Hybrid automaton

In automata theory, a hybrid automaton (plural: "hybrid automata" or "hybrid automatons") is a mathematical model for precisely describing systems in which digital computational processes interact with analog physical processes. A hybrid automaton is a finite state machine with a finite set of continuous variables whose values are described by a set of ordinary differential equations. This combined specification of discrete and continuous behaviors enables dynamic systems that comprise both digital and analog components to be modeled and analyzed.
Examples.
A simple example is a room-thermostat-heater system where the temperature of the room evolves according to laws of thermodynamics and the state of the heater (on/off); the thermostat senses the temperature, performs certain computations and turns the heater on and off. In general, hybrid automata have been used to model and analyze a variety of embedded systems including vehicle control systems, air traffic control systems, mobile robots, and processes from systems biology.
Formal Definition.
An Alur-Henzinger hybrid automaton formula_1 comprises the following components:
So this is a labeled multidigraph.
Related models.
Hybrid automata come in several flavors: The Alur-Henzinger hybrid automaton is a popular model; it was developed primarily for algorithmic analysis of hybrid systems model checking. The HyTech model checking tool is based on this model. The Hybrid Input/Output Automaton model has been developed more recently. This model enables compositional modeling and analysis of hybrid systems. Another formalism which is useful to model implementations of hybrid automaton is the lazy linear hybrid automaton.
Decidable Subclass of Hybrid Automata.
Given the expressiveness of hybrid automata it is not surprising that simple reachability questions are undecidable for general hybrid automata. In fact, a straightforward reduction from Counter machine to three variables hybrid automata (two variables for storing counter values and one to restrict spending a unit-time per location) proves the undecidability of the reachability problem for hybrid automata. A sub-class of hybrid automata are timed automata where all of the variables grow with uniform rate (i.e., all continuous variables have derivative 1). Such restricted variables can act as timer variables, called clocks, and permit modeling of real-time systems. Other notable decidable subclasses include initialized rectangular hybrid automata, one-dimensional piecewise-constant derivatives (PCD) systems, priced timed automata, and constant-rate multi-mode systems.