  Computing a stabilizing static output-feedback (SOF) controller is an NP-hard
problem, in general. Yet, these controllers have amassed popularity in recent
years because of their practical use in feedback control applications, such as
fluid flow control and sensor/actuator selection. The inherent difficulty of
synthesizing SOF controllers is rooted in solving a series of non-convex
problems that make the solution computationally intractable. In this note, we
show that SOF synthesis is a convex problem for the specific case of systems
with a lossless (i.e., energy-conserving) nonlinearity. Our proposed method
ensures asymptotic stability of an SOF controller by enforcing the lossless
behavior of the nonlinearity using a quadratic constraint approach. In
particular, we formulate a bilinear matrix inequality~(BMI) using the approach,
then show that the resulting BMI can be recast as a linear matrix inequality
(LMI). The resulting LMI is a convex problem whose feasible solution, if one
exists, yields an asymptotically stabilizing SOF controller.
