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Complex fractional-order (CFO) transfer functions, being more generalized versions of their real-order counterparts, lend greater flexibility to system modeling. Due to the absence of commercial complex-order fractance elements, the implementation of CFO models is challenging. To alleviate this issue, a constrained optimization approach that meets the targeted frequency responses is proposed for the rational approximation of CFO systems. The technique generates stable, minimum-phase, and real-valued coefficients based approximants, which are not always feasible for the curve-fitting approach reported in the literature. Stability and performance studies of the CFO proportional-integral-derivative (CFOPID) controllers for the Podlubny's, the internal model control, and the El-Khazali's forms are considered to demonstrate the feasibility of the proposed technique. Simulation results highlight that, for a practically reasonable order, all the designs achieve good agreement with the theoretical characteristics. Performance comparisons with the CFOPID controller approximants determined by the Oustaloup's CFO differentiator based substitution method justify the proposed approach. [ABSTRACT FROM AUTHOR]

5 demonstrate that: (i) coefficient I u i HT ht increases as HT ht is increased and I u i HT ht approaches 2 as I M i approaches 3. If I M i is an irrational number or it leads to the occurrence of a recurring decimal for HT ht , then HT ht may be truncated to three digits after the decimal point (viz., I H i ( I s i ) for I M i = 2.5, 2.8). Various approaches have been reported to emulate the frequency response of HT ht , [[5]]. [Extracted from the article]

This paper presents a new approach for reduction of commensurate fractional-order single-input-single-output systems. The minimization in the frequency response error of the reduced order model (ROM) relative to the original system is carried out in the F-plane. A constrained optimization technique is introduced to satisfy the angle criteria for F-domain stability of the proposed ROM. Significant improvements in both the time- and frequency-responses over the recently published literature are illustrated using several numerical examples. [ABSTRACT FROM AUTHOR]

B Publisher Correction: Fractional Calculus and Applied Analysis (2023) 26:1566-1593 b https://doi.org/10.1007/s13540-023-00168-x The original online version of this article was revised: Figure 6 was replaced with Figure 7, making Figure 7 serve for both 6 and 7. Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. [Extracted from the article]