Sequential stabilizing spline algorithm for linear systems: Eigenvalue approximation and polishing

Abstract

The sequential stabilizing spline (SSS) algorithm is a remarkable algorithm for identifying linear dynamical systems. It can guarantee the stability of an estimated model by polishing the maximum modulus roots of an equation. Therefore, the root structure of an equation plays an important role in the SSS algorithm. Although the traditional power iterative method can be applied to approximate the extremal eigenvalues which have the largest modulus, it has two limitations: (1) the extremal eigenvalues must be real numbers, and (2) the structures of the extremal eigenvalues should be known a priori. In this paper, a novel power iterative method is proposed to approximate the extremal eigenvalues. Compared with the traditional power iterative method, the method in this paper can (1) determine the types of the extremal eigenvalues without prior knowledge of the matrix; (2) approximate the true values of the extremal eigenvalues regardless of their type; (3) become a worthy addition to SSS algorithm and gradient descent algorithm. Simulation examples demonstrate the effectiveness of the proposed method.