**Nonlinear Control Systems **

**IST-DEEC PhD Course - Springer
Semester 2011/2012**

*Posted on May 15, 2012*

The homework #8 [HW#8] has been posted online.

The due date is Monday, May 21, 2012.

*Posted on May 7, 2012*

The homework #7 [HW#7] has been posted online.

The due date is Monday, May 14.

The slides of part 7- Feedback Linearization can be downloaded here [Slides#7]

*Posted on April 24, 2012*

The homework #6 [HW#6] has been posted online.

The due date is Wednesday, May 2.

The slides of part 6- Nonlinear Design can be downloaded here [Slides#6]

*Posted on April 2, 2012*

The homework #5 [HW#5] has been posted online.

The due date is Wednesday, April 11.

The slides of part 5- Input-Output Stability can be downloaded here [Slides#5]

*Posted on March 23, 2012*

The homework #4 [HW#4] has been posted online.

The due date is Wednesday, March 28.

*Posted on March 14, 2012*

The homework #3 [HW#3] has been posted online.

The due date is Wednesday, March 21.

*Posted on March 8, 2012*

The slides of part 3- Fundamental properties can be
downloaded here [Slides#3]

The slides of part 4- Lyapunov stability can be
downloaded here [Slides#4]

The homework #2 [HW#2] has been posted online.

The due date is Wednesday, March 14.

*Posted on March 2, 2012*

The slides of part 1- Introduction to Nonlinear Systems can be
downloaded here [Slides#1]

The slides of part 2- Mathematical review can be downloaded here [Slides#2]

The homework #1 [HW#1] has been posted online.

The due date is Monday, March 12.

Schedule:

Monday 16:00 – 17:30 room C10

Wednesday 16:00 – 17:30 room E1

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

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

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.

Basic
knowledge of calculus, linear algebra and ordinary differential equations is
assumed.

Mondays
– 16h00-17h30 room C10

Wednesdays
– 16h00-17h30 room E1

*Instructor:*

·
Antonio Pedro Aguiar

Office: 8.13,
North Tower, IST

Phone: +351 21 841 8056

Email: pedro@isr.ist.utl.pt

Web: http://users.isr.ist.utl.pt/~pedro/

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

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

*Complementary:*
Collection of journal papers.

Homeworks – 30%

Final
Project – 40%

Final
Exam – 30% (24h take home)

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

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

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. *L*_{2} Gain.
Feedback Systems: The Small-Gain Theorem

6. Passivity

Memoryless Functions. State Models. Positive Real Transfer
Functions. *L*_{2} 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.

3 hours/week (theoretical lectures)