Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: \\\webshare\eecd\Ecourses\ARM blue-yellow.jpg

Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: bismillah


Nonlinear Control Course


Dr. Hamid D. Taghirad


  Description   Contents   References   Assignments    Evaluation   Projects   Documents    Software   Papers   Links


Page Quick Access:


Course Description


Course Contents










Related papers


Related Links





Related pages:


 Dr. Taghirad Homepage


 Control Group page


 E-courses page


 Dr. Taghirad E-courses






 Nonlinear Control



 Robust Control






 Modern Control



 Industrial Control



 Linear Control



 Digital Control



Theory of linear systems



Advanced dynamics



Advanced industrial control



Digital and Nonlinear Control



Signals and Systems
















Course Description



Course Name:

 Nonlinear Control

Course No.

 EE - 43071


 Dr. Hamid D. Taghirad


 Fall 93

Room and Time:

 Mon-Wed: 9:00-10:30   Room 314

Office Hours:

 Mon-Wed: 10:30-12:00  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 input-state, and input-output 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:


     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, input-state feedback linearization, feedback linearizability, involutivity, and controllability conditions.

Week 11

Feedback Linearization: input-state feedback linearization algorithm, normal forms, diffeomorphism, comprehensive examples.

Week 12

Feedback Linearization: input-output 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.





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


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


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


Nonlinear System Analysis, M. Vidyasagar, Prentice-Hall, 1993.


Selected papers.

E-books on this subject are available in my \ebooks\Control Engineering\Nonlinear.

Contact me if you need one.


Assignments (pdf)

Projects (pdf)

Exams (pdf)

Assignment 1



List of projects, Student list


Assignment 2



Guideline for Research and report


Assignment 3



Part one: 03/09/93  Part two: 6/11/93 at 14:00


Assignment 4



Selected Weekly Projects:


Assignment 5



Chaotic systems:Tamjidi, Sharbafi, Sharafi


Assignment 6



Region of Attraction: Shams



Frequency Domain method, Mokhtari


Course Evaluation:

Dear Students: Please kindly provide me your precious feedback by completing this survey.


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.


Phase portrait programs:

Matlab: pplane7, pplane8, pptool, quiver

Maple: Guide


Mathworks Matlab:

 Nonlinear Controller Design (NCD) Toolbox Guide

 Matlab Toolboxes user manuals (pdf):

 NCD toolbox 

Related Papers


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. 765-769, June 2003, Montreal, Canada.


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.


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.


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.


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 19-26, May 1998, Tehran


H.D. Taghirad and P.R. Belanger, Robust friction compensation for harmonic drive system, Proceedings of IEEE international Conference on Control Application, pp 547-551, 1997, Trieste, Italy.

Related Links

UK Nonlinear Dynamics Groups

CalTech Control And Dynamical Systems Group

CalTech control engineering virtual library

ICambridgeU. Control Group




 Description   Contents   References   Assignments    Evaluation   Projects   Documents    Software   Papers   Links