, his contributions to fractional calculus, to boundary value problems for fractional order operators, to the investigation of properties of the Queen function of Fractional Calculus (the Mittag-Leffler function), to integral transforms’ theory has to be understood on a better level. Unfortunately, most of his works are not enough popular as in that time were published in Russian. The aim of this survey is to fill in the gap in the clear recognition of M.M. Djrbashian’s results in these areas. For same purpose, we decided also to translate in English one of his basic papers [21] of 1968 (joint with A.B. Nersesian, “Fractional derivatives and the Cauchy problem for differential equations of fractional order”), and were invited by the “FCAA” editors to publish its re-edited version in this same issue of the journal.

This paper addresses a control frequency scheme of the microgrid system using a fractional order PID controller. The proposed Microgrid system is consisted of a Photovoltaic System, Wind Turbine Generator, Diesel Engine Generator, Fuel Cell, and different storage systems like Battery Energy Storage Systems, and Flywheel Energy Storage Systems. The principal objective of the present paper is to limit the frequency and power deviations by the application of the proposed controller which has five parameters to be determined through optimization techniques. Krill Herd algorithm is used for determining the optimum fractional order PID controller parameters using the Integral of Squared Error. A comparison between the Genetic Algorithm and Krill Herd is done, and the obtained simulation results presents that the investigated controller-based Krill Herd outperforms the Genetic Algorithm in terms of fewer fluctuations in power and frequency deviation.

Motivated by the importance of study on the complex behaviors, which may be exhibited by fractional order differential equations, this review paper focuses on dynamical fractional order systems exhibiting chaotic behaviors. The review begins with a brief history on the first publications on the above-mentioned subject. Then, the review is continued by investigating the recent progresses relevant to fractional order chaotic systems. Furthermore, a summary on some applications for such systems, which have been reported in the literature, is presented. Finally, the paper is closed by discussing some open problems on the aforementioned research subject. These open problems, as future challenges for further study on fractional order chaotic systems, can specify some direction lines for continuing the research on that subject.