An algebra-geometric approach for model reduction

This paper presents an algebra-geometric approach for determining the time-domain reduced-order models and frequency-domain reduced-degree models of large-scale multivariable systems. First, the structures of the canonical state-space representations and corresponding matrix fraction descriptions of general multivariable systems are introduced, and the associated characteristic ??-matrices are defined. Next, the divisors and spectral decomposition theorems for the nonsingular characteristic ??-matrices, which may not be regular or monic, are developed by using the algebraic and geometric properties of multivariable system structures. Then, the derived algebra-geometric theorems are used to develop a frequency-domain aggregation method and a time-domain aggregation method for the model reduction of large-scale multivariable systems. Finally, the newly developed matrix sign functions [1] in conjunction with the aggregation method are used to obtain the reduced-order and reduced-degree models of large-scale multivariable systems without assuming that the eigenvalues of the original systems are known and/or the singularly-perturbed models are available.