Application of fractional order transfer functions to modeling of high - order systems Krzysztof Oprzedkiewicz AGH University of Science and Technology al. A Mickiewicza 30 Krakow, Poland Email: kop@agh.edu.pl Wojciech Mitkowski AGH University of Science and Technology al. A Mickiewicza 30 Krakow, Poland Email: wojciech.mitkowski@agh.edu.pl Edyta Gawin High Vocational School in Tarnow al. A Mickiewicza 8 Tarnow, Poland Email: e gawin@pwsztar.edu.pl Abstract—The paper presents an application of elementary fractional order transfer funtions to efective modeling of high order control plants. An example of modeling experimental heat plant is also given. I. AN I NTRODUCTION Non integer order calculus has been introduced and considered by de l’Hospital (1661-1704), Leibniz (1646- 1716), Newton (1643-1727) and Euler (1707-1783) and Laplace (1749-1827). Further evolution of theory and applications was run in XIX and XX centuries. Rcently new, powerful computation tools allow many Au- thors [1], [2], [8], [9], [14], [16], [19] to apply the fractional - order calculus to describe a number of physical phenomena: heat and mass transfer ( [15], diffusion, supercapacitors [18] [25] and so on [10], [11], [22]. A transfer function applying the Laplace trasform: ()= ∞ ∫ 0 () − is a very useful model of many dynamic systems. In the case of fractional order systems this transfer function is a function of complex variable where ∈ . The usefulness of this model in MATLAB is determined by possibility of its modeling with the use of integer order and finite dimensional approximations. In the paper the following problems will be presented: ∙ An idea of fractional - order transfer function, ∙ Integer-order approximations of the fractional order trans- fer function, ∙ Hybrid fractional order models proposed by Authors, ∙ An example. II. AN IDEA OF FRACTIONAL - ORDER TRANSFER FUNCTION An idea of fractional order transfer function is analogical to an idea of integer order transfer function. Let us consider dynamic system shown in figure 1. Let () and () are Laplace transforms of input and output signals and all initial conditions are equal zero. Then the fractional order transfer function of the above system can be expressed as underneath: ()= () () (1) Fig. 1: A fractional-order dynamic system Known approximations are dedicated to any elementary frac- tional order transfer functions, the first one describes the fractional order inertial plant: ()= 1 ( 1 + 1) 1 (2) where 0 < 1 < 1. The next one describes an elementary differential plant: ()= 2 (3) where 2 ∈ . Notice, that the both elementary transfer functions given above can be applied to describe much more complicated plants. Additionally, very interesting model can be built as a combination of integer - order and fractional - order trasfer functions. Examples of these models will be presented in further sections. III. I NTEGER- ORDER APPROXIMATIONS OF THE FRACTIONAL ORDER TRANSFER FUNCTION Integer order approximations of the fractional order systems were presented by many Authors, for example: [3] [7], [4]. Modeling fractional order transfer functions with the use of MATLAB/SIMULINK requires us to apply models: integer order and finite-dimensional.In this paper the approximations proposed by Charef and Oustaloup, dedicated to approximate