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# Lecture-14 - Main Points Lecture-14 Kalman Filter Very...

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Copyright Mubarak Shah 2008 Lecture-14 Kalman Filter Copyright Mubarak Shah 2008 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 an estimate of state over time. ! It is a sequential state estimator. Copyright Mubarak Shah 2008 State-Space Model State-transition equation Measurement (observation) equation State Vector Measurement Vector State model error With covariance Q(k) Observation Noise with covariance R(k) ) ( ) 1 ( ) 1 , ( ) ( k k k k k w z z ! " " # \$ ) ( ) ( ) ( ) ( k k k k v z H y ! \$ Copyright Mubarak Shah 2008 Kalman Filter Equations State Prediction Covariance Prediction Kalman Gain State-update Covariance-update ) 1 ( ˆ ) 1 , ( ) ( ˆ " " # \$ k k k k a b z z ) ( ) 1 , ( ) 1 ( ) 1 , ( ) ( k k k k k k k T a b Q P P ! " # " " # \$ 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 " \$

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Copyright Mubarak Shah 2008 Two Special Cases Steady State Recursive least squares R R H H Q Q \$ \$ \$ # \$ " # ) ( ) ( ) ( ) 1 , ( k k k k k 0 ) ( ) 1 , ( \$ \$ " # k k k Q I Copyright Mubarak Shah 2008 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. Copyright Mubarak Shah 2008 Extended Kalman Filter T aylor series ) ( )) 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 Copyright Mubarak Shah 2008
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Lecture-14 - Main Points Lecture-14 Kalman Filter Very...

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