Global heuristic methods for reduced-order modelling of fractional-order systems in the delta domain: a unified approach

In this paper, global optimization-based schemes have been presented to reduce fractional-order (FO) systems in the discrete-delta domain. The delta transform theory brings about the fusion of the continuous and the discrete domains at high sampling rates. The technique can be considered a generalized approach for approximation of fractional (FO) systems, commensurate or non-commensurate. The fractional-order (FO) system is initially transformed into an integer-order (IO) system using the Oustaloup approximation. The lower-order systems of the comparable integer-order structure have been developed with the help of hybrid firefly algorithms employing applicable constraints. The proposed approach is suitably supported by two numerical problems. It is revealed from the examples that the step and Bode responses of the reduced systems generated from the considered techniques are relatively closer to that of the Oustaloup-approximate higher-order model. The efficacy of the recommended methods is also illustrated by the comparison of several performance indicators with some of the latest techniques published in academia. A handful number of techniques have been employed for comparison. The statistical measures also validate the superiority of the advocated techniques. The percentage improvement is formulated to highlight the efficacy of the suggested methods. The non-parametric tests are also performed to test the significance of the methods proposed methods. The techniques can further be extended to carry out the reduction of the fractional-order multi-input multi-output (MIMO) systems. An attempt to diminish fractional-order systems having non-rational powers will also be taken up in future.