## a matrix problem, a robust control problem

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):

$$S:=\begin{bmatrix} -\xi P &* &* &* &*\\\\ 0 &-\Lambda &* &* &*\\\\ 0 &0 &\frac{\alpha}{4}(\xi_k-1) &* &*\\\\ A+BK &B_p\Lambda &0 &-P^{-1} &*\\\\ C_q+D_{qu}K &0 &C_qz+D_{qu}Kz & 0 &-\Lambda \end{bmatrix}\leq 0,$$

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What are R and Q (which do not appear in (2))? Same question concerns $\alpha,z$ and $\xi_k$. And do we assume that X is your variable? Are there other variables? – Jacques Carette Mar 28 2010 at 3:12