The non-symmetric discrete algebraic Riccati equation and canonical factorization of rational matrix functions on the unit circle
A rational \(m\times m\) matrix function \(R\) with no poles on the unit circle is said to admit a right canonical factorization if \(R\) can be factored as \(R(z)=\Psi(z)\Theta(z)\), where \(\Theta\) and \(\Psi\) are regular \(m\times m\) matrix rational functions such that \(\Theta(z)\) and \(\Theta(z)^{-1}\) have no poles inside the unit disc, while \(\Psi(z)\) and \(\Psi(z)^{-1}\) have no poles outside the unit disc. The main result of the paper says that if \(R\) has a so-called stable representation, then \(R\) admits a right canonical factorization with respect to the unit circle if and only if the algebraic Riccati equation associated with the stable representation has a stabilizing solution. In this case, the right canonical factorization is explicitly given. When the values of \(R\) on the unit circle are hermitian matrices, the corresponding associated Riccati equation is a symmetric one originated from stochastic realization theory and \(R(z)\) plays the role of the so called Popov function. As an application of the main result, the case when the corresponding Toeplitz operator attached to \(R\) is tri-diagonal and the case when \(R\) has no poles on the closed unit disc are analysed. Also, to illustrate how one can use the main result of the paper, an example is given. Riccati iteration and the finite sections method is used to analyze the invertibility of the block Toeplitz operator attached to \(R\).
scientific article; zbMATH DE number 5774250
scientific article