how to proof the results of matrix inequality (1) is the solution of matrix inequality (2),
$X=P^{-1}$, $K=YX^{-1}$, $P>0$ is a matrix, $0<\xi\leq 1$, $\Lambda=\hbox{diag}(\lambda_{1}I,\cdots,\lambda_{m}I)>0$ such that
inequality (1):
$$ \begin{bmatrix} -X &* &* &* &* \\ R^{\frac{1}{2}}Y &-I &* &* &* \\ Q^{\frac{1}{2}} &0 &-I &* &* \\ C_qX+D_{qu}Y &0 &0 &-\Lambda &* \\ AX+BY &0 &0 &0 &-X+B_p\Lambda B_p^T \end{bmatrix}\leq 0, $$
inequality (2):
$$ S:=\begin{bmatrix}
-\xi P &* &* &* &*\\\\
0 &-\Lambda &* &* &*\\\\
0 &0 &\frac{\alpha}{4}(\xi_k-1) &* &*\\\\
A+BK &B_p\Lambda &0 &-P^{-1} &*\\\\
C_q+D_{qu}K &0 &C_qz+D_{qu}Kz & 0 &-\Lambda
\end{bmatrix}\leq 0,
$$
