Lecture-17 - Lecture-17 Kalman Filter Main Points Very...

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Lecture-17 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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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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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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Extended Kalman Filter ) ( )) 1 ( ( ) ( k k k w z f z + = ) ( )) ( ( ) ( k k k v z h y + = )) 1 ( ˆ - 1) - k ( ( ) 1 ( )) 1 ( ( )) 1 ( ˆ ( )) 1 ( ( + k k k k k a a z z z z f z f z f )) 1 ( ˆ ) ( ( ) ( )) ( ( )) ( ˆ ( )) ( ( + k k k k k k b b z z z z h z h z h T aylor series Extended Kalman Filter ) ( ) ( ) 1 ( ) 1 , ( ) ( k k k k k k w u z
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