Difference between revisions of "CDS 131, Fall 2020"

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The primary course texts are
The primary course texts are
* [FBS2e] K. J. Astrom and Richard M. Murray, [http://fbsbook.org ''Feedback Systems: An Introduction for Scientists and Engineers''], Princeton University Press, Second Edition*, 2020.
* [FBS2e] K. J. Astrom and Richard M. Murray, [http://fbsbook.org ''Feedback Systems: An Introduction for Scientists and Engineers''], Princeton University Press, Second Edition*, 2020.
* [FBS2s] Richard M. Murray, ''{{cds131 fa20 pdf|fbs-linsys_11Oct2020.pdf|Feedback Systems:  Notes on Linear Systems Theory}}'', 2020. (Updated 11 Oct 2020)
* [FBS2s] Richard M. Murray, ''{{cds131 fa20 pdf|fbs-linsys_30Oct2020.pdf|Feedback Systems:  Notes on Linear Systems Theory}}'', 2020. (Updated 30 Oct 2020)
* [DFT] J. Doyle, B. Francis and A. Tannenbaum, [http://www.control.utoronto.ca/people/profs/francis/dft.pdf ''Feedback Control Theory''], Dover, 2009 (originally published by Macmillan, 1992).
* [DFT] J. Doyle, B. Francis and A. Tannenbaum, [http://www.control.utoronto.ca/people/profs/francis/dft.pdf ''Feedback Control Theory''], Dover, 2009 (originally published by Macmillan, 1992).
* [OBC] R. M. Murray, "Optimization-Based Control", 2010. [http://www.cds.caltech.edu/~murray/amwiki/index.php?title=OBC:Main_Page Online access]
* [OBC] R. M. Murray, "Optimization-Based Control", 2010. [http://www.cds.caltech.edu/~murray/amwiki/index.php?title=OBC:Main_Page Online access]

Revision as of 14:43, 30 October 2020

Linear Systems Theory


  • Richard Murray (CDS/BE), murray@cds.caltech.edu
  • Lectures: MWF, 2-3 pm, via Zoom

Teaching Assistants

  • Apurva Badithela (CDS), Rachel Gehlhar (ME)
  • Office hours: Fri, 4-5 pm and Tue, 4-5 pm via Zoom

This is the course homepage for CDS 131, Fall 2020. This course is intended for first year graduate students in controls, advanced undergraduates in EE, ChE, and ME who have taken a basic controls course (e.g., CDS 110, ChE 105, EE 113), and motivated graduate students in other disciplines would would like to learn more about linear systems and control. All students taking the course should also have a good understanding of (matrix) differential equations and linear algebra, in addition to some familiarity with control system design.

Access information

  • Lectures (Zoom): MWF, 2-3 pm. See Caltech Canvas for Zoom link (registered students only)
  • Lecture recordings (Google Drive): See Caltech Canvas for link (registered students only)
  • Class discussion (Piazza): See Caltech Canvas for Piazza link (registered students only)

Catalog Description

CDS 131. Linear Systems Theory. 9 units (3-0-6); first term. Prerequisites: Ma 1b, 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. Instructor: Murray.

Lecture Schedule

There will be 2-3 one hour lectures per week, with the specific days varying from week-to-week. The lecture days for each week will be announced in class and posted here at least 1 week in advance.


  • Opt = optional reading (useful if you are confused and trying to understand the basic concepts)
  • Rec = recommended reading (this is what the homework is based on)
  • Adv = advanced reading (more detailed results, useful if you are interested in learning more)


  • 30 Oct 2020: updated FBS2s (Ch 4, 5 bugs fixed)
  • 11 Oct 2020: updated FBS2s posted (Ch 3 typo fixes)
  • 26 Sep 2020: updated syllabus
  • 13 Sep 2020: Added access information
  • 6 Sep 2020: Minor updates
  • 17 May 2020: Syllabus created
Date Topic Reading Homework
Week 1

30 Sep
2 Oct

Introduction and review
  • Course logistics
  • Review: feedback systems
  • Intro to input/output dynamical systems
  • Opt: FBS2e Ch 1, 2
  • Rec: FBS2e Ch 9, 10 (review)
  • Adv: FBS2s Ch 1, Sontag, Ch 2
HW #1

Out: 30 Sep
Due: 7 Oct
Solns (Caltech only)

Week 2

5 Oct
7 Oct
9 Oct

Linear I/O systems
  • Differential and difference equations (with inputs and outputs, including disturbances and noise)
  • Linearized system dynamics
  • Stability of equilibrium points, I/O stability
  • Convolution equation, impulse response
  • Opt: FBS2e Ch 3
  • Rec: FBS2e Sec 5.1‑5.3, 6.1‑6.3; FBS2s Ch 2
  • Adv: Sontag Sec C.4, 2.6
HW #2

Out: 7 Oct
Due: 14 Oct
Solns (Caltech only)

Week 3

12 Oct
14 Oct
16 Oct

  • Definitions (reachability, stabilizability)
  • Characterization and rank tests (Grammian, PBH)
  • Decomposition into reachable/unreachable subspaces
  • Eigenvalue placement theorem
  • Rec: FBS2e Sec 7.1, 7.2; FBS2s Ch 3
  • Adv: FBS2e Sec 7.3; Sontag Sec 3.1‑3.3, 3.5
HW #3

Out: 14 Oct
Due: 21 Oct
Solns (Caltech only)

Week 4

19 Oct
21 Oct
23 Oct

State feedback
  • Optimization and optimal control
  • Linear quadratic regulator (including Ricatti equation)
  • Opt: FBS2e Sec 7.5
  • Rec: FBS2s Ch 4 (= OBC Ch 2)
  • Adv: Sontag Sec 8.1‑8.3, 9.1, 9.2
HW #4

Out: 21 Oct
Due: 28 Oct

Week 5

26 Oct
28 Oct
30 Oct

Observability and state estimation
  • Definitions (observability, observable subspace)
  • Characterization and rank tests
  • Kalman decomposition
  • Linear observers (full-state)
  • Rec: FBS2e Sec 8.1-8.3; FBS2s Ch 5
  • Adv: Sontag Sec 6.1‑6.3, 7.1
HW #5

Out: 28 Oct
Due: 4 Nov

Week 6

2 Nov
4 Nov
6 Nov

Signals and Systems
  • Norms of signals in continuous (and discrete) time
  • I/O systems, LTI systems
  • Induced system norms
  • Opt: TBD
  • Rec: DFT Sec 2.1‑2.4
  • Adv: TBD
HW #6

Out: 4 Nov
Due: 11 Nov

Week 7

9 Nov
11 Nov
13 Nov

Frequency domain analysis
  • Internal stability
  • Tracking, disturbance rejection
  • I/O performance
  • Opt: FBS2e Sec 9.1, 9.2 and 9.5, Sec 10.1-10.2, Sec 12.1-12.2
  • Rec: DFT Ch 3
  • Adv: Lewis Ch 5-8
HW #7

Out: 11 Nov
Due: 18 Nov

Week 8

16 Nov
18 Nov
15 Nov

Uncertainty and robustness
  • Types of uncertainty: parametric, operator, disturbances/noise
  • Robust stability and robust performance
  • Opt: FBS2e Sec 10.3, Sec 13.1-13.3
  • Rec: DFT Ch 4
HW #8

Out: 18 Nov
Due: 25 Nov

Week 9

23 Nov
30 Nov
2 Dec

Fundamental limits
  • Algebraic limits
  • Bode's integral formula
  • Maximum modulus principle
  • Opt: FBS2e Sec 14.1, 14.2. 14.4
  • Rec: DFT Ch 6
  • Adv: Lewis, Ch 9
HW #9

Out: 25 Nov
Due: 4 Dec (Fri)

Week 10

4 Dec

Review for final Final

Out: 4 Dec
Due: 11 Dec


The final grade will be based on homework sets, a midterm exam, 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 (4 Dec) 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

* Please make sure to use the second edition [FBS2e].

The following additional references may also be useful:

  • [Lew03] A. D. Lewis, A Mathematical Approach to Classical Control, 2003. Online access.

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.