# CDS 131, Fall 2019

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 Linear Systems Theory Instructors Richard Murray (CDS/BE), murray@cds.caltech.edu Lectures: MWF, 2-3 pm, 213 Annenberg Teaching Assistants TBD Office hours: TBD Piazza course page

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

### 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.

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 have a good understanding of (matrix) differential equations and linear algebra.

### 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)
 Date Topic Reading Homework Week 1 30 Sep 2 Oct 4 Oct Introduction and review Course logistics Norms of signals in continuous and discrete time (later) I/O systems, LTI systems Induced system norms Opt: FBS2e Ch 1 and 2; DFT Ch 1 Rec: DFT Sec 2.1‑2.4 Adv: Sontag, Ch 2 HW #1 Out: 2 Oct Due: 9 Oct Week 2 7 Oct 9 Oct 11 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; DFT Sec 2.6 Rec: FBS2e Sec 5.1‑5.3, 6.1‑6.3 Adv: Sontag Sec C.4, 2.6 HW #2 Out: 9 Oct Due: 16 Oct Week 3 14 Oct 16 Oct 18 Oct Reachability Definitions (reachability, stabilizability) Characterization and rank tests (Grammian, PBH) Decomposition into reachable/unreachable subspaces Eigenvalue placement theorem Rec: FBS2e Sec 7.1, 7.2; Sontag Sec 3.1‑3.3 Adv: FBS2e Sec 7.3; Sontag Sec 3.5 HW #3 Out: 16 Oct Due: 23 Oct Week 4 21 Oct 23 Oct 25 Oct State feedback Optimization and optimal control Linear quadratic regulator (including Ricatti equation) Opt: FBS2e Sec 7.5 Rec: OBC Ch 2 Adv: Sontag Sec 8.1‑8.3, 9.1, 9.2 HW #4 Out: 23 Oct Due: 30 Oct Week 5 28 Oct 30 Oct 1 Nov 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 Adv: Sontag Sec 6.1‑6.3, 7.1 HW #5 Out: 30 Oct Due: 6 Nov Week 6 4 Nov 6 Nov 8 Nov Frequency domain modeling Control system transfer functions State space realizations, minimal realizations Poles and zeros Opt: FBS23 Ch 2 Rec: FBS2e Ch 9; DFT Sec 2.6 Adv: Lewis Ch 3 and 4 HW #6 Out: 6 Nov Due: 13 Nov Week 7 11 Nov 13 Nov 15 Nov Frequency domain analysis Internal stability Tracking, disturbance rejection I/O performance Opt: FBS2e Sec 10.1-10.2, Sec 12.1-12.2 Rec: DFT Ch 3 Adv: Lewis Ch 5-8 HW #7 Out: 13 Nov Due: 20 Nov Week 8 18 Nov 20 Nov 22 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: 20 Nov Due: 27 Nov Week 9 25 Nov 28 Nov* 30 Nov 2 Dec Fundamental limits Algebraic limits Bode's integral formula Maximum modulus principle Opt: FBS2e Sec 14.3-14.5 Rec: DFT Ch 6 Adv: Lewis, Ch 9 HW #9 Out: 27 Nov Due: 6 Dec (Fri) Solns (Caltech only) Week 10 4 Dec 6 Dec Review for final Final

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 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).
• 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:

• [Son98] E. D. Sontag, Mathematical Control Theory, Springer, 1998. Online access
• [Lew03] A. D. Lewis, A Mathematical Approach to Classical Control, 2003. Online access.
• [OBC] R. M. Murray, "Optimization-Based Control", 2010. Online access