lect1109h - Kalman Filter Main Points Very useful tool. It...

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1 Kalman Filter Main Points • Very useful tool. • It produces an optimal estimate of the state vector based on the noisy measurements (observations). • For the state vector it also provides confidence (certainty) measure in terms of a covariance matrix . • It integrates estimate of state over time. • It is a sequential state estimator.
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2 State-Space Model ) ( ) 1 ( ) 1 , ( ) ( k k k k k w z z + - - Φ = ) ( ) ( ) ( ) ( k k k k v z H y + = State-transition equation Measurement (observation) equation State Vector Measurement Vector State model error With covariance Q(k) Observation Noise with covariance R(k) Kalman Filter Equations ) 1 ( ˆ ) 1 , ( ) ( ˆ - - Φ = k k k k a b z z ) ( ) 1 , ( ) 1 ( ) 1 , ( ) ( k k k k k k k T a b Q P P + - Φ - - Φ = 1 )) ( ) ( ) ( ) ( )( ( ) ( ) ( - + = k k k k k k k T b T b R H P H H P K )] ( ˆ ) ( ) ( )[ ( ) ( ˆ ) ( ˆ k k k k k k b b a z H y K z z - + = ) ( ) ( ) ( ) ( ) ( k k k k k b b a P H K P P - = State Prediction Covariance Prediction Kalman Gain State-update Covariance-update
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3 Two Special Cases R R H H Q Q = = = Φ = - Φ ) ( ) ( ) ( ) 1 , ( k k k k k 0 ) ( ) 1 , ( = = - Φ k k k Q I Steady State Recursive least squares Comments • In some cases, state transition equation and the observation equation both may be non- linear. • We need to linearize these equation using Taylor series.
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4 Extended Kalman Filter ) ( )) 1 ( ( ) ( k k k w z f z + - = ) ( )) ( ( ) ( k k k v z
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This note was uploaded on 06/12/2011 for the course CAP 6411 taught by Professor Shah during the Spring '09 term at University of Central Florida.

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lect1109h - Kalman Filter Main Points Very useful tool. It...

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