Kalman Filter: Do we need to measure each entry of the output-sequence, or can we derive them?

Question

A Kalman-Filter gets applied to a state-space model in order to obtain estimates of the state-vector.

Example: Assume a car moving on a straight line in x-direction (this is a 1-dimensional problem). The State-space model could be described by:

$$ \begin{bmatrix} x_k \\ \dot{x}_k \\ \ddot{x}_k \end{bmatrix} = \begin{bmatrix} 1 & T & \frac12\,T^2 \\ 0 & 1 & T \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x_{k-1} \\ \dot{x}_{k-1} \\ \ddot{x}_{k-1} \end{bmatrix} + w_{k-1} $$

$$ \begin{bmatrix} y_k \\ \dot{y}_k \\ \ddot{y}_k \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x_k \\ \dot{x}_k \\ \ddot{x}_k \end{bmatrix} + v_k $$

The y-vector is our measurement-sequence. And here comes my question: For the Kalman-Filter to work, is it necessary to measure each of the entries of the y-vector?

Coming back to the car: I can measure the number of rotations of the driving-shaft using a wheel-encoder. From this I can infer the distance travelled, which is my x-coordinate assuming a movement on a straight line in x-direction.

I cannot measure the velocity directly. But I can infer the velocity from the number of edges detected within a certain time-gap, using the wheel encoder. And I can derivate velocity in order to obtain acceleration.

So, what I am saying is: In order to obtain each entry of the y-vector, I am using but one sensor: The wheel encoder. Is this valid when applying the Kalman-Filter equations?

Answer 1

If you formulate your Kalman filter with the output matrix as eye(3) - $$ \left[\begin{array} \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array}\right] $$

then you need to provide all those measurements, but you could alternatively define the output matrix as eye(1,3) -

$$ \left[\begin{array} \\ 1 & 0 & 0 \\ \end{array}\right] $$

and then that should let you supply just the feedback measurement on encoder position and have the Kalman filter provide the speed and acceleration estimates. This would be the preferred way to do it, instead of trying to manually derive the speed and then passing that speed "measurement" to the Kalman filter.