CDS 112/Ae 103a, Winter 2022

Optimal Control and Estimation
Instructors Richard Murray (CDS/BE), murray@cds.caltech.edu Lectures: MWF, 2-3 pm, 213 ANB Office hours: Wed, 3-3:30 pm (at CDS tea)	Teaching Assistants Apurva Badithela (CDS), Ayush Pandey (CDS) Office hours: Fri 4-4:45 / Mon 3-3:45 in 213 ANB; Fri 4:45-5:15 / Mon 3:45-4:15 via Zoom

This is the course homepage for CDS 112 (and Ae 103a), Winter 2022. This course is intended for undergraduates and graduate students interested in optimization-based methods in control. After completion of the course, students will understand the key principles of state-space based controller design, including optimal estimation and control techniques.

Catalog Description

CDS 112. Optimal Control and Estimation. 9 units (3-0-6): second term. Prerequisites: CDS 110 (or equivalent) and CDS 131. Optimization-based design of control systems, including optimal control and receding horizon control. Introductory random processes and optimal estimation. Kalman filtering and nonlinear filtering methods for autonomous systems.

Ae 103 a. Aerospace Control Systems. 9 units (3-0-6): second term. Prerequisites: CDS 110 (or equivalent), CDS 131 or permission of instructor. Optimization-based design of control systems, including optimal control and receding horizon control. Introductory random processes and optimal estimation. Kalman filtering and nonlinear filtering methods for autonomous systems.

Lecture Schedule

Date	Topic	Reading	Homework
Week 1 3 Jan 5 Jan 7 Jan	Introduction and review Course introduction and logistics Overview: control architectures Python Control Systems Library	OBC, Chapter 1 (PDF) Lecture notes: Mon, Wed Jupyter notebook: W1_intro_to_python-control.ipynb Feedback Control Theory video tutorial (Simons Institute)	HW #1 servomech-python_template.ipynb diskdrive.py Out: 5 Jan Due: 12 Jan Solns (Caltech only)
Week 2 10 Jan 12 Jan 19 Jan	Two degree of freedom control design Trajectory generation Differential flatness Implementation in Python Gain scheduling	OBC, Chapter 2 (PDF) Background: FBS2e, Section 8.5 Theory: LST, Sections 3.1, 3.2 Jupyter notebooks: W2_flatness.ipynb, W2_gainsched.ipynb	HW #2 Out: 12 Jan Due: 19 Jan Solns (Caltech only)
Week 3 17 Jan 19 Jan 21 Jan	Optimal control Maximum principle Dynamic programming Examples and applications Implementation in Python	OBC, Chapter 3 (PDF) Theory: Liberzon, 2010 Jupyter notebooks: W3_optimal.ipynb, W3_linquad.ipynb, vehicle.py	HW #3 Out: 19 Jan Due: 26 Jan pvtol.py Solns (Caltech only)
Week 4 24 Jan 26 Jan 28 Jan*	Linear quadratic regulators Problem formulation and solution Choosing LQR Weights Incorporating integral feedback Implementation in Python Dynamic programming	OBC, Chapter 3 (PDF) Theory: Liberzon, 2010 Jupyter notebooks (Wed): W4_pvtol-lqr.ipynb, W4_lqr-tracking.ipynb, ctrlutil.py, pvtol.py Jupyter notebooks (Fri): value_iteration_discrete.ipynb, transys.py	HW #4 Out: 26 Jan Due: 2 Feb Solns (Caltech only)
Week 5 31 Jan 2 Feb 4 Feb	Receding horizon control Problem formulation and solution Receding horizon control using differential flatness Example: Caltech ducted fan Implementation in Python	OBC, Chapter 4 (PDF) Theory: Model Predictive Control (2012) by Rawlings, Mayne, Diehl Jupyter notebooks: W5_rhc-doubleint.ipynb Lecture slides: Fri	HW #5 Out: 2 Feb Due: 9 Feb Solns (Caltech only)
Week 6 7 Feb 9 Feb 11 Feb	Stochastic systems Review of random variables Introduction to random processes Continuous-time, vector-valued random processes Linear stochastic systems Random processes in the frequency domain	OBC, Chapter 5 (PDF) Lectures notes: Mon, Wed/Fri Jupyter notebook: W6_stochresp.ipynb	HW #6 pvtol.py (updated) Out: 9 Feb Due: 16 Feb Solns (Caltech only)
Week 7 14 Feb 16 Feb 18 Feb*	Kalman filtering Linear quadratic estimators Extensions of the Kalman filter LQG control Example: vectored thrust aircraft Implementation in Python	OBC, Chapter 6 (PDF) Jupyter notebooks: W7_pvtol.ipynb, ctrlutil.py (updated), pvtol.py (updated), W7_Bayesian_Inference.ipynb	HW #7 Out: 16 Feb Due: 23 Feb Solns (Caltech only)
Week 8 21 Feb 23 Feb 25 Feb	Sensor fusion Discrete-time stochastic systems Kalman filters in discrete time Predictor-corrector form Combining information from multiple sensors Information filters Implementation in Python	OBC, Chapter 8 (PDF) Jupyter notebook: W8_kincar-fusion.ipynb, ctrlutil.py (updated), vehicle.py	HW #8 Out: 23 Feb Due: 2 Mar Solns (Caltech only)
Week 9 28 Feb 2 Mar 4 Mar	Autonomous systems Advanced estimation: information filter, UKF, MHE, particle filter Multi-layer control stack for autonomous systems Introduction to discrete decision-making Introduction to safety-critical systems Challenges and open problems	Lecture slides: Mon, Wed, Fri Appendix from Ben Grocholsky's thesis on information filter Rawlings, Mayne, Diehl: Model Predictive Control: Theory, Computation, and Design (2nd edition) Synthesis of Control Protocols for Autonomous Systems, N. Wongpiromsarn, U. Topcu and R. M. Murray. Unmanned Systems, 2013 Can We Really Use Machine Learning in Safety Critical Systems? (IPAM talk)	HW #9 Out: 2 Mar Due: 9 Mar Solns (Caltech only)
Week 10 7 Mar 9 Mar	Review for final		Final Out: 9 Mar Due: 16 Mar, 5 pm

Grading

The final grade will be based on homework sets and a final exam:

• Homework (70%): Homework sets will be handed out weekly and due on Wednesdays by 2 pm using GradeScope. Each student is allowed up to two extensions of no more than 2 days each over the course of the term. Homework turned in after Friday at 2 pm or after the two extensions are exhausted will not be accepted without a note from the health center or the Dean. MATLAB/Python code and SIMULINK/Modelica diagrams are considered part of your solution and should be printed and turned in with the problem set (whether the problem asks for it or not).

The lowest homework set grade will be dropped when computing your final grade.

• Final exam (30%): The final exam will be handed out on the last day of class (9 Mar) and due at the end of finals week. It will be an open book exam and computers will be allowed (though not required).

Collaboration Policy

Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor, but you cannot consult homework solutions from prior years and you must cite any use of material from outside references. All solutions that are handed in should be written up individually and should reflect your own understanding of the subject matter at the time of writing. Any computer code that is used to solve homework problems is considered part of your writeup and should be done individually (you can share ideas, but not code).

No collaboration is allowed on the final exam.

Course Text and References

The primary course texts are

• [OBC] R. M. Murray, "Optimization-Based Control", 2022. Online access

The following additional references may also be useful:

• [FBS2e] K. J. Astrom and Richard M. Murray, Feedback Systems: An Introduction for Scientists and Engineers, Princeton University Press, Second Edition*, 2020.
• [LST] Richard M. Murray, Feedback Systems: Notes on Linear Systems Theory, 2020. (Updated 30 Oct 2020)

Note: the only sources listed here are those that allow free access to online versions. Additional textbooks that are not freely available can be obtained from the library.