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QinetiQ — Thales Research — Technical Presentation (March 2001)

23 slides from QinetiQ's technical presentation, Mar 7, 2001

1. Slide 1: Kalman filters Applications And Pitfalls

   Kalman filters Applications And Pitfalls IEE / DERA / ASPC Adaptive Signal Processing Club One Day Discussion Meeting Heckfield Place, near Reading Wednesday 7th March 2001 Hosted by Thales Research Ltd, Reading and Thales Defence Systems Ltd, Crawley

2. Slide 2: Number of emails in each month

   Number of emails in each month 0 10 20 30 40 50 60 70 80 90 7 March 2001 Oct-99 Nov-99 Dec-99 Jan-00 Feb-00 Mar-00 Apr-00 May-00 Jun-00 Da te Jul-00 Aug-00 Sep-00 E V Stansfield / Thales Research Ltd / Kalman Filter Tutorial Oct-00 Nov-00 Dec-00 Jan-01 Feb-01 Mar-01 Pre-meeting organisation … Slide

3. Slide 3: Kalman Filters: A tutorial

   Kalman Filters A tutorial Edward Stansfield Thales Research Ltd Reading 7 March 2001 E V Stansfield / Thales Research Ltd / Kalman Filter Tutorial Slide 3

4. Slide 4: Example Kalman filter application

   Example Kalman filter application x2 x1 7 March 2001 E V Stansfield / Thales Research Ltd / Kalman Filter Tutorial Slide 4

5. Slide 5: Kalman filter models

   Kalman filter models Target motion  x1 (t + T )   1 0 T 0   x1 (t )   w 1 (t )   f1 (t )            x 2 (t + T )  0 1 0 T   x 2 (t )  w 2 (t )  f2 (t )  x (t + T )  =  0 0 1 0 x 1 (t )    + 1 (t ) w  +  f (t )  1        1   x (t + T )  0 0 0 2 (t

6. Slide 6: State and observation models

   State and observation models State (plant) model Observation model Forcing Transition function matrix Observation fk Φk τk matrix Observation y Hk True state x wk vk Plant Observation noise noise E[wkwkT] E[vkvkT] 7 March 2001 E V Stansfield / Thales Research Ltd / Kalman Filter Tutorial Slide 6

7. Slide 7: Notation

   Notation The following notation is common in estimation and prediction x k = x (t ) at time t = t k x k | j = best estimate of x (t ) at time t = t k based on observations up to time t = tj • With k < j we have smoothing • With k = j we have estimation • With k > j we have prediction x̂ k = x k | k

8. Slide 8: Kalman filter equations

   Kalman filter equations State and measurement equations zk = y k − Hk ~ xk Innovation = measurement - prediction x̂ k = ~ xk + K k zk State estimate ~ x k +1 = Φ k x̂ k + fk +1 State prediction Kalman gain and error covariance equations Kk = [ ~ T ~ T PkHk HkPkHk + R k −1 ] Kalman gain P̂k = [I − K

9. Slide 9: Kalman filter estimator

   State estimator Forcing function f Φk Transition Kalman filter Predicted state x Observation matrix Hk τk estimator matrix H Observation y Kk Estimated state x Innovation z Correction E[vkvkT] ( )-1 Observation Initialise x0 Noise covariance R Hk( )HkT E[wkwkT] PkHkT( ) Kalman gain Kk Plant noise Pr

10. Slide 10: Kalman filter process

    Observation State model model Predicted state x State estimator Forcing Transition Observation Transition function matrix matrix H matrix Observation fk Φk τk matrix Hk Φk τk Observation y Hk Kk Estimated state x True state x wk vk Plant Observation Innovation z Correction noise noise E[wkwkT] E[vkv

11. Slide 11: Degenerate Kalman filter

    Degenerate Kalman filter Alpha-Beta Filter x̂ k = x k + α(y k − ~ ~ xk ) Position estimate β v̂ k = v̂ k −1 + (yk − ~xk ) Velocity estimate T ~ x k +1 = x̂ k + Tv̂ k Position prediction Measurement matrix Coefficient (gain) matrix  α 0     1 0 0 0  0 α  H =   K= 0 1 0 0 β/T 0    

12. Slide 12: Derivation of basic Kalman filter equations

    Derivation of basic Kalman filter equations Recursive estimator with following key properties • Linear • Unbiased • Minimum variance Optimality is assured if the • Plant noise and • Observation noise are normal, spectrally white and independent 7 March 2001 E V Stansfield / Thales Research Ltd / Kal

13. Slide 13: Linear unbiased estimator

    Linear unbiased estimator x̂ k = L k ~ xk + K k yk Linear state estimator êk = x̂ k − x k Estimation error = Lk ~xk + K k y k − x k = Lk (~ek + x k ) + K k y k − x k êk = L k (~ ek + x k ) + K k (Hk x k + v k ) − x k Substituting for yk indicates = L ~e + (L + K H − I) x + K v k k k k k k k k êk

14. Slide 14: Minimum variance estimator

    Minimum variance estimator P̂ = E(ê ê ) P = E(~ e ) ~ k ke ~ T Error covariances k k k T k [ P̂k = E {(I − K kHk ) ~ ek + K k v k } {(I − K kHk ) ~ ek + K k v k } T ] Substitute for ê and [ ] k = E (I − K kHk ) ~ ekT (I − K kHk ) + K k v k v kTK kT rearrange. ek ~ T Second [ + E (I − K kHk ) ~ ek

15. Slide 15: Minimum prediction error covariance

    Minimum prediction error covariance ~ x k +1 = Φ k x̂ k Best prediction ~ ek +1 = Φ k x̂ k − x k +1 Minimised prediction error = Φ k (êk + x k ) − x k +1 = Φ k êk + Φ k x k − x k +1 = Φ k êk − w k ~ Pk +1 = E ~ ek +1~[ ekT+1 ] Minimised prediction error covariance [ = E (Φ k êk − w k )(Φ k êk −

16. Slide 16: Minimum estimation error covariance

    Minimum estimation error covariance Problem Choose Kalman gain K to minimise Trace[P] where −KH)S(I− P = (I− −KH)T + KRKT Solution P, S and R are symmetric, non-negative definite covariances P = S + KCKT − KB − (KB)T where B = HS and C = HSHT+R. Tr[P] is linear Tr[P] = Tr[S] + Tr[KCKT] − 2 Tr[KB] Tr

17. Slide 17: Kalman filter equations

    Kalman filter equations State and measurement equations zk = y k − Hk ~ xk Innovation = measurement - prediction x̂ k = ~ xk + K k zk State estimate ~ x k +1 = Φ k x̂ k + fk +1 State prediction Kalman gain and error covariance equations Kk = [ ~ T ~ T PkHk HkPkHk + R k −1 ] Kalman gain P̂k = [I − K

18. Slide 18: One-dimensional example

    One-dimensional example Position (m) Position error Position error 1750 10 1500 8 s -0.5 10 20 30 1250 6 1000 -1 750 4 -1.5 500 2 -2 250 s -2.5 s 10 20 30 10 20 30 -2 Velocity (m/s) Velocity error Velocity error 80 2 20 60 1 40 10 s 20 s 10 20 30 10 20 30 -1 s 10 20 30 -10 -2 Acceleration (m/s/s) Ac

19. Slide 19: Properties of innovation - Property 1

    Properties of innovation - 1 Property 1 E[z z ] = R + H P ~ T k k H where z = y −H ~ k x k k T k k k k k Proof z k = (Hk x k + v k ) − Hk (x k + ~ ek ) = v k − Hk ~ ek Hence [ E z k z kT ] = E[v v ] + E[H ~e ~e H ] − E[H ~e v ] − E[v ~e H ] k T k k k T k T k k k T k k T k T k yielding [ ] ~ E z k zk

20. Slide 20: Properties of innovation - Property 2

    Properties of innovation - 2 E z (R + H P ~ H ) z  = Dim(y ) Universal Most T T −1 Property 2  k k k k k k  Powerful Test Statistic Proof With zk = v k − Hk ~ ek we get   T ( ~ T −1   ) ~ T −1  E zk Rk + HkPkHk zk = E v k Rk + HkPkHk v k  T  ( )   ~ T T ( ~ T −1 ~ ~ ) ~ T −1  +

21. Slide 21: Example using UMPTS

    Example using UMPTS Position (m) Position error Position error 1750 10 1500 8 s 1250 6 -0.5 10 20 30 1000 750 4 -1 500 -1.5 250 2 -2 s s -2.5 10 20 30 -2 10 20 30 Velocity (m/s) Velocity error Velocity error 80 2 20 60 1 40 10 s 20 s 10 20 30 10 20 30 -1 s -10 10 20 30 -2 Acceleration (m/s/s) Accele

22. Slide 22: GPS composite clock Kalman filter

    GPS composite clock Kalman filter Clock Clock spoiling / re-initialisation GPS noise clock covariance maintenance Q ~ ~ R Measurement P = ΦP̂Φ T + Q HPHT Φ ( ) ΦT Noise covariance Clock ΦP̂Φ T Bias estimate Estimation Kalman gain Φ( ) ΦT ( )-1 Error computation (R + HP~H ) Covariance −1 T decomposit

23. Slide 23: Extended Kalman filter (EKF)

    Extended Kalman filter (EKF) If state or measurement equation is non-linear, then EKF can be used. • based on target reference trajectory and a linearised model • Kalman filter updates state estimate relative to the reference EKF enables sequential estimation of non-linear systems, • optimality of t

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