ℒp-stability with respect to sets applied towards self-triggered communication for single-integrator consensus

In this paper, we formulate and study the concept of ℒ_p-stability with respect to a set. This robustness concept generalizes the standard ℒ_p-stability notion towards control systems designed to steer the system state into the vicinity of a set rather than of a point. We focus on stable LTI systems with the property that all eigenvalues with zero real part are located in the origin. Employing the Real Jordan Form, we devise a mechanism for computing upper bounds associated with ℒ_p-stability and ℒ_p to ℒ_p detectability with respect to the equilibrium manifold. Notable examples of this class of LTI systems arise from consensus control. In a self-triggered realization of consensus control problems, each agent broadcasts its state only when necessary in order to achieve consensus. Bringing together ℒ_p-stability with respect to the consensus manifold and the small-gain theorem, we develop self-triggering for single-integrator consensus with fixed and switching network topology. In addition, we show that this consensus problem is Input-to-State Stable with respect to the consensus manifold. Finally, our results are corroborated by numerical simulations.