CDS 131, Fall 2018

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Introduction to Feedback Control Systems


  • Richard Murray (CDS/BE),
  • Lectures: MWF, 2-3 pm, 213 Annenberg

Teaching Assistants

  • TBD
  • Office hours: TBD

This is the course homepage for CDS 131, Fall 2018.

WARNING: This web page is in preliminary form. It will be updated prior to the start of class

Course Syllabus

Basic system concepts; state-space and I/O representation. Properties of linear systems, including stability, performance, robustness. Reachability, observability, minimality, state and output-feedback.

Intended audience and expected background:

  • First year graduate students in controls - assume they have already had at least one control course (could be frequency domain only). This course will provide the mathematical basis for CDS 231 (John’s course; not quite sure what it covers) and CDS 232 (nonlinear systems).
  • Advanced undergraduates in EE and ChE who have taken EE 113 or ChE 105 - this course could serve as a second course for students who have access to a basic controls course in their discipline.
  • Motivated graduate students in CMS, BE, or other disciplines - students who have taken CDS 141 (John’s other course) and are motivated to learn more about linear systems and control could take the course, assuming they have a decent mathematical background (differential equations, linear algebra).

Lecture Schedule

With the exception of the first week, there will be two 1-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.

Date Topic Reading Homework
Week 1

1 Oct
3 Oct
5 Oct

Introduction and review
  • System modeling, I/O systems, LTI systems
  • Linearized system dynamics
  • Norms of signals in continuous and discrete time
FBS2e, Ch 3 HW #1
Week 2

8 Oct
10 Oxct
12 Oct

Linear I/O systems
  • Differential and difference equations (with inputs and outputs, including disturbances and noise)
  • Convolution equation, impulse response
  • Stability of equilibrium points, I/O stability
FBS2e, Ch 4 HW #2
Week 3

15 Oct
17 Oct
19 Oct


Definitions (reachability, stabilizability) Characterization and rank tests (Grammian, PBH) Decomposition into stable/unstable and reachable/unreachable subspaces

FBS2e, Ch 4 (concepts) HW #3
Week 4

22 Oct
24 Oct
26 Oct

State feedback
  • Eigenvalue placement theorem
  • Linear quadratic regulator (including Ricatti equation)
FBS2e, Ch 5 (concepts) HW #4
Week 5

29 Oct
31 Oct
2 Nov

Observability and state estimation
  • Definitions (observability, observable subspace)
  • Characterization and rank tests
  • Kalman decomposition
  • Linear observers (full-state)
FBS2e, Ch 6 (concepts) Midterm
Week 6

5 Nov
7 Nov
9 Nov

Frequency domain representations
  • Exponential (frequency) response of a linear input/output system
  • State space realizations, minimal realizations
  • Control systems in the frequency domain; Youla parameterization
HW #5
Week 7
HW #6
Week 8
HW #7
Week 9
HW #8
Week 10
Final Review + Demos None


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

  • Homework (50%): Homework sets will be handed out weekly and due on Wednesdays by 2 pm either in class or in the labeled box across from 107 Steele Lab. 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).
  • Midterm exam (20%): A midterm exam will be handed out at the beginning of midterms period (28 Oct) and due at the end of the midterm examination period (3 Nov). The midterm exam will be open book and computers will be allowed (though not required).
  • 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. MATLAB/Python scripts and plots are considered part of your writeup and should be done individually (you can share ideas, but not code).

No collaboration is allowed on the midterm or final exams.

Course Text and References

The primary course text is

This book is available via the Caltech online bookstore or via download from the companion web site. Note that the second edition of this book is in preparation for publication and will serve as the primary text for the course (but almost all of the material we will cover is also in the first edition).

The following additional references may also be useful:

  • A. D. Lewis, A Mathematical Approach to Classical Control, 2003. Online access.
  • J. Distefano III, A. R. Stubberud and Ivan J. Williams (Author), Schaum's Outline of Feedback and Control Systems, 2nd Edition, 2013.

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.