Toward a Rational Matrix Approximation of the Parameter-Dependent Riccati Equation Solution

Abstract

This paper considers the problem of solving parameter-dependent Riccati equations. In this paper, a tractable iterative scheme involving mainly additions and multiplications is developed for finding solutions to arbitrary accuracy. It is first presented in the parameter-independent case and then extended to the parametric case. It hinges upon two results: (i) a palindromic quadratic polynomial matrix characterization of the matrix sign and square root functions. (ii) a particular representation of parameterdependent matrices with negative and positive power series with respect to parameters. Several numerical examples are given throughout the paper to prove the validity of the proposed results.

Mixed matrix sign function/DFT inversion method for solving parameter-dependent Riccati equation

“…Furthermore, some recent works have tried to extend these results to the parameter-dependent case (Rice et al, 2010), (Guerra et al, 2012) and (Guerra et al, 2014). Specifically, the method developed in (Rice et al, 2010) consists of using the main iteration for the matrix sign function, which is Newton's method associated with a linear fractional transformation (LFT) parametric dependence.…”

“…In (Guerra et al, 2014), a promising palindromic quadratic polynomial matrix characterization of the matrix sign is used. Although efficient, this method is limited to those cases where a spectral radius condition is verified.…”