Nonlinear Systems
Code  Completion  Credits  Range  Language 

BE3M35NES  Z,ZK  6  2p+2c 
 Lecturer:
 Sergej Čelikovský (guarantor)
 Tutor:
 Sergej Čelikovský (guarantor)
 Supervisor:
 Department of Control Engineering
 Synopsis:

The goal of this course is to introduce basics of the modern approaches to the theory and applications of nonlinear control. Fundamental difference when dealing with nonlinear systems control compared with linear case is that the state space approach prevails. Indeed, the frequency response approach is almost useless in nonlinear control. State space models are based mainly on ordinary differential equations, therefore, an introduction to solving these equations is part of the course. More importantly, the qualitative methods for ordinary differential equations will be presented, among them Lyapunov stability theory is crucial. More specifically, the focus will be on Lyapunov function method enabling to analyse stability of nonlinear systems, not only that of linear ones. Furthemore, stabilization desing methods will be studied in detail, among them the socalled control Lyapunov function concept and related backstepping method. Special stress will be, nevertheless, given by this course to introduce and study methods how to transform complex nonlinear models to simpler forms where more standard linear methods would be applicable. Such an approach is usually refered to as the socalled exact nonlinearity compensation. Contrary to the wellknown approximate linearization this method does not ignore nonlinearities but compensates them up to the best possible extent. The course introduces some interesting case studies as well, e.g. the planar vertical take off and landing plane („planar VTOL“), or a simple 2dimensional model of the walking robot. Finally, the course introduces basics of chaotic systems theory and some their examples.
 Requirements:

Prerequisites are: knowledge of basics of control theory (frequency response, feedback, stability, PID controllers, etc.), finishing advanced course on linear systems introducing notions like controllability, observability, minimal realization. Last but not least, a good knowledge ol linear algebra (eigenvalues, singular decompostion, etc.) and of mathematical analysis (multivariable differential calculus).
 Syllabus of lectures:

1. State space description of the nonlinear dynamical system. Specific nonlinear properties and typical nonlinear phenomena. Examples of natural and technological systems modelled using nonlinear systems.
2. Mathematical basics of the state space methods for the nonlinear systems. Definition of stability and its investigation methods. Approximate linearization method and Lyapunov function method.
3. Invariant sets and LaSalle principle. Exponential stability. Analysis of additive perturbations influence on asymptotically and exponentially stable nonlinear systems.
4. Feedback stabilization of nonlinear systems based on control Lyapunov function. „Backstepping“.
5. Control design using structural methods: introduction, basic notions and definition of the exact system transformations.
6. Structural methods and various types of the exact linearization. Zero dynamics and minimum phase property.
7. Singleinput singleoutput systems: relative degree, inputoutput linearization, zero dynamics computation and minimum phase property test.
8. Single input single output systems: examples.
9. Multiinput multioutput systems: vector relative degree, inputoutput linearization and decoupling.
10. Multiinput multioutput systems: zero dynamics computation and minimum phase property test.
11. Multiinput multioutput systems: examples.
12. Multiinput multioutput systems: dynamical feedback, example of its application in the case study of the planar vertical takeoff and landing plane. Further examples of the practical applications of the exact linearization.
13. Chaotical systems and further complex nonlinear phenomena.
 Syllabus of tutorials:

1. Solving ordinary differential equations. Examples of nonlinear dynamical systems, their control based on exact linearization. Comparision of the exact linearization and aproximate linearization based control designs.
2. Nonlinear dynamical systems stability analysis. Lyapunov function and LaSalle principle.
3. Control using Lyapunov function. Backstepping.
4. Lie derivative and its computation.
5. Exact feedback linearization of singleinput singleoutput nonlinear dynamical systems.
6. Exact feedback linearization of multiinput multioutput nonlinear dynamical systems.
 Study Objective:
 Study materials:

S. Čelikovský, Nelineární systémy, Vydavatelství FEL ČVUT, 2006. (in Czech).
Available in the ČVUT bookstore.
H.K. Khalil, Nonlinear Systems, Third Edition, Prentice Hall, 2002.
Available in library.
 Note:
 Timetable for winter semester 2018/2019:
 Timetable is not available yet
 Timetable for summer semester 2018/2019:
 Timetable is not available yet
 The course is a part of the following study plans:

 Cybernetics and Robotics  Cybernetics and Robotics (compulsory elective course)
 Cybernetics and Robotics  Robotics (compulsory elective course)
 Cybernetics and Robotics  Senzors and Instrumention (compulsory elective course)
 Cybernetics and Robotics  Systems and Control (compulsory course of the specialization)
 Cybernetics and Robotics  Aerospace Systems (compulsory elective course)