An algorithm for coprime matrix fraction description using Sylvester matrices
The problem of coprime matrix fraction description (MFD) of transfer function matrices is considered: given a \(p \times q\) transfer function matrix \(H(z)\) with a left MFD \(A^{-1}(z)B(z)\), find a right coprime MFD \(N(z)M^{-1}(z)\), where \(A(z)\), \(B(z)\), \(N(z)\) and \(M(z)\) are \(p \times p\), \(p \times q\), \(p \times q\) and \(q \times q\) polynomial matrices respectively, \(A(z) = \sum_{k=0}^{a} A_k z^{a-k}\), \(B(z) = \sum_{k=0}^{b} B_k z^{b-k}\), \(N(z) = \sum_{k=0}^{n} N_k z^{n-k}\), \(M(z) = \sum_{k=0}^{m} M_k z^{m-k}\). The problem then reduces to finding \(N_k\), \(0 \leq k \leq n\), and \(M_k\), \(0 \leq k \leq m\), which is equivalent to solving \(S_{m,n}(B,A)X=0\), where \(S_{m,n}\) is the Sylvester resultant matrix of \(A(z)\) and \(B(z)\). The authors explore the properties of \(S_{m,n}(B,A)\) in order to determine the smallest permissible values for \(m\) and \(n\), and propose an algorithm for the computation of matrices \(X\) which lead to coprime polynomial factorization. The algorithm is based on singular value decomposition and therefore avoids problems of numerical ill-conditioning. Numerical examples illustrate the advantages of the proposed method. The same approach can be used to compute column reduced equivalents of polynomial matrices.
scientific article; zbMATH DE number 1083212
scientific article