Contents I. Introduction
The Proportional-Integral (PI) controller is one of the most widely used controllers in the control industry and is universally popular in motion control, process control, power electronics, hydraulics, pneumatics, and manufacturing [1]–[3]. In these industrial practice, controller designs are performed based on an approximate model of the actual process. Generally, the approximated model is First Order Plus Time Delay (FOPTD) or Second Order Plus Time Delay (SOPTD). It is essential to design a control system which will exhibit a robust performance because the physical systems can vary with operating conditions and time. Gain Margin (GM) and Phase Margin (PM) are well known index for evaluating the control performance of a control system [4]. And there are lots of research work on the design of robust PI controller based on these two parameters [5]–[10]. However, the use of GM and PM together still cannot ensure the stability of the control system, since they only add constraints to two points of the Nyquist locus . There may exist such a situation that GM and PM are already satisfied, but part of the Nyquist locus is still very close to (-1, j0) point and stability is not good. It is known that when the Nyquist locus is very close to (-1, j0) point, there bound to be a large error value of unit step input response, and this could be dealt with the error integration indices.
