I have the state-space model
$$\dot x = Ax + Bu + W$$ $$y = Cx + Du + E$$
where $E\in \mathbb{R}^{n \times (N-1)}$ is a noise vector and $W\in \mathbb{R}^{p \times (N-1)}$ a disturbance vector.
What I need to do is to find the covariance matrix of $W$ and $E$:
$$\begin{bmatrix} \Sigma_{ww} & \Sigma_{we} \\ \Sigma_{ew} & \Sigma_{ee} \end{bmatrix} = \frac{1}{N-1}\begin{bmatrix} WW^T & WE^T \\ EW^T & EE^T \end{bmatrix}.$$
Assume that we calling they for these matrices instead.
$$ S = \Sigma_{we} \in \mathbb{R}^{n\times p}$$ $$ Q = \Sigma_{ww} \in \mathbb{R}^{n\times n}$$ $$ R = \Sigma_{ee} \in \mathbb{R}^{p\times p}$$
Question:
If I want to solve the discrete algebraic Riccati equation for computing the Kalman gain matrix.
$$X = \tilde A^TX\tilde A - E^TXE - (\tilde A^TX\tilde B + S)(\tilde B^TX\tilde B + R)^{-1}(\tilde A^TX\tilde B + S)^T + Q$$
Where $E = 0, \tilde B = C^T, \tilde A = A^T$
But here is the problem. $X$ is a square matrix of the size $n \times n$ and this part $(\tilde A^TX\tilde B + S)^T$ result a matrix of the size $p \times n$ where $(\tilde B^TX\tilde B + R)^{-1}$ have the size $n \times n$. $\tilde (A^TX\tilde B + S)$ have the size $n\times p$. They cannot be multiplied...because $S$ is not square as $A$ is.
So how can I solve the algebraic Riccati equation if $S$ is not square?
Update:
Here is the source of the equations and formulas. It's from the book Subspace Methods for System Identification by Tohru Katayama, ISBN-10: 1852339810
