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ECE/ME/GE 428: Analysis of Nonlinear Systems (Spring 2002)

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 347) is recommended.

Schedule: Tue Thu 1:30-2:50, 169 Everitt Lab.

Prerequisites: ECE 415 (Linear Systems) and Math 285 or 341 (Differential Equations).

Instructor: Daniel Liberzon
Office: 144 CSL
Email: liberzon at
Office hours: Tue 4:00-5:30, Wed 12:00-1:30

Required text: H. K. Khalil, Nonlinear Systems, 3rd edition. Prentice Hall, 2002.
Supplementary text (on reserve in Grainger library): E. D. Sontag, Mathematical Control Theory, 2nd edition. Springer, 1998.

Assignments and grading policy: There will be approximately 5-6 problem sets, one midterm exam, and one final exam. These three components will contribute equally to the final grade. Homework solutions will be discussed in class. Homework will be due at the beginning of class. If you turn it in late but within 24 hours, you will lose 25% of the grade. No homework will ever be accepted more than 24 hours after the due time. The lowest homework score will be dropped.


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).