Kalman filter

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1. predict state x^(n+1, )
2. calculate predict covariance P(n+1, n) = E[e(n+1,n)e(n+1,n)^T]=E[(x(n)-x^(n+1,n)) (x(n)−x^(n+1,n)^T)]
3. Correct estimation using measurement z(n)
4. calculate covariance of the correction x^(n,n). P(n,n) = E[e(n,n)e(n,n)^T]=E[(x(n)-x^(n,n)) (x(n)−x^(n,n)^T)]
5. Optimize Kalman gain by solving differential equation [sum of a trace of Covariance matrix P(n,n)]

https://www.kalmanfilter.net/background2.html#covExp

THE ESTIMATE UNCERTAINTY

Pn+1,n=E((x^n+1,n−μxn+1,n)(x^n+1,n−μxn+1,n)T)

Uncertainty is the covariance of estimation x^(n+1). P(n+1,n) = COV(X^(n+1,n))