Regular implementability of nD systems: Results and open problems

We study the problem of finding necessary and sufficient conditions for regular implementability by partial interconnection for nD system behaviors. In [3] such conditions were obtained in the context of 1D systems. In the present paper we show that the conditions obtained in [3] are no longer valid in general in the nD context. We also show that under additional assumptions, the conditions still remain relevant. We also reinvestigate the conditions for regular implementability by partial interconnection in terms of the canonical controller that were obtained in [11]. Using the geometry of the underlying modules we generalize a result on regular implementability from the 1D to the nD case. Also, we study how, in the 1D context, the conditions from [3] and [11] are connected. Finally, we pose some open problems.