### Nonlinear Systems and Chaos

##### B232 - Summer 23/24
This is a grouped Moodle course. It consists of several separate courses that share learning materials, assignments, tests etc. Below you can see information about the individual courses that make up this Moodle course.

#### Nonlinear Systems and Chaos - B3M35NES

##### Main course
 Credits 6 Semesters Winter Completion Assessment + Examination Language of teaching Czech Extent of teaching 2P+2C
Annotation
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 so-called 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 so-called exact nonlinearity compensation. Contrary to the well-known 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 2-dimensional model of the walking robot.
Course outlines
1. State space description of the nonlinear dynamical system. Specific nonlinear properties and typical nonlinear phenomena. Nonlinear control techniques outlook.

2. Stability of equilibrium points. 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 using control Lyapunov function. Backstepping.

5. Control design using structural methods. Definition of system transformations using the state and input variables change.

6. Control design using structural methods. Exact feedback linearization. Zero dynamics and minimum phase property.

7. Structure of single-input single-output systems. Exact feedback linearization, relative degree, partial and input-output linearization, zero dynamics computation and minimum phase property test. Examples.

8. Structure of multi-input multi-output systems. Vector relative degree, input-output linearization and decoupling, zero dynamics computation and minimum phase property test.

9. Structure of multi-input multi-output systems.. Examples, dynamical feedback, example of its application in the case study of the planar vertical take-off and landing plane.

10. Further examples of the practical applications of the exact feedback linearization.
Exercises outlines
1.Examples of natural and technological systems modelled using nonlinear systems. Comparision of the exact linearization and aproximate linearization based control designs.
2. Nonlinear dynamical systems stability analysis. Lyapunov function and LaSalle principle.
3. Lyapunov-based control and the backstepping.
4. Lie derivative and its computation.
5. Exact feedback linearization of single-input single-output nonlinear dynamical systems.
6. Exact feedback linearization of multi-input multi-output nonlinear dynamical systems.
Literature
H.K. Khalil, Nonlinear Control, Global Edition, PEARSON, 2015.
Available in library
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. Last but not least, a good knowledge ol linear algebra (eigenvalues, eigenvectors, equivalence of matrices, canonical forms of matrices, etc.) and of mathematical analysis (multi-variable differential calculus, ordinary differential equations).

#### Nonlinear Systems - R35NES

 Credits 6 Semesters Winter Completion Assessment + Examination Language of teaching English Extent of teaching 2P+2C

#### Nonlinear systems - XP35NES1

 Credits 4 Semesters Undefined Completion Exam Language of teaching Czech Extent of teaching 2P+2C
Annotation
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 so-called differential-geometric 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. Differential-geometric 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.
Course outlines
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, small-time local controllability and local-local 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.
Literature
Compulsory literature:
H. K. Khalil, Nonlinear Systems. Third edition. Prentice Hall 2002. ISBN-13: 978-0130673893
A. Isidori. Nonlinear Systems: Third Edition, Springer Verlag, Heidelberg, 1995. ISBN 978-1-4471-0549-7

Recommended literature:
M. Vidyasagar, Nonlinear Systems Analysis, Second Edition. SIAM Classics in Applied Mathematiacs 42. SIAM 2002. ISBN 0-89871-526-1.
R. Marino and P. Tomei: Nonlinear Control Design. Geometric, Adaptive and Robust Approach, Prentice Hall, Englewood Cli_s, NJ 1995. ISBN 0-13-342635-1

#### Nonlinear Systems - BE3M35NES

 Credits 6 Semesters Winter Completion Assessment + Examination Language of teaching English Extent of teaching 2P+2C
Annotation
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 so-called 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 so-called exact nonlinearity compensation. Contrary to the well-known 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 2-dimensional model of the walking robot.
Course outlines
1. State space description of the nonlinear dynamical system. Specific nonlinear properties and typical nonlinear phenomena. Nonlinear control techniques outlook.

2. Stability of equilibrium points. 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 using control Lyapunov function. Backstepping.

5. Control design using structural methods. Definition of system transformations using the state and input variables change.

6. Control design using structural methods. Exact feedback linearization. Zero dynamics and minimum phase property.

7. Structure of single-input single-output systems. Exact feedback linearization, relative degree, partial and input-output linearization, zero dynamics computation and minimum phase property test. Examples.

8. Structure of multi-input multi-output systems. Vector relative degree, input-output linearization and decoupling, zero dynamics computation and minimum phase property test.

9. Structure of multi-input multi-output systems.. Examples, dynamical feedback, example of its application in the case study of the planar vertical take-off and landing plane.

10. Further examples of the practical applications of the exact feedback linearization.
Exercises outlines
1.Examples of natural and technological systems modelled using nonlinear systems. Comparision of the exact linearization and aproximate linearization based control designs.
2. Nonlinear dynamical systems stability analysis. Lyapunov function and LaSalle principle.
3. Lyapunov-based control and the backstepping.
4. Lie derivative and its computation.
5. Exact feedback linearization of single-input single-output nonlinear dynamical systems.
6. Exact feedback linearization of multi-input multi-output nonlinear dynamical systems.
Literature
H.K. Khalil, Nonlinear Control, Global Edition, PEARSON, 2015.
Available in library
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 (multi-variable differential calculus).