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

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

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

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

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

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

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

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

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

System 1

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

System 1

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

System 2

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

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

Our Square Roots

State Error

Observations

Measurement Equation

Measurement Update Then, by definition

Updating for Free?

Error Part 2

Time Updating

Paige’s Filter

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

System 3

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