Nonlinear CONTROL Systems

Springer Semester 2008/2009

 

 

  Software: Microsoft Office

Computer simulation of an autonomous underwater vehicle following a desired 3D trajectory 

Objective

The main goal of this course is to provide to the students a solid background in analysis and design of nonlinear control systems.

 

Course Description

Many control systems of practical importance are inherently nonlinear. A common practice for control system design is to linearize the system to be controlled around some equilibrium or operating point through small perturbation state approximations. The key assumption is that the range of operation is restricted to a small region around the equilibrium on which the linear model remains valid. As a consequence, adequate control is only guaranteed in a neighborhood of the selected operating points. Moreover, performance can suffer significantly when the required operating range is large, such as when controlling an autonomous vehicle that executes maneuvers that emphasize its nonlinearity and cross-couplings.

 

This course covers the analysis and design of nonlinear control systems and is suitable for post-graduate students in science and engineering. The course begins with an introduction to nonlinear system theory and stability analysis. Topics include Lyapunov stability analysis techniques, stability of perturbed systems with vanishing and non-vanishing perturbations, input-to-state stability, input-output stability and passivity. The last part of the course is dedicated to nonlinear control design tools such as feedback linearization, sliding mode control, Lyapunov redesign, backstepping, passivity based control and nonlinear adaptive control (if time permits). Emphasis is placed upon application of the theory to systems of interest to the students.

The lectures will be in English.

 

Prerequisites

Basic knowledge of calculus, linear algebra and ordinary differential equations is assumed. Familiarity with MATLAB is recommended.

 

Instructors

António Pedro Aguiar (pedro@isr.ist.utl.pt)

António Pascoal (antonio@isr.ist.utl.pt)

Phone:  (ext) 2056

Office: room 8.13, Torre Norte, IST.

Office hours: Please email or phone in advance to schedule an appointment.

 

Textbook

Khalil, H. K. Nonlinear Systems, 3rd Edition, Prentice Hall, Upper Saddle River, NJ, 2002.

 

Grading Policy

Homeworks – 40%

Final Project – 60%

 

Projects

The following two types of projects are possible in this course:

  1. Solution of a research problem relevant to the student’s area of research which makes use of methods described in the course.
  2. Independent study of a topic not covered in-depth in class (e.g., reading a paper or book chapter).

 

Course Topics:

1.   Introduction to nonlinear systems

         Nonlinear systems vs. linear systems, multiple isolated equilibrium points, finite escape times, limit cycles.

2.   Mathematical preliminaries

         Normed vector spaces. Induced norms. Mean value and implicit function theorems. Gronwall-Bellman inequality. Lipschitz condition.

3.   Fundamental properties        

         Local and global existence and uniqueness of solutions. Continuity with respect to initial conditions. Comparison Principle.

4.   Lyapunov Stability

        Autonomous Systems. The Invariance Principle. Linear Systems and Linearization. Comparison Functions. Nonautonomous Systems. Linear Time-Varying Systems and Linearization. Converse Theorems. Boundedness and Ultimate Boundedness. Input-to-State Stability.

5.   Input–Output Stability

         L Stability. L Stability of State Models. L2 Gain. Feedback Systems: The Small-Gain Theorem

6.   Passivity

         Memoryless Functions. State Models. Positive Real Transfer Functions. L2 and Lyapunov Stability. Feedback Systems: Passivity Theorems.

7.   Frequency Domain Analysis of Feedback Systems

         Absolute Stability: Circle Criterion, Popov Criterion.

8.   Advanced Stability Analysis

         The Center Manifold Theorem. Region of Attraction. Invariance-like Theorems. Stability of Periodic Solutions

9.   Stability of Perturbed Systems

         Vanishing Perturbation. Nonvanishing Perturbation. Comparison Method.

10. Feedback Linearization

         Motivation. Input–Output Linearization. Full-State Linearization. State Feedback Control. Stabilization. Tracking.

11. Nonlinear Design Tools

        Sliding Mode Control: Motivating Example, Stabilization, Tracking, Regulation via Integral Control. Lyapunov Redesign: Stabilization, Nonlinear Damping. Backstepping. Passivity-Based Control. Basic nonlinear adaptive control design.