Nonlinear Control Systems

Course Description

Dynamic effects due to nonlinearities in systems. Common nonlinear elements in control systems. Phase-plane analysis. Linearization of nonlinear systems. Harmonic linearization. Describing function. Self-oscillations (limit-cycles). Forced oscillations. Harmonic lubrication by dither signal. Stability of nonlinear systems. Feedback linearization. Sliding mode control. Unconventional control methods.

General Competencies

Student will learn effects of nonlinearities in control systems. They will be acquinted with methods for analysis of nonlinear control systems as well as how to diminish negative effects of nonlinearities.

Learning Outcomes

  1. recognize common nonlinear control problems
  2. apply some powerfull analysis methods
  3. apply some practical design methods
  4. discover how nonlinearities can improve system dynamics
  5. apply stability analysis methods to real systems
  6. identify and remove negative effects that appear in nonlinear control systems

Forms of Teaching


Mixed type (blackboard+ppt presentations)


One midterm and one final exam in written form, or an exam in a written and oral form.


Examples are given during lectures.


Planned with students.

Grading Method

Continuous Assessment Exam
Type Threshold Percent of Grade Threshold Percent of Grade
Homeworks 0 % 8 % 0 % 8 %
Mid Term Exam: Written 0 % 16 % 0 %
Final Exam: Written 0 % 16 %
Final Exam: Oral 60 %
Exam: Written 0 % 40 %
Exam: Oral 60 %

Week by Week Schedule

  1. Introduction in nonlinear systems theory; Basic mathematical and structural models of nonlinear systems; Basic specific and fundamental properties of nonlinear systems
  2. The basic properties of nonlinear functions; Typical nonlinear elements; Atypical nonlinear systems
  3. Issues of stability; Equilibrium states in the phase plane; The concept of stability of a nonlinear system; Stability of nonlinear systems based on linearized models;
  4. Lyapunov stability criterion; Absolute stability of nonlinear systems
  5. Geometrical interpretation of the V. M. Popov stability criterion; Stability of forced and unforced systems
  6. Linearization methods: Analytical linearization, graphic linearization, harmonic linearization; Describing function; Statistical linearization; Dual describing function
  7. Dynamic analysis of nonlinear systems using phase trajectories; Phase trajectories of nonoscillatory processes and periodic solution
  8. Midterm exam
  9. Dynamic analysis of nonlinear systems using the describing function; Determination of symmetrical and asymmetrical self-oscillations; Stability of periodic solutions
  10. Application of the describing function in determination of symmetrical and asymmetrical forced oscillations; Determination of the resonant jump;
  11. Control processes in nonlinear systems exhibiting periodical oscillations; Vibrational linearization
  12. Determination of transients near a periodic solution; Transient quality diagrams
  13. Lie algebra; Feedback linearization - input/output linearization and input/state linearization; Examples
  14. Sliding-mode control; Examples
  15. Final exam.

Study Programmes

University graduate
Control Engineering and Automation (profile)
Theoretical Course (2. semester)


Z. Vukić, Lj. Kuljača, D. Đongalić, S. Tešnjak (2004.), Nonlinear Control Systems, New York: Marcel Dekker
J. J. Slotine, W. Li (1991.), Applied Nonlinear Control, New York: Prentice Hall
M. Vidyasagar (2002.), Nonlinear Systems Analysis, SIAM
Khalil, H. K. (2002.), Nonlinear Systems (3rd ed.), New York: Prentice Hall

Associate Lecturers

For students


ID 34366
  Summer semester
L2 English Level
L1 e-Learning

Grading System

90 Excellent
80 Very Good
70 Good
60 Acceptable