Time-Deterministic Hybrid Transition Systems

Hybrid transition systems in their full generality describe continuous behaviour by a set of equations in each mode { an algebraic or di erential equation for each state variable in terms of inputs and other state variables. Each discrete transition may be taken according to a (non-deterministic) time constraint. In this paper we restrict this model to time-deterministic discrete transitions. Thus, every transition is guarded by a condition g and has a xed delay t. Di erent transitions may have di erent delays (including zero), but progress is enforced after the delay. Using this restriction and a composition operator which uses union of mode sets we then prove certain compositionality properties. In particular, that the parallel composition of two subsystems produces a system whose semantics is de ned in terms of semantics of its constituents provided that it has a run. The restriction is well-justi ed in a large class of control applications where the complex mode-changing software is realised as a synchronous program.