Circle criterion in linear control design

This paper presents a method for higher-order controller design with improved low-frequent disturbance rejection properties. Given a default PID-based controller, the circle criterion is used to construct an extra controller part that deals with resonances resulting from closing the loop. The advantage is twofold. First, a closed-loop shaping method is obtained that is straightforward, graphically-supported, and intuitively clear. Second, it applies to existing control designs and therefore exploits (or even further extends) the design freedom and low-frequency performances therein. The effectiveness of the method is demonstrated using examples from the wafer scanning industry.