Optimization of relative and absolute thresholding parameters in event-triggered control

In this work, we consider the event-based implementation of a linear state-feedback controller. In the first part of the paper, we consider the relative thresholding mechanism and formulate an optimization problem to find the largest threshold that the closed-loop system allows, beyond which the stability cannot be guaranteed. We note that all positive definite matrix pairs satisfying the Lyapunov equation are equivalent with respect to guaranteeing the qualitative property of stability. However, the ranges of thresholds derived from such matrix pairs are different, which necessitates to search over all such pairs, to find the optimal pair that yields the largest relative threshold. In the second part, we make the relative and absolute thresholds state-dependent and use S-Procedure to find the optimal matrix pair at each sampling instant. In both the cases, a convex fractional program is solved to find the best pair and the corresponding threshold. Simulations illustrating the applicability of the proposed optimized thresholds are presented.