A standard state-space for Kalman filter is: \begin{align} x_{t+1}&= F x_{t} + Gw_t\\ y_t&= Hx_t + v_t. \end{align}
We know that there exists a similarity transform (i.e., an invertible matrix) such that an equivalent input-output relation can be described by the state-space \begin{align} x_{t+1}&= \begin{pmatrix}F_{obs}&0 \\ F_{mixed}& F_{Non-Obs} \end{pmatrix} x_{t} + \tilde{G} w_t\\ y_t&= \begin{pmatrix}H_{obs} &0 \end{pmatrix}x_t + v_t. \end{align} The objective is to construct an optimal estimator of $x_{t+1}$ based on $y_1,\dots,y_{t}$.
I have two questions:
- Why not omitting the unobservable part of the state from the state-space? Since the unobservable part does not effect the measurements $y_t$, their best estimation is independent of the measurements.
- My guess is that, in most cases, the state vector has some physical interpretation so that people do not want to omit this part from the problem.
- In the more general setup, we can estimate a linear combination of the state. If we aim to estimate $Hx_t$ rather than $x_t$, is it reasonable to assume that the system is fully observable?
For convenience for further use I first define the equivalent system as
\begin{align} \begin{bmatrix} x_{1,t+1} \\ x_{2,t+1} \end{bmatrix} &= \begin{bmatrix} F_1 & 0 \\ F_2 & F_3 \end{bmatrix} \begin{bmatrix} x_{1,t} \\ x_{2,t} \end{bmatrix} + \begin{bmatrix} G_1 \\ G_2 \end{bmatrix} w_t, \\ y_t &= \begin{bmatrix} H_1 & 0 \end{bmatrix} \begin{bmatrix} x_{1,t} \\ x_{2,t} \end{bmatrix} + v_t. \end{align}
The prediction and correction step of the Kalman filter can be written as
\begin{align} \begin{bmatrix} \hat{x}_{1,t+1|t} \\ \hat{x}_{2,t+1|t} \end{bmatrix} &= \begin{bmatrix} F_1 & 0 \\ F_2 & F_3 \end{bmatrix} \begin{bmatrix} \hat{x}_{1,t|t} \\ \hat{x}_{2,t|t} \end{bmatrix}, \\ \begin{bmatrix} \hat{x}_{1,t+1|t+1} \\ \hat{x}_{2,t+1|t+1} \end{bmatrix} &= \begin{bmatrix} \hat{x}_{1,t+1|t} \\ \hat{x}_{2,t+1|t} \end{bmatrix} + \begin{bmatrix} K_1 \\ K_2 \end{bmatrix} \left(y_{t+1} - \begin{bmatrix} H_1 & 0 \end{bmatrix} \begin{bmatrix} \hat{x}_{1,t+1|t} \\ \hat{x}_{2,t+1|t} \end{bmatrix}\right). \end{align}
Here the correction gains $K_1$ and $K_2$ might be time varying. Substituting the prediction step in the correction step and splitting the expressions for $\hat{x}_1$ and $\hat{x}_2$ yields
\begin{align} \hat{x}_{1,t+1|t+1} &= (I - K_1\,H_1)\,F_1\,\hat{x}_{1,t|t} + K_1\,y_{t+1}, \\ \hat{x}_{2,t+1|t+1} &= (F_2 - K_2\,H_1\,F_1)\,\hat{x}_{1,t|t} + F_3\,\hat{x}_{2,t|t} + K_2\,y_{t+1}. \end{align}
From this it can be observed that the observable part of the state, $\hat{x}_1$, is only affected by the previous estimate of the observable part of the state and the measured output. Therefore, the best estimate of $H_1\,x_{1,t}$ would be $H_1\,\hat{x}_{1,t|t}$, whose dynamics thus doesn't depend on $\hat{x}_2$. So using a reduced observable model by omitting $\hat{x}_2$ from the state of the initial model would yield the same estimating performance regarding $\hat{x_1}$ and thus $H_1\,\hat{x}_{1,t|t}$.
The unobservable part of the state, $x_2$, might however still be of interest in some cases. For example when that part of the state is controllable and you do want to drive it to some desired reference. Even though that part of the state is not observable, it can still be detectable if $F_3$ is stable/Schur. Therefore, it can still be possible to drive $x_2$ (close) to such desired reference, but usually when a state is not observable its associated estimation error covariance matrix can be quite large and thus such tracking performance of the desired reference might not be great.