Analysis and Design of Hybrid Systems

Doctoral thesis, 1999

Many physical systems today are modeled by interacting continuous and discrete event systems. Such hybrid systems contain both continuous and discrete states that influence the dynamic behavior. There has been an increasing interest in these types of systems during the last decade, mostly due to the growing use of computers in the control of physical plants but also as a result of the hybrid nature of physical processes. Hybrid system models, suitable for describing the essential dynamics of a fairly large class of physical systems in control engineering applications, are proposed in this thesis. The continuous dynamics is described by differential equations whose evolution depends on continuous states and inputs as well as discrete states. The discrete dynamics is modeled by discrete event systems dependent on discrete and continuous states and inputs. It is shown that hybrid systems can be constructed by modular decompositions. A closed-loop hybrid system structure consisting of an open-loop hybrid plant and a hybrid controller, suitable for the description of control engineering systems, is proposed. Stability is one of the most important properties of dynamic systems. A large portion of this thesis is focused on conditions ensuring stability of hybrid systems. The stability results are extensions of Lyapunov theory where the existence of an abstract energy function satisfying certain properties verifies stability. It is shown how the search for such functions can be formulated as linear matrix inequality (LMI) problems, where solutions can be found by computerized methods. Stability robustness dealing with the possibility to guarantee stability despite the presence of model uncertainties is also treated. A large number of examples illustrating different approaches is given. There are many controller structures in the industry consisting of local controllers and the design task is to decide the appropriate switching among these. A related problem occurs in the case of having discrete actuators in continuous processes. The design part of this thesis addresses the problem of how to switch between different continuous vector fields guaranteeing stability of the closed-loop system.