# Advanced Nonlinear Control Systems

Instructor: Jafar Ghaisari

Course Description:

This course is a second graduate course in nonlinear systems. The course is structured to emphasize some of the recent research activity in nonlinear analysis and control. We will use concepts from differential geometry, however the course is self contained in that this mathematics will be taught as part of the course. The class starts with the development of the mathematical background necessary to study nonlinear systems. This is followed by the geometric theory of nonlinear control, including linearization by state feedback for SISO and MIMO systems, zero dynamics of nonlinear systems, decoupling by state feedback. We finally discuss the Input-Output stability concept and Drift-Free control systems.

Course Outline:

**1. Mathematical Preliminaries**

(Chapter 3 of the textbook)

� Vector Spaces, Subspaces, Norms.

� Contraction Mapping Theorem

� Differential Equations, Vector Fields

� Relaxation Techniques for integration of differential equations

� Degree Theory

**2. Basic of Differential Geometry **

(Chapter 8 of the textbook)

� Tangent Space

� Distributions and Co-distributions

� Frobenius Theorem

� Matrix Group: Matrix Lie Group, Lie Algebra, Exponential Map, Canonical Coordinates on Matrix Lie Group

� Left-Invariant Control Systems on Matrix Lie Group

**3. Linearization by State Feedback: Theory and Applications**

(Chapter 9 and Chapter 10 of the textbook)

� SISO systems: Input Output Linearization

� SISO systems: Full State Linearization

� Zero Dynamics

� Inversion, tracking, stabilization

� MIMO systems: linearization by static state feedback

� Full state linearization of MIMO systems

� Dynamic Extension

� Sliding Mode Control and Robust Linearization

� Nonlinear Observers

� Design Examples: Ball and Beam, Nonlinear Flight Control

**4. Input �Output Analysis and Stability**

(Chapter 4 of the textbook)

� Definitions of Input - Output Stability

� Small Gain Theorems

� Passivity and passivity theorems

� Connections between Input - Output and State Space Stability

**5. Geometric Nonlinear Control**

(Chapter 11 of the textbook)

� Drift-Free Control Systems

� Steering of Drift-Free Nonholonomic Systems

� Steering Model Control Systems Using Sinusoid

� Zero Dynamic Algorithm

**Textbook: **

S. S. Sastry, Nonlinear Systems: Analysis, Stability, and Control, Springer-Verlag, 1999.

**References:**

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

M. Vidyasagar, Nonlinear Systems Analysis, 2nd Edition, Prentice-Hall, 1993.

H. K. Khalil, Nonlinear Systems, 3rd Edition, Prentice-Hall, 2002.

Prerequisites:

Some introduction to linear systems theory, mathematical analysis, and Linear Algebra;

Grading:

Homework: 20 %

Simulations: 15 %

Lab. Experimental: 20 %

Seminar: 15 %

Final Exam: 30 %