CDS 110/ChE 105, Spring 2024

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Analysis and Design of Feedback Control Systems


  • Richard Murray (CDS/BE),
  • Lectures: MWF, 2-3 pm, 106 Spalding
  • Office hours: Wed, 3-4 pm, Annenberg Lounge

Teaching Assistants

  • Natalie Bernat (CMS), Manisha Kapasiawala (BE)
  • Office hours:
      Mon, 3-4 pm, 111 Keck
      Tue, 4-5 pm, 110 Steele

This is the course homepage for CDS 110, Spring 2024. This course is co-taught with ChE 105 (Dynamics and Control of Chemical Systems).

Course Syllabus

An introduction to analysis and design of feedback control systems in the time and frequency domain, with an emphasis on state space methods, robustness, and design tradeoffs. Linear input/output systems, including input/output response via convolution, reachability, and observability. State feedback methods, including eigenvalue placement, linear quadratic regulators, and model predictive control. Output feedback including estimators and two-degree of freedom design. Input/output modeling via transfer functions and frequency domain analysis of performance and robustness, including the use of Bode and Nyquist plots. Robustness, tradeoffs and fundamental limits, including the effects of external disturbances and unmodeled dynamics, sensitivity functions, and the Bode integral formula.

Lecture Schedule

Date Topic Reading Homework
Week 1

1 Apr
3 Apr
5 Apr

Introduction and review
  • Course overview and logistics
  • Introduction to feedback and control
  • Introduction to python-control
FBS2e 1.1-1.5 (skim), 2.1-2.4 HW #1
  • Out: 3 Apr 2024
  • Due: 10 Apr 2024, 2 pm
  • Solutions (Caltech only)
Week 2

8 Apr*
10 Apr
12 Apr

Modeling, Stability
  • State space models
  • Continuous and discrete time systems
  • Phase portraits and stability
FBS2e 3.1-3.2, 4.1, 5.1-5.3 HW #2
  • Out: 10 Apr 2024
  • Due: 17 Apr 2024, 2 pm
  • Solutions (Caltech only)
Week 3

15 Apr
17 Apr
19 Apr

Linear Systems
  • Input/output response of LTI systems
  • Matrix exponential, convolution equation
  • Linearization around an equilibrium point
FBS2e 6.1-6.4 HW #3
  • Out: 17 Apr 2024
  • Due: 24 Apr 2024, 2 pm
  • Solutions (Caltech only)
Week 4

22 Apr
24 Apr
26 Apr*

State Feedback
  • State feedback and eigenvalue placement
  • Integral action
  • Linear quadratic regulators (LQR)
FBS2e 7.2-7.5 HW #4
  • Out: 24 Apr 2024
  • Due: 1 May 2024, 2 pm
  • Solutions (Caltech only)
Week 5

29 Apr
1 May*
3 May

State estimation
  • Observers, observability
  • Control using estimated state
  • Kalman filtering (intro)
FBS2e 8.1-8.4 HW #5
  • Out: 2 May 2024
  • Due: 8 May 2024, 2 pm
  • Solutions (Caltech only)
Week 6

6 May
8 May
10 May

Trajectory generation and tracking
  • Two degree of freedom design
  • Gain scheduling
  • Receding horizon/model predictive control
FBS2e, 7.1, 8.5
OBC, Ch 1, 2.1, 2.2, 4.3
HW #6
  • Out: 9 May 2024
  • Due: 15 May 2024, 2 pm
  • Solutions (Caltech only)
Week 7

13 May
15 May
17 May

Frequency domain analysis
  • Bode and Nyquist plots
  • Stability margins
FBS2e 9.1-9.4, 10.1-10.3 HW #7
  • Out: 16 May 2024
  • Due: 22 May 2024, 2 pm
  • Solutions (Caltech only)
Week 8

20 May
22 May
24 May*

Robustness and fundamental tradeoffs
  • Sensitivity functions, performance specifications
  • Bode integral formula
  • Limits due to RHP poles/zeros via maximum modulus theory
FBS2e, 12.1-12.3, 13.3, 14.2, 14.4 HW #8
  • Out: 23 May 2024
  • Due: 31 May 2024, 2 pm
  • Solutions (Caltech only)
Week 9

27 May
29 May*
31 May
3 Jun

PID control
  • Frequency domain design concepts
  • Windup and anti-windup
FBS2e, 11.1-11.4 HW #9 (Sophomores, Juniors)
  • Out: 31 May 2024
  • Due: 7 Jun 2024, 2 pm
  • Solutions (Caltech only)
Week 10

5 Jun
7 Jun

Final review and applications
  • Final exam review (Wed)
Final exam (Seniors, Graduate Students)
  • 7 Jun (Fri), 2-3 pm
Finals week (Sophomores, Juniors)
None Final exam (Sophomores, Juniors)
  • 12 Jun (Wed), 2-3 pm


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

  • Homework (60%): Homework sets will be handed out weekly and due on Wednesdays by 2 pm 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 2 pm or after the two extensions are exhausted will receive 50% credit. Python (or MATLAB) code is considered part of your solution and should be printed and turned in with the problem set (whether the problem asks for it or not).
(Sophomores and juniors only) The lowest score on your homework sets will be dropped.
  • Final exam (40%): The final exam will be a 1-2 hour, in-class, closed-book exam.
    • Seniors and graduate students: the final exam will be on 7 Jun (Fri), 2-4 pm
    • Sophomores and juniors: the final exam will be on 12 Jun (Wed), 2-4 pm

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).

ChatGPT and other 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 written up individually. You cannot use online tools to generate solutions for coding problems (cutting and pasting from templates or materials handed out in class and editing them as appropriate is OK).

No collaboration is allowed on the final exam.

Course Text and References

The primary course text is

This book is available via free download.

The following additional references, also available for free, may also be useful:

In addition to the books above, the textbooks below may also be useful. They are available in the library (non-reserve), from other students, or you can order them online.

  • B. Friedland, Control System Design: An Introduction to State-Space Methods, McGraw-Hill, 1986.
  • G. F. Franklin, J. D. Powell, and A. Emami-Naeni, Feedback Control of Dynamic Systems, Addison-Wesley, 2002.