Sliding motion of discontinuous dynamical systems described by semi-implicit index one differential algebraic equations

We have proposed an approach to derive a continuous system of differential algebraic equations (DAE) of index one that is dynamically equivalent to a discontinuous index one DAE system. This involves augmenting the convex combination of the ordinary differential equations with the algebraic equations from individual models. This result is proved by using the implicit function theorem. This procedure is illustrated with the help of an ideal gas–liquid system in which the algebraic variables can be expressed as explicit functions of differential variables. It is also demonstrated with an example from a soft-drink manufacturing process, in which, it is difficult to express the algebraic variables as explicit functions of differential variables. Through computer simulation, it is shown that the equivalent dynamic DAE system and the discontinuous DAE system have identical solutions. The proposed method is several orders of magnitude more efficient than the procedure that works with the discontinuous system of DAEs.