Proper exponential approximation of non proper compensators: the MIMO case

In some control problems, it may be convenient, at least from the analysis point of view, to use non proper compensators. However, as far as their implementation is concerned, proper approximations have to be designed. In the present paper, we first show how exponential approximations can be rather easily designed (Lemma 1). Then, we characterize, in geometric terms, the external properness of an implicit description (Theorem 1 and Corollary 1). Finally, the combination of those two results solves the problem of proper exponential approximation and generalizes to the MIMO case a previous result from Bonilla and Lozano (Theorem 2).