Data fitting using solutions of differential equations: Fractional-order model versus integer-order model

Abstract

The present paper deals with using different mathematical tools and approaches for the data fitting. A comparison of two types of mathematical models, which solutions are used to fit experimental data is done, where the error-of-fit and the computation time are taken as the fitting benchmarks. Both types of the defined models consist of differential equations, one uses fractional, the other integer orders of differentiation. The first advantage of the fractional-order models is the fact that fractional-order differential equation (FDE) has one degree of freedom more (the order of differentiation lies in the interval (0,1)), whereas the integer-order differential equation has the order of differentiation constant, equal to 1. The other advantage besides the “freedom in order” is that FDEs provide a powerful instrument for description of memory and hereditary properties of systems in comparison to the integer-order models, where such effects are neglected or difficult to incorporate. The first aspect of this work consist in choosing a suitable optimization method for finding the parameters of the defined models based on minimizing chosen fitting criterion. The Medium-scale optimization, using the tools of sequential quadratic programming, Quasi-Newton and line-search algorithms (performed by the MATLAB function fmincon) is compared with the evolutionary - genetic algorithm (performed by the MATLAB function ga). The second aspect lies in using two different approaches in the formulation of the optimization criterion. For the time domain identification the classical least squares method (LSM) and so the sum of vertical offsets will be used. The state-space identification will use the total least squares method (TLSM or the so-called orthogonal distance fitting criterion - ODF), which uses the sum of orthogonal (perpendicular) distances between the experimental points and the fitting curve. Several examples are presented in the form of figures. The efficiency of computation and the values of the error-of-fit using different approaches are compared in the form of tables.