fLyapext – IQClab

The function $X_\mathrm{e}=fLyapext(X,Y,type)$ constructs the extended Lyapunov matrix $X_\mathrm{e}$ in various fashions. Given the symmetric and nonsignular matrices $X$ and $Y$ (perturb if necessary), the function can be extended the Lyapunov function $X_e$ such that:

\left(\begin{array}{cc}Y&I\\I&X\end{array}\right)\succ0

The different types are constructed as follows

\[X_\mathrm{e}=\left(\begin{array}{cc}X&I\\I&(X-Y^{-1})^{-1}\end{array}\right)\]

\[X_\mathrm{e}=\left(\begin{array}{cc}X& X-Y^{-1} \\ X-Y^{-1} &X-Y^{-1}\end{array}\right)\]

Types	Description
1-6	For the coupling condition $\left(\begin{array}{cc}Y&I\\I&X\end{array}\right)\succ0$ construct the extended Lyapunov matrix $X_\mathrm{e}$ such that $X_\mathrm{e}^{-1}=\left(\begin{array}{cc}X&\star\\ \star&\star\end{array}\right)^{-1}=\left(\begin{array}{cc}Y&\star\\ \star&\star\end{array}\right)$
7-12	For the coupling condition $\left(\begin{array}{cc}Y&I\\I&X\end{array}\right)\succ0$ construct the extended Lyapunov matrix $X_\mathrm{e}$ such that $X_\mathrm{e}^{-1}=\left(\begin{array}{cc}X&\star\\ \star&\star\end{array}\right)^{-1}=\left(\begin{array}{cc}Y^{-1}&\star\\ \star&\star\end{array}\right)$
13-15	For the coupling condition $\left(\begin{array}{ccc}Y_{11}+X_{11}&Y_{12}&-X_{12}\\ Y_{12}^T&Y_{22}&I\\-X_{12}^T&I&X_{22}\end{array}\right)\succ0$ with $X=\left(\!\!\begin{array}{cc}X_{11}&X_{12}\\ X_{12}^T&X_{22}\end{array}\!\!\right),\ Y=\left(\!\!\begin{array}{cc}Y_{11}&Y_{12}\\ Y_{12}^T&Y_{22}\end{array}\!\!\right)$ construct the extended Lyapunov matrix $X_\mathrm{e}$ such that $X_\mathrm{e}^{-1}=\left(\begin{array}{cc}X&\star\\ \star&\star\end{array}\right)^{-1}=\left(\begin{array}{cc}Y&\star\\ \star&\star\end{array}\right)$

Type	Description
1	$X_\mathrm{e}=\left(\begin{array}{cc}X&I\\I&(X-Y^{-1})^{-1}\end{array}\right)$
2	$X_\mathrm{e}=\left(\begin{array}{cc}X& X-Y^{-1} \\ X-Y^{-1} &X-Y^{-1}\end{array}\right)$
3	If $Y\succ0$ , let $Y=Y_\mathrm{c}^TY_\mathrm{c}$ (by means of a Cholexki factorization) and let $Y_\mathrm{ci}= Y_\mathrm{c}^{-1}$ . Then $X_\mathrm{e}=\left(\begin{array}{cc}X& X Y_\mathrm{c}^T- Y_\mathrm{ci}\\ Y_\mathrm{c}X- Y_\mathrm{ci}^T & Y_\mathrm{c}XY_\mathrm{c}^T -I\end{array}\right)$ If $Y$ is indefinite, let $Y=M^TJM$ with $J=\mathrm{diag}(I,-I)$ and $M_\mathrm{i}=M^{-1}$ . Then $X_\mathrm{e}=\left(\begin{array}{cc}X& X M^T- M_\mathrm{i}J\\ MX- JM_\mathrm{i}^T & MXM^T -J\end{array}\right)$
4	$X_\mathrm{e}=\left(\begin{array}{cc}X& XY-I \\ YX-I &YXY-Y\end{array}\right)$
5	If $X$ and $Y$ are positive definite, let $X=M^TM$ and $Y=N^TN$ (by means of a Choleski factorization) and let $M_\mathrm{i}=M^{-1}$ , $N_\mathrm{i}=N^{-1}$ , and $U\Sigma V^T=N_\mathrm{i}^TM_\mathrm{i}-NM^T$ . Then $X_\mathrm{e}=\left(\begin{array}{cc}M_\mathrm{i}^T&0\\N&U\Sigma^{\frac{1}{2}}\end{array}\right)^{-1} \left(\begin{array}{cc}M&V\Sigma^{\frac{1}{2}}\\N_\mathrm{i}^T&0\end{array}\right)$ If $X$ or $Y$ is indefinite, let $X=M^TJ_xM$ and $Y=N^TJ_yN$ with $J_x=\mathrm{diag}(I,-I)$ and $J_y=\mathrm{diag}(I,-I)$ . Also let $M_\mathrm{i}=M^{-1}$ , $N_\mathrm{i}=N^{-1}$ and $U\Sigma V^T=N_\mathrm{i}^TM_\mathrm{i}-J_yNM^TJ_x$ . Then $X_\mathrm{e}=\left(\begin{array}{cc}M_\mathrm{i}^T&0\\J_yN&U\Sigma^{\frac{1}{2}}\end{array}\right)^{-1} \left(\begin{array}{cc}J_xM&V\Sigma^{\frac{1}{2}}\\N_\mathrm{i}^T&0\end{array}\right)$
6	If $X\succ0$ , let $X=X_\mathrm{c}^TX_\mathrm{c}$ (by means of a Cholexki factorization) and let $X_\mathrm{ci}= X_\mathrm{c}^{-1}$ . Then $X_\mathrm{e}=\left(\begin{array}{cc}X& X_\mathrm{c}^T- Y^{-1}X_\mathrm{ci}\\ X_\mathrm{c}-X_\mathrm{ci}^TY^{-1} &I-X_\mathrm{ci}^TY^{-1}X_\mathrm{ci}\end{array}\right)$ If $X$ is indefinite, let $X=M^TJM$ with $J=\mathrm{diag}(I,-I)$ and $M_\mathrm{i}=M^{-1}$ . Then $X_\mathrm{e}=\left(\begin{array}{cc}X&M^TJ-Y^{-1}M_\mathrm{i}\\JM- M_\mathrm{i}^TY^{-1} & J-M_\mathrm{i}^TY^{-1}M_\mathrm{i}\end{array}\right)$
7	$X_\mathrm{e}=\left(\begin{array}{cc}X&I\\I&(X-Y)^{-1}\end{array}\right)$
8	$X_\mathrm{e}=\left(\begin{array}{cc}X& X-Y \\ X-Y&X-Y\end{array}\right)$
9	If $Y\succ0$ , let $Y=Y_\mathrm{c}^TY_\mathrm{c}$ (by means of a Cholexki factorization) and let $Y_\mathrm{ci}= Y_\mathrm{c}^{-1}$ . Then $X_\mathrm{e}=\left(\begin{array}{cc}X& XY_\mathrm{ci}-Y_\mathrm{c}^T\\ Y_\mathrm{ci}^TX-Y_\mathrm{c}&Y_\mathrm{ci}^TXY_\mathrm{ci} -I\end{array}\right)$ If $Y$ is indefinite, let $Y=M^TJM$ with $J=\mathrm{diag}(I,-I)$ and $M_\mathrm{i}=M^{-1}$ . Then $X_\mathrm{e}=\left(\begin{array}{cc}X& XM_\mathrm{i}-M^TJ\\ M _\mathrm{i}^TX-JM& M_\mathrm{i}^TXM_\mathrm{i}-J\end{array}\right)$
10	$X_\mathrm{e}=\left(\begin{array}{cc}X&XY^{-1}-I \\Y^{-1} X-I &Y^{-1}XY^{-1} -Y^{-1} \end{array}\right)$
11	If $X$ and $Y$ are positive definite, let $X=M^TM$ and $Y=N^TN$ (by means of a Choleski factorization) and let $M_\mathrm{i}=M^{-1}$ , $N_\mathrm{i}=N^{-1}$ , and $U\Sigma V^T=N^TM_\mathrm{i}-N_\mathrm{i}^TM^T$ . Then $X_\mathrm{e}=\left(\begin{array}{cc}M_\mathrm{i}^T&0\\N_\mathrm{i}^T&U\Sigma^{\frac{1}{2}}\end{array}\right)^{-1} \left(\begin{array}{cc}M&V\Sigma^{\frac{1}{2}}\\N&0\end{array}\right)$ If $X$ or $Y$ is indefinite, let $X=M^TJ_xM$ and $Y=N^TJ_yN$ with $J_x=\mathrm{diag}(I,-I)$ and $J_y=\mathrm{diag}(I,-I)$ . Also let $M_\mathrm{i}=M^{-1}$ , $N_\mathrm{i}=N^{-1}$ and $U\Sigma V^T=NM_\mathrm{i}-J_yN_\mathrm{i}^TM^TJ_x$ . Then $X_\mathrm{e}=\left(\begin{array}{cc}M_\mathrm{i}^T&0\\J_yN_\mathrm{i}^T&U\Sigma^{\frac{1}{2}}\end{array}\right)^{-1} \left(\begin{array}{cc}J_xM&V\Sigma^{\frac{1}{2}}\\N&0\end{array}\right)$
12	If $X\succ0$ , let $X=X_\mathrm{c}^TX_\mathrm{c}$ (by means of a Cholexki factorization) and let $X_\mathrm{ci}= X_\mathrm{c}^{-1}$ . Then $X_\mathrm{e}=\left(\begin{array}{cc}X&X_\mathrm{c}^T- YX_\mathrm{ci}\\ X_\mathrm{c}-X_\mathrm{ci}^TY&I-X_\mathrm{ci}^TYX_\mathrm{ci}\end{array}\right)$ If $X$ is indefinite, let $X=M^TJM$ with $J=\mathrm{diag}(I,-I)$ and $M_\mathrm{i}=M^{-1}$ . Then $X_\mathrm{e}=\left(\begin{array}{cc}X&M^TJ-YM_\mathrm{i}\\JM- M_\mathrm{i}^TY&J-M_\mathrm{i}^TYM_\mathrm{i}\end{array}\right)$
13	$X_\mathrm{e}=\left(\begin{array}{ccc}X_{11}&X_{12}&X_{13}\\ X_{12}^T&X_{22}&X_{23}\\ X_{13}^T&X_{23}^T&X_{33} \end{array}\right)$ where $X_{13}=\left(\begin{array}{cc}M_{12}&I\end{array}\right)$ , $X_{23}=\left(\begin{array}{cc}M_{22}&0\end{array}\right)$ , $X_{33}=\left(\begin{array}{cc}M_{22}&0\\0&M_{33}\end{array}\right)$ , $M_{12}=X_{12}+Y_{12}Y_{22}^{-1}$ , $M_{22}=X_{22}-Y_{22}^{-1}$ , $M_{3}=X_{11}+Y_{11}-Y_{12}Y_{22}^{-1}Y_{12}^T-M_{12}M_{22}^{-1}M_{12}^T$
14	$X_\mathrm{e}=\left(\begin{array}{ccc}X_{11}&X_{12}&X_{13}\\ X_{12}^T&X_{22}&X_{23}\\ X_{13}^T&X_{23}^T&X_{33} \end{array}\right)$ where $X_{13}=\left(\begin{array}{cc}A&C\end{array}\right)$ , $X_{23}=\left(\begin{array}{cc}B&A^T\end{array}\right)$ , $X_{33}=\left(\begin{array}{cc}B&A^T\\A&C\end{array}\right)$ , $A=X_{12}+Y_{12}Y_{22}^{-1}$ , $B=X_{22}-Y_{22}^{-1}$ , $C=X_{11}+Y_{11}-Y_{12}Y_{22}^{-1}Y_{12}^T$
15	$X_\mathrm{e}=\left(\begin{array}{ccc}X_{11}&X_{12}&X_{13}\\ X_{12}^T&X_{22}&X_{23}\\ X_{13}^T&X_{23}^T&X_{33} \end{array}\right)$ where $X_{13}=\left(\begin{array}{cc}A&B\end{array}\right)$ , $X_{23}=\left(\begin{array}{cc}C&D\end{array}\right)$ , $X_{33}=\left(\begin{array}{cc}E&F\\F^T&G\end{array}\right)$ , $A=X_{12}Y_{22}+Y_{12}$ , $B=Y_{11}+X_{12}Y_{12}^T$ , $C=X_{22}Y_{22}-I$ , $D=X_{22}Y_{12}^T$ , $E=Y_{22}X_{22}Y_{22}-Y_{22}$ , $F=C^TY_{12}^T$ , $G=Y_{12}X_{22}Y_{12}^T-Y_{11}$