A paradigmatic example of a mixed discrete-continuous system is a digital controller of an analog plant. The discrete state of the controller is modeled by the vertices of a graph (control modes), and the discrete dynamics of the controller is modeled by the edges of the graph (control switches). The continuous state of the plant is modeled by points in Rn, and the continuous dynamics of the plant is modeled by ow conditions such as differential equations. The behavior of the plant depends on the state of the controller: each control mode determines a flow condition, and each control switch may cause a discrete change in the state of the plant, as determined by a jump condition. Dually, the behavior of the controller depends on the state of the plant: each control mode continuously observes an invariant condition of the plant state, and by violating the invariant condition, a continuous change in the plant state will cause a control switch.

A hybrid automaton H consists of the following components: variables, control graph (control modes=V, control switches=E), initial, invariant, and flow conditions, jump conditions (one for each control switch), and events.