On the convexity of static output feedback control synthesis for systems with lossless nonlinearities

TL;DR

This paper demonstrates that static output feedback control synthesis becomes a convex problem for systems with lossless nonlinearities, enabling easier design of stabilizing controllers through a reformulation into linear matrix inequalities.

Contribution

The paper shows that SOF synthesis is convex for systems with lossless nonlinearities by reformulating the problem as an LMI, facilitating controller design.

Findings

SOF synthesis is convex for lossless nonlinear systems.

The BMI can be recast as an LMI, making the problem tractable.

The method guarantees asymptotic stability of the controller.

Abstract

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.