# Nonlinear Control Theory

## Introduction to Nonlinear Control Theory

The course covers analysis and design of nonlinear control systems using Lyapunov theory and geometric methods. Includes properties of solutions of nonlinear systems, Lyapunov stability analysis, feedback linearization, and nonlinear control design tools. Prerequisite(s): basic knowledge of ordinary differential equations; basic knowledge of linear algebra and basic knowledge of linear control theory. Enrollment restricted to graduate students or permission of instructor.

Text Book: Nonlinear Systems, H. Khalil, Prentice-Hall, 3rd Ed., 2002 or 2nd Ed., 1996

Reference: Nonlinear Systems Analysis, M. Vidyasagar, Prentice-Hall, 2nd Ed., 1996

Nonlinear Control Systems, A. Isidori, Springer, 3rd Ed., 1995

Lectures: Monday, Wednesday and Friday 12:30PM - 1:40PM, Soc Sci 1 149

Office Hours: Tuesday, 9:00am - 11:00am, BE 361A.

Grading: Homework 60%, Final Exam 40%.

Tentative Schedule

Week 1: Introduction of nonlinear systems (Chapter 1), phase-plane for 2D linear and nonlinear systems (Section 2.1, 2.3), limit cycles (Section 2.4), and existence of periodic orbits (Section 2.6).

Week 2: Mathematical preliminaries including Banach space, Contraction Mapping Theorem (Appendix A & B); Local Existence and Uniqueness Theorem (Theorem 3.1).

Week 3: Discussion on Lipschitz continunity, Gronwell inequality (Appendix A), and Global Existence and Uniqueness Theorem (Theorem 3.2).

Week 4: Continuous dependence on initial conditions and parameters (Section 3.2), sensitivity equations (Section 3.3), and Definition of Lyapunov Stability for autonomous systems (Section 4.1).

Week 5: Lyapunov stability theory for autonomous systems (Section 4.1), Lasalle’s Invariance Principle (Section 4.2), linearization and linear state feedback control (Section 4.3), Brockett’s necessary conditions.

Week 6: Lyapunov stability theory for time-varying nonlinear systems, comparison functions, uniform stability, exponential stability  (Section 4.4, 4.5).

Week 7: Time-varying systems and linearization (Section 4.6), and Converse Theorems (Section 4.7).

Week 8. Input-state feedback linearization and input-output linearization (Chapter 14).

Week 9. Lyapunov-based nonlinear control design. Backstepping for lower triangular systems (Section 14.3).

Week 10. Adaptive backstepping (linear and nonliner parameterization), nonlinear observer design and output-feedback control.