|
Vol: 52(66) No: 1 / March 2007 Stability Criteria for Positive Linear Systems Mihaela-Hanako Matcovschi Department of Automatic Control and Applied Informatics, Faculty of Automatic Control and Computer Science Technical University Gh. Asachi of Iasi, Bd. Mangeron 53A, 700050 Iasi, Romania, phone: +40-232-230751, e-mail: mhanako@delta.ac.tuiasi.ro Octavian Pastravanu Department of Automatic Control and Applied Informatics, Faculty of Automatic Control and Computer Science Technical University Gh. Asachi of Iasi, Bd. Mangeron 53A, 700050 Iasi, Romania, e-mail: opastrav@delta.ac.tuiasi.ro Keywords: positive linear system, asymptotic stability, Stein inequality, Lyapunov inequality, Holder norm, matrix measure. Abstract The paper develops novel criteria for the stability analysis of positive linear systems. Unlike traditional approaches focusing on quadratic-type Lyapunov functions, we consider Lyapunov functions defined by arbitrary vector norms. We address the cases of both discrete-time and continuous-time dynamics. We derive necessary and sufficient conditions formulated as matrix norm or matrix measure inequalities. Our results generalize the classical Stein or Lyapunov matrix inequalities in the sense of association with non-quadratic Lyapunov functions. References [1] L. Farina, S. Rinaldi, Positive Linear Systems: Theory and Applications, Pure and Applied Mathematics: A Wiley-Interscience Series of Text, Monographs, and Tracts, John Wiley & Sons, New York, 2000. [2] T. Kaczorek, Positive 1D and 2D Systems, Springer-Verlag, London, 2002. [3] L. Farina, “Positive Systems in the State Space Approach: Main Issues and Recent Results”, 15-th Int. Symp. on Mathematical Theory of Networks and Systems, 2002. [4] G. James, V. Rumchev, “Stability of Positive Linear Discrete-Time Systems”, Bulletin of the Polish Academy of Sciences, Technical Sciences, vol. 53, no. 1, pp. 1 – 8, 2005. [5] L. Benvenuti, A. De Santis, L. Farina (Eds.): Positive Systems, Proceedings of the First Multidisciplinary International Symposium on Positive Systems: Theory and Applications (POSTA 2003), Lecture Notes in Control and Information Sciences 294, Springer-Verlag, Heidelberg, 2003. [6] C.A. Desoer and M. Vidyasagar, Feedback Systems: Input-Output Properties, New York Academic, 1975. [7] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985. [8] J. Stoer, C. Witzgall, “Transformations by diagonal matrices in a normed space”, Numerische Mathematik, vol. 4, 1962, pp. 158 – 171. [9] A. Michel, K. Wang, Qualitative Theory of Dynamical Systems, Marcel Dekker, Inc., New York-Bassel-Hong Kong, 1995. |