Due Thursday 2/22 11:59 PM Eastern.

In this homework, we are going to explore LQR, convex trajectory optimization, and convex model predictive control. Here is an overview of the problems:

1. Solve for the finite-horizon LQR controls as a convex optimization problem, then solve for the optimal feedback policy with the Ricatti recursion. Show that these are equivalent.

2. Here we will use TVLQR to track a cartpole swingup, as well as infinite horizon LQR to stabilize the cartpole in the inverted position.

3. In this question we will design controllers using LQR, convex trajectory optimization, and convex MPC for controlling the SpaceX Dragon spacecraft as it rendezvous with the ISS.

NOTE: UNIT TESTS ARE NOT FULLY COMPREHENSIVE!! PASSING THEM DOES NOT GUARANTEE FULL CREDIT IF CODE IMPLEMENTATION IS INCORRECT.

Before you submit your completed homework PDF, remember to complete all the checklist items below:

• Make sure text and code, including special characters, are completely legible.
• Make sure code is completely visible (i.e., no cropped lines). If we can't see it, then we can't grade it... Look at section below for details on remedying this.
• Make sure plots and unit tests have been outputted and rendered properly. Meshcat visuals are exempt.
• DO NOT ALTER UNIT TEST CASES. We check and can tell...
• Assign questions to each respective page of your PDF in Gradescope, including the text of the problem.

If your code is cropped in the PDF, then there are multiple remedies for this:

1. Split line around binary operator (e.g., +):

L_grad = FD.gradient(f, x) +
                transpose(FD.jacobian(c, x))*λ

2. Split line around parenthesis:

L_grad = (FD.gradient(f, x)
                + transpose(FD.jacobian(c, x))*λ)

3. Split line after assignment operator (e.g., =):

L_grad = 
    FD.gradient(f, x) + transpose(FD.jacobian(c, x))*λ

4. Combine 1-3 to split into more lines:

L_grad = 
    FD.gradient(f, x) +
    transpose(FD.jacobian(c, x))*λ

5. Assign terms to more variables:

cost_grad = FD.gradient(f, x)
constraint_jac_T = transpose(FD.jacobian(c, x))

L_grad = cost_grad + constraint_jac_T*λ

Feel free to use any method you'd like to export your Jupyter notebook as a PDF (with all checklist items completed) and submit on gradescope.

We recommend the following method of converting your Jupyter notebook to a PDF because it requires no additional installs (hopefully). It's slightly involved, but it is the most consistent in our experience.

1. Open the Jupyter notebook in your favorite web browser (not VS Code) with IJulia.
2. Go through the submission checklist, and make sure all relevant items are completed.
3. In the top left corner of the Jupyter menu bar, do File -> Save and Export Notebook As -> HTML. It should download an HTML file.
4. Open the downloaded HTML file in your favorite web browser.
5. Open up the browser's print menu and select Save as PDF.
6. Save and combine PDFs.
7. Submit on Gradescope, and make sure to assign the right pages to all questions.

If HTML to PDF does not work, feel free to try other methods: https://mljar.com/blog/jupyter-notebook-pdf/.