An iterative LMI approach for H∞ synthesis of multivariable PI/PD controllers for stable and unstable processes

An iterative linear matrix inequality (LMI) approach for designing multi-input multi-output (MIMO) PI/PD controller for stable/unstable multivariable processes is proposed in this paper. For this purpose, the matrix gains of controller are calculated such that the closed-loop system be stable, and simultaneously, the infinity norm of the weighted sensitivity function is minimized. This problem is mathematically formulated using the well-known bounded real lemma (BRL). The matrix inequality of the BRL is nonlinear because of multiplication of the variable of Lyapunov equation and gains of controller. To remove this nonlinearity, first a solution to the Lyapunov LMI is calculated using some necessary-type LMIs developed for this purpose. Then, this solution is substituted in the BRL to arrive at an LMI whose solution determines the gains of a stabilizing MIMO PI/PD controller which also minimizes the infinity norm of the weighted sensitivity function. If the resulting controller was not satisfactory, one can use the proposed iterative algorithm to improve its performance. The proposed method is used for tuning MIMO PI/PD for four stable/unstable MIMO processes.