STATISTICAL ORBIT DETERMINATION Coordinate Systems and Time Kalman Filtering ASEN 5070 LECTURE 21 10/16/09

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Numerical Considerations for the Kalman Filter Time and measurement update Of the estimation error covariance matrix

Numerical Considerations for the Kalman Filter

Equivalence of Joseph and Conventional Formulation of the Measurement Update of P Need to show that First derive the Joseph formulation subst. subtract from both sides and rearrange

Equivalence of Joseph and Conventional Formulation of the Measurement Update of P where since is independent of

Equivalence of Joseph and Conventional Formulation of the Measurement Update of P Next, show that this is the same as the conventional Kalman subst. for

Example Illustrating Numerical Instability of Sequential (Kalman) Filter (see 4.7.1) The following example problem from Bierman (1977) illustrates the numerical characteristics of several algorithms we have studied to date. Consider the problem of estimating and from scalar measurements and Where and are uncorrelated zero mean random variables with unit variance. If the did not have the above traits we could perform a whitening transformation so that they do. In matrix form

Example Illustrating Numerical Instability of Sequential (Kalman) Filter The a priori covariance associated with our knowledge of is assumed to be Where and . The quantity is assumed to be small enough such that computer round-off produces This estimation problem is well posed. The observation provides an estimate of which, when coupled with should accurately determine . However, when the various data processing algorithms are applied several diverse results are obtained.

Example Illustrating Numerical Instability of Sequential (Kalman) Filter Let the gain and estimation error covariance associated with be denoted as and respectively. Similarly the measurement is associated with and respectively. Note that this is a parameter (constant) estimation problem. Hence, We will process the observation one at a time. Hence,

Example Illustrating Numerical Instability of Sequential (Kalman) Filter Note: Since we will process the observations one at a time, the matrix inversion in Eq (4.7.21a) is a scalar inversion. The exact solution is where The estimation error covariance associated with processing the first data point is

Example Illustrating Numerical Instability of Sequential (Kalman) Filter where Processing the second observation: where

Example Illustrating Numerical Instability of Sequential (Kalman) Filter The exact solution for is given by: The conventional Kalman filter yields (using )

Example Illustrating Numerical Instability of Sequential (Kalman) Filter Note that is no longer positive definite. The diagonals of a matrix must be positive. However does exist since . Compute P2 using the Conventional Kalman Now is not positive definite ( ) nor does it have positive terms on the diagonal. In fact, the conventional Kalman filter has failed for these observations.

Example Illustrating Numerical Instability of Sequential (Kalman) Filter The Joseph formulation yields The Batch Processor yields

Example Illustrating Numerical Instability of Sequential (Kalman) Filter P2 for the batch To order , the exact solution for is

Example Illustrating Numerical Instability of Sequential (Kalman) Filter Summary of Results Exact to order Conventional Kalman Joseph Batch