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Description
The computation of Greatest Common Right Divisors (GCRDs) is a cornerstone problem in control theory, finding applications in identifying controllable and uncontrollable subsystems, computing intersection of behaviors, and calculating the radius of uncontrollability. The methods to compute the Greatest Common Divisors (GCDs) of scalar polynomials has been researched extensively, but the study of GCRDs remains relatively limited. It is important to come up with fast and reliable ways to compute both exact and approximate GCRD, especially when dealing with approximate cases where the data is corrupted by noise.
We address the problem of computing both an exact and approximate GCRD for a given set of univariate polynomial matrices $B_1(s), \dots, B_t(s)$. We establish a key theoretical result—proving that the rank deficiency of a particular Trimmed Sylvester matrix associated with $P(s)$, where $P(s)$ is obtained by stacking $B_1(s), \dots, B_t(s)$ one below the other, corresponds to the degree of the determinant of their GCRD. Further, we propose an algorithm based on $QR$ decomposition of the Trimmed Sylvester matrix extract the coefficients of a GCRD. The Trimmed Sylvester matrices are proved to be a computationally advantageous alternative to standard generalized Sylvester matrices and we demonstrate the effectiveness of this approach through numerical examples, showing significant computational advantage compared to standard generalized Sylvester matrix methods and recent method based on state space approach when the column degrees vary a lot. Furthermore, we extend the framework to the approximate case by integrating the Trimmed Sylvester structure with Structured Total Least Norm (STLN) algorithm. Both algorithms are simple to develop, and convenient to implement.