Devising Numerical Methods for Simulation, Identification and Control of Fractional Order Processes

Abstract

This thesis is devoted to develop novel numerical methods based on triangular functions for simulation, identification and control of fractional order processes demonstrating fractional order dynamics. Fractional calculus (FC) is an active branch of mathematical analysis that deals with the theory of differentiation and integration of arbitrary order. It has been emerging as an indubitably crucial subject for applied mathematicians, applied scientists, and engineers for better comprehending numerous physical processes in diverse applied areas of science and engineering. FC has broad applications in modelling and control. Abel's integral equation, one of the very first integral equations seriously studied, investigated by Niels Henrik Abel in 1823 and by Liouville in 1832 as a fractional power of the operator of anti-derivation, are encountered in inversion of seismic travel times, stereology of spherical particles, spectroscopy of gas discharges (more generally: "tomography" of cylindrically or spherically symmetric objects like e.g. globular clusters of stars), determination of the refractive index of optical fibers and electro chemistry. The physical process models involving Abel’s integral equations do not have closed form solutions, thus, their numerical solutions have been the subject of pure and applied mathematicians. In this thesis, we devise simple and efficient numerical algorithm using triangular functions, which are the foundations of most of the numerical methods developed in the thesis. The proposed method is rigorously tested on examples as well as applications of Abel’s integral equations, and the obtained results are compared with the results of existing methods. It is found that the proposed method exhibits superior performance over those existing methods. Some physical processes such as magnetic field induction in dielectric media, anomalous diffusion, micro and nanotechnology, velocity fluctuation of a hard core Brownian particle, transfer of dust and fog particles in atmosphere, viscoelasticity, optimal control, heat transfer, thermodynamic and electrical conduction ofpolymers need to be described by fractional order integro-differential equations (FIDEs). Abel’s integral equations cannot describe the above mentioned processes (which require to be modelled by fractional order integro-differential equation), hence, FIDEs have to be treated independently. One chapter is devoted to study of the existence of unique solution to FIDEs and as well as their numerical solution. Similar to Abel’s integral equations, most FIDEs have no known solutions and the process models representing the above mentioned processes viii possess no exact solutions. Without the availability of exact solutions or numerical solutions, it is very problematic to gain insights into those real processes exhibiting fractional order dynamics. Based on the triangular functions, an efficient numerical method is developed and tested on a wide variety of FIDEs and applications of FIDEs. A comparison study is carried
out to highlight that the proposed method works better than some of the existing methods used for the same purpose. It has been proved that numerous physical processes in various applied areas of science and engineering such as electrochemistry, physics, geology, astrophysics, seismic wave analysis, sound wave propagation, psychology and life sciences, biology, etc. can be better described by the mathematical models involving stiff or non-stiff fractional differential equations (FDEs) or fractional order differential-algebraic equations (FDAEs). It is surprisingly noticed that most extensively employed semi-analytical techniques such Adomian decomposition method, homotopy analysis method, fractional differential transform method etc. are not able to provide stable approximate to stiff FDEs or FDAEs, at least, in the neighborhood of the initial time point 0. To fill this gap, triangular functions
based numerical method, which owns larger convergence region in compared to the semianalytical methods, is proposed in this thesis. The proposed method is found to be far superior to most numerical methods reported in the literature. Formulating mathematical models, which involve operators of fractional calculus, for real world problems is not an easy task. The geometric and physical interpretation of the operators of fractional calculus is not as distinct as that of integer calculus, thus, it is difficult to model real systems as fractional order system directly based on mechanism analysis. Therefore, system identification method is a practical way to model a fractional order system. However, the existing system identification methods (developed for integer order system identification) cannot be directly applied to estimate parameters of fractional order mathematical models
from experimental or simulated data. Therefore, an arbitrary order system identification method is formulated to estimate parameters of linear and nonlinear fractional as well as integer order mathematical models from simulated data. The obtained results are compared with system identification methods devised based on the piecewise constant basis functions such as Haar wavelets and block pulse function, and orthogonal polynomials. The proposed method yields better results.ix In addition to modelling, FC has significant applications in control theory. Theoretical and experimental results have been shown that the fractional order PID controller can better control fractional order systems as well as some integer order systems than the classical PID controller. From this perspective, a robust fractional order PID controller tuning method is proposed using triangular strip operational matrices. The proposed method of control system design is implemented in heating furnace temperature control, automatic voltage regulator
system and some integer and fractional order process models. Fractional PIλ, fractional PDμ, PIλDμDμ2, fractional PID with fractional order filter and series form of fractional PID controller are designed as optimal controllers using the triangular strip operational matrix based control design method. The performance of the proposed fractional order controller tuning technique is found to be better than the performance of some fractional order controller tuning methodologies reported in the literature. Triangular strip operational matrices proposed from the perspective of Mathematics (for the solution of fractional differential and partial differential equation) finds its elegant application in the presently proposed method of control system design.