Tracking Moving Objects with Kalman Filters-- Over-fitting over time?

Question

I've been learning about Kalman Filters, and the classic example given is tracking an object via radar/gps. My issue here is that each time you get a new data point, you update the error in the estimate, with the error always decreasing. Does this mean that Kalman Filters can't be used long-term to track a moving object?

For example, with a satellite, each measurement via GPS of the satellite location would decrease our error. Because of this and the fact that the satellite is in continuous motion, the measurement of the satellite location would slowly "fall behind" its current location. This seems to be an even bigger problem if the object in motion doesn't have a constant speed or velocity (i.g. a truck).

In more generic mathematical terms, is a Kalman Filter designed to track a set parameter, or can it be used to track a parameter that moves over time? It seems like I either have a fundamental misunderstanding about what the Kalman filter is used for or a fundamental misunderstanding of the equations.

Answer 1

I did not completely get your example, but Kalman filter can absolutely be used for parameters that are changing over time. The idea behind Kalman filter is to combine two sets of knowledge. One is the underlying dynamics of the system or the knowledge of how parameters change over time. The latter is the measurements of the parameters or a function of the parameters. The two sets of information are balanced with process noise and measurement noise. Kalman filter equations show how to fuse these two pieces of information in an "optimal" manner.

Take a look at this example and see if it makes sense. In this example, the goal is to track 1D position and velocity. Velocity is assumed to be random noise and thus, we can write the process/dynamic equations as

$$x_{1}[k] = x_{1}[k-1] + x_{2}[k-1]dt + w{1}$$ $$x_{2}[k] = x_{2}[k-1] + w_{2}$$

Basically, the process equations show how we expect the parameters to change plus some noise to account for uncertainties. This is the measurement equation:

$$z_{1}[k] = x_{1}[k] + n_{1}$$

Like here, we assume that we are directly measuring the position and not the velocity.