Design of FOPID Controllers by Using 3D Plots

Abstract

A three-dimensional (3D) graphical method is developed to tune the gain parameters of the fractional-order proportional-integral-derivative (FOPID) controller knowing fractional orders. The controlled plant itself is also fractional order (FO) and it may have a time delay. For special cases, the method is applicable for integer-order PID controller design for integer and fractional-order systems with or without time delay. Some graphical design tools are used beneficially in literature using 2D plots. Although the use of some minor 3D plots has appeared, they are based on shifting 2D plots to the third dimension, which makes the approach semi-or quasi-3D. This paper presents the mathematical formulations of five design specifications in accordance with the 3D drawing with programming implementations by MATLAB. For designing controllers by using the introduced 3D graphical method, system design specifications such as phase margin (PM), gain margin (GM), phase flatness (PF), low-frequency output disturbance rejection (LFODR) and high-frequency noise rejection (HFNR) are considered, and their important characteristics are shown. The requirements are mapped in the 3D Euclid space (K-p, K-i, K-d) is an element of R-3 by 3D surfaces and/or lines so that the proportional, integral, derivative control coefficients K-p, K-i, K-d can be chosen to meet the given specifications in an optimum way and to allow trade-off or compromise.