Time domain design of fractional differintegrators using least-squares

Abstract

In this paper we propose the use of the least-squares based methods for obtaining digital rational approximations (IIR filters) to fractional-order integrators and differentiators of type s a , a R. Adoption of the Pade´, Prony and Shanks techniques is suggested. These techniques are usually applied in the signal modeling of deterministic signals. These methods yield suboptimal solutions to the problem which only requires finding the solution of a set of linear equations. The results reveal that the least-squares approach gives similar or superior approximations in comparison with other widely used methods. Their effectiveness is illustrated, both in the time and frequency domains, as well in the fractional differintegration of some standard time domain functions.

Fractional control of heat diffusion systems

“…Thus, we obtain an approximation that has a perfect match to the desired impulse response h α (k) for the first m + n + 1 values of k [8]. Note that the above Padé approximation is obtained by considering the Euler operator but the determination process will be exactly the same for other types of discretization schemes, such as the Tustin scheme.…”

“…Fig. 2 Root-locus of G(j ω) for 1 <α < 2, K ≥ 0 ous operators of type s α adopts the Euler, Tustin, and Al-Alaoui generating functions [6][7][8].…”

“…The solution of system (8) in the time domain yields: . A rational-type approximation can be obtained through a Padé approximation to the impulse response sequence (5) h α (k), yielding the discrete transfer function:…”

“…Due to the fact that these operators are defined by irrational continuous transfer functions, in the Laplace domain, or infinite dimensional discrete transfer functions, in the Z domain, we often encounter evaluation problems in the simulations. Therefore, when analyzing fractionalorder systems, we usually adopt continuous or discrete integer-order approximations of fractional-order operators [5,8]. The following two subsections provide a background for the remaining of the article by giving the fundamental aspects of the FC, and the discrete integer-order approximations of fractional-order operators.…”

Optimal Controllers with Complex Order Derivatives

“…Nevertheless, many aspects of this mathematical tool are still to be explored and further research efforts are necessary for the development of practical models. Furthermore, fractional derivatives (FDs) are more elaborated than their integer counterparts and their calculation requires some type of approximation [20][21][22][23][24][25][26].…”