Hybrid and Switched Systems
Hybrid systems combine continuous dynamics with discrete transitions; switched systems select among continuous modes according to a switching signal.
Continuous meets discrete
A hybrid system evolves continuously most of the time but occasionally jumps: the state flows according to differential equations, and at certain events it resets or the governing dynamics switch. Switched systems are the important special case where a switching signal selects which of several continuous subsystems is active, without state resets. Thermostats, gear changes, and mode-based controllers are everyday examples.
Stability is subtle
A crucial fact is that switching among individually stable subsystems can produce an unstable trajectory, and switching among unstable subsystems can be stabilized by a clever schedule. Stability therefore depends on both the subsystems and the switching signal. A common Lyapunov function for all modes guarantees stability under arbitrary switching. When none exists, multiple Lyapunov functions with constraints at switch times, or a dwell-time condition requiring modes to persist long enough, secure stability.
Analysis tools
Average dwell time relaxes the strict dwell-time condition, allowing occasional fast switches if the average interval is long enough. For state-dependent switching, where the mode depends on which region the state occupies, piecewise-quadratic Lyapunov functions found via LMIs certify stability. Hybrid automata formalize the interplay of flow, jump, and guard conditions.
- Continuous flow punctuated by discrete jumps or switches
- Stable modes can combine into instability, and vice versa
- Common Lyapunov function gives arbitrary-switching stability
- Dwell-time and average dwell-time conditions relax this
Zeno behavior, infinitely many switches in finite time, is a pathology hybrid models must exclude or handle, since it has no physical realization and breaks simulation. Well-posed hybrid models rule it out.
Many practical control architectures are hybrid: supervisory logic selecting among controllers, or protection systems overriding normal operation. For a design-stage plant with distinct operating phases and protective interlocks, hybrid-systems theory provides the tools to certify safe transitions in simulation.