Estimation and the Kalman Filter

1. Estimation and the Kalman Filter David Johnson

2. The Mean of a Discrete Distribution • “I have more legs than average”

3. p ( x ) ~ Ν ( μ , Σ ) : 1 1 - t 1 - - - x μ Σ x μ ( ) ( ) = p ( x ) e 2 1 / 2 p d / 2 ( 2 ) Σ Gaussian Definition m s 2 p ( x ) ~ N ( , ) : 2 - m 1 ( ) x 1 m - = 2 s 2 p ( x ) e p s 2 Univariate -s s m Multivariate

4. Back to the non-evolving case • Two different processes measure the same thing • Want to combine into one better measurement • Estimation

5. Estimation Estimator Data + noise Data + noise Data + noise Estimation What is meant by estimation? z estimate H ŷ Stochastic process

6. A Least-Squares Approach • We want to fuse these measurements to obtain a new estimate for the range • Using a weighted least-squares approach, the resulting sum of squares error will be • Minimizing this error with respect to yields

7. A Least-Squares Approach • Rearranging we have • If we choose the weight to be we obtain

8. A Least-Squares Approach • For merging Gaussian distributions, the update rule is Show for N(0,a) N(0,b)

9. What happens when you move? derive

10. Moving • As you move • Uncertainty grows • Need to make new measurements • Combine measurements using Kalman gain

11. The Kalman Filter • OPTIMAL: • Linear dynamics • Measurements linear w/r to state • Errors in sensors and dynamics must be zero-mean (un-bias) white Gaussian • RECURSIVE: • Does not require all previous data • Incoming measurements ‘modify’ current estimate DATA PROCESSING ALGORITHM: The Kalman filter is essentially a technique of estimation given a system model and concurrent measurements (not a function of frequency) “an optimal recursive data processing algorithm”

12. Estimate the state of a discrete-time controlled process that is governed by the linear stochastic difference equation: with a measurement: The random variables wk and vk represent the process and measurement noise (respectively). They are assumed to be independent (of each other), white, and with normal probability distributions In practice, the process noise covariance and measurement noise covariance matrices might change with each time step or measurement. The Discrete Kalman Filter (PDFs)

13. The Discrete Kalman Filter State transition Control signal State prediction “prior” estimate First part – model forecast: prediction Process noise covariance Error covariance prediction Prediction is based only the model of the system dynamics.

14. The Discrete Kalman Filter actual measurement “prior” state prediction state correction Kalman gain predicted measurement “posterior” estimate Second part – measurement update: correction update error covariance matrix (posterior)

15. The Discrete Kalman Filter variance of the predicted states = ------------------------------------------------------------ variance of the predicted + measured states measurement sensitivity matrix measurement noise covariance The Kalman gain, K:“Do I trust my model or measurements?”

16. Estimate a constant voltage • Measurements have noise • Update step is • Measurement step is

17. Results

18. Variance

19. Parameter tuning

20. More tuning

21. Kalman Filter Extended Kalman Filter The Extended Kalman Filter (EKF) • The Extended Kalman (EKF) is a sub-optimal extension of the original KF algorithm • The EKF allows for estimation of non-linear processes or measurement relationships • This is accomplished by linearizing the current mean and covariance estimates (similar to a first order Taylor series approximation) • Suppose our process and measurement equations are the non-linear functions

22. Linearity Assumption Revisited

23. Non-linear Function

24. EKF Linearization (1)

25. EKF Linearization (2)

26. EKF Linearization (3)