# Robust Control

 Credits 4 Semesters Winter Completion Exam Language of teaching undefined Extent of teaching 2P+2C
Annotation
Advanced course on selected topics in robust control.
Study targets
Motivation and goal for the course is to build a solid understanding of derivation of some classical methods for robust control design, namely Hinf-optimal control, in order to be able to modify the algorithms for systems with special structure. Several approaches for solving Hinf optimal control problem will be presented: state-space approach (based on solving two coupled Riccati equations), interpolation method (based on some fundamental results from complex analysis), polynomial approach (taking advantage of having the so-called "polynomial school of systems and controls" started here in Prague) and LMI-approach (revolving around the optimization-intensive framework of linear matrix inequalities). Related issues such that positiveness and dissipativeness of systems will also be discussed.
Course outlines
1. Parametric uncertainties: classification, Kharitonov's theorem for interval plants, Bialas' theorem for one-parameter uncertainties, zero exclusion principle, guardian maps, more complex parameter uncertainty structures.

2. Hankel, Toeplitz a mixed Hankel-Toeplitz operator, Nehari's theorem.

3. Forumlation of a general problem of Hinf optimal control design: generalized plant, linear fractional transformation (LFT), 4 basic problems: full information (FI), disturbance feedforward (DF), full control (FC), output estimation (OE).

4. Solution to the Hinf problem leading to 2 coupled Riccati equations.

5. Robust stabilization of a system with coprime factor uncertainty.

6. Basic linear matric inequalities (LMI) in control: Bounded real lemma, KYP lemma. Solving the Hinf problem using LMIs.

7. Interpolation approach to control design: Nevanlinna-Pick problem

8. Design of robust controllers of a fixed order.

9. Linear parameter varying (LPV) control.

10. Passivity vs. robustness, dissipative systems.

11. Riccati equation: analysis, numerical solution, spectral factorization, positive real functions, inner functions, inner-outer factorization, J-spectral factorization.

12. Reduction of order of a model and controller: truncation and residualization for balanced realization, diverse methods of balancing, minimization of Hankel norm of the approximation error. Lyapunov equation: properties and numerical solution.
Exercises outlines
Exercises follow the lectures.
Literature
G. E. Dullerud, F. Paganini. A Course in Robust Control Theory. Springer; 1 edition, 2005.

K. Zhou, J. C. Doyle, K. Glover. Robust and Optimal Control.Prentice Hall, 1st edition, 1995.

B. A. Francis, A Course in H Control Theory, Springer, 1987.

M. Green and D. J. N. Limebeer. Linear Robust Control. Prentice Hall, London, 1994.

S. P. Bhattacharyya, H. Chapellat, L. H. Keel. Robust Control - The Parametric Approach. Prentice-Hall, 1996.
Requirements
Students must be familiar with basic concepts of optimal and robust control. These could be obtained, for example, in an equally named graduate course (ORR) at FEE CTU. This advance course will not teach how to design a robust controller anymore. Instead, it will teach how to develop a computationally efficient and reliable algorithmic procedure for such design.