Real and complex stabilization: Stability and performance

This paper presents recent results and algorithms that can be used to generate the entire set of stabilizing PID controllers for single-input single-output 1) continuous-time rational plants of arbitrary order, 2) discrete-time rational plants of arbitrary order, and 3) continuous-time first order plants with time delay. These algorithms follow from substantial theoretical advances on PID stabilization that have been reported by us in recent years. They display the rich mathematical structure underlying the topology of PID stabilizing sets. By presenting these algorithms without the highly technical details of the underlying theory, the paper seeks to make the results accessible to as wide an engineering audience as possible. Examples are presented throughout the paper to clarify the steps involved in implementing the different algorithms. We believe that these algorithms can significantly complement the current techniques for industrial PID design, many of which are adhoc in nature. In particular, the graphical displays of feasible design regions using 2-D and 3-D graphics should appeal to control designers and are very suitable for computer aided design where several performance objectives have to be overlaid and intersected. Specific design problems where these algorithms can be profitably used are discussed.