Lowering the order of input shifts in multi-input discrete-time generalized state-space systems

The problem of lowering the order of input forward shifts in multi-input generalized state space systems by generalized state transformations is studied through the language of differential forms. Necessary and sufficient conditions are given for the local existence of such transformations. These conditions have been formulated in terms of integrability of certain subspaces of one-forms, classified according to their relative degree. The sufficiency part of the proof gives a constructive procedure (up to the integration of one-forms) for finding these generalized state transformations. In a particular case, these conditions show when it is possible to transform a generalized state space representation into the classical state equations