Nonlinear Systems and Chaos

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. Finally, the course introduces basics of chaotic systems theory and some their examples.

Code
BE3M35NES
Semester
Winter
Range
2p+2c
Capacity
18
Occupied
4
Credits
6
Completion
Assessment + Examination
Language
English
Lectures contents

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. Single-input single-output systems: relative degree, input-output linearization, zero dynamics computation and minimum phase property test.
8. Single input single output systems: examples.
9. Multi-input multi-output systems: vector relative degree, input-output linearization and decoupling.
10. Multi-input multi-output systems: zero dynamics computation and minimum phase property test.
11. Multi-input multi-output systems: examples.
12. Multi-input multi-output systems: dynamical feedback, example of its application in the case study of the planar vertical take-off and landing plane. Further examples of the practical applications of the exact linearization.
13. Chaotical systems and further complex nonlinear phenomena.

Tutorials contents

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 single-input single-output nonlinear dynamical systems.
6. Exact feedback linearization of multi-input multi-output nonlinear dynamical systems.