Newton’s method for the positive solution of the coupled algebraic Riccati equation applied to automatic control

Abstract

In this paper, we discuss the existence of the positive solution for the coupled algebraic Riccati equation which is usually encountered in control theory. When this equation has a positive solution, Newton's method is presented to find the minimal positive solution. Furthermore, we show some properties of Newton's iteration method. Finally, we give corresponding numerical examples to demonstrate the effectiveness of the derived iteration method.

On the Iterative Methods for the Solution of Three Types of Nonlinear Matrix Equations

“…We propose a different iterative method to compute the minimal nonnegative solution and derive convergence properties of the new iteration. We apply some properties of M-matrices in the proof and show that the new iteration method is faster than Newton's method investigated in [16] by Liu, Zhang, and Luo.…”

“…Authors Liu, Zhang, and Luo [16] investigated the Newton method to compute the positive minimal solution to the set of Riccati Equation (1).…”

“…The proof of the convergence for (5) is derived by Liu, Zhang, and Luo in [16], whereas the iterations (6) and (7) are used by them as an empirical experiment. The idea of applying approximation X (k+1) i for computing X (k+1) j , j > i, as in (6), is an effective one.…”

“…Remark 1. The existence of a nonnegative solution for the set of matrix Equation (1) is commented by [16]. Two assumptions are necessary in [14], which involve the existence of a nonnegative matrices sequence Z 1 , .…”