Multivariable Proportional-Integral-Derivative Controller Tuning Via Linear Matrix Inequalities Based on Minimizing the Nonconvexity of Linearized Bilinear Matrix Inequalities

This paper proposes an iterative linear matrix inequality (LMI) method for tuning the parameters of multi-input multi-output (MIMO) proportional–integral–derivative (PID) controllers. The proposed method calculates the parameters of controller such that the singular values (SVs) of the sensitivity function are shaped according to the given weight function. For this purpose, first using bounded real lemma (BRL), this problem is represented as a bilinear matrix inequality (BMI) where one of the matrix variables is the variable of Lyapunov equation and the other is structured and contains the matrices of the state-space representation of the controller. This BMI is solved approximately using a novel iterative procedure which linearizes the BMI around an initial solution to arrive at an LMI. The point around which the BMI is linearized is updated automatically at each iteration and the linearized BMI has the nice property that it is obtained by minimizing the amplitude of nonconvex terms.