An improved Toeplitz algorithm for polynomial matrix null-space computation

Abstract

In this paper we present an improved algorithm to compute the minimal null-space basis of polynomial matrices, a problem which has many applications in control and systems theory. This algorithm takes advantage of the block Toeplitz structure of the Sylvester matrix associated with the polynomial matrix. The analysis of algorithmic complexity and numerical stability shows that the algorithm is reliable and can be considered as an efficient alternative to the well-known pencil (state-space) algorithms found in the literature.

The Generalized Schur Algorithm and Some Applications

“…As described in [8,19,20], the above problem is equivalent to that of computing the null-space of a related block-Toeplitz matrix. Algorithms to solve this problem are proposed in [8,19] but they do not explicitly exploit the structure of the involved matrix. Algorithms to solve related problems have also been described in the literature, e.g., in [8,19,21,22].…”

“…In this paper, we propose an algorithm for computing the null-space of polynomial matrices based on a variant of the GSA for computing the null-space of a related band block-Toeplitz matrix [8].…”

“…wherem = m(δ + n b ),n = nn b , with n b = γ + 1 the number of block columns of T, that can be determined, e.g., by the algorithm described in [8]. Hence, the problem of computing the null-space of the polynomial matrix in Equation (12) is equivalent to the problem of computing the null-space of the matrix in Equation (13).…”

“…As already mentioned in Section 5.1, the number of desired blocks n b of the matrix T in Equation (13) can be computed as described in [8]. For the sake of simplicity, in the considered examples, we choose n b large enough to compute the null-space of T.…”

“…Finally, we propose a GSA-based approach for computing a null-space basis of a polynomial matrix, which is an important problem in several systems and control applications [7,8]. For instance, the computation of the null-space of a polynomial matrix arises when solving the column reduction problem of a polynomial matrix [9,10].…”