Derivation and Reduction of the Singular Fornasini-Marchesini State-Space Model for~the~Class of Multidimensional Systems

This study is devoted to derive the singular Fornasini–Machesini (F-M) state-space model for a class of MIMO multidimensional (n-D) systems represented by non-causal multivariate transfer function matrices. It is shown, first, that the singular n-D F-M realisation can be generated by using the realisation methods of the standard n-D F-M models. Since the resulted realisations are usually non-minimal and deriving a minimal realisation is an extremely difficult problem, a model-order reduction method is developed. In particular, the common eigenvector approach is proposed, which significantly reveals the relationship among the reducibility, the eigenvalues and the eigenvectors associated with a given singular F-M state-space model. An example of spatially interconnected, hexagonal circuit is also given to illustrate some details and confirm the effectiveness of the proposed method.