Hybrid Dynamical Systems, or HDS: The Ultimate Switching Experience Michael S. Branicky * Laboratory for Information and Decision Systems Massachusetts Institute of Technology Abstract In previous work I have concentrated on formalizing the notion of a hybrid system as switching among an indexed collection of dynamical systems. I have also studied in some depth the modeling, analysis, and control of such systems. Here, I give a quick overview of the area of hybrid systems. I also briefly review the formal definition and discuss the main approaches taken in the study of hybrid systems. Finally, I elucidate issues in each of the research areas in light of previous results. 1 Introduction Many complicated control systems today (e.g., those for flight control, man- ufacturing systems, and transportation) have vast amounts of computer code at their highest level. More pervasively, programmable logic controllers are widely used in industrial process control. We also see that today’s products incorporate logical decision-making into even the simplest control loops (e.g., embedded systems). Thus, virtually all control systems today issue continuous- variable controls and perform logical checks that determine the mode—and hence the control algorithms—the continuous-variable system is operating un- der at any given moment. As such, these “hybrid control” systems offer a challenging set of problems. Hybrid systems involve both continuous-valued and discrete variables. Their evolution is given by equations of motion that generally depend on both. In turn these equations contain mixtures of logic, discrete-valued or digital dy- namics, and continuous-variable or analog dynamics. The continuous dynam- ics of such systems may be continuous-time, discrete-time, or mixed (sampled- data), but is generally given by differential equations. The discrete-variable dynamics of hybrid systems is generally governed by a digital automaton, or input-output transition system with a countable number of states. The continuous and discrete dynamics interact at “event” or “trigger” times when the continuous state hits certain prescribed sets in the continuous state space. See Figure 1(a). Hybrid control systems are control systems that involve both continuous and discrete dynamics and continuous and discrete controls. The continuous dynamics of such a system is usually modeled by a controlled vec- tor field or difference equation. Its hybrid nature is expressed by a dependence on some discrete phenomena, corresponding to discrete states, dynamics, and controls. The result is a system as in Figure 1(b). * Supported by Army Research Office and Center for Intelligent Control Systems, grants DAAL03-92-G-0164/-0115. MIT, 35-415, Cambridge, MA 02139. branicky@lids.mit.edu