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 x 0  Go to x G Minimize or maximize some quantity (time, energy, etc.)

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

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

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

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

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

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

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

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

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

12. System 1

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

17. System 1

21. System 2 (Stiffness) X  2, Large Eigenvalue Spread Condition number around 10 9 Large sampling time (big steps)

22. 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 r T r=R Use on R w, R v, 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

40. Conclusions Called Paige’s filter but really Paige and Saunders developed O(n 3 ) and about 60% faster than regular square root Current interests: faster, special structures, robustness