From LQR to Static Output Feedback: a New LMI Approach

This paper proposes a new Linear Matrix Inequality (LMI) for static output feedback control assuming that a Linear Quadratic Regulator (LQR) has been previously designed for the system. The main idea is to use a quadratic candidate Lyapunov function for the closed-loop system parameterized by the unique positive definite matrix that solves the Riccati equation. A converse result will also be proved guaranteeing the existence of matrices verifying the LMI if the system is static output feedback stabilizable. The proposed method will then be extended to the design of static output feedback for the H1 control problem. Besides being a sufficient condition for which a converse result is proved, there are another four main advantages of the proposed methodology. First, it is computationally tractable. Second, one can use weighting matrices and obtain a solution in a similar way to LQR design. Third, the proposed method has an extremely simple LMI structure when compared with other LMI methods proposed in the literature. Finally, for the cases where the output is equal to the state it is shown that the LQR solution verifies the proposed LMI. Therefore, the static output feedback includes the LQR solution as a special case when the state is available, which is a desired property. Several examples show that the method is consistently successful and works well in practice.