In many situations the initial state is not on the manifold and a projection is required. Instead of applying a mathematically straightforward minimum norm technique [5], this paper derives a physically consistent projection based on previously derived conservation principles [7,8,9]. The projection is implicit in the system equations and requires integration of the time-derivatives of the constrained state variables. Combined with the algebraic equations that define the manifold, this system of equations can be solved.
In many cases, system variables are completely determined by exogeneous variables and conservation principles may be violated. In other situations, exogeneous variables may be of an impulsive nature (possibly modulated by the system) and propagate into the system. Though no dependency is present, integrating the impulses returns nonzero values and their areas have to be accounted for in the conserved variable values.
The described approach is the dual to solving the state variable gradients on the manifold. Instead of differentiating the algebraic equations, the time-derivative dependencies are integrated. Note that the algebraic equations have to be given. If these are omitted after deriving the behavior evolution gradient on the manifold, the projection cannot be derived.
The methodology is restricted to semi-explicit DAEs with linear constraints. A number of examples have demonstrated its applicability to physical system models that can be rather complex. In general, linear physical system models derived from the bond graph modeling methodology [4,15] can be handled, which constitutes a large class of dynamic physical systems.