Hybrid systems describe the dynamical interaction between continuous and discrete signals in one common framework (see Figure 16.1). In this chapter we focus our attention on mathematical models of hybrid systems that are particularly suitable for solving finite time-constrained optimal control problems.
Models of Hybrid Systems
The mathematical model of a dynamical system is traditionally expressed through differential or difference equations, typically derived from physical laws governing the dynamics of the system under consideration. Consequently, most of the control theory and tools address models describing the evolution of real-valued signals according to smooth linear or nonlinear state transition functions, typically differential or difference equations. In many applications, however, the system to be controlled also contains discrete-valued signals satisfying Boolean relations, if-then-else conditions, on/off conditions, etc., that also involve the real-valued signals. An example would be an on/off alarm signal triggered by an analog variable exceeding a given threshold. Hybrid systems describe in a common framework the dynamics of real-valued variables, the dynamics of discrete variables, and their interaction.
In this chapter we will focus on discrete-time hybrid systems, which we will call discrete hybrid automata (DHA), whose continuous dynamics is described by linear difference equations and whose discrete dynamics is described by finite state machines, both synchronized by the same clock . A particular case of DHA is the class of piecewise affine (PWA) systems . Essentially, PWA systems are switched affine systems whose mode depends on the current location of the state vector, as depicted in Figure 16.2. PWA and DHA systems can be translated into a form, denoted as mixed logical dynamical (MLD) form, that is more suitable for solving optimization problems. In particular, complex finite time hybrid dynamical optimization problems can be recast into mixed-integer linear or quadratic programs as will be shown in Chapter 17.
In Section 16.7 we will introduce the tool HYSDEL (HYbrid Systems DEscription Language), a high level language for modeling and simulating DHA. Therefore, DHA will represent for us the starting point for modeling hybrid systems. We will show that DHA, PWA, and MLD systems are equivalent model classes, and in particular that DHA systems can be converted to an equivalent PWA or MLD form for solving optimal control problems.