Standard CARE and DARE data, equations, and conventions

Let $F\in \{\mathbb{R},\mathbb{C}\}$, $A\in F^{n\times n}$, $B\in F^{n\times m}$, $C\in F^{p\times n}$, and $R=R^{\ast }\in F^{m\times m}$ be positive definite. Write $Q=C^{\ast }C$ and $G=B{R}^{-1}{B}^{\ast }$. The inverse $R^{-1}$ is well-defined: A matrix admits a Cholesky factorisation with positive diagonal exactly when it is Hermitian positive definite, and that factor is unique supplies an invertible factor $R=L{L}^{\ast }$, so $R^{-1}=L^{-\ast }{L}^{-1}$. Thus $v^{\ast }Gv=\mathrm{\parallel }{L}^{-1}{B}^{\ast }v{\mathrm{\parallel }}_2^2\ge 0$ and $v^{\ast }Qv=\mathrm{\parallel }Cv{\mathrm{\parallel }}_2^2\ge 0$. The symbol ${}^{\ast }$ is transpose over $\mathbb{R}$ and conjugate transpose over $\mathbb{C}$.

The continuous algebraic Riccati equation (CARE) is $A^{\ast }X+XA-XGX+Q=0.$ Its residual is $F_c(X)=A^{\ast }X+XA-XGX+Q$, its feedback is $K_c(X)=R^{-1}{B}^{\ast }X$, and its closed-loop matrix is $A_c(X)=A-B{K}_c(X)=A-GX$.

For $X=X^{\ast }⪰0$, $S_X:=R+B^{\ast }XB$ is positive definite because $v^{\ast }{S}_{X}v=v^{\ast }Rv+(Bv{)}^{\ast }X(Bv)>0$ for every nonzero $v$. The discrete algebraic Riccati equation (DARE) is $X=Q+A^{\ast }XA-A^{\ast }XB(R+B^{\ast }XB{)}^{-1}{B}^{\ast }XA\mathrm{.}$ Its residual is $F_d(X)=Q+A^{\ast }XA-A^{\ast }XB(R+B^{\ast }XB{)}^{-1}{B}^{\ast }XA-X$. Whenever $X=X^{\ast }$, the matrix $R+B^{\ast }XB$ is positive definite if $X⪰0$. Its feedback and closed-loop matrix are $K_d(X)=(R+B^{\ast }XB{)}^{-1}{B}^{\ast }XA,A_c(X)=A-B{K}_d(X)\mathrm{.}$ A CARE solution is stabilizing when it is Hermitian and $A_c(X)$ is Hurwitz. A DARE solution is stabilizing when it is Hermitian, $R+B^{\ast }XB\succ 0$, and $A_c(X)$ is Schur stable. No inverse of $A$ is part of either definition.

For Hermitian $X$ with $S_X$ invertible, multiplication gives $(I+GX{)}^{-1}=I-B{S}_X^{-1}{B}^{\ast }X,X(I+GX{)}^{-1}=X-XB{S}_X^{-1}{B}^{\ast }X\mathrm{.}$ In either product, the additional coefficient of $B$ is $R^{-1}-S_X^{-1}-S_X^{-1}(B^{\ast }XB)R^{-1}=0$, because $S_X=R+B^{\ast }XB$. Thus the DARE is equivalently $X=Q+A^{\ast }X(I+GX{)}^{-1}A$, and its closed-loop matrix is $A_c(X)=(I+GX{)}^{-1}A$.

The associated homogeneous symplectic pencil is $N=\left[\begin{array}{cc}A& 0\\ -Q& I\end{array}\right],M=\left[\begin{array}{cc}I& G\\ 0& A^{\ast }\end{array}\right],\mathcal{P}(\alpha ,\beta )=\beta N-\alpha M\mathrm{.}$ The homogeneous-coordinate convention is that a finite eigenvalue $\lambda$ is $[\alpha :\beta ]=[\lambda :1]$ and $[1:0]$ denotes infinity.