I need help to convert the following discrete-time Riccati eqn. into an LMI with two decision variables $(K, P)$:
$$ (A+BK)^T P (A+BK) - P +Q + K^T R K \prec 0$$
The trick, at least for the continuous-time variant here, seems to be with a congruence transformation, i.e., multiply both LHS and RHS of the above by $Y = P^{-1}$, then, define $L = KY$ and apply Schur. I did the first two pieces, but, wasn't able to do the Schur part. (This also seems to be a useful reference)
Also, as I do not have the control systems toolbox for Matlab, can someone provide me (or point me) with (to) a Matlab implementation for solving the DARE eqn, i.e., the second eqn. here?
You have $Y - YQY - (AY + BL)^TY^{-1}(AY+BL) - L^TRL \succeq 0$.
Apply Schur three times, or see it as $$Y - \begin{bmatrix}AY + BL\\L\\Y\end{bmatrix}^T\begin{bmatrix}Y & 0 &0\\ 0 & R^{-1} & 0 \\0 & 0 & Q^{-1}\end{bmatrix}^{-1}\begin{bmatrix}AY + BL\\L\\Y\end{bmatrix}\succeq 0$$
and apply Schur once