CDS 131, Winter 2027

Linear Systems Theory
Instructors Richard Murray (CDS/BE), murray@cds.caltech.edu Lectures: MW, 1-2:30 pm, room TBD Office hours: TBD	Teaching Assistants TBD Office hours: TBD

This is the course homepage for CDS 131, Winter 2027.

Catalog Description

CDS 131. Linear Systems Theory. 12 units (3-0-9): second term. Prerequisites: Ma 1 b, Ma 2, ACM/IDS 104 or equivalent (may be taken concurrently). Basic system concepts; state-space and I/O representation. Properties of linear systems, including stability, performance, robustness. Reachability, observability, minimality, state and output-feedback. Brief introduction to optimal control and control of networked and nonlinear systems. Motivating case studies from tech, biology, neuroscience, and medical systems.

Lecture Schedule

Date	Topic	Reading	Homework
Week 1 4 Jan 6 Jan	Introduction and state-space systems Course logistics and review of feedback systems State-space and input/output dynamical systems Differential and difference equations with inputs, outputs, disturbances and noise Nonlinear and learned dynamics as motivation	Opt: FBS2e Ch 1, 3 Rec: FBS2e Ch 2, Sec 5.1-5.3, 6.1-6.3; LST Ch 2 Adv: LST Ch 1 Lecture materials:	HW #1 Out: 4 Jan Due: 13 Jan, 10 am
Week 2 11 Jan 13 Jan	Linear systems and linearization Linearized system dynamics Solutions of linear systems, convolution equation, impulse response	Opt: DFT Sec 2.6 Rec: FBS2e Sec 5.1-5.3, 6.1-6.3; LST Ch 2 Lecture materials:	HW #2 Out: 11 Jan Due: 20 Jan, 10 am
Week 3 18 Jan 20 Jan	Stability Stability of equilibrium points, I/O stability Lyapunov functions and the Lyapunov equation	Rec: FBS2e Sec 5.1-5.3; LST Ch 2 Lecture materials:	HW #3 Out: 18 Jan Due: 27 Jan, 10 am
Week 4 25 Jan 27 Jan*	Reachability and state feedback Definitions (reachability, stabilizability) Rank tests (Gramian, PBH) Decomposition into reachable/unreachable subspaces Eigenvalue placement theorem	Rec: FBS2e Sec 7.1, 7.2; LST Ch 3 Adv: FBS2e Sec 7.3 Lecture materials:	HW #4 Out: 25 Jan Due: 3 Feb, 10 am
Week 5 1 Feb 3 Feb	Optimal control Optimization and optimal control Linear quadratic regulator and the Riccati equation Optimization-based control (details TBD)	Opt: FBS2e Sec 7.5 Rec: LST Ch 4 (= OBC Ch 2) Lecture materials:	HW #5 Out: 1 Feb Due: 10 Feb, 10 am
Week 6 8 Feb 10 Feb	Observability and state estimation Definitions (observability, observable subspace) and rank tests Kalman decomposition Linear observers (full-state) and output feedback Kalman filter	Rec: FBS2e Sec 8.1-8.3; LST Ch 5 Lecture materials:	HW #6 Out: 8 Feb Due: 17 Feb, 10 am
Week 7 15 Feb 17 Feb	Signals and systems Norms of signals in continuous and discrete time I/O systems, LTI systems and induced system norms (L2 gain, H2 and H-infinity norms)	Rec: DFT Sec 2.1-2.4 Opt: ROC Ch 4 Lecture materials:	HW #7 Out: 15 Feb Due: 24 Feb, 10 am
Week 8 22 Feb 24 Feb*	Uncertainty and robustness Uncertainty models: parametric, unstructured, and learned components as bounded-gain blocks Small gain theorem and robust stability Robust performance H-infinity control as a game against a worst-case disturbance	Opt: FBS2e Sec 10.3, 13.1-13.3 Rec: ROC Ch 9 (small gain); DFT Ch 4 (selected) Lecture materials:	HW #8 Out: 22 Feb Due: 3 Mar, 10 am
Week 9 1 Mar 3 Mar	Learning-based approaches to control Input-to-state stability Contraction and incremental stability Robustness of learning algorithms	Notes (TBD) Lecture materials:	HW #9 Out: 1 Mar Due: 10 Mar, 10 am
Week 10 8 Mar 10 Mar	Review and special topic 8 Mar: Review for the final exam 10 Mar: Special topic (TBD)	TBD Lecture materials:
Finals 15 Mar 17 Mar	Final exam period		Final exam

Grading

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

• Homework (15%): Homework sets will be handed out weekly and due on Wednesdays by 10 am via 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 10 am or after the two extensions are exhausted will receive 50% credit.

• Final exam (85%): The final exam will be either an oral exam or a 1.5 hour, in-class, closed-book exam.

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. Python (or MATLAB) scripts and plots are considered part of your writeup and should be done individually (you can share ideas, but not code).

AI tools may be used in the same manner as a fellow student in the class: you are allowed to consult online tools and use them to understand the topics, but all solutions should be guided by you.

No collaboration is allowed on the final exam.

Course Text and References

The primary course text is

• [LST] Richard M. Murray, Feedback Systems: Notes on Linear Systems Theory, 2020. (Updated 30 Oct 2020)

Additional references, used for selected readings:

• [FBS2e] K. J. Astrom and R. M. Murray, Feedback Systems: An Introduction for Scientists and Engineers, Princeton University Press, Second Edition, 2020.
• [DFT] J. Doyle, B. Francis and A. Tannenbaum, Feedback Control Theory, Dover, 2009 (originally published by Macmillan, 1992).
• [ROC] K. Zhou, J. C. Doyle and K. Glover, Robust and Optimal Control, Prentice Hall, 1996.
• [OBC] R. M. Murray, Optimization-Based Control, 2010.