A New Method for Getting Rational Approximation for Fractional Order Differintegrals

Abstract

In this paper a new algorithm is presented to calculate the poles and zeros to approximate a fractional order (FO) differintegral (s±α,α∈(0,1)) by a rational function on a finite frequency band ω∈(ωl,ωh). The constant phase property of the FO differintegral is the basis for development of the algorithm. Interlacing of real poles and zeros is used to achieve the constant phase. The calculations are done using the asymptotic Bode phase plot. A brief investigation is made to get a good approximation for the Bode phase plot. Two design parameters are introduced to keep the average phase close to the desired phase angle and to keep the error within the allowed bounds. A study is done to empirically understand the relationship between the error and the design parameters. The results thus obtained help in the further calculations. The algorithm is computationally simple and inexpensive, and gives a fairly good approximation of fractance frequency response on the specified frequency band.

Balanced Truncation Model Order Reduction in Limited Frequency and Time Intervals for Discrete-Time Commensurate Fractional-Order Systems

“…Consider an asymptotically stable discrete-time commensurate fractional-order state space system (1), with the Grünwald-Letnikov fractional-order difference (2). Then the infinite controllability and observability Gramians of the fractional-order system are respectively given as…”

“…The first approach can be implemented by either determination of the fractional-order derivative/difference approximators involved in a fractional-order system [1,2] or by selection of integer-order approximators to the whole fractional-order systems [3][4][5][6][7][8]. In both approaches, a very high integer-order model is usually obtained, which is not effective from the computational point of view due to large memory requirements and long simulation times.…”

Optimal fractional order PIλDμ controller for stabilization of cart‐inverted pendulum system: Experimental results

“…Sufficient conditions for exponential stability of fractional order (FO) systems has been introduced in . A new method to approximate differ‐integrals by rational function and tuning of PI λ D μ controller of time delayed systems has been proposed. Fractional order PI λ , PD μ and PI λ D μ controllers have been used for precise control to achieve desired dynamic behavior of a system within safe and stable bounds.…”

Computation of controllability and observability Gramians in modeling of discrete‐time noncommensurate fractional‐order systems

“…The main problem encountered in implementation of fractional-order systems is infinite time-complexity of incorporated fractional-order derivatives (or differences). The main concepts in practical implementation of fractional-order systems are based on either determination of approximators to the fractional-order derivative (or difference) which is involved in a fractional-order system [24][25][26] or selection of integer-order approximations to the whole fractional-order system. In the second case, there exist a number of methods for approximation of fractional-order systems by integer-order models involving various techniques, e.g.…”