When should I use a kalman filter, if the observability function is known?

Solution 1:

As stated by the second sentence of the quoted text, in this case a Kalman filter can still be beneficial to minimize the impact of noise. Noises can both act on the state dynamics as the measurements. If there is no noise on the measurements then directly using the measurements will be best. If there is no noise on the dynamics then a Kalman filter will eventually use the measurements less and less and more and more rely on the prediction steps. However, in practice both will act to some degree on actual systems. Though, it might be that the not all assumptions need for a Kalman filter are satisfied, such as that noises are Gaussian zero mean white noise and that dynamics can be described by a linear model. But in many cases those assumptions give a close enough approximation of the actual system and thus Kalman filters at least yield some improvement over just using the output directly in this case.