An analytical synthesis of fractional order PI[formula omitted]D[formula omitted] controller design

Abstract

This paper proposed a comprehensive synthesis of fractional order PIλDμ (FOPIλDμ) controller analytical design, which is illustrated through the typical first order plus normalized time delay (FOPNTD) systems, with fulfilling five frequency-domain specifications simultaneously: phase margin (ϕm), gain margin (Am), phase crossover frequency (ωpc), gain crossover frequency (ωgc) and a “flat phase” constraint. The control loop shape can be adjusted with wide freedom according to the five design specifications, which can all be beneficial on optimizing the control system: ωgc represents the system control bandwidth and response speed, ϕm and Am guarantee the stability, and flat phase constraint keeps the system with iso-damping property on robustness of loop gain variations. The impact of ωpc adjustment is thoroughly discovered via frequency-domain analysis and also time-domain analysis with low-frequency disturbance and high-frequency noise. The frequency response functions are presented to show the loop-shaping advantages of the proposed synthesis scheme. A further in-depth study on designing guideline is also presented: the feasible region of four specifications, e.g. ωgc, ωpc, ϕm and Am, can all be collected and visualized in the multi-dimensional graphics. This feasible region gives users prior information and great flexibility before the controller design. Simulation results using the designed FOPIλDμ controller are carried out to demonstrate the performance advantages over the optimized integer-order PID, three-parameter FOPID, fractional filter-fractional order PID and Ziegler–Nichols FOPID controllers.