Switched systems are described by a set of continuous state-space models together with conditions that decide which model of this set is valid for the current continuous state. As an extension of the classical linear or affine state-space representations of dynamical systems, this modelling formalism has been thoroughly investigated, as this chapter shows. The identification of the model parameters, observability, and stability analysis as well as methods for stabilization and control of switched systems are surveyed. As shown in the last section, many analysis and design problems for switched systems have a high computational complexity or are even undecidable.

Definition of the system class

Switched systems represent a type of model of hybrid systems that has been studied extensively. The reason for this research activity is given by the fact that this class of systems is very close to “non-hybrid” systems and an extension of the theory of continuous systems towards hybrid systems is, therefore, rather straightforward. Nevertheless, this system class already exhibits several important phenomena of hybrid dynamical systems.

The basic representation format is the state-space model

which describes the dynamical behavior of the system for the input u ∈ ℝm and the operation mode q ∈ Q. The vector field f and the output function g are assumed to be Lipschitz continuous with respect to x and u so that for a fixed operation mode q solutions to the state-space model exist.