Let \documentclass[12pt]{minimal}
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$\tilde{X}$\end{document} be a Hermitian matrix which approximates the unique Hermitian positive semi-definite solution \documentclass[12pt]{minimal}
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$X$\end{document} to the discrete-time algebraic Riccati equation (DARE) \documentclass[12pt]{minimal}
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\[ X-F^{\rm H}XF+F^{\rm H}XG_1(G_2+G_1^{\rm H}XG_1)^{-1}G_1^{\rm H}XF+C^{\rm H}C=0, \] \end{document} where \documentclass[12pt]{minimal}
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$F \in{\cal C}^{n \times n}$\end{document}, \documentclass[12pt]{minimal}
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$C_2 \in{\cal C}^{m \times m}$\end{document} is Hermitian positive definite, \documentclass[12pt]{minimal}
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$G_1 \in{\cal C}^{n \times m}, C \in{\cal C}^{r \times n}$\end{document}, the pair \documentclass[12pt]{minimal}
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$(F,G_1)$\end{document} is stabilizable, and the pair \documentclass[12pt]{minimal}
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$(C,F)$\end{document} is detectable. Assume that \documentclass[12pt]{minimal}
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$I+G\tilde{X}$\end{document} is nonsingular, and \documentclass[12pt]{minimal}
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$(I+G\tilde{X})^{-1}F$\end{document} is stable. Let \documentclass[12pt]{minimal}
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$G=G_1G_2^{-1}G_1^{\rm H}, H=C^{\rm H}C$\end{document}, and let \documentclass[12pt]{minimal}
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\[ \hat{R}=\tilde{X}-F^{\rm H}\tilde{X}(I+G\tilde{X})^{-1}F-H \] \end{document} be the residual of the DARE with respect to \documentclass[12pt]{minimal}
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$\tilde{X}$\end{document}. Define the linear operator \documentclass[12pt]{minimal}
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$\vec L$\end{document} by \documentclass[12pt]{minimal}
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\[ {\vec L}W=W-F^{\rm H}(I+\tilde{X}G)^{-1}W(I+G\tilde{X})^{-1}F,\;\;\;\;\; W=W^{\rm H} \in{\cal C}^{n \times n}. \] \end{document} The main result of this paper is: If \documentclass[12pt]{minimal}
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\[ \epsilon \equiv \|{\vec L}^{-1}\hat{R}\| \leq \frac{l}{\gamma(2\phi^2+2\phi\sqrt{\phi^2+l}+l)}, \] \end{document} where \documentclass[12pt]{minimal}
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$\|\;\|$\end{document} denotes any unitarily invariant norm, and \documentclass[12pt]{minimal}
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\[ l=\|{\vec L}^{-1}\|^{-1},\;\;\;\; \phi=\|(I+G\tilde{X})^{-1}F\|_2,\;\;\;\; \gamma=\|(I+G\tilde{X})^{-1}G\|_2, \] \end{document} then \documentclass[12pt]{minimal}
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\[ \|\tilde{X}-X\| \leq \frac{2l\epsilon}{(1+\gamma\epsilon)l +\sqrt{(1+\gamma\epsilon)^2l^2-4(\phi^2+l)\gamma l\epsilon}} \leq \frac{2\|{\vec L}^{-1}\hat{R}\|} {1+\gamma\|{\vec L}^{-1}\hat{R}\|}. \] \end{document}
