Design of Improved Fractional Order Integrators using Indirect Discretization Method

Abstract

This paper has adopted rational approximation based on Regular Newton method, for frequency domain fitting of transfer functions of fractional order integrators (FOIs). Further, different discretized mathematical models of one-half, one-third and one-fourth order integrators based on Al-Alaoui operator and New optimized four segment operator have been also developed using these rational approximations by indirect discretization technique. All the proposed models of FOIs are found to be stable when investigated for stability. Simulation results of magnitude responses, phase responses and absolute magnitude errors show that the proposed FOIs obtained by approximations based on Regular Newton method, clearly outperform the other existing approximation techniques which have been used for designing fractional order operators. Results of absolute magnitude errors for all proposed fractional order integrators have been reported to be as low as 0.01, in range 0.35 ≤ ω ≤ 1 π radians of full band of normalized frequency. Among the proposed FOIs, the one-half, one-third and onefourth order models based on Al-Alaoui operator (for 2 nd iterations) are noticeable with tremendously improved results with absolute magnitude errors of ≤ 0.004 in complete normalized frequency range. General TermsFractional order integrators, indirect discretization KeywordsRegular Newton method, Al-Alaoui operator, New optimized four segment operator.

Experimental verification of on‐chip CMOS fractional‐order capacitor emulators

“…Fig. 4 c shows the response of the integrator to a square‐wave input where a 33.7 mV pp 1/2 integrated output is obtained [12]. Spice simulation results show excellent agreement predicting a 37 mV pp 1/2 integrated output.…”

Discretization of Fractional-Order Differentiators and Integrators

“…There are two discretization techniques; direct and indirect discretization. In the indirect method, the fractional-order operators are first approximated by a rational finite-order transfer functions; i.e., (Gupta, and Yadav, 2012;Oustaloup et al, 2000;Lubich, 1986), and then the resulting form is discretized using any existing technique such as Tustin method, Al-Alaoui operator, or a linear combination of other discrete-time operators. For example, the Al-Alaoui integral operator is simply a linear interpolation of the backward rectangular rule and the trapezoidal rule, i.e., , where (Al-Alaoui, 2006).…”

“…For example, the Al-Alaoui integral operator is simply a linear interpolation of the backward rectangular rule and the trapezoidal rule, i.e., , where (Al-Alaoui, 2006). A similar technique was also used to develop a dynamic hybrid digital integrator using a linear combination of Simpson's and the Trapezoidal integrators (Chen et al, 2003;Gupta et al, 2012). All these operators reduce the frequency deviation over limited range of the frequency bands, while they deteriorate their phase responses.…”

Optimal design of wideband fractional order digital integrator using symbiotic organisms search algorithm

“…Design of FODIs is reported based on the least squares method [18]. In the indirect discretisation method [19][20][21][22], the frequency domain mapping is performed in the continuous time domain and the resultant transfer function is discretised. Discretisation of particle swarm optimisation (PSO) algorithm optimised integer order digital integrators has been reported recently in [23].…”