Residual bounds of approximate solutions of the discrete-time algebraic Riccati equation

Abstract.

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