  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.
