Suppose I have 1-dimensional process with no dynamics:
$x_{n+1} = x_{n}$,
e.g. a gyroscope which measures angular velocity.
Angular velocity measured by a gyroscope for the step $ n $:
(1) $ x^{m}_{n} = x^{t}_{n} + x^{b}_{n} + x^{w}_{n} $
(2) $ x^{b}_{n} = x^{b}_{n-1} + x^{w}_{n} $
where
$x^{w}_n = \sigma_{gn}*w_{n}, \{ \sigma_{gn} = \sigma_{n}*\frac{1}{\sqrt{dt}}, w_{n} = N(0,1) \}$
$x^{b}_{n} = \sigma_{gb}*w_n, \{ \sigma_{gb} = \sigma_{b}*\sqrt{dt}, w_{n} = N(0,1) \} $
In other words, gyroscope's measured value is a sum of:
- True angular velocity - $x^{t}_{n}$.
- Bias (or "random walk") - $x^{b}_{n}$.
- White noise - $x^{w}_{n}$.
Bias is an "integral" or a "sum" of white noise, and values $ \sigma_{n}, \sigma_{b} $ are determined from a datasheet or an experiment setup.
In addition to the gyroscope, which measures angular velocity with high rate, I have visual odometry sensor which measures angular velocity with relatively low rate:
$ y_{n} = x^{t}_{n} + v^{w}_{n}$, where $ v^{w}_{n}$ is a visual odometry measurement noise.
My final goal is to estimate $ X = \begin{bmatrix} x^{t}_{n} \\ x^{b}_{n} \end{bmatrix}$, or equivalently $ X = \begin{bmatrix} (x^{m}_{n} - x^{b}_{n}) \\ x^{b}_{n} \end{bmatrix}$
In order to do it, I construct Kalman filter equations:
STEP 1. Build a model.
I don't have dynamics equation for the angular velocity. I have only bias dynamics.
So equation becomes:
$X_{n} = X_{n-1} + Q_{n-1} $, or equivalently:
$\begin{bmatrix} x^{t}_{n} \\ x^{b}_{n} \end{bmatrix} = \begin {bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} (x^{m}_{n-1} - x^{b}_{n-1}) \\ x^{b}_{n-1} \end{bmatrix} + \begin{bmatrix} 1 \\ 1 \end{bmatrix} x^{w}_{n-1}$
And here I stuck. This matrix equation is not consistent with (1) and (2) because I can't propagate previous state to the current state.
Does it mean I can't use Kalman filter here?