Hybrid dynamical systems with hybrid inputs: Definition of solutions and applications to interconnections

Funding information Air Force Office of Scientific Research, Grant/Award Number: FA9550-16-1-0015; Air Force Research Laboratory, Grant/Award Number: FA9453-16-1-0053; CITRIS and the Banatao Institute; National Science Foundation, Grant/Award Number: CNS-1544396, ECS-1450484, and ECS-1710621 Summary In this paper, we define solutions for hybrid systems with prespecified hybrid inputs. Unlike previous work where solutions and inputs are assumed to be defined on the same domain a priori, we consider the case where intervals of flow and jump times of the input are not necessarily synchronized with those of the state trajectory. This happens in particular when the input is the output of another hybrid system, for instance, in the context of observer design or reference tracking. The proposed approach relies on reparametrizing the jumps of the input in order to write it on a common domain. The solutions then consist of a pair made of the state trajectory and the reparametrized input. Our definition generalizes the notions of solutions of continuous-time and discrete-time systems with inputs. We provide an algorithm that automatically performs the construction of solutions for a given hybrid input. In the context of hybrid interconnections, we show how the solutions of the individual systems can be linked to the solutions of a closed-loop system. Examples illustrate the notions and the proposed algorithm.