Hybrid system

Definition

Several formal models of hybrid systems are in use; a common one is the hybrid automaton. A hybrid automaton with continuous state space $\R^n$ consists of the following data.

1. A finite set $Q$ of discrete states, called modes.
2. For each mode $q \in Q$, a vector field $f_q \colon \R^n \to \R^n$ and a set $D_q \subseteq \R^n$, the domain of $q$.
3. A set $E \subseteq Q \times Q$ of edges, that is, allowed mode switches.
4. For each edge $e \in E$, a guard $G_e \subseteq \R^n$ and a reset map $R_e$ that gives a nonempty set $R_e(x) \subseteq \R^n$ for each $x \in G_e$.
5. A set $\mathrm{Init} \subseteq Q \times \R^n$ of initial states.

An execution is a finite or infinite sequence of intervals $[\tau_0, \tau_0'], [\tau_1, \tau_1'], \ldots$ with $\tau_i' = \tau_{i+1}$. On the interval $[\tau_i, \tau_i']$ it has a mode $q_i$ and a curve $x_i$ that solves $x' = f_{q_i}(x)$ and stays in $D_{q_i}$. The start satisfies $(q_0, x_0(\tau_0)) \in \mathrm{Init}$. At each switch, $e = (q_i, q_{i+1}) \in E$, $x_i(\tau_i') \in G_e$ and $x_{i+1}(\tau_{i+1}) \in R_e(x_i(\tau_i'))$. A hybrid system is a system modelled by such data, or by an equivalent formalism with flows and jumps.