Page 296 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

COMPUTATIONAL ISSUES 285

The computational cost of the update of the information matrix is only
determined by the inversion of Y(iji) (in the time-invariant case) because
T
1
the term H C H is constant and can be kept outside the loop. The
v
2
1
3
1
number of required operations is about M þ M þ M.
2 2
In order to develop the expression of the predicted state information
matrix, the covariance of the process noise is factored as follows
T
C w ¼ GG . As mentioned in Section 8.2.3, such a factorization is
T
obtained by an eigenvector–eigenvalue decomposition C w ¼ V w w V .
w
The diagonal elements of w contain the eigenvalues of C w .Ifsomeof
the eigenvalues are zero, we remove the rows and columns in which
these zero eigenvalues appear. Also, the corresponding columns in V w
are removed. Suppose that the number of nonzero eigenvalues is K,
then w becomes a K K matrix and V w an M K matrix. Conse-
quently, G ¼ V w 1/2 is also an M K matrix.
w
Furthermore, we define the matrix A(i) as:
AðiÞ¼ F 1 T YðijiÞF 1 ð8:35Þ
def
1
T
In other words, A (i) ¼ FC(iji)F is the predicted covariance matrix in
the absence of process noise. Here, we have silently assumed that the
matrix F is invertible (which is the case if the time-discrete system is an
approximation of a time-continuous system).
The information matrix of the predicted state follows from (8.2):

T T 1
Yði þ 1jiÞ¼ðFCðijiÞF þ GG Þ
(predicted state information)
1 T 1
¼ðA ðiÞþ GG Þ
ð8:36Þ

Using the matrix inversion lemma, this expression can be moulded in the
easier to implement form:

T 1 T
Yði þ 1jiÞ¼ AðiÞ AðiÞGðG AðiÞG þ IÞ G AðiÞ ð8:37Þ

This completes the Ricatti loop. The number of required operations of
3
2
2
3
(8.36) and (8.37) is 2M þ 2M K þ 2MK þ K .
As mentioned above, the information filter can represent the situ-
ation where no information about some states is available. Typically,
this occurs at the initialization of the filter, if no prior knowledge is