Inexact Kleinman-Newton Method for Riccati Equations

In this paper we consider the numerical solution of the algebraic Riccati equation using Newton's method. We propose an inexact variant which allows one control the number of the inner iterates used in an iterative solver for each Newton step. Conditions are given under which the monotonicity and global convergence result of Kleinman also hold for the inexact Newton iterates. Numerical results illustrate the efficiency of this method. 1. Introduction. The numerical solution of Riccati equations for large scale feedback control systems is still a formidable task. In order to reduce computing time in the context of Kleinman–Newton methods, it is mandatory that one uses iterative solvers for the solution of the linear systems occurring at each iteration. In such an approach, it is important to control the accuracy of the solution of the linear systems at each Newton step in order to gain efficiency, but not to lose the overall fast convergence properties of Newton's method. This can be achieved in the framework of inexact Newton's methods. In his classical paper, Kleinman [13] applied Newton's method to the algebraic Riccati equation, a quadratic equation for matrices of the type: