Multi-loop PID tuning strategy based on non-iterative linear matrix inequalities

Chemical processing plants usually have a control architecture composed of several single-paired loops. This type of control system is also called a multi-loop or decentralized control system. In this context, tuning PID controllers in a multi-loop system has become more important in recent decades. This is due to the need to ensure that the closed-loop system is stable or to achieve the expected dynamic performance over a wide range of possible operational conditions. To do this, several authors in the control theory field have used methods based on the Lyapunov stability criteria using linear matrix inequalities (LMI) to tune PID controllers in multi-loop systems. These methods solve the static output feedback (SOF) problem for systems represented by state spaces. This tuning problem is originally bilinear, and some authors have suggested iterative approaches that split the optimization into two layers to turn the problem into a convex one. However, these approaches may lead to high computational costs, depending on the initial guess for the decision variables. This work presents a strategy where only the control gain matrices are used as decision variables, and the Lyapunov matrix is expressed as a function of the control gain matrices. This makes quadratic matrix terms arise, which are handled by the congruency property and an $S$-procedure along with a slack variable. This strategy results in a non-iterative LMI-based SOF tuning approach. To illustrate the approach, a SOF problem that maximizes the system’s Lyapunov function decay rate with an upper bound on $H_{\infty }$ norm was used.