Riccati equation in matlabs icare solver

Question

I'm using the icare riccati equation solver from MATLAB to solve lqr problems were the constraint is given by the equation

$My^{\prime} = Ay+Bu$

where for context M (mass matrix), A (stiffness matrix) and B come from a fem discretization. For the objective I use the standard form

$r(y,u) = y^{\top}Qy + u^{\top}Ru$

The icare solver in MATLAB solves the riccati equation of the following form

$A^{\top}XE + E^{\top}XA + E^{\top}XGXE -(E^{\top}XB+S)R^{-1}(B^{\top}XE+S^{\top})+Q=0$

To fit my problem I set $E = M$, $S=0$ and $G = 0$. What I don't understand is how the equation evolves from the classical CARE? So I'm not sure what's the trick or if there is a deeper derivation?

Answer 1

I found the trick you just have to substitute

$\hat{y} = My$

then it's analogous to the derivation of the classical CARE.