Declaration of unknowns in DAE-based hybrid system specification

The majority of hybrid languages are based on the assumption that discontinuities in differential variables at discrete events are modeled by explicit mappings. When there are algebraic equations restricting the allowed new values of the differential variables, explicit remapping of differential variables forces the modeler to solve the algebraic equations. To overcome this difficulty, hybrid languages use many different language elements. This article shows that only one language element is needed for this purpose: an unknown declaration, which allows the explicit declaration of a variable as unknown. The syntax and semantics of unknown declarations are discussed. Examples are given, using the Chi language, in which unknown declarations are used for modeling multi-body collision, steady-state initialization, and consistent initialization of higher index systems. It is also illustrated how the declaration of unknowns can help to clarify the structure of the system of equations, and how it can help the modeler detect structurally singular systems of equations.