Nonlinear systems
Code  Completion  Credits  Range  Language 

XP35NES1  ZK  4  2P+2C  Czech 
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 Department of Control Engineering
 Synopsis:

The goal of this course is to help student develop a deeper and broader perspective on theory and applications of nonlinear systems. At the hearth of the course will be the socalled differentialgeometric approach, which can be used for controllability and observability analysis of nonlinear systems, characterization of various types of exact feedback linearization and many other tasks. Great attention is paid to analysis of the structure of nonlinear systems from the perspective of control design. It follows from the state description of nonlinear systems and uses state transformations of the nonlinear model into a simpler form that is usable for control design. Differentialgeometric conditions for existence of these transformations are studied in this course. Concepts of nonlinear controllability and observability are introduced in this course and their relation to stabilization and reconstruction is analyzed because it is not as clear as for linear systems. Some additional topics such nonsmooth stabilization and discontinuous stabilization will be covered. Examples of use of the presented theories in underactuated robotic walking, nonholonomic systems and optimization of biosystems will be given.
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 Syllabus of lectures:

Mathematical foundations: vector fields, Lie derivative of a function with respect to a vector field, Lie bracket, Lie algebras and their properties.
Controllability of nonlinear systems. Reachability, strong reachability, controllability, global controllability, local controllability, smalltime local controllability and locallocal controllability.
Lie algebra of reachability and strong reachability. Conditions for various types of reachability and controllability and the properties of Lie algebras of reachability and strong reachability.
Observability of nonlinear systems. Definitions of observability and its shortcomings in the nonlinear case.
Algebra of observability and conditions of observability. Nonlinear canonical form of observability. Conditions for transformation of a nonlinear system into this form.
Nonlinear observer canonical form. Conditions for transformation of a nonlinear system into this form.
Necessary and sufficient conditions for exact feedback linearization, Relative degree of a nonlinear system with a single input and a single output and its vector version for systems with multiple inputs and multiple outputs. The problem of choosing an „auxiliary“ linearizing output for exact feedback linearization.
Distribution, its involutivity and integrability, Frobenius theorem.
Using Frobenius theorem for determining necessary conditions of exact feedback linearization. Differential forms, exact differential forms, their relation with involutive distributions and use for search for an „auxiliary“ linearizing output.
Other open problems of theory of nonlinear control and examples of use. Nonsmooth and discontinous stabilization of nonlinear systems.
Brockett condition of smooth and continuous stabilization. Controllability vs. stabilizatility for nonlinear systems.
Nonholonomic systems, their controllability and stabilizability
Using partial exact linearization for control of underactuated mechanical systems. The problem of walking robots.
Optimal control of nonlinear systems. Pontryagin principle of maximum for the problem with a free right end. An application to the optimal production of algae.
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Compulsory literature:
H. K. Khalil, Nonlinear Systems. Third edition. Prentice Hall 2002. ISBN13: 9780130673893
A. Isidori. Nonlinear Systems: Third Edition, Springer Verlag, Heidelberg, 1995. ISBN 9781447105497
Recommended literature:
M. Vidyasagar, Nonlinear Systems Analysis, Second Edition. SIAM Classics in Applied Mathematiacs 42. SIAM 2002. ISBN 0898715261.
R. Marino and P. Tomei: Nonlinear Control Design. Geometric, Adaptive and Robust Approach, Prentice Hall, Englewood Cli_s, NJ 1995. ISBN 0133426351
 Note:
 Further information:
 No timetable has been prepared for this course
 The course is a part of the following study plans:

 Doctoral studies, daily studies (compulsory elective course)
 Doctoral studies, combined studies (compulsory elective course)
 Cybernetics and Robotics (compulsory elective course)