Canonical lossless state-space systems: staircase forms and the Schur algorithm

A new finite atlas of overlapping balanced canonical forms for multivariate discrete-time loss-less systems is presented. The canonical forms have the property that the controllability matrix ispositive upper triangular up to a suitable permutation of its columns. This is a generalization of asimilar balanced canonical form for continuous-time lossless systems. It is shown that this atlas isin fact a finite sub-atlas of the infinite atlas of overlapping balanced canonical forms for losslesssystems that is associated with the tangential Schur algorithm; such canonical forms satisfy certaininterpolation conditions on a corresponding sequence of lossless transfer matrices. The connectionbetween these balanced canonical forms for lossless systems and the tangential Schur algorithmfor lossless systems is a generalization of the same connection in the SISO case that was notedbefore. The results are directly applicable to obtain a finite sub-atlas of multivariate input-normalcanonical forms for stable linear systems of given fixed order, which is minimal in the sense thatno chart can be left out of the atlas without losing the property that the atlas covers the manifold.