# Basis Functions for a Transient Analysis of Linear Commensurate Fractional-Order Systems
## 2 Problem Formulation  
The purpose of this article is to solve the following problem:  
Consider a linear time-invariant IOS, described with a proper \(s\) plane transfer function  
\[K(s)=\frac{\sum\limits_{i=0}^{m}b_{i}s^{i}}{\sum\limits_{i=0}^{n}a_{i}s^{i}}, \tag{2}\]  
where \(a_{i}\), \(b_{i}\in\mathbb{R}\), \(a_{n}\
eq 0\), \(m\), \(n\in\mathbb{N}+\) and \(m<n\). Let us define an associated commensurate FOS with the transfer function \(K_{F}\), obtained from transfer function (2) by substituting  
\[s\to s^{\alpha},\alpha\in\mathbb{R}+, \tag{3}\]  
thus  
\[K_{F}(s^{\alpha})=\frac{\sum\limits_{i=0}^{m}b_{i}s^{\alpha i}}{\sum\limits_{i =0}^{n}a_{i}s^{\alpha i}}. \tag{4}\]  
Denote the impulse responses of systems (2) and (4), i.e., the Laplace inverse of (2) and (4), as \(g(t)\) and \(g_{F}(t)\), respectively. Let us solve the following problem:  
Suppose we know the analytic formula for \(g(t)\) as a linear combination of the basis functions of time. Let us find a procedure for deriving \(g_{F}(t)\) from \(g(t)\) for all possible configurations of the coefficients \(a_{i}\), \(b_{i}\) from (2) and (4), respectively. It will be shown below that \(g_{F}(t)\) can be obtained from \(g(t)\) such that to each known basis function of the IOS, the corresponding basis function of the FOS will be assigned. This result can be arrived at in the following steps: transfer function (1) will be decomposed into partial fractions, and each fraction will be converted into a basis function of time according to whether the corresponding pole is real or complex and simple or multiple. In the next step, substitution (2) will be applied to the formula of partial fraction decomposition, and the Laplace inversion will be used to arrive at the basis functions of associated system (3). By comparing the results from the two steps, we come to unambiguous correspondences between the basis functions in the integer-order and fractional-order domains. The above procedure leads to the main contribution of this paper, which is the definition of a complete set of basis functions of commensurate FOS with transfer function (4). These functions allow an analytical description of the natural responses of all the above FOSs for an arbitrary set of their eigenvalues. This automatically implies the possible use of this analytical modeling for an accurate transient analysis of FOS without the need for a numerical solution of fractional-order differential equations, while the accuracy is guaranteed for both stable and unstable systems. Another possible use arises from revealing the above correspondence between the basis functions of the FOS and the IOS: it can be used for an effective computer-aided transient analysis of commensurate FOSs with a direct utilization of the well-known algorithms for finding analytic formulas of responses of IOSs, with the proviso that in the final stage, the classical basis functions are replaced by new basis functions of associated FOS (3). In this paper, we will point out another interesting use of the formulas of impulse responses of FOS constructed from basis functions: reflecting them back into the space of IOSs, one can arrive at hitherto unpublished or little-known correspondences between the Laplace transforms and their time-domain representations from the world of classical IOS. This paper is structured as follows: in Section 3, following this section, the mathematical prerequisites for constructing the basis functions are summarized. In Sections 4 and 5, the Laplace inversions of the transfer function of a commensurate proper FOS for real and complex, single and multiple poles are performed with the aim of expressing the corresponding basis functions in a unified way. Section 6 proposes the formalism for writing these functions and the function generating all the proposed basis functions, together with an unambiguous correspondence between the basis functions for IOS and FOS.