Dear Guido Magnani,

Without prior information we can start the UKF with initial value of X0 and P0 equal to zero. Also we can ajust the estimation of filter with different weights of the matrix R.

For more information about this subject, i suggest you to see links and attached files on topic.

https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.106.5877&rep=rep1&type=pdf

Article Unscented Kalman filter with parameter identifiability analy...

Article Unscented Kalman filter (UKF) based nonlinear parameter esti...

Best regards

Dear Guido Magnani

Please read my preprint:

LAPLACE-TRANSFORM; BEYOND TRADITIONAL FILTERING

available as a PDF file in:

Preprint LAPLACE-TRANSFORM; BEYOND TRADITIONAL FILTERING

In practice performance of any Kalman filter implementation will always be affected by initial values of parameters. Sometimes you might have relevant data about signal noise, but the problem is that with covariances you are not capturing just signal noise but also other inaccuracies. Sometimes it help to set covariances to very low values or to very high values. Usually you need to test various values for covariance and initial values to get the filling for what you can do with your signal pipeline.

In your case it seems that the Kalman filter becomes unstable, you might have under-determined model of the process. Simple case is this: you want to determine values of a and b parameters, and you measurement model is c = a+b, where c is your measurement. So a and b can be any value, and you can only estimate the difference between a and b. This is very simple example but in my experiences this can happen often. Sometimes you think that you are estimating a value of parameter, but all kind of unmodeled errors can creep into your estimated value of parameter.

Another example is estimating the bias of y gyroscope in IMU unit. Bias can actually be very small, but the scaling parameters of the other sensors might be wrong and the Kalman filter smuggles the errors of those parameters into the bias estimation. Estimated bias then affects the estimation of the orientation, and then you get the estimation of orientation which is worse then what you could get with simpler model without bias estimation.

All sort of things can happen. You need to understand the procces and the measurement far better than what you actually put into the models for Klaman filter.

Dear Mr. Magnani,

I agree with Janez Podobnik. I assume you are using the Joint Unscented Kalman Filter for this purpose. In this case, the parameter is considered as a state and all the initial conditions, noise and measurement covariances will affect the estimation of the parameter. In 2019, I have proposed a different approach for this purpose and I am attaching the article link below:

Article A Novel and Computationally Efficient Joint Unscented Kalman...

The main advantage of this approach is you can determine weights for the measurements (a Jacobian approach) for parameter estimation. If one of the measurements significantly affects the parameter estimation, then, you can increase the weight of that measurement. The only drawback of this approach is that the estimated parameter must be linear in terms of measurements. The system itself can be a nonlinear system, but parameter(s) must be linear in terms of measurements. For frequency estimation, we have modified this approach with my colleagues (link below). In this case, the parameter is no longer linear in terms of measurements. However, we haven't tested this approach for other nonlinear systems in which the parameter(s) is nonlinear in terms of measurements.

Article Non-Contact Respiratory Rate Estimation in Real-Time With Mo...

You may try this approach and it can be useful for your aim.

Best regards,

Altan