# ENEE 661: Nonlinear Control Systems

## Course Goals:

The goal of this course is to introduce the student to the analysis of the qualitative behavior of nonlinear systems and the design and synthesis of controllers for such systems. Techniques include the Lyapunov's direct method, linearization, frequency domain stability analysis, and functional analysis methods.

## Course Prerequisites:

ENEE 460 or equivalent is a prerequisite. Corequisite ENEE 663 or permission of instructor. A prior course in advanced calculus (e.g., MATH 410 and 411) is recommended.

## Topic Prerequisite:

The student should be familiar with basic concepts and tools from linear systems theory including the Nyquist Criterion, matrix exponentials and the variation of constants formula, controllability, observability, and stabilizability. It would be helpful (but not essential) to be familiar with normed vector spaces, the Inverse Function Theorem, and the Implicit Function Theorem.

## References:

The probable textbook is: H.K. Khalil, Nonlinear Systems, Macmillan, New York, (1992). An alternative book which covers similar materials is: M. Vidyasagar, Nonlinear Systems Analysis (second edition), Prentice-Hall, Englewood Cliffs, New Jersey (1993). The mathematical background can be found in any standard textbook on advanced calculus. The prerequisite material on frequency domain linear systems can be found in: G.F. Franklin, J.D. Powell, and A. Emami-Naeini, Feedback Control of Dynamic Systems, Addison Wesley, Reading, Massachusetts (1991). The prerequisite material on state space linear systems can be found in either of the following books: C.-T. Chen, New York (1984); W.J. Rugh, Linear System Theory, Prentice-Hall, Englewood Cliffs, New Jersey (1993).

## Core Topics:

1. Existence, uniqueness, and continuous dependence on initial conditions for ordinary differential equations.
2. Lyapunov's direct method for both time-invariant and time-varying systems. Stability and instabililty results. Lasalle's Invariance Principle.
3. Regions of attraction.
4. Linearization Theorem for both time-invariant and time-varying systems. Stability and instability results.
5. Linearization Theorem for periodic orbits of time-invariant systems.
6. Input-output stability and the Small Gain Theorem.
7. Absolute stability (including Circle and Popov Criteria).
8. Stabilization using state feedback (via linearization).

## Optional Topics:

Center manifolds. Feedback linearization. Relative degree and zero dynamics. Bifurcations. Perturbation theory and averaging. Nonsmooth dynamics. Singular perturbations. Discrete-time nonlinear systems.