Local Stability of Observer-Based Output Feedback Control Systems with the SOS Method for Discrete-Time Single-Input Bilinear Systems

This paper investigates the stability problem for discrete-time single-input bilinear systems when a stabilizing nonlinear state feedback control law and an observer are applied. We suppose that the state feedback control law is designed via the Sum of Squares (SOS) method with state and input constraints, while we suppose that the observer for the discrete-time bilinear system with input constraints is designed through the Linear Matrix Inequalities (LMI). Taking advantage of a special feature in the error dynamics of the observer estimation, together with the property of the Lyapunov function associated with the stabilizing state feedback, it is shown that the origin of the observer-incorporated closed-loop system is locally asymptotically stable. In addition, the proposed analysis framework is also extended to accommodate the case without explicit control input constraints.