Switched system
High school
A dynamical system is called a switched system if it exhibits instantaneous switches between several sets of rules for how it changes over time.
Undergraduate
A switched system is a family of continuous-time systems $\dot{x} = f_{p}(x)$, $p \in \lbrace 1, \dots, m \rbrace$, together with a switching signal $\sigma(t)$ that selects the active one: $\dot{x} = f_{\sigma(t)}(x)$.
Properties
- A switched system can be unstable even if every subsystem $\dot{x} = f_{p}(x)$ is asymptotically stable, because of how the switching is timed. A common Lyapunov function for all subsystems guarantees stability under arbitrary switching.
Notes
In contrast to a hybrid system, the state of a switched system is continuous at switches.