I am interested in solving the multi-model discrete-time LQG problem, that is to synthesize a stabilizing output feedback controller that stabilizes all systems with matrices $A_i, B_i,C_i,D_i$ with $i=1, ...,n$ . The problem for the single model case requires solving two discrete Riccati equations of the form:
$A_x^TXA_x-X-A_x^TXB_x(I+B_x^TXB_x)^{-1}B_x^TXA_x+Q_x = 0$
and
$A_y^TYA_y-Y-A_y^TYC_y(I+C_y^TYC_y)^{-1}C_y^TYA_y+Q_y = 0$.
(more information about how these 2 Riccati equations lead to a solution for the single model discrete-time LQG problem at Zhou, K., Doyle, J. C., & Glover, K. (1996). Robust and optimal control (Vol. 40, p. 146). New Jersey: Prentice hall. Paragraph 21.5)
Is there a way to convert both discrete Riccati equations to two LMIs, so that we can stabilize all $i$ models, by performing an optimization constrained by $2*i$ LMI's?