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The “  ” Paige in Kalman Filtering

The “  ” Paige in Kalman Filtering. K. E. Schubert. Kalman’s Interest . State Space (Matrix Representation) Discrete Time (difference equations). Optimal Control Starting at x 0  Go to x G Minimize or maximize some quantity (time, energy, etc.). Why Filtering?.

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The “  ” Paige in Kalman Filtering

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  1. The “” Paige in Kalman Filtering K. E. Schubert

  2. Kalman’s Interest • State Space (Matrix Representation) • Discrete Time (difference equations) • Optimal Control • Starting at x0  Go to xG • Minimize or maximize some quantity (time, energy, etc.)

  3. Why Filtering? • State (xi) is not directly known • Must observe through minimum measurements • Observer Equation • Want to reconstruct the state vector

  4. y=ax+b Random Variables • Process and observation noise • Independent, white Gaussian noise

  5. Complete Problem • Control and estimation are independent • Concerned only with observer • Obtain estimate:

  6. Predictor-Corrector Measurements Correct (Measurement Update) Predict (Time Update)

  7. To Err Is Kalman! • How accurate is the estimate? • What is its distribution?

  8. Predictor-Corrector Measurements Correct (Measurement Update) Predict (Time Update)

  9. No random variable You don’t know it Predict Eigenvalues must be <1 (For convergence) Distribution does effect error covariance

  10. Kalman Gain Innovations (What’s New) Oblique Projection Correct

  11. System 1 (Basic Example) • X 2, • Companion Form • Nice but not perfect numerics and stability

  12. System 1

  13. System 1

  14. System 1

  15. System 1

  16. System 1 (Again) • X 2, • Companion Form • Nice but not perfect numerics and stability

  17. System 1

  18. System 1

  19. System 1

  20. System 1

  21. X 2, Large Eigenvalue Spread Condition number around 109 Large sampling time (big steps) System 2 (Stiffness)

  22. System 2

  23. System 2

  24. Trouble in Paradise • Inversion in the Kalman gain is slow and generally not stable • A is usually in companion form • numerically unstable (Laub) • Covariance are symmetric positive definite • Calculation cause P to become unsymmetric then lose positivity

  25. Square Root Filters • Kailath suggested propegating the square root rather than the whole covariance • Not really square root, actually Choleski Factor • rTr=R • Use on Rw, Rv, P

  26. Our Square Roots

  27. State Error

  28. Observations

  29. Measurement Equation

  30. Measurement Update • Then, by definition

  31. Updating for Free?

  32. Error Part 2

  33. Time Updating

  34. Paige’s Filter

  35. System 3 (Fun Problem) • X 20, • Known difficult matrix that was scaled to be stable

  36. System 3

  37. System 3

  38. System 3

  39. System 3

  40. Conclusions • Called Paige’s filter but really Paige and Saunders developed • O(n3) and about 60% faster than regular square root • Current interests: faster, special structures, robustness

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