## ECE 528: Analysis of Nonlinear Systems (Spring 2018)

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. Math 414 may also be useful for practice in writing rigorous proofs.

Announcements:

• The midterm will be held in class on Thursday, Mar 8
• The first class will be on Thursday, Jan 18

Homework:

Homework 1 (posted Jan 29, due Feb 8) | Solution (posted Feb 19)
Homework 2 (posted Feb 19, due Mar 1) | Solution (posted Mar 12)
Homework 3 (posted Mar 23, due Apr 5) | Solution (posted Apr 11)
Homework 4 (posted Apr 9, due Apr 19) | Solution (posted Apr 30)

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, 2015 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: Ivan Abraham
Email: itabrah2 at illinois.edu
Office hours: Tue 7:00-8:00pm in ECEB 3034. (If you can't make this time, Ivan will also be available Tue 4:00-5:00pm in CSL 164.)

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, Thu Mar 8) - 30%, final exam (take-home, during the week of May 7) - 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): center manifold theorem, averaging, singular perturbations.