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 $\dot{x} = f_{p}(x)$, $p \in \setdef{1, \dots, m}$, together with a switching signal $\sigma(t)$ that selects the active one:
\[\dot{x} = f_{\sigma(t)}(x).\]Examples
Examples
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A thermostat that switches a heater on and off
High school
The thermostat is a switched system because the heater mode switches while the temperature changes continuously.
Non-examples
Things that may be mistaken for examples.
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A bouncing ball
Undergraduate
A bouncing ball is not a switched system, because the state of the system (its velocity) changes suddenly each time it hits the ground—it does not merely switch into a different mode.
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A sampled-data control system
Undergraduate
A sampled-data control system is not a switched system, because the held input jumps, so its state is not continuous.
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.