My friend is now working on a simulation problem in control theory with MATLAB, where a linear matrix inequality (LMI) feasiblity problem needs to be solved at first. However, in this LMI problem, there is an equality constraint shown as follows \begin{align} M^TN=I-XY, \end{align} where $M, N$ are all nonsingular matrix, $X>0, Y>0$, and these four variables are all to be determined in the LMI, not constant matrices.
We have no idea how to express this condition with LMI solver in MATLAB, and can't find any instructions on Internet. Anyone can help us out? Thanks.