I am familiar with the equations of basic kalman filter. I was reading a paper where the author's proposed a variant of kalman filter and in the prediction step they had more terms than standard kalman filter. I am referring to equation 10 of the paper "Vehicular Node Localization Using Received-Signal-Strength Indicator" where author has added a term gamma multiplied with dt^2
https://www.comm.utoronto.ca/~valaee/Publications/07-Parker-TVT.pdf. I understand the idea behind term. I do not how to justify the factor one can add new terms affecting the uncertainty of prediction step of kalman filter.
My specific question how can I justify or prove that the new terms in error covariance equation equation 10 (inside attachment as well is correct)
Dear Varun Garg
This lecture might clear your doubts, have a look
https://atmos.washington.edu/~breth/classes/AS552/lect/lect26.pdf
Kind Regards
Qamar Ul Islam
Hello Varun Garg,
Equation (10) is a result of the non-additive noise formulation for the EKF:
https://en.wikipedia.org/wiki/Extended_Kalman_filter#Non-additive_noise_formulation_and_equations
The usual simplification is to add noise to the state after the dynamic/prediction step. Often, we have a better knowledge where noise in the system is present (in the paper example the noise on the velocity measurement ist known). In such cases, you can use the non-additive noise formulation.
If you want to prove/check covariance equations in general you can refer to covariance computation rules:
https://en.wikipedia.org/wiki/Covariance#Covariance_of_linear_combinations
The equations (9) and (10) of the paper calculate the expected value and covariance of the distribution propagated through the dynamic model (2).
Solving:
A_k|k-1=E(A_{k-1|k-1}+Ts * u_{k-1}+Ts w_{k-1})
P_k|k-1 = Cov(A_{k-1|k-1}+Ts * u_{k-1}+Ts w_{k-1})
, where E(...) computes the expected value results in (9) and (10).
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