Optimal approximation of analog PID controllers of complex fractional-order

Abstract

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.

Processing characteristics correction of measuring systems by means a differintegral of variable order

“…The fractional calculus have found a range of applications, in particular, modelling of process dynamics and physical effects whose modelling with classic mathematical apparatus has not always been faithful to reality, e.g. modelling of such effects as memory process, PID controllers, robust control, heat transfer process, electrical drive, voltage regulator, charging and discharging of supercapacitors, robot manipulators, cell growth dynamics, biomedical engineering, image processing, chemical reaction processes, dynamics of automatic or electronic systems, photovoltaic systems, hybrid power systems or such non-technical issues as analysis of financial processes [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. The fractional calculus seems an ideal tool for modelling of nonlinear and highly complex effects and processes.…”