Nonlinear Systems and Chaos - B3M35NES

Credits 6
Semesters Winter
Completion Assessment + Examination
Language of teaching Czech
Extent of teaching 2P+2C
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.
H.K. Khalil, Nonlinear Control, Global Edition, PEARSON, 2015.
Available in library
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).