Nominal and robust stability regions of optimization-based PID controllers

In recent decades, several optimization-based methods have been developed for the proportional-integral-derivative (PID) controller design, and the common feature of these methods is that the controller has only one adjustable parameter. To keep the closed-loop systems stable is an essential requirement for the optimization-based PID controllers. In almost all these methods, however, no exact stability region for the single adjustable parameter was sketched. In this paper, using the proposed analytical procedure based on the dual-locus diagram technique, explicit stability regions of the optimization-based PID controllers are derived for stable, integrating, and unstable processes with time delay in the nominal and perturbed cases, respectively. It is revealed that the proposed analytical procedure is effective for the determination of the nominal and robust stability regions and it offers simplicity and ease of mathematical calculations over other available stability analysis methods. The results in this paper provide some insight into the tuning of the optimization-based PID controllers.