Stability analysis of shift-invariant multidimensional systems

In a state-space model of a class of linear discrete-time shift-invariant multivariable multidimensional dynamical systems, the problem of internal stability of a system is studied from various aspects. A sequence of equivalent statements is presented to characterize the necessary and sufficient conditions for the internal stability of a multidimensional dynamics. These statements are generalized to further enhance ones for meeting the stability of a mixed multidimensional-and-multicircular dynamic, while they degenerate into the stability condition of a circulant matrix when the underlining structure entirely degenerates. As a related topic, a model degree reduction problem is studied by the balancing realization method in a class of linear shift-invariant multivariable multidimensional systems.