# Basis Functions for a Transient Analysis of Linear Commensurate Fractional-Order Systems
###### Abstract
For example, formulas of responses to initial conditions or impulse or step responses as functions of time can be generated. The Laplace transform of the response, for example, the transfer function as the Laplace transform of the impulse response, is decomposed into partial fractions. A basis function of time is assigned to each fraction. The impulse response is then expressed as a linear combination of the basis functions. Once the basis functions are quantified, the waveform of the corresponding response is directly obtained with an algebraic evaluation, not with a classical numerical solution of differential equations. The accuracy of the result is determined with the accuracy of the pole calculation, the accuracy of the algorithm of partial fraction decomposition and the accuracy in computing the basis functions. This procedure also has the advantage of offering insight into the dynamics of the system through the analytical formula of the response. Simulation programs for a so-called symbolic and semi-symbolic circuit analysis work on this principle [14]. As far as numerical methods for calculating the standard responses of linear systems are concerned, particularly the impulse or step responses, either standard simulation programs such as SPICE or another platform for scientific and technical calculations such as MATLAB can be used. Of course, programs like MATLAB also allow an analysis using various numerical algorithms that are not available in classical simulation programs [15; 16]. A review of numerical methods for solving linear fractional differential equations and a discussion of specific problems associated with them are given in [17]. Additional proceduresare published in [15] together with codes for implementation in MATLAB. The method based on closed-form solutions to linear fractional-order differential equations starts from the approximate Grunwald-Letnikov definition of the fractional-order derivative. However, the accuracy of the transient analysis strongly depends on the choice of the computational step, so the method is of little use when solving practical problems. In [18], a generalization of the classical integer-order integrator approach for the numerical solution of the free (zero-input) response is made by elaborating the concept of the fractional integrator. In [19], the concept of LFD (Local Fractional Derivative) is used to compute zero-input responses. Numerical algorithms of the Laplace inversion can also be used for a transient analysis [20; 21]. An excellent discussion of the selected algorithms is given in [22; 23]. Procedures utilizing analytical preprocessing of results are also popular. The analytical solution of a linear homogeneous fractional-order differential equation was first published in the classic work [13]. However, the corresponding formula contains infinite sums of derivatives of two-parameter MLFs, and is thus of little use for practical calculations. Interesting possibilities of a transient analysis are offered for the so-called commensurate systems, where all non-integer powers of the operator \(s\) in their transfer functions are rational numbers. For commensurate systems, one can use a procedure leading to expressing their impulse or step responses in terms of a finite sum of MLFs [15; 24; 25]. The algorithmic generation of waveform formulas, constructed from basis functions, is an interesting alternative to the above numerical methods because it completely eliminates the numerical solution of differential equations of a non-integer order. The above summary of the state of the art indicates the usefulness of extending the concept of basis functions to the domain of linear fractional-order systems. In the following section, the problem to be solved is specified, and new pieces of knowledge arising from its solution as well as their application potential are pointed out.