Research On Discretization Methods Of Fractional Order Model

Recently, with the rapid development of computer science and the increasing ability of calculation, the realization of fractional calculus becomes feasible and fractional calculus was gradually applied to various engineering fields. Moreover, with more and more widespread application of the fractional calculus theory in the actual control system, discretization methods of fractional calculus are being got increasingly concerned about the vast numbers of experts and scholars.At present, the discretization methods can be broadly divided into two kinds: direct and indirect discrete method. Indirect discrete methods require first frequency-domain matching in a continuous time domain, then discrete processing to the matched S-function. Discrete Methods use generating functionω( z?1) to represent the fractional calculus operator s±r, and get a discrete-time domain transfer function, because this transfer function is irrational function, a rational function is needed for its approximation.For the direct discrete method, both the generating function and the rational function approximation methods are studied. The main research work of this dissertation is given as follows:(1) Systematically summarizes the fundamental theories of fractional calculus, introducing several kinds of defintion and the conversion between each other.(2) Original discretization method is detailed and specific studied. First, several major rational function approximation methods (Power Series Expansion, Muir-recursion and Continuous Fractional Expansion)of the fractional caculus operator are researched and analyzed, then, the four kinds of generating function (Euler,Tustin,Simpson and Al-Alaoui) were analyzed and compared, and summarized advantages and disadvantages of all kinds of rational function approximation and discrete methods, derived both for time-domain and the frequency characteristics, approximation results of Al-Alaoui + CFE in general is relatively better.(3) Improves the generating function Al-Alaoui, and approximation results of amplitude and phase frequency characteristics have improved.(4) The Chebyshev-Padéalgorithm is applied to achieve rational function approximation of fractional calculus operators, and Simulation results show that, in the same order of the transfer function case, approximation results have been significantly improved compared with the CFE.(5)The Remez algorithm is successfully applied to achieve rational function approximation of fractional calculus operators. Based on introducing the Remez algorithm, a detailed specific simulation is discussed, and the simulation results show that the use of this algorithm has got relatively good approximation results in the time-domain and frequency characteristics.The innovative points of the dissertation are the improvements to the generating function Al-Alaoui,as well as the Chebyshev-Padéalgorithm and Remez algorithm are successfully applied to achieve rational function approximation of fractional calculus operators.Studies have shown that the use of these discrete and rational function approximation method made approximation effects better than the original method, especially the Remez rational approximation method obtains a satisfactory approximation results.