An effective approach of approximation of fractional order system using real interpolation method

Fractional-order controllers are recognized to guarantee better closed-loop performance and robustness than conventional integerorder controllers. However, fractional-order transfer functions make time, frequency domain analysis and simulation signi cantly di cult. In practice, the popular way to overcome these difculties is linearization of the fractional-order system. Here, a systematic approach is proposed for linearizing the transfer function of fractionalorder systems. This approach is based on the real interpolation method (RIM) to approximate fractional-order transfer function (FOTF) by rational-order transfer function. The proposed method is implemented and compared to CFE high-frequency method; Carlson's method; Matsuda's method; Chare 's method; Oustaloup's method; least-squares, frequency interpolation method (FIM). The results of comparison show that, the method is simple, computationally efcient, exible, and more accurate in time domain than the above considered methods.


Introduction
The concept of fractional calculus has appeared long time ago but due to its complexity, it could not be used in many applications. It is only in the recent years with rapid development of hardware and software applications in computer and electronics elds that fractional calculus theory has been widely used in many applications of science and engineering, including acoustics [1], [2], robotics [3], [4], biomedical engineering, control systems [5], [6], [7] and signal processing [8], [9].
In fact, one could argue that real world processes are fractional order systems in general [10], [11].
Fractional-order models are innite dimensional, and more adequate for the description of dynamical systems than the integer-order models. In technical literature, fractional-order differential equations are mostly analyzed using Laplace transform techniques [10]. However, the signals involved in these applications are charac-  [12]. There are also some methods based on Mittag-Leer functions, Grunwald-Letnikov fractional derivative and Gamma functions for computation of the impulse and step responses of commensurateorder system [13], [14]. However, the solution methods using Mittag-Leer functions and Gamma function are time consuming and highly inaccurate, occurring in solving complicated and high fractional-order dierential equation.
One possible approach to modelling fractional order system is based on numerical approximation of the non-integer order operator [15], [16], [17].
which assigns the image function F (δ) in accordance to the original function f (t) as a function of the real variable δ.

Approximation of FOTs
Using Real

Interpolation Method
In this paper we consider the following approximation task of fractional-order systems. The FOTF is given by the following expression: where p, q − interger and β i , α i − real numbers.
Let us consider rational transfer function: where m ≤ n; m, n are the integer, which should be used to approximate transfer function G(s) of linear fractional order system. For (G (0) = 0, b 0 = 1) or (G (0) = 0, a 0 = 1) there are N = n + m + 1 real coecients which should be determined from N equations obtained from the condition of overlapping the numerical characteristics in the corresponding discrete points, or for G (0) = 0, a 0 = 1 one obtained a n δ n i G(δ 1 ) + ...
For xed δ i both numerator and denominator polynomials are linear combinations of the unknown process parameters. Thus, the set of equations (9) represents a linear system of equations having N linear equations, one obtains N coecients of the rational approximation Eq. 3.
The obtained Eq. 5 are conveniently rewritten in the following matrix form, which is easily solved using some of the modern computer algebra packages, in particular, introducing one easily obtains the desired system of linear where X is the vector of unknown parameters, It is important to mention that the selected set of points δ ∈ [ δ 1 , δ 2 , .., δ N ] can produce a singular matrix from the set of equations. In such a case, another, more appropriate set of points should be used. It is also signicant to note that it is also possible to use more than n incident points in the selected set. The exact solution cannot be found in such a case.     where: Arg 1 (ω) -exact phase response; Arg 1cf e (ω) -phase response by CFE method; Arg 1mat (ω) -phase response by Matsuda's method; Arg 1ls (ω) phase response by leastsquares method; Arg 1R' (ω) -phase response by    The exact time response of the fractional order system, as well as those of the approximation models, are presented in Fig. 7. In additional approximation errors are illustrated in Fig. 8.   As presented in the Fig. 7 and Fig. 9, it clearly shows that, new method provides a well-tting.