Fractional Order Digital Integrators Research Articles

Efficient design of wideband digital fractional order differentiators and integrators using multi-verse optimizer

In this paper, a novel method is proposed based on combining L1-norm optimally criterion with a recently-proposed metaheuristic called multi-verse optimizer (MVO) to design 2nd–4th order stable, minimum phase and wideband infinite impulse response (IIR) digital fractional order differentiators (DFODs) for the fractional order differentiators (FODs) of one-half, one-third and one-fourth order. To confirm the superiority of the proposed approach, we conduct comparisons of the MVO-based designs with the real-coded genetic algorithm (RCGA) and particle swarm optimization (PSO)-based designs in terms of accuracy, robustness, consistency, and efficiency. The transfer functions of the proposed designs are inverted to obtain new models of digital fractional order integrators (DFOIs) of the same order. A comparative study of the frequency responses of the proposed digital fractional order differentiators and integrators with the ones of the existing models is then conducted. The results demonstrate that the proposed designs yield the optimal magnitude responses in terms of absolute magnitude error (AME) with flat response profiles.

Improved IIR-type fractional order digital integrators using cat swarm optimization

Design of wideband infinite impulse response (IIR) digital fractional order integrators (DFOIs) based on a bio-inspired metaheuristic optimization approach called the cat swarm optimization (CSO) algorithm is presented in this paper. To investigate the efficiency of the proposed approach, the CSO-based DFOIs are evaluated against those of the approximations designed using real-coded genetic algorithm (RGA), standard particle swarm optimization (PSO), and differential evolution (DE) by different magnitude and phase response error metrics. Simulation results reveal the better frequency response of the CSO-based DFOIs in comparison with the competing designs. Both parametric and nonparametric statistical hypothesis tests validate the performance consistency of CSO. Comparisons with the cited literature confirm the efficacy of the proposed models.

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

Optimal design of digital rational approximations with α-dependant coefficients to model the fractional order integrator of any arbitrary order α, where α e (0, 1), is presented in this work. The analytical expressions of the coefficients for the proposed fractional order digital integrators (FODIs) are derived by a two-step method: (a) the coefficients of FODIs for α varying from 0.01 to 0.99 in steps of 0.01 are determined by a meta-heuristic optimisation algorithm called symbiotic organisms search (SOS) and (b) curve fitting is applied on the SOS-optimised coefficients to obtain their generalised expressions. Previous works dealing with the design of FODI based on various meta-heuristic optimisers have considered only a few specific fractional orders; hence, the practical usability of such designs is restricted. This gap provides the motivation for conducting this research. Design quality robustness and convergence consistency of SOS are extensively compared with three other well-known meta-heuristic algorithms. The superior modelling accuracy of the proposed designs is justified by comparing with the recent literature. Simulation results validate the effectiveness of the proposed models as a fractional order proportional–integral controller.

Optimal design of fractional-order digital integrators:An evolutionary approach

This paper presents an optimal approach to design Fractional-Order Digital Integrators (FODIs) using a metaheuristic technique, called Hybrid Flower Pollination Algorithm (HFPA). HFPA is a hybrid approach which combines the exploitation and exploration capabilities of two di erent evolutionary optimization algorithms, namely, Particle Swarm Optimization (PSO) and Flower Pollination Algorithm (FPA). The proposed HFPA based designs are compared with the designs based on Real Coded Genetic Algorithm (RGA), PSO, Di erential Evolution (DE), and FPA. Simulation results demonstrate that HFPA based FODIs of all the di erent orders consistently achieve the best magnitude responses. The proposed technique yields FODIs which surpass all the designs based on both classical and evolutionary optimization approaches reported in recent literature.

Optimal design of Fractional Order Digital Integrators – An Evolutionary Approach

The optimal design of grid-connected Hybrid Renewable Energy Systems (HRESs) is studied by using multi-objective evolutionary algorithm in this paper. With the total system cost and fuel emissions to be minimized, a two-objective optimization model of the hybrid system is established. Then, a modifi ed preference-inspired co-evolutionary algorithm is, for the first time, applied to fi nd the optimal con guration of a grid-connectedhybrid system. As an example, a grid-connected hybrid system, including PV panels, wind turbines, and diesel generators, has been designed and good results are obtained which show that the proposed method is effective.

Artificial Raindrop Algorithm-Based Design of Wideband IIR Fractional Order Digital Integrators

A new technique to design accurate, stable, and wideband infinite impulse response, fractional order digital integrators (FODIs) for the fractional order integrators based on a metaheuristic optimization technique called artificial raindrop algorithm (ARA) is proposed in this paper. ARA is inspired by the process of natural rainfall, and achieves the global optimal solution by identifying the raindrop which occupies the lowest altitude. To investigate the efficiency of the proposed approach, comparisons have been carried out with real coded genetic algorithm, particle swarm optimization, and differential evolution-based FODIs. Simulation results demonstrate that ARA-based FODIs consistently accomplish better magnitude responses in the least number of iteration cycles. The proposed FODIs also outperform all the recently reported designs.

Optimal design of IIR digital FOI using IPSO

ABSTRACTThis paper presents an efficient approach to design digital Fractional-Order Integrators (FOIs) exhibiting an accurate magnitude response by employing a metaheuristic algorithm called Improved Particle Swarm Optimisation (IPSO). The proposed infinite impulse response (IIR) filter type IPSO-based digital FOIs have been judged against the Real-coded Genetic Algorithm, Particle Swarm Optimisation (PSO), and Differential Evolution -based digital FOIs with respect to various performance metrics. The superior performance of the IPSO-based designs is also emphasised using different statistical indices. Parametric and non-parametric statistical hypothesis tests demonstrate the reliable performance of IPSO-based designs. MATLAB simulation results confirm the accurate magnitude response of the proposed digital FOIs as compared with the published literature.

Optimal design of wideband infinite impulse response fractional order digital integrators using colliding bodies optimisation algorithm

This study presents a novel approach to design wideband infinite impulse response fractional order digital integrators (FODIs) for the half and one-fourth order integrators based on a parameter independent metaheuristic algorithm called colliding bodies optimisation (CBO). The performance of CBO-based FODIs have been compared with the designs based on three well-known benchmark evolutionary optimisation algorithms namely, real coded genetic algorithm (RGA), particle swarm optimisation (PSO), and differential evolution (DE) in terms of robustness, consistency, parameter sensitivity, convergence speed, and computational time. Simulations results confirm that the proposed CBO-based designed FODIs achieve consistently superior magnitude responses in a computationally efficient manner as compared with the designs based on RGA, PSO, and DE. The proposed CBO-based FODIs also significantly outperform all state-of-the-art designs reported in literature in terms of two different magnitude response error metrics. This research work underlines the potential of CBO as an efficient optimisation tool for the design of accurate digital approximations to the fractional order integrators.

Impulse invariance-based method for the computation of fractional integral of order [formula omitted

This paper presents a simple and efficient method for the design of recursive digital fractional order integrator when the order of integration is a real number between 0 and 1. The proposed method is based on the impulse invariance method. First the initial value theorem is used for the selection of the initial value of the impulse response and then any of the well-established signal modeling techniques can be employed for the parameterization of the discrete impulse response by pole-zero models. For a given model order, the approximation accuracy greatly depends on the initial value selected. Illustrative examples are presented to demonstrate the effectiveness and usefulness of the proposed method.