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.. . Factors that contribute to such ill-conditioning include the following:

1. Large uncertainties in the values of the matrix paraIIleters , Q, H, or R. Such modeling errors are not accounted for in the derivation of the Kalman filter.

2. Large ranges of the actual values of these matrix parameters, the measurements, or the state variables-all of which can result from poor choices of scaling or dimensional units.

3. Ill-conditioning of the intermediate result R* =H P RT +R for inversion in the Kalman gain formula.

4. Ill-conditioned theoretical solutions of the matrix Riccati equation-without considering numerical solution errors. With numerical errors, the solution may become indefinite-which can destabilize the filt~r estimation error.

5. Large matrix dimensions. The number of arithmetic operations grows as the square or cube of matrix dimensions, and each o",ration can introduce roundoff errors. I

I

6. Poor machine precision, which makes the roundoff ettors larger. . ... --_....._. ". .. ."-l'I

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There is a great difference between theory and p*tice. Giacomo Antonelli (1806-~876)1

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Uses of Cholesky decomposition In Kalman filtering. Cholesky decom

position methods produce triangular matrix factors (Cholesky factors), and the sparse

ness ofthese factors can be exploited in the implementation of the Kalman filter equa

tions. These methods are used for the following purposes:

1. In the decomposition of covariance matrices (P, R, and Q) for implementa

tion of "square root" filters.

2. In "decorrelating" measurement errors between components of vector

valued measurements, so that the components may be processed sequentially

as independent scalar-valued measurements. (See page 218.)

3. As part of a numerically stable method for computing matrix expressions

containing the factor (HPHT+R) -1 in the conventional form of the Kalman

filter. (This matrix inversion can be obviated by the decorrelation methods,

hOWever.)

4. In Monte Carlo analysis of Kalman filters by simulation. Cholesky factors

are used for generating independent random sequences of vectors with pre

specified means and covariance matrices. (See §3.4.8 on page 70.)

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