  A new finite atlas of overlapping balanced canonical forms for multivariate
discrete-time lossless systems is presented. The canonical forms have the
property that the controllability matrix is positive upper triangular up to a
suitable permutation of its columns. This is a generalization of a similar
balanced canonical form for continuous-time lossless systems. It is shown that
this atlas is in fact a finite sub-atlas of the infinite atlas of overlapping
balanced canonical forms for lossless systems that is associated with the
tangential Schur algorithm; such canonical forms satisfy certain interpolation
conditions on a corresponding sequence of lossless transfer matrices. The
connection between these balanced canonical forms for lossless systems and the
tangential Schur algorithm for lossless systems is a generalization of the same
connection in the SISO case that was noted before. The results are directly
applicable to obtain a finite sub-atlas of multivariate input-normal canonical
forms for stable linear systems of given fixed order, which is minimal in the
sense that no chart can be left out of the atlas without losing the property
that the atlas covers the manifold.
