Output feedback control for quadratic systems: A Lyapunov function approach

This article deals with the design of quadratic and linear dynamic output feedback controllers for quadratic systems to ensure, on the one hand, local exponential stability of the origin in the closed-loop form and, on the other hand, to increase an estimate of the basin of attraction of the origin as large as possible. The introduction of a quadratic term in the controller allows us to consider a controller in the same class of dynamics as the studied system. Here, our approach is to consider a Lyapunov function with respect to the extended state gathering the states of the system and the controller. The induced nonlinear inequalities are treated thanks to an auxiliary vector repeating the extended state adequately and by combining distinct linearization techniques to finally obtain linear matrix inequalities. For comparison purposes, we also provide another approach characterizing the extended state as belonging to a polytope in the state space. Numerical examples illustrate the results.