E6602 Modern Control Theory

E6602 is typically taught once per year in the Spring semester. The information below is meant to provide a snapshot of the material covered.

Overview

Course description

Students will learn to recognize, model, formulate and solve optimal control problems through the lens of convex optimization, and in particular, linear matrix inequalities. Example problem instances from a diverse range of applications including; circuits, mechanics, robotics, finance, etc. will serve as motivation. The primary object of interest will be linear time-invariant dynamical systems. We begin with a primer on convex optimization and least squares problems, before moving on to state space models, system analysis (stability, controllability, observability, etc.), feedback control via linear matrix inequalities (H-infinity and H2 optimal control), and uncertainty modeling. Finally, time permitting, we study more advanced topics such as system identification (learning a dynamic model from data), and distributed control.

Lecture notes
Spring’25 update: I am currently revising the slides, topics beyond lecture 8 will be made available towards the end of the semester.

1. Overview
2. Least squares problems
3. Linear dynamical systems
4. Autonomous systems
5. Convex optimization - a primer
6. Semidefinite programming for control
7. Controllability, stabilizability, and state feedback
8. An operator theory perspective of controllability
9: Observability and the Luenberger observer
10: BIBO stability and well-posed interconnections
11: H2 optimal control
12: LFTs
13: KYP lemma
14: H-infinity state feedback
15: H-infinity output feedback
16: Fundamental performance limits
17: Robustness and uncertainty
18: Optional topics if time permits

Additional material

a. Fundamental theorem of linear algebra
b. LQR via dynamic programming
c. Disciplined convex programming

Intended audience

This class is intended for students with an interest in the mathematical foundations of modern systems and control theory and those wishing to deploy optimal control algorithms to real applications. We particularly encourage students from non-engineering disciplines such as finance, biology, and physics to attend.

Textbook

The course doesn't follow one specific book; all necessary material will be provided. However, we will draw heavily on material from:

• A course in robust control theory: a convex approach
  Dullerud & Paganini
  Vol. 36. Springer Science & Business Media, 2013.

• LMIs in control systems
  Duan & Yu
  CRC press, 2013.

An additional useful reference is

• Linear matrix inequalities in system and control theory
  Boyd, El Ghaoui, Feron,& Balakrishnan
  SIAM, 1994.

Sample course projects

The projects below are a sample of past student projects:

• Anaesthesia control, Patrick Munar

• Control of a double inverted pendulum using a neural network, Ketian Wang

• Self-balancing cube robot, Qixiao Zhang

Course organization

Prerequisites

• Linear algebra: APMA E3101 linear algebra or comfortable with the material here

• Signals and systems: E3801 or equivalent

• Control Any introduction to control course, or, permission from the instructor

• Optimization: any optimization class (E4650 is ideal but not necessary, E6616 is being taught this semester)

In addition, familiarity with basic programming will be necessary to complete some homework questions. One of CVX (Matlab), CVXPY (Python), CVXR (R), or Convex.jl (Julia) will be used to write simple scripts. No prior knowledge of CVX is assumed, a recitation on this topic will be provided.

Students are required to use LaTeX to typeset their homework (undergraduates are encouraged to use LaTeX, but it is not a requirement). Templates and examples will be provided.

Grading

• Homework 40%

• Midterm exam 20%

• Project 40%

Homework

There will be approximately 6 homework exercises. All homeworks are due ten days after they are released and must be submitted via Gradescope.

Apart from homework 1, students must typeset homework using LaTeX. A template will be provided. Students should read the style guide (written by Stephen Boyd for his EE364a class at Stanford) before submitting homework assignments.

Midterm

Closed-book midterm held during class time. Date tba.

Project

Students will develop their own system models, with the aim of simulating, analyzing, and designing feedback controllers. The project will consist of 3 components:

• Proposal

• Midterm progress report

• Final report (conference style paper)

Details will be provided in class.