# Basis Functions for a Transient Analysis of Linear Commensurate Fractional-Order Systems
###### Abstract  
In this paper, the possibilities of expressing the natural response of a linear commensurate fractional-order system (FOS) as a linear combination of basis functions are analyzed. For all possible types of \(\delta^{n}\)-domain poles, the corresponding basis functions are found, the kernel of which is the two-parameter Mittag-Leffler function \(E_{\kappa,\beta}\), \(\beta=\alpha\). It is pointed out that there are mutually unambiguous correspondences between the basis functions of FOS and the known basis functions of the integer-order system (IOS) for \(\alpha=1\). This correspondence can be used to algorithmically find analytical formulas for the impulse responses of FOS when the formulas for the characteristics of IOS are known. It is shown that all basis functions of FOS can be generated with Podlubny's function of type \(\epsilon_{k}\) (\(t\), \(c\); \(\alpha\), \(\alpha\)), where \(c\) and \(k\) are the corresponding pole and its multiplicity, respectively. Mittag-Leffler function; commensurate fractional-order system; basis function; impulse response +
Footnote †: journal: _Algorithms_  
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multiplied by the functions \(t^{k}\), \(k=0\), \(1\),..., \(m-1\), \(m\in\) N, \(a\in\) R, \(b\in\) R+ in the case of \(m\)-tuple poles. All these basis functions can be expressed by the real and imaginary components of the exponential function of the complex argument, which will be hereinafter referred to as the generating basis function \(f\in\) C:  
\[f(t)=t^{k}e^{(a+\mathrm{i}b)t}. \tag{1}\]  
The above basis functions play an important role in the concept of _modes_ of motion of a linear system [3], which is known from classical mechanics [4]. The linear combination of the basis functions can be used to model not only the natural response, but also selected types of forced responses, which include the impulse responses of so-called proper systems, i.e., systems of an integer order whose transfer functions have a polynomial of a lower order in the numerator than in the denominator. In the opposite case of improper systems, the impulse response contains, in addition to the basis functions mentioned above, the Dirac impulses and their derivatives, which penetrate the output due to the Dirac impulse exciting the system input. The concept of basis functions can be extended to the class of linear fractional-order systems (FOSs). It is well known that the Mittag-Leffler functions (MLFs) with one, two or three parameters play an important role in the description of these systems, and can thus be viewed as a generalization of the classical exponential function from the domain of IOS [5]. It can be therefore expected that basis functions from the world of FOS are a generalization of the above basis functions and thus they could be derived from some known kinds of the MLFs. On the other hand, the MLFs are not the only possible candidates: for example, several different definitions of exponential and goniometric functions generalized to the fractional domain are known [6; 7; 8; 9; 10; 11]. Dictionaries of the Laplace transform for a fractional domain include functions such as Dawson, arc, ercf, Bessel, extended Bessel, hypergeometric, Hermite polynomials [12] or, for example, the function introduced by [PERSON] in [13] for solving fractional differential equations. It is necessary to consider which of these or similar functions are appropriate to be selected for the set of basis functions. Furthermore, one should analyze the modification of these functions for the case of multiple poles and try to find an equivalent to the above function \(t^{k}\) for FOS. The notional culmination of this work should be to find a generating basis function from which all basis functions for the fractional domain can be generated, i.e., an analogue of function (1) for the integer-order domain. The concept of basis functions can be used to algorithmically generate equations of waveforms of system responses via an analysis program.