Event-triggered Stabilization for Nonlinear Systems with Center Manifolds

In this work, we consider event-based implementation of control laws designed for local stabilization of nonlinear systems with center-manifolds. The systems being considered possess linearised models with uncontrollable modes on the imaginary axis. The controller chosen decides both the structure of the center-manifold and its stability. Although the control of systems with center-manifolds is well studied, event-based control of such systems is yet to be probed. This involves the exploration of input-to-state stability (ISS) properties of such systems, with respect to measurement errors. Considering the most general structure for the controller, we prove that a controller that locally asymptotically stabilizes the dynamics on the center-manifold, also renders the overall system locally input-to-state stable (LISS) and find the comparison functions involved in the Lyapunov characterization of ISS. This general characterization required a nonlinear change of variables, involving the center-manifold, which can only be approximately determined in most cases. Because of this, it is found to be unsuitable for designing event-triggered controllers. We then explore an approach that does not resort to this change of variables and present our findings. We discuss the possibility of a simpler relative thresholding mechanism and present simulation results for an illustrative example.