


Course Name: 
Nonlinear Control 
Course No. 
EE  43071 
Professor: 
Dr.
Hamid D. Taghirad 
Semester: 
Fall 83 
Room and Time: 
Mon and Wed: 8:0010:00 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 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 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, 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. 


1 
Nonlinear
systems, H. Khalil, Prentice Hall, QA427.K48, 1996. 
2 
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. 


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: 123132, 1976. 
2. 
V.R. Saksena, J. O'Reilley, and P.V. Kokotovich,
Singular Perturbations
and Timescale methods in control theory: Survey 19761983, Automatica 20:
273293, 1984.. 
3. 
J. E. Slotine, The robust control
of robot manipulators, International Journal of Robotics Research, Vol
4, No. 2, pp 4964, 1985. 
4. 
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.

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. 
6. 
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 1926, 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 547551, 1997, Trieste, Italy. 
