Convex equations and differential inclusions in hybrid systems

Differential equations with discontinuous right hand sides enable modeling and analysis of control systems with switching elements at a high level of abstraction. Solutions of these differential equations are based on the Filippov, Utkin or similar solution concepts. These solution concepts are in general inconvenient for modeling and verification using formal languages, because they lead to ambiguities in differential algebraic equations. This paper introduces convex equations to avoid such ambiguities in formal languages. Convex equations integrate the functionality of the Filippov solution concept with much of the Utkin solution concept in general differential algebraic equations. A formal semantics of convex equations is given, and an example model is specified using a combined discrete-event/continuous-time formalism.