Data-Driven Structured Controller Design Using the Matrix S-Procedure

This paper focuses on the data-driven optimal structured controller design for discrete-time linear time-invariant (LTI) systems, considering both the $H_2$ performance and the $H_\infty$ performance. Specifically, we consider three scenarios: (i) the model-based structured control, (ii) the data-driven unstructured control, and (iii) the data-driven structured control. For the $H_2$ performance, we primarily investigate cases (ii) and (iii), since case (i) has been extensively studied in the literature. For the $H_\infty$ performance, all three scenarios are considered. For the structured control, we introduce a linearization technique that transforms the original nonconvex problem into a semidefinite programming (SDP) problem. Based on this transformation, we develop an iterative linear matrix inequality (ILMI) algorithm. For the data-driven control, we describe the set of all possible system matrices that can generate the sequence of collected data. Additionally, we propose a sufficient condition to handle all possible system matrices using the matrix S-procedure. The data-driven structured control is followed by combining the previous two cases. We compare our methods with those in the existing literature and demonstrate our superiority via several numerical simulations.