Model Predictive Control

Chapter 14
Models of Hybrid Systems
Hybrid systems describe the dynamical interaction between continuous and discrete
signals in one common framework (see Figure 14.1). 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.
14.1 Hybrid models
The mathematical model of a dynamical system is traditionally associated with
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 are based on 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
passing over 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, that 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 [243]. A particular case of DHA
is the popular class of piecewise affine (PWA) systems [233]. Essentially, PWA
systems are switched affine systems whose mode depends on the current location
of the state vector, as depicted in Figure 14.2. 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 15.