I understand how to use Kalman filter for fusing two estimates and that essentially it fuses predicted state estimate produced by system dynamics update and observable state estimate produced by taking measurement.

However what if I have more than one measurement source for any of the state variables? The only way I can think of is to maintain two measurement models each of them having it's own covariance matrix and for each prediction step calculate two measurement updates using the following sequence:

  1. Predict state and covariance matrix using system dynamics model.
  2. Obtain state posterior estimate using measurement update.
  3. Use state posterior obtained in step 2 for the second measurement update.

Unfortunately I have not enough confidence in math in order to check the validity of this assumption.


  • $\begingroup$ Can't you just define one vector containing all your measurements and define a covariance matrix for that vector? $\endgroup$ Commented May 21, 2019 at 9:37
  • $\begingroup$ @KwinvanderVeen oh.. and just define observation matrix H to map same state variable for two different measurement. So obvious. Thanks! Would you post it as an answer? $\endgroup$
    – Long Smith
    Commented May 21, 2019 at 9:50
  • $\begingroup$ Could you solve the problem with the suggested approach? I think the better way would be to either use the sequential Kalman filter and feed every new measurement from different sensors one after another, or use a small preprocessing filter to fuse the multiple measurements of the same kind. I would be appreciated to hear your feedback. $\endgroup$
    – Anton
    Commented Jun 4, 2019 at 19:37

1 Answer 1


The advantage of using a state space representation of your system is that a Kalman filter can, besides the state vector you try to estimate, deal with all outputs of the system as one vector as well. For this vector representation of the measurement of the state vector one can define one (bigger) covariance matrix, after which one can use the usual equations for a Kalman filter.


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