Third quadrant Nyquist point for the autotuning of PI controllers

The original relay autotuning method of Astrom and Hagglund [1] is based on the Nyquist point at the phase angle of -square (the critical frequency). Recently Friman and Waller [8] have shown that the critical frequency is too high for PI controllers and proposed an autotuning method that finds a Nyquist point at the third quadrant through the two-channel relay and tunes PI/PID controllers. Here, we provide other reasons that Nyquist points at the third quadrant are better to tune PI controllers. In addition, methods to identify Nyquist points at the third quadrant which can be applied to noisy processes and to tune PI controllers are proposed.