Control of Hybrid Dynamical Systems: An Overview of Recent Advances

Hybrid systems are dynamical systems exhibiting both continuous and discrete behavior. Having states that can evolve continuously or discretely, hybrid dynamical systems permit modeling and simulation of systems in a wide range of applications including robotics, automotive systems, power systems, biological systems, to just list a few. Key motivation for studying hybrid systems comes from the recognition of the capabilities of hybrid feedback in robust stabilization of nonlinear systems. Numerous frameworks for modeling and analysis of hybrid systems have appeared in the literature. These include the work of Tavernini [26], Michel and Hu [11], Lygeros et al. [10], Aubin et al. [2], and van der Schaft and Schumacher [28], among others. In this paper, we consider the hybrid systems framework in [7, 6], where the continuous dynamics (or flows) of a hybrid system are modeled using differential inclusions while the discrete dynamics (or jumps) are captured by difference inclusions. Trajectories to a hybrid system conveniently use two parameters: an ordinary time parameter t ∈ [0,+∞), which is incremented continuously as flows occur, an a discrete time parameter j ∈ {0, 1, 2, . . .}, which is incremented at unitary steps when jumps occur. The conditions determining whether a trajectory to a hybrid system should flow or jump are captured by subsets of the state space and input space. In simple terms, given an input (t, j) 7→ u(t, j), a trajectory (t, j) 7→ x(t, j) to a hybrid system satisfies, over intervals of flow,