On the regular implementability of nD systems

In this short paper we study the problem of finding necessary and sufficient conditions for regular implementability by partial interconnection for nD system behaviors. In [M.N. Belur, H.L. Trentelman, Stabilization, pole placement and regular implementability, IEEE Trans. Automat. Control 47(5) (2002) 735–744.] such conditions were obtained in the context of 1D systems. In the present paper we show that the conditions obtained in [M.N. Belur, H.L. Trentelman, Stabilization, pole placement and regular implementability, IEEE Trans. Automat. Control 47(5) (2002) 735–744.] are no longer valid in general in the nD context.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 [P. Rocha, Canonical controllers and regular implementation of nD behaviors, Proceedings of the 16th IFAC World Congress, Prague, Czech Republic, 2005.]. Using the geometry of the underlying modules we generalize a result on regular implementability from the 1D to the nD case. Finally, we study how, in the 1D context, the conditions from [M.N. Belur, H.L. Trentelman, Stabilization, pole placement and regular implementability, IEEE Trans. Automat. Control 47(5) (2002) 735–744; P. Rocha, Canonical controllers and regular implementation of nD behaviors, Proceedings of the 16th IFAC World Congress, Prague, Czech Republic, 2005.] are connected.