The purpose of self-optimizing control (SOC) is minimizing the steady-state economic loss of chemical processes in the presence of disturbances and measurement noises by keeping selected controlled variables (CVs) at constant set-points. In self-optimizing control, by defining a desired objective/loss function and selecting the appropriate combination of process measurements, the average loss, the worst-case loss, or both can be minimized. In general, the optimization problem of self-optimizing control is a non-convex problem and there exist some approaches to change it to a convex form by adding another constraint to the optimization problem, using branch and bound algorithm or mixed integer quadratic programming method to solve the SOC problem. Linear Matrix Inequalities (LMIs) are one of the popular and powerful tools to solve convex optimization problems and changing the optimization problems to the LMI form is gaining popularity. In parallel, for some problems that are non-convex and cannot be transformed to the LMI form, Bilinear Matrix Inequalities (BMI) have been developed. In this paper, we present; first a method to change the convex form of SOC problem to the LMI form and second, reformulate the main and non-convex SOC problem to a BMI form and then change it to the LMI form. The proposed methods are then evaluated on three benchmark processes: a binary distillation column, an evaporator, and a Kaibel column. The LMI/BMI methods are implemented using LMI Control Toolbox and PENBMI of YALMIP toolbox of MATLAB® software. Results show that the proposed algorithm outperforms other methods in the case of structured measurement matrix H. The main benefit of LMI approach is that the desired structure of matrix H can be directly implemented in the optimization method.

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