Reduced-Order Algorithm for Eigenvalue Assignment of Singularly Perturbed Linear Systems

Abstract

In this paper, we present an algorithm for eigenvalue assignment of linear singularly perturbed systems in terms of reduced-order slow and fast subproblem matrices. No similar algorithm exists in the literature. First, we present an algorithm for the recursive solution of the singularly perturbed algebraic Sylvester equation used for eigenvalue assignment. Due to the presence of a small singular perturbation parameter that indicates separation of the system variables into slow and fast, the corresponding algebraic Sylvester equation is numerically ill-conditioned. The proposed method for the recursive reduced-order solution of the algebraic Sylvester equations removes ill-conditioning and iteratively obtains the solution in terms of four reduced-order numerically well-conditioned algebraic Sylvester equations corresponding to slow and fast variables. The convergence rate of the proposed algorithm is Oε, where ε is a small positive singular perturbation parameter.