AE6505_HW2_Spr2024

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University of Notre Dame *

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Electrical Engineering

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Feb 20, 2024

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Homework #2 AE6505 Kalman Filtering, Spring 2024 Prof. Gunter Assigned: 1-29-24 Due: 2-12-24 (2-19-24 for DL Students) Homework is due by 11:59p on the indicated due date, and should be submitted electronically via Canvas. Late homework assignments may be turned in within 48 hours of the original due date for half credit. Any homework turned in after this is not counted. In-class verbal due date announcements override projected dates in the lecture plan. Please submit your materials as two files. The first should be a writeup of your solutions, complete with any figures, explanations, etc., in .pdf form. This is the document that will be graded, i.e., do not embed solutions in your code, or require the grader to run your code to get any results. Homework should be professional, legible, indicate units, and sufficiently describe all important steps in a solution. Your final answer for each problem should be boxed or clearly indicated. You are welcome to scan any pages that are handwritten, but please make sure any such pages are clear and legible. Deductions will be made for incomplete solutions and improper formats. In addition to the .pdf file, upload any Matlab files that you have developed to generate the results described in your writeup as a single .zip file. The Matlab code you submit should be able to be run without modification, so do not include hardcoded file paths. 1. Exercises 3.5, 3.9, 3.13, from the text by Simon. For 3.5, show that you achieve the same results using both the LUMVE w/apriori formulation derived in class, as well as the recursive approach shown in Sec. 3.3 of Simon. For 3.13, write your own curve fitting routine (i.e., don’t use built-in routines like polyfit , although you can use them to verify the correctness of your own code), and use the following time series instead for the steel production from 1946 to 1956: [50.1 74.5 80.4 72.3 66.4 78.9 86.6 95.3 90.8 111.2 121.5] 2. Exercises 4.4, 4.6, 4.8, and 4.11 from the text by Simon. For 4.11, run the simulation for 7 seconds using the following initial parameters instead: P 0 = 3 , f = - . 25 , q c = 0 . 75 3. Using the following expression of a vector derivative, where u is a 1xm vector, x is a nx1 vector, and v is a 1xm vector, show how we go from Eqn. 3.5 to Eqn. 3.6 in the text. ∂vu x = v ∂u ∂x + u T ∂v T ∂x Eqn. 3.6 does not fully verify that ˆ x is a minimum. To do this, now find the second derivative of the cost function, J, and show that it is positive definite (and under what conditions for the matrix H). 4. You are in charge of assessing the performance of a lidar ranging sensor for a self-guided drone application. In an effort to calibrate the sensor, you gather a set of 400 measurements at 1 second intervals from a fixed distance (see file ”data.txt”). The figure below shows the result after the known range is removed, i.e., the plot shows measurement residuals, in units of mm. There is clearly some remnant signal in the measurements, which you would like to remove as much as possible, because you believe it to be systematic error in the ranging system. 1

(a) First compute the pre-fit RMS of the residuals. This will provide a baseline as to whether future error modeling improves the results. (b) Now estimate a constant bias and linear trend term to the residuals, e.g., y = Ax + B, using a weighted least squares approach. With the estimates, now compute a post-fit RMS value to determine if removing these terms from the data reduces the overall RMS. (c) Even after removing a bias and trend, you still see some periodic signal in the resid- uals. Run a spectral analysis on the residuals to determine the dominant frequencies involved, and plot the results. For this, it’s fine to use a simple fft approach, such as the one here: https://www.mathworks.com/help/matlab/math/basic-spectral-analysis.html or even a Matlab toolbox, as long as you can clearly plot and identify the primary frequencies in the signal. (d) Based on the spectral analysis, now develop a modified error model that would target these frequencies. Write out the model and rerun your weighted least squares code with the new model, and compare the post-fit RMS values to the earlier values obtained using just the bias and trend. Plot a histogram of the residuals. Do you suspect there is still signal that could be removed, or is it Gaussian noise? (e) What is the final, post-calibrated measurement uncertainty that you will pass on to your other project team members to use for this ranging sensor in their real-time positioning system? 2

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