# Jeffery J. Leader's MA 3390 Page

The Mathematics of Kalman Filtering

Class meets on Mondays and Thursdays for the first half of the quarter (10 meetings total). There will be five homework assignments (60%), due on Mondays, and a final exam (40%). Your presence is expected in class. If you are absent more than twice then a grade penalty may be applied.

Homework:

HW1 (due date 9 Dec.)

1. Give examples of two physical applications of the finite-dimensional linear system described in class.

2. Give an example of marginally Gaussian random variables that are not jointly Gaussian.

3. Let X be a random 2-vector with each component following the standard normal p.d.f., and Y a random 2-vector with each component following the continuous uniform p.d.f. on [-1,1]. Compute the cross-covariance of X and Y.

HW2 (due date 16 Dec.)

4. Explain why the Kalman filter system diagram is an accurate representation of the Kalman filter state equations.

5. Simulate the simple random walk 10 times using 100 time steps each time. Comment on your results.

6. First, simulate step-by-step the application of the Kalman filter to the constant voltage problem in order to generate the delayed and current Kalman estimates. Take the measurement process observations to be 0.39,0.50,0.48,0.29,0.25,0.32,0.34,0.48,0.41,0.45 (10 time steps). Second, repeat this (via computer simulation) with an improved estimate of the initial system state based on the outcome of your previous trial and the same measurements. Compute the error covariance matrices also. Third, simulate applying the Kalman filter to this problem to generate the delayed and current Kalman estimates from an initial state of zero. Do this 5 times, using 100 time steps for each run. (Explain how you used the input and output noise process parameters to generate measurement process observations.) Comment on your results.

HW3 (due date 6 Jan.)

7. Verify that the Kalman filter plant is the transition matrix for the actual error in the estimate; that is, derive eq. (*).

8. Verify the formula for computing the delayed Kalman estimate from the previous current Kalman estimate; that is, derive eq. (**).

9. Demonstrate the optimality of the Kalman filter among linear filters by applying to the constant voltage problem both the Kalman filter and another filter of the form in eq. (***) that has a gain matrix close to, but not the same as, the Kalman gain matrix.

HW4 (due date 13 Jan.)

10.a. Give an example of a strict-sense stationary process that is not i.i.d. (and which has a nontrivial autocovariance). Compute the autocovariance function and write it in terms of the lag.

10.b. Give an example of a wide-sense stationary process that is not strict-sense stationary.

11. Show that for a strict-sense stationary process, the autocovariance depends only on the lag.

12.a. Consider again the constant-voltage problem but now take G(k)=1+1/(k+1) and R(k)=.1+2/(k+1) to mimic a process that must settle-in after being started. Leave the other parameters as before. Simulate 10 runs of the Kalman filter for 100 time steps, then 10 runs of the (suboptimal) time-invariant Kalman filter. Compute the (traces of the) error covariance matrices as well. How does the suboptimal filter perform compared to the optimal filter?

b. Repeat part a. but for k=50 to k=70 use R(k)=.3 instead.

HW5 (due date 20 Jan.)

13.Implement the time-invariant and time-varying Kalman filters at this page using the Matlab Kalman filter toolbox. Include plots in your submission, and discuss your results.

14.Implement and explain the Kalman filter example here.

15.Discuss the relationship between ARMA models, state-space models, and the Kalman filter.