I need to solve discrete time Lur'e equations using MATLAB. These equations are as shown below: $$ P-A^TPA=H^TH....(1) $$ $$ C^T-A^TPB=H^TJ....(2) $$ $$ D+D^T-B^TPB=J^TJ...(3) $$ The Known matrices are ($A,B,C,D$) whereas ($P,H,J$) are to be found.
EDIT:
From equation (2) we can find:
$$
H^T=(C^T-A^TPB)J^{-1}...(4)
$$
Putting (4) in (1) and using (3) we get:
$$
A^TPA-P+(C^T-A^TPB)(R-B^TPB)^{-1}(C^T-B^TPBA)=0....(A)
$$
But the problem is that in MATLAB the "dare" command solves following equation:
$$
A^TXA-X−(A^TXB+S)(B^TXB+R)^{−1}(B^TXA+S^T)+Q=0
$$
How can I solve equation (A) using MATLAB. Any help will be appreciated. Thanks in advance