Kalman Filtering Tutorial

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Matrix Riccati Equation The covariance calculations are independent of state (not so for EKF later) => can be performed offline and are given by: 20 [ ] −1 ⎧⎪ P( k| k −1) − P( k| k −1) H( k)' H( k) P( k| k − 1) H( k)' + R( k) ⎫⎪ P( k + 1| k) = F( k) ⎨ ⎬F( k)' + Q( ⎩⎪ . H( k) P( k| k −1) ⎭⎪ This is the Riccati equation and can be obtained from the <strong>Kalman</strong> filter equations above. The solution of the Riccati equation in a time invariant system converges to steady state (finite) covariance if the pair {F, H} is completely observable (ie the state is visible from the measurements alone).
{F, H} is completely observable if and only if the observability matrix has full rank of nx. Q 0 ⎡F ⎢ FH = ⎢ ⎢ ... ⎢ ⎣ 21 FH n x The convergent solution to the Riccati equation yields the steady state gain for the <strong>Kalman</strong> Filter. −1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦
Page 1 and 2: Understanding and Applying Kalman F
Page 3 and 4: What is a Kalman Filter and What Ca
Page 5 and 6: Word examples: • Determination of
Page 7 and 8: The covariance of a column vector x
Page 9 and 10: Error Ellipses corresponding to 50
Page 11 and 12: Formulating a Kalman Filter Problem
Page 13 and 14: FALLING BODY EXAMPLE Consider an ob
Page 15 and 16: ⎡1 1⎤ ⎡0.5⎤ x(k +1) = ⎢ x
Page 17 and 18: State Estimation 0. Known are x$( k
Page 19: Page 219 Bar-Shalom ANATOMY OF KALM
Page 23 and 24: 101 99 97 95 93 91 89 87 True posit
Page 25 and 26: Kalman Filter Extensions • Valida
Page 27 and 28: Sequential Measurement Processing I
Page 29 and 30: Extended Kalman Filter (EKF) Many p
Page 31 and 32: Iterated Extended Kalman Filter (IE
Page 33 and 34: Kalman filtering classes, for defin
Page 35 and 36: Odometry Error Covariance Estimatio
Page 37: Scanned Monocular Sonar Sensing Sma

Kalman Filtering Tutorial

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