Adaptive and Variable Structure Systems

I FREQUENCY DOMAIN

1. Static characteristics
2. Description function
3. Oscillations and limit cycles
4. Method of harmonic balance
5. Absolute stability
6. Positive real functions, the Kalman-Yakubovich-Popov-lemma
7. The Circle Criterion
8. The Popov Criterion

II TIME DOMAIN

9. Gain scheduling and variable structure control
10. Switched linear systems
11. Stability for arbitrary switching
12. Common quadratic Lyapunov functions
13. Piecewise linear systems
14. Sliding mode control

• Assignment 1
• Assignment 2
• Assignment 3
• Assignment 4
• Assignment 5

Texts marked with ** are also downloads. These downloads are intended exclusively for students and staff of the TU Ilmenau. You will be asked for your login.

• Elementary Characteristics
• Stability in the sense of Lyapunov
• Strict positive real functions - MIMO case
• Small Matrix primer
• Definiteness of matrices
• Converse theorems for the stability of switched linear systems
• Fitt's counterexample (Aizerman's conjecture), Matlab demo : m-file**, mdl-file**
• Example: local stabiliy via Circle Criterion and estimate of the region of attractionm-file**
• Matlab demo** switching system

Christmas puzzle: Consider the switching order {system 2, system 1} with a dwell time of 1.11s. Simulate the solution once for 6 periods and once for 20 periods. Is the solution limited? How do you explain the observed behaviour?

The module includes an internship consisting of two experiments. To be credited for the module, these attempts must be successfully completed, but they are not a prerequisite for the examination. Binding registration for the internship via Thoska-System until 31.10.

ASR 1

ASR 2

Please check the German version of this page for the current schedule of the lab assignments.

Slot allocation (will be activated shortly) via our online portal

In the following some suggestions, in which important elements of the lecture can be read.

1. Switched systems

• Daniel Liberzon. Switching in Systems and Control. Birkhäuser, Boston, 2003.
• Mikael Johansson. Piecewise Linear Control Systems. Springer publishing house, Berlin, Heidelberg, 2003.
• R. Shorten, F. Wirth, O. Mason, K. Wulff, and C. King. Stability Criteria for Switched and Hybrid Systems. SIAM Review, 49(4):545-592, 2007 (URL) (Scientific essay, not a textbook. So only for people who want to know exactly!)

2. Stability of linear systems (stability theory)

• W. J. Rugh. Linear System Theory. Second Edition. Prentice Hall, Upper Saddle River, New Jersey, 1996.

3. Nonlinear systems (stability theory and classical methods in frequency domain)

• Ch. Desoer, M. Vidyasagar. Feedback Systems: Input-Output Properties, Academic Press, London, 1975.
• M. Vidyasagar. Nonlinear Systems Analysis. Second Edition. Prentice Hall, Englewood Cliffs, New Jersey, 1993.
• H. K. Khalil. Nonlinear Systems. Third Edition. Prentice Hall, Upper Saddle River, New Jersey, 2002.
• O. Föllinger. Nonlinear control systems 1st 7th edition. Oldenbourg, Munich, 1993.
• O. Föllinger. Nonlinear control systems 2nd. 7th edition. Oldenbourg, Munich, 1993.

4. Adaptive systems

• P. A. Ioannou, J. Sun. Robust Adaptive Control, Prentice Hall, 1996.

5. Sliding mode control

• Y. Shtessel, C. Edwards, L. Fridman, A. Levant. Sliding Mode Control and Observation. Birkhäuser, Basel, 2013.