Achievable PID performance using sums of squares programming

In addition to process elements like time delay, the PID structure of the controller can pose fundamental limitations on the achievable control performance. A key difficulty in characterizing the limitations due to controller structure is the non-convexity of the resulting optimization problem. In this paper, we present a global lower bound on the achievable PID performance, defined in terms of output variance, by solving a series of convex programs using sums of squares programming. This result is also extended to minimize the weighted sum of the variances of input rate and output. The tightness of the proposed bounds is demonstrated using several benchmark examples drawn from literature.