Considering the following control system
${\dot {\mathbf {x} }}(t)=A(t)\mathbf {x} (t)+B(t)\mathbf {u} (t)+\mathbf {v} (t), \\ {\displaystyle \mathbf {y} (t)=C(t)\mathbf {x} (t)+\mathbf {w} (t),}$
where $\mathbf{x}$ is the state, $\mathbf{y}$ is the observation, $\mathbf{u}$ is the control input, $\mathbf{v}$ is the process disturbance and $\mathbf{w}$ is the obsevation noise.
The controller is given by
${\displaystyle {\dot {\hat {\mathbf {x} }}}(t)=A(t){\hat {\mathbf {x} }}(t)+B(t){\mathbf {u} }(t)+L(t)\left({\mathbf {y} }(t)-C(t){\hat {\mathbf {x} }}(t)\right)}\\ {\displaystyle {\mathbf {u} }(t)=-K(t){\hat {\mathbf {x} }}(t).}$
When the control gain $K$ and innovation gain $L$ are given and assumed to be time-invariant, can I obtain the covariance of $\mathbf{x}$ and $\mathbf{u}$ directly from the continuous-time model?