Simplified Covariance Update Equation

In many textbooks, you will find a simplified form of the Covariance Update Equation:

\[ \boldsymbol{ P_{n,n} = \left( I - K_{n}H \right) P_{n,n-1} } \]

In order to derive a simplified form of the Covariance Update Equation, substitute the Kalman Gain Equation into the Covariance Update Equation.

Notes
\( \boldsymbol{P_{n,n}} = \boldsymbol{ P_{n,n-1} - P_{n,n-1}H^{T}K_{n}^{T} - K_{n}H P_{n,n-1}} + \\ + \color{#7030A0}{\boldsymbol{ K_{n} }} \boldsymbol{ \left( H P_{n,n-1}H^{T} + R_{n} \right) K_{n}^{T} } \) Covariance Update Equation after expansion
\( \boldsymbol{P_{n,n}} = \boldsymbol{ P_{n,n-1} - P_{n,n-1}H^{T}K_{n}^{T} - K_{n}H P_{n,n-1}} + \\ + \color{#7030A0}{\boldsymbol{ P_{n,n-1}H^{T}\left( HP_{n,n-1}H^{T} + R_{n} \right)^{-1} }} \boldsymbol{ \left( H P_{n,n-1}H^{T} + R_{n} \right)} \boldsymbol{ K_{n}^{T} } \) Substitute the Kalman Gain Equation
\( \boldsymbol{P_{n,n}} = \boldsymbol{ P_{n,n-1} - P_{n,n-1}H^{T}K_{n}^{T} - K_{n}H P_{n,n-1}} + \\ + \boldsymbol{P_{n,n-1}H^{T} } \boldsymbol{ K_{n}^{T} } \) \( \boldsymbol{\left( HP_{n,n-1}H^{T} + R_{n} \right)^{-1} \times \\ \times \left( HP_{n,n-1}H^{T} + R_{n} \right) } = 1 \)
\( \boldsymbol{P_{n,n}} = \boldsymbol{ P_{n,n-1} - K_{n}H P_{n,n-1}} \)
\( \boldsymbol{P_{n,n}} = \boldsymbol{ P_{n,n-1} \left( I - K_{n}H \right)} \)
Warning! This equation is much more elegant and rememberable. In many cases it performs well, however, even the smallest error in computing the Kalman Gain (due to round off) can lead to huge computation errors. The subtraction \( (I - K_{n} H) \) can lead to nonsymmetric matrixes results due to the floating-point errors. This equation is numerically unstable!

For more details, see: “Bucy, R. S. and Joseph, P. D. (1968). Filtering for Stochastic Processes with Applications to Guidance. Interscience, New York”, Chapter 16, “Roundoff errors” section.

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