
Course Description


Course
Name:

Nonlinear
Control

Course
No.

EE
 43071

Professor:

Dr. Hamid D. Taghirad

Semester:

Fall
93

Room
and Time:

MonWed:
9:0010:30 Room 314

Office
Hours:

MonWed:
10:3012:00 Dept. Chair

Course Contents
This course aims to introduce the analysis of nonlinear system, and the
common nonlinear control schemes. The course is divided into two parts,
namely analysis and synthesis. In the analysis part, the state space
description of nonlinear system is introduced, and the phase portrait
analysis of the second order system is elaborated. Stability analysis of the
nonlinear system, based on linearization method, and direct method of
Lyapunov is explained next, while the stability analysis is completed with Lasalle's theorem, absolute stability notion, Popov, and
circle criteria, and the stability analysis of time varying nonlinear
systems. finally, the analysis of limit cycles is
thoroughly elaborated using describing functions. In the synthesis part,
after introducing of Lie Algebra, and required mathematics, Feedback
linearization methods for inputstate, and inputoutput cases are described,
and backstepping method and sliding mode control is
introduced next. To evaluate the expertise of the student in nonlinear
control analysis and synthesis, a thorough and comprehensive design task
is performed by students in a series of the assignment exercises using Matlab
simulations. Moreover, a research project is assigned to each student to
further study the topics which are less emphasized throughout the course as a
term project.
The tentative course contents are as following:
Time:

Teaching Contents

Week
1

Introduction: Common nonlinear systems, state space
representation, equilibrium point, common behaviors of nonlinear systems,
and limit cycles.

Week
2

Phase plane
Analysis: 2nd order nonlinear
systems, phase portrait graphical representation, singular points.

Week
3

Phase plane
Analysis: Graphical and numerical methods of phase portrait generation,
stability analysis of linear systems via phase portrait, stability analysis
of nonlinear system with phase portraits.

Week
4

Stability
Analysis: Different definition of
stability for nonlinear systems, Lyapunov linearization method, Lyapunov
direct method, globally asymptotically stability analysis.

Week
5

Stability
Analysis: Lyapunov direct method
extensions, Lasalle's theorem, time varying
nonlinear systems stability theorems, instability theorems

Week
6

Stability
Analysis: Absolute stability
theorems, Sector nonlinearity, Popov and circle criteria, Lyapunov based
controller synthesis.

Week
7

Describing
Functions: Limit cycle definition
and characteristics, existance theorems,
describing function definitions.

Week
8

Describing
Functions: Describing function
for saturation, relay, dead zone and hysteresis, limit cycle analysis by
describing function, limit cycle stability analysis.

Week
9

Midterm Exam

Week
10

Feedback
Linearization: Background mathematics,
Lie algebra, inputstate feedback linearization, feedback linearizability, involutivity,
and controllability conditions.

Week
11

Feedback
Linearization: inputstate
feedback linearization algorithm, normal forms, diffeomorphism,
comprehensive examples.

Week
12

Feedback
Linearization: inputoutput
feedback linearization algorithm, internal dynsmics,
zero dynamics, asymptotically minimum phase nonlinear systems,
comprehensive example.

Week
13

Back Stepping: Controller general description, required
conditions, Back stepping method, controller characteristics, comprehensive example.

Week
14

Sliding mode: General description, sliding surfaces, switching
mode controller law, sliding mode controller structure, comprehensive
example.

Week
15

Sliding mode: Chattering problem, boundary layer description,
sliding condition extension, fixed threshold boundary layer, variable
boundary layer, comprehensive example.



References
1

Nonlinear
systems, H. Khalil, Prentice Hall, 3rd Edition, QA427.K48, 2002.

2

Applied Nonlinear Control, J.J. Slotine and W. Li, Prentice Hall, 1991.

3

Nonlinear Control Systems, A. Isidori, Springer Verlag, 1995.

4

Nonlinear
System Analysis, M. Vidyasagar, PrenticeHall,
1993.

5

Selected papers.


Ebooks
on this subject are available in my \ebooks\Control
Engineering\Nonlinear.


Contact
me if you need one.

Assignments
Course Evaluation:
Dear Students: Please kindly provide me your precious feedback by
completing this survey.
Projects
Selected
Projects reports  Fall 87
Course Documents:
Project References


Extra Problems

Set 1, Set 2, Set 3, Set 4, Set 5


Course Notes:

Chapter1, Chapter2, Chapter3, Chapter4, Chapter5, Chapter6, Chapter7

Updated and each slide in one page:

Chapter1, Chapter2, Chapter3, Chapter4, Chapter5, Chapter6, Chapter7

Exercise
From web (pdf):

Part 1, part 2, part 3, part 4, part 5, part 6, part 7.

Exams
From web (pdf)

Exam1, solution, exam2, solution, exam3, solution, exam4, solution, exam5 with solution, exam6 with solution

Other
handouts (pdf):

Back stepping method: Tank example, Sattelite example,
Inverted pendulum example.

Software
Related Papers
1.

M. Tavakoli, H.
D. Taghirad and M. Abrishamchian,
Parametric
and Nonparametric Identification and Robust Control of a
Rotational/Translational Actuator, In Proceedings of The Fourth
International Conference on Control and Automation (ICCA'03), pp. 765769,
June 2003, Montreal, Canada.

2.

H.D. Taghirad and M.A. Khosravi, Stability analysis
and robust composite controller synthesis for flexible joint robots,
submitted to the IEEE International Conference on Intelligent and Robotic
Systems, 2002.

3.

H.D. Taghirad,
N. Abedi, and E. Noohi,
A New
Vector Control Method for Sensorless Permanent
Magnet Synchronous Motor Without Velocity Estimator, in the proceeding
of the IEEE International Workshop on Motion Control, Slovenia, July 2002.

4.

H.D. Taghirad and E. Noohi, A New Lyapunov based
control method for Vector Control of Permanent Magnet Synchronous Motor,
in the proceeding of 11th International conference of Electrical
Engineering, Tabriz, 2002.

5.

H.D. Taghirad, M. Abrishamchian and R. Ghabcheloo,
Electromagnetic
levitation system: An experimental approach, Proceedings of the 7th international
Conference on Electrical Engineering, Power System Vol,
pp 1926, May 1998, Tehran

6.

H.D. Taghirad and P.R.
Belanger, Robust
friction compensation for harmonic drive system, Proceedings of IEEE
international Conference on Control Application, pp
547551, 1997, Trieste, Italy.

Related Links
