A max-plus based synchronization controller for multiple spring-mass hoppers

This paper introduces a new event-driven centralized synchronization controller for a well-known class of cyclic, underactuated hybrid systems: the spring-mass hoppers. We show that using a supervisory reference generator, formulated in the max-plus algebra, providing low-level controllers with reference event instants induces synchronized behaviors. The reference generator updates the reference signals every time events occur or are expected to occur according to the state feedback of the systems. The synchronicity is achieved while (i) there is no direct interconnection between the individual low-level controllers; and (ii) there are uncontrollable modes in the hybrid system considered. The effectiveness of the proposed approach is demonstrated via a synchronization simulation of multiple vertical hoppers, starting from different initial conditions and under external disturbances.