• Automatic Control, Semester 2, position 2

• Introduction to nonlinearities in the plants and processes. • Introduction to basic concepts of analysis of nonlinear systems. • Understanding and using the basic tools for testing the stability of nonlinear systems. • Analysis of nonlinear systems using Matlab/Simulink programming software.

On successful completion of the course the students should be able to: Knowledge and understanding of: • Recognize and understand nonlinear problems and phenomena in the processes and plants. • Determine mathematical models of nonlinear systems. • Apply basic methods for analyzing nonlinear systems in time domain and state space. • Perform simulations, analysis and control of nonlinear systems using software package Matlab/Simulink.

Introduction to nonlinear systems and control. Typical nonlinear problems and phenomena. Types of nonlinearity.Types of nonlinear systems. State space. Solution of nonlinear diferential equations, existence and uniqueness of solutions, Lipschitz function, comparison principle. Equilibrium points. Phase-plane analysis: phase portrait, limit cycles, stability domain, classification of singular points. Poincaré-Bendixson criterion. Lyapunov stability concepts. Lyapunov stability and instability theorems of equilibrium points. LaSalle's theorem, invariance principle and Chetaev’s theorem. Lyapunov’s direct and indirect methods. Krasovski's criterion.

PA: Nonlinear mathematical models of dynamic systems. Determination of equilibrium points. Phase-plane analysis: phase portrait, limit cycles, stability domain, stability and attraction of equilibrium points. Determining the system stability by applying indirect and direct method of Lyapunov. PL: Practice and experiments: verification of non-linear mathematical models of different objects using a PC; experimental determination of nonlinear static characteristics and analysis of the dynamic behavior of different objects of automatic control (DC servo motor, heat flow experiment, coupled tanks experiment) using the programming software Matlab/Simulink.

Defined by curriculum of the study programme.

• Lj.Grujić, D.Lazić, Nonlinear Systems, Lecture notes in electronic form. • Radiša Jovanović, Nonlinear Systems 1, Lecture notes. • Modular educational real time control system with various control plants (DC servo motor, inverted pendulum, double inverted pendulum, heat flow experiment, coupled water tanks experiment), with acquisition hardware and software. • PC and PC Embedded controllers, Siemens Simatic PLC, National Instruments controllers. • Installation for control system testing and acquisition of electrical variables. • Intelligent Control Systems Laboratory, Control Systems Laboratory.

Total assigned hours: 75

New material: 20
Elaboration and examples (recapitulation): 10

Auditory exercises: 20
Laboratory exercises: 10
Calculation tasks: 0
Seminar paper: 0
Project: 0
Consultations: 0
Discussion/workshop: 0
Research study work: 0

Review and grading of calculation tasks: 3
Review and grading of lab reports: 3
Review and grading of seminar papers: 0
Review and grading of the project: 0
Test: 4
Test: 0
Final exam: 5

Activity during lectures: 5
Test/test: 50
Laboratory practice: 5
Calculation tasks: 0
Seminar paper: 0
Project: 0
Final exam: 40
Requirement for taking the exam (required number of points): 30

Hassan K. Khalil, Nonlinear Systems, 3rd Edition, Prentice-Hall, 2002.; Jean-Jacques E. Slotine, Weiping Li, Applied Nonlinear Control, Englewood Cliffs, NJ: Prentice-Hall, 1991.; Radiša Jovanović, Mаtlab and Simulink in Automatic Control, Faculty of Mechanical Engineering, Belgrade, ISBN 978-86-7083-896-3,2021.