PID Design via LMIs: Improved Transient Response with Robustness to Uncertain Time-Delay

Abstract

The history of proportional-integral-derivative (PID) control has come a long way so far and many clues indicate that it should last for long time. Despite its simplicity, PID control still is the most dominant strategy for feedback control nowadays. Nevertheless, despite the variety of strategies for PID tuning many open issues remain even for simple SISO (Single Input Single Output) systems, specially when the time-delay, or dead-time, is present in the process. This chapter is devoted to a systematic method of PID design for process that can be modeled as second order plus time-delay (SOPTD) transfer functions. Although the process model is described in Laplace domain the problem is formulated in such a way that the closed-loop control system can be studied in state-space form. This allows the authors to tackle the PID design with the Lyapunov-Krasovskii theory. As consequence, the controller design is carried out by means of convex optimization problems written in the form of Linear Matrix Inequalities (LMIs), that can be solved by several numerical packages widely available. The innovative aspect in the proposed method is that robustness issues associated with time-delay are investigated. The time-delay is allowed to vary within prescribed bounds, deviating from the model nominal value. Moreover, another advantage in the proposed approach is the possibility to impose a decay rate in the transient response. Therefore compromise solutions between robustness and performance can be achieved by the designer. Numerical experiments are provided to illustrate the possibilities of this new method.