Sensitivity analysis of the discrete-time algebraic Riccati equation
The paper deals with the perturbation analysis of the real discrete-time algebraic Riccati equation \[ A^TXA-X- (A^TXB+S) (R+B^TXB)^{-1}(B^TXA+ S^T)+Q=0, \] where \(A\), \(Q\) and the solution \(X\) are \(n\times n\) matrices with \(Q\) and \(X\) symmetric, \(B\) and \(S\) are \(n\times m\) matrices and \(R=R^T\) is an \(m\times m\) matrix. Let the coefficient matrices be subject to perturbations \(A\to A+\Delta A\), etc., which gives rise to a perturbed solution \(X+\Delta X\). The problem is to find asymptotic as well as non-local bounds for the norm of the perturbation \(\Delta X\) as functions of the norms of the perturbations in the coefficient matrices. The analysis is done without the assumption that \(R\) is positive definite and is based on an appropriate representation of the perturbed equation. Residual bounds for an approximate solution are also given. The results obtained are illustrated by examples. Thus previous results of the reviewer, \textit{P. Hr. Petkov} and \textit{N. D. Christov} [Kybernetika 29, No. 1, 18-29 (1993; Zbl 0792.93053)] are generalized. Note that by using the technique of homogeneous first order bounds proposed in [the reviewer, \textit{M. O. Stanislavova} and \textit{P. Hr. Petkov}, Linear Algebra Appl. 285, 7-31 (1998; Zbl 0933.15025)], the asymptotic results of the type presented in Theorem 3.2 on p. 603 may be improved.
real discrete-time algebraic Riccati equation
scientific article; zbMATH DE number 1354734
scientific article