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Newton's Method for a Rational Matrix Equation Occurring in Stochastic Control

Authors: Tobias Damm, Diederich Hinrichsen

Published: 2001 (Journal Paper)

Source: Linear Algebra and its Applications

Algorithm: Newton's Method

DOI: 10.1016/S0024-3795(00)00144-0

Summary

Develops a unified analytical framework for a family of rational matrix equations generalizing both the CARE and DARE, arising in stochastic H-infinity control. Shows that Newton's method converges from any stabilizing initial point by exploiting the fact that the derivative of the generalized Riccati operator is a generalized Lyapunov operator, combined with resolvent positivity arguments.

Abstract

We consider a general class of rational matrix equations, which contains the continuous (CARE) and discrete (DARE) algebraic Riccati equations as special cases. Equations of this type have been encountered in earlier work on H-infinity-type disturbance attenuation for stochastic linear systems. We develop a unifying framework for the analysis of these equations based on the theory of (resolvent) positive operators and show that they can be solved by Newton's method starting at an arbitrary stabilizing matrix.

Primary

Paper sciencedirect.com

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DOI 10.1016/S0024-3795(00)00144-0

Tags

  • Algebraic Riccati equation

  • Newton's method

  • Stochastic control

  • Rational matrix equations

  • Lyapunov equations

  • Positive operators

  • H-infinity control