## ECE 528: Analysis of Nonlinear Systems (Spring 2015)

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 447) is strongly recommended. Math 414 may also be useful for practice in writing rigorous proofs.

Announcements:

• Final exam solutions are available here. Have a nice summer!

Homework:

Homework 1 (posted Jan 29, due Feb 10) | Solution (posted Feb 23)

Homework 2 (posted Feb 19, due Mar 3) | Solution (posted Mar 17)

Homework 3 (posted Mar 19, due Apr 7) | Solution (posted Apr 23)

Homework 4 (posted Apr 9, due Apr 21) | Solution (posted Apr 30)

Lecture notes (courtesy of James Schmidt)

Schedule: Tue Thu 11:00-12:20, 3015 ECE Building.

Prerequisites: ECE 515 (Linear Systems) and Math 285 or 441 (Differential Equations).

Instructor: Daniel Liberzon
Office: 144 CSL
Email: liberzon at illinois.edu
Office hours: Fri 11:00-12:30 unless posted otherwise (no homework questions, please)

Homework TA: Guosong (Oliver) Yang
Email: yang150 at illinois.edu
Office hours: Mon 2:30-4:00pm in 368 CSL

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) - 50% of the grade, final exam - 50%.

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.