Impulse controllability and observability of rectangular descriptor systems

In this note, the presence of impulsive responses in descriptor systems and how it relates to impulse controllability and impulse observability is considered. It is shown that the equivalence between impulse controllability (observability) and the existence of an impulse eliminating semistate feedback (output injection) gain, although true for square descriptor systems, does not hold for rectangular descriptor systems in general. Hence, necessary and sufficient conditions for the existence of an impulse eliminating semistate feedback (output injection) gain are presented.