ECE 528
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ECE 528: Analysis of Nonlinear Systems (Spring 2024)

This is a fundamental first-year graduate course in nonlinear systems.
It covers properties of solutions of nonlinear dynamical
systems, Lyapunov stability analysis techniques, effects of perturbations,
and basic nonlinear control design tools. Proofs of most of the results
are presented in a rigorous mathematical style. Familiarity with real
analysis (on the level of Math 444 or 447) is important. A course like Math 347 or Math 414 may also be useful
for practice in writing rigorous proofs.

**Announcements: **

- The lectures will be in-person (with a possible occasional Zoom) and will NOT be recorded.
- Use entry code
**XX5BWW** to add the course on Gradescope where you will be submitting assignments. It is best to use your Illinois e-mail to create an account.

**Homework: **

Homework 1 (posted Jan 25, due Feb 8) | Solutions (posted Feb 10)

Optional additional problems on math background

Homework 2 (posted Feb 8, due Feb 22) | Solutions (posted Feb 27)

Homework 3 (posted Mar 20, due Apr 4) | Solutions (posted Apr 12)

Homework 4 (posted Apr 4, due Apr 18) | Solutions (posted Apr 26)

Lecture notes typeset by students (created by James Schmidt in Spring 2015, updated by Adriano Lima Abrantes in Spring 2018; I have not checked the notes and cannot guarantee their correctness)

**Schedule: **Tue Thu 11:00-12:20, 3020 ECE Building.

**Prerequisites:** ECE 515 (Linear Systems) and Math 444 or 447
(Real Analysis).

**Instructor:** Daniel Liberzon

Office: 144 CSL

Email: liberzon at illinois.edu

Office hours: please see me after class

**Homework TA:** Emre Eraslan

Email: emree2 at illinois.edu

Office hours: Wed 4:00-5:30pm, 2036 ECEB (subject to change)
sign-up link

**Required text:**
H. K. Khalil, * Nonlinear Systems*, 3rd edition. Prentice Hall,
2002.

** Supplementary text**:
E. D. Sontag, *Mathematical Control Theory*, 2nd edition. Springer, 1998.
Available from the author's
website.

**Assignments and grading policy:**
Homework (4-5 problem sets) - 30% of the grade, midterm exam (in class) - 30%, final exam (take-home) - 40%. Note: this information is tentative and subject to change.

**Brief course outline:**

1. Mathematical background.

2. Fundamental properties of dynamical systems:
existence and uniqueness of solutions,
continuous dependence on initial conditions and parameters, comparison
principles.

3. Stability analysis: Lyapunov stability of autonomous and
nonautonomous
systems, LaSalle's invariance principle, converse Lyapunov theorems,
stability of feedback systems, effects of perturbations.

4. Systems with inputs and outputs: input-to-state stability and related
notions, Lyapunov characterizations.

5. Nonlinear control: control Lyapunov functions, universal formulas
for feedback stabilization and disturbance attenuation.

6. Advanced topics (time permitting): sampled-data control, center manifold theorem, averaging, entropy.