Multi-input and multi-output proportional-integral-derivative controller design via linear quadratic regulator-linear matrix inequality approach

This study considers the problem of designing a multi-input and multi-output (MIMO) proportional-integral-derivative (PID) controller via direct optimal or suboptimal linear quadratic regulator (LQR) approach. To design the controller, first the MIMO PID design problem is transformed into a state feedback control and then the gains of the state feedback controller are chosen through an optimal or suboptimal LQR design. Given a minimal state space representation (A, B, C) of the plant, a necessary and sufficient condition (based on matrices A, C) for which the optimal problem (i.e. PID design via optimal LQR) is solvable is obtained. When this optimal problem is not solvable, a suboptimal solution (i.e. PID design via suboptimal LQR), if exists, is obtained by converting the problem into trace minimisation one, which is solved using linear matrix inequality-based method. Suitable examples are considered to illustrate the approaches.