Factorization principles in linear systems: theory and computation

Dr Cristian Oara (CUED)

Factorization of rational matrices is a key technical tool throughout linear systems control and identification. However, many of the useful factorizations although known to exist under some general conditions, can be computed only by imposing additional hypotheses which restrict their applicability in solving concrete problems. Such hypotheses usually restrict the location of poles/zeros, and impose conditions on the rank of the rational matrix to be factored.

We introduce some powerful factorization principles which can be used to dislocate the structure of a completely arbitrary rational matrix: poles, zeros, and singular structure. We recast several well known factorization problems formulated for linear systems as problems of dislocating simultaneously or successively parts of the structure of a rational matrix, and apply the dislocation technique to obtain solutions for inner outer, spectral, coprime, normalized coprime, and J-spectral factorizations, under the most general conditions. The proposed methods are completely general being applicable for rational matrices which may have any location of poles/zeros, arbitrary rank, and may be even polynomial or improper. All the developments are based on descriptor state-space computations and orthogonal transformations leading thus to computable formulas and numerically-sound algorithms.

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