ICANNGA'07 tutorial on

Selected Mathematical Equations of Contemporary Technology

Tadeusz Kaczorek

Institute of Control and Industrial Electronics

Warsaw University of Technology

Warsaw, Poland

The talk will be consisting of two parts. The first part will be devoted to some recent developments of selected mathematical matrix equations. A special attention will be focused on the matrix Sylvester equation, the matrix Lyapunov equation and the matrix Riccati equation. The polynomial matrix equations will be also discussed. The numerical aspects of solutions to the equations will be considered and illustrated by numerical examples. Some applications of the equations in engineering problems will be also demonstrated.
The second part will be devoted to some new extensions of the classical Cayley-Hamilton theorem for n-D polynomial matrices, the block matrices, the coefficients matrices of the impulse response matrices, the right and left inverse matrices and for time-varying and nonlinear systems. It will be shown that the extensions are useful not only in systems and control systems theory but they can be also applied in solving many engineering problems.

Basic references

1. B.N.Datta, Numerical Methods for Linear Control Systems, Amsterdam, Elsevier 2003.

2. T.Kaczorek, Theory of Polynomial and Rational Matrices, Springer-Verlag, London 2006

3. T.Kaczorek, Generalizations of Cayley-Hamilton Theorem for n-D Polynomial Matrices, IEEE Trans.Autom. Contr.vol.50,No.5,2005,pp.671-674

4. T.Kaczorek, An Extension of the Cayley-Hamilton Theorem for Nonlinear Time-Varying Systems, Int. J. Appl. Math. Comput. Sci. Vol.16,No.1,2006, pp. 141-145.

5. T.Kaczorek, New Extensions of the Cayley-Hamilton Theorem with Applications, Pro. 19th European Conf. on Modelling and Simulation June 1-4, 2005, Riga, Latvia, pp.119-124.
    

Tadeusz Kaczorek received the MSc., PhD and DSc degrees from Electrical Engineering of Warsaw University of Technology in 1956, 1962 and 1964, respectively. In the period 1968 - 69 he was the dean of Electrical Engineering Faculty and in the period 1970 - 73 he was the prorector of Warsaw University of Technology. Since 1971 he has been professor and since 1974 full professor at Warsaw University of Technology. In 1986 he was elected a corresp. member and in 1996 full member of Polish Academy of Sciences. In the period 1988 - 1991 he was the director of the Research Centre of Polish Academy of Sciences in Rome. In June 1999 he was elected the full member of the Academy of Engineering in Poland. In May 2004 he was elected the honorary member of the Hungarian Academy of Sciences. He was awarded by the title doctor honoris causa by the University of Zielona Góra (2002), the Technical University of Lublin (2004), the Technical University of Szczecin (2004), Warsaw University of Technology 2004. His research interests cover the theory of systems and the automatic control systems theory, specially, singular multidimensional systems, positive multidimensional systems and singular positive 1D and 2D systems. He has initiated the research in the field of singular 2D and positive 2D linear systems.  He has published 17 books (5 in English) and over 600 scientific papers in journals like IEEE Transactions on Automatic Control, IEEE Transactions on Neural Networks, Multidimensional Systems and Signal Processing, International Journal of Control, Bull. Pol. Acad. Sciences, etc. and proceedings of conferences. He has presented more than 80 invited papers on international conferences and world congresses. He has given invited lectures in more than 50 universities in USA, Canada, UK, Germany, Italy, France, Japan, Greece etc. He has been a member of many international committees and programme committees. He supervised over 60 Ph.D. theses. More than 20 of this PhD students became professors in USA, UK and Japan. He is editor-in-chief of Bulletin of the Polish Academy of Sciences, Techn. Sciences and editorial member of about ten international journals.