New Improved Fractional Order Differentiator Models Based on Optimized Digital Differentiators

Abstract

Different evolutionary algorithms (EAs), namely, particle swarm optimization (PSO), genetic algorithm (GA), and PSO-GA hybrid optimization, have been used to optimize digital differential operators so that these can be better fitted to exemplify their new improved fractional order differentiator counterparts. First, the paper aims to provide efficient 2nd and 3rd order operators in connection with process of minimization of error fitness function by registering mean, median, and standard deviation values in different random iterations to ascertain the best results among them, using all the abovementioned EAs. Later, these optimized operators are discretized for half differentiator models for utilizing their restored qualities inhibited from their optimization. Simulation results present the comparisons of the proposed half differentiators with the existing and amongst different models based on 2nd and 3rd order optimized operators. Proposed half differentiators have been observed to approximate the ideal half differentiator and also outperform the existing ones reasonably well in complete range of Nyquist frequency.

A Comparative Analysis of Laguerre-Based Approximators to the Grünwald-Letnikov Fractional-Order Difference

“…Thus, our Laguerrebased approach is highly competitive in terms of computational efficiency, in addition to a very high approximation accuracy. Also, our discretization approach is computationally superior to the optimization-based competitors of [14]. It is also worth mentioning that another Laguerre-based discretization approach of [20] is related to the Tustin operator which will be shown essentially inferior to our approximation concept.…”

“…The above introductory reference review is, deliberately, far from completeness. We refer the reader to the excellent surveys of the state of the art in discretization of fractionalorder derivatives [9][10][11][12][13][14][15], providing a broad spectrum of the discretization machinery. For space saving reasons, we refrain from repeating the discretization principles and technologies covered therein.…”

Design and implementation of fractional‐order microwave differentiator

“…It has been observed that the proposed FOMD design closely follows the ideal behaviour of FOMD in the frequency range 0.8 GHz ≤ f ≤ 6.6 GHz. Along with the magnitude response analysis, the performance of the proposed FOMD design has been evaluated on the basis of magnitude error parameters defined in (18) and (19). The measured response for the scattering parameter S 21 ( f ) over the proposed frequency range 0.8 GHz ≤ f ≤ 6.6 GHz, gives the value of ARE max = − 26.01 dB and ARE mean = − 43.62 dB.…”

“…Rana et al [18] used a Nelder-Mead simplex algorithm to design fractional-order differential operators. Gupta and Yadav [19] have used evolutionary algorithms such as particle swarm optimisation (PSO), genetic algorithm (GA) and hybrid PSO-GA optimisation to model FODDs. Along with the various FOD designs discussed so far, lots of work have also been done for the design of the integer order digital differentiators [20][21][22][23][24][25][26].…”

Optimal and accurate design of fractional‐order digital differentiator – an evolutionary approach

“…Classical optimisation approach based on Nelder-Mead simplex algorithm was explored by Rana et al to design half, one-third, and one-fourth-order differentiators in [20]. Particle swarm optimisation (PSO), genetic algorithm (GA), and PSO-GA hybrid optimisation algorithm have been applied to design digital integer order derivative operators which are then employed to design FODDs in [21] by Gupta and Yadav. Design of half and one-fourth order FODs were reported by using CFE-based indirect discretisation scheme in [22] by Yadav and Gupta. Improved IIR filter approximations to the FODs were proposed by employing power series expansion (PSE)-signal modelling approach by using a second-order operator called new mapping function (NMF) in [23] by Leulmi and Ferdi. The optimal design of wideband FODDs in terms of improved magnitude response involves the minimisation of the absolute magnitude error between the ideal FOD and its designed counterpart over a wide frequency band.…”