Kalman filter is an optimal state estimator, in terms of minimum-variance error reconstruction. It has the structure of a state observer. Let the linear system
\[\begin{split}\begin{cases}
\dot{\mathbf{x}} = \mathbf{A} \mathbf{x} + \mathbf{B} \mathbf{u} + \mathbf{B}_d \mathbf{d} \\
\mathbf{y} = \mathbf{C} \mathbf{x} + \mathbf{D}_u \mathbf{u} + \mathbf{D}_d \mathbf{d} + \mathbf{D}_r \mathbf{r} \ ,
\end{cases}\end{split}\]
\[\begin{split}\begin{cases}
\dot{\mathbf{o}} = \mathbf{A} \mathbf{o} + \mathbf{B} \mathbf{u} + \mathbf{L} ( \mathbf{y} - \mathbf{y}_o ) \\
\mathbf{y}_o = \mathbf{C} \mathbf{o} + \mathbf{D}_u \mathbf{u}
\end{cases}\end{split}\]
31.2.7.1. Error dynamics
Let the error be defined as \(\mathbf{e} := \mathbf{x} - \mathbf{o}\), its dynamical equation reads
(31.10)\[\begin{split}\begin{aligned}
\dot{\mathbf{e}} & = \left( \mathbf{A} - \mathbf{L} \mathbf{C} \right) \mathbf{e} + \left( \mathbf{B}_d - \mathbf{L} \mathbf{D}_d \right) \mathbf{d} - \mathbf{L} \mathbf{D}_r \mathbf{r} \\
\dot{\mathbf{e}} & = \mathbf{A}_e \mathbf{e} + \mathbf{B}_{ed} \mathbf{d} + \mathbf{B}_{er} \mathbf{r} \\
\dot{\mathbf{e}} & = \mathbf{A}_e \mathbf{e} + \boldsymbol\xi \ ,
\end{aligned}\end{split}\]
with the definition of the noise input in the dynamical equation of the error, \(\boldsymbol\xi = \left( \mathbf{B}_d - \mathbf{L} \mathbf{D}_d \right) \mathbf{d} - \mathbf{L} \mathbf{D}_r \mathbf{r}\).
\[\begin{split}\begin{aligned}
\dot{\mathbf{e}}
& = \dot{\mathbf{x}} - \dot{\mathbf{o}} = \\
& = \mathbf{A} \mathbf{x} + \mathbf{B} \mathbf{u} + \mathbf{B}_d \mathbf{d} - \left( \mathbf{A} \mathbf{o} + \mathbf{B} \mathbf{u} + \mathbf{L} ( \mathbf{y} - \mathbf{y}_o )\right) = \\
& = \left( \mathbf{A} - \mathbf{L} \mathbf{C} \right) \mathbf{e} + \left( \mathbf{B}_d - \mathbf{L} \mathbf{D}_d \right) \mathbf{d} - \mathbf{L} \mathbf{D}_r \mathbf{r}
\end{aligned}\end{split}\]
Let the exogenous input \(\mathbf{d}\) and the measurement input \(\mathbf{r}\) be random signals, thus the input \(\boldsymbol\xi\) of the error dynamical equation (31.10) is a random signal with correlation
\[\begin{split}\begin{aligned}
\mathbf{R}_{\boldsymbol\xi \boldsymbol\xi}(t,\tau) := \mathbb{E}\left[ \boldsymbol\xi(t) \boldsymbol\xi^*(\tau) \right]
& = \mathbf{B}_{ed} \mathbf{R}_{\mathbf{d}\mathbf{d}}(t,\tau) \mathbf{B}_{ed}^*
+ \mathbf{B}_{ed} \mathbf{R}_{\mathbf{d}\mathbf{r}}(t,\tau) \mathbf{B}_{er}^* + \\
& + \mathbf{B}_{er} \mathbf{R}_{\mathbf{r}\mathbf{d}}(t,\tau) \mathbf{B}_{ed}^*
+ \mathbf{B}_{er} \mathbf{R}_{\mathbf{r}\mathbf{r}}(t,\tau) \mathbf{B}_{er}^*
\end{aligned}\end{split}\]
Using the general results for linear systems for white noise as an input \(\boldsymbol\xi\), the covariance matrix \(\mathbf{P}_{\mathbf{e}}(t) := \mathbf{R}_{\mathbf{e}\mathbf{e}}(t,t)\) of the error - governed by eqaution (31.10) - satisfies the Lyapunov equation (23.3)
\[\dot{\mathbf{P}}_{\mathbf{e}} = \mathbf{A}_e \mathbf{P}_{\mathbf{e}} + \mathbf{P}_{\mathbf{e}} \mathbf{A}_e^* + \boldsymbol\sigma_{\xi}^2 \ ,\]
being
\[\boldsymbol\sigma_{xi}^2 = \dots \]
