Nonlinear Control Course
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Course Name:  Nonlinear Control
Course No.  EE - 43071
Professor:  Dr. Hamid D. Taghirad
Semester:  Fall 83
Room and Time:  Mon and Wed: 8:00-10:00 Room 204
Office Hours:  Mon:    15:00:17:00

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 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 Alegebra, 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 them as a term project using Matlab simulations.

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, 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.



1 Nonlinear systems, H. Khalil, Prentice Hall, QA427.K48, 1996.

ترجمة كتاب فوق توسط دكتر غلامعلي منتظر، انتشارات دانشگاه تربيت مدرس.

3 Applied Nonlinear Control, J.J. Slotine and W. Li, Prentice Hall, 1991.
4 Nonlinear Control Systems, A. Isidori, Springer Verlag, 1995.
5 Selected papers.

Assignments (pdf) Projects (pdf) Exams (pdf)
Assignment 1  Solution  Part 1 Solution   Midterm
Assignment 2  Solution  Part 2   Final
Assignment 3  Solution  Part 3 (doc)   Quizz 1
Assignment 4  Solution      MidTerm scores
Assignment 5  Solution Industrial Proj   KNTU  PUT
Assignment 6  Solution     Final Grades
Assignment 7  Solution     KNTU

 Exercise From web (pdf):
 Part 1, part 2, part 3, part 4, part 5, part 6, part 7.
 Chap 2, Chap 3,Chap 4,Chap 5,Chap 6,Chap 7,Chap 12,Chap 13,Chap 14
 Assign 2, solution, Assign 3, solution (From Hassan Khalil)
 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 in Matlab (6.5 or higher)
pplane6.m, dfield6.m, pplane.pdf, pplane advanced.pdf
Matlab Premier:
chap1, chap2, chap3, chap4, chap5, chap6, chap7, chap8, chap9, chap10, chap11, chap12, chap13
 Mathworks Matlab:
 Nonlinear Controller Design (NCD) Toolbox Guide
 Matlab Toolboxes user manuals (pdf):
 NCD toolbox 

1. P.V. Kokotovich, R.E. O'Malley, and P. Sannuti, Singular Perturbations and Order Reduction in control Theory, Automatica 12: 123-132, 1976.
2. V.R. Saksena, J. O'Reilley, and P.V. Kokotovich,  Singular Perturbations and Time-scale methods in control theory: Survey 1976-1983, Automatica 20: 273-293, 1984..
3. J. E. Slotine, The robust control of robot manipulators, International Journal of Robotics Research, Vol 4, No. 2, pp 49-64, 1985.

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.

5. 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.

7. 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.
8. 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
9. 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.

UK Nonlinear Dynamics Groups
CalTech Control And Dynamical Systems Group
CalTech control engineering virtual library
Cambridge U. Control Group
Nonlinear E-course in Lund University
The Joy of Feedback (1991 Bode Prize Lecture by P. Kokotovic)



Copyright Dr. Hamid D. Taghirad
K.N. Toosi University of Technology
Last Updated Saturday January 15, 2005