Extended Kalman Filter - 2024.1 English

Vitis Libraries

Release Date
2024-05-30
Version
2024.1 English

The Kalman filter estimates the state vector in a linear model. If the model is nonlinear, then a linearization procedure is performed to obtain the filtering equations. The Kalman filter so obtained will be called the Extended Kalman filter. A state-space description of non-linear system can have a non-linear model of the form:



7.073625 x k + 1 = ( x k ) + T k ( x k ) ξ k



6.754124999999998 z k =   h k ( x k ) + η k


Where fk and hk are valued functions with ranges in Rn and Rq, respectively. 1≤q≤n, and Tk a matrix-valued function with range in RnxRq such that for each k the first order partial derivatives of fk (x:sub:k) and hk (x:sub:k) with respect to all the components of xk are continuous. We consider zero-mean Gaussian white noise sequences image122 and image123 with ranges in Rp and Rq respectively, 1≤p, q≤n.

The real-time linearization process is carried out as shown in the following equations. In the lines of the linear model, the initial estimate and predicted position are chosen to be:

11.835 x ^ 0 = E   x 0 ,   x ^ 1 | 0 = f 0   x ^ 0


Then, 8.436824999999999 x ^ k = x ^ k | k , consecutively, for k=1,2,…, use the predicted positions.

11.835 x ^ k | k - 1 = f k - 1 x ^ k - 1


Note

  1. 28.615470000000002 f k ( x k ) = f k 1 ( x k ) f k n ( x k ) , where 28.615469999999995 x k = x k 1 x k n , k is a time index and superscript is row index and 50.40188999999998 f k (   x k )   x k   = f k 1 (   x k ) x k 1 f k 1 (   x k ) x k n f k n (   x k ) x k 1 f k n (   x k ) x k n
  2. 3 R m is a space of column vectors 6.599699999999999 x = [ x 1 . x m ] T
The equation for time update computations is as follows:

26.025 P k | k - 1 = f k - 1 x ^ k - 1 x k - 1 P k - 1 | k - 1 f k - 1 x ^ k - 1 x k - 1 T +   T k - 1 x ^ k - 1 Q k - 1 T k - 1 x ^ k - 1 T


17.13495 = F k - 1 P k - 1 | k - 1 F k - 1 T +   T k - 1 x ^ k - 1 Q k - 1 T k - 1 x ^ k - 1 T
The equation for measurement update computations is as follows:

image134



11.835 x ^ k | k = x ^ k | k - 1 + G k v k - h k x ^ k | k - 1



26.025 P k | k = I - G k h k x ^ k | k - 1 x k P k | k - 1 = I - G k H k P k | k - 1