Stability of a Special Class of Switched Positive Systems
This paper is concerned with the existence of a linear copositive Lyapunov function(LCLF) for a special class of switched positive linear systems(SPLSs) composed of continuousand discrete-time subsystems. Firstly, by using system matrices, we construct a special kind of matrices in appropriate manner. Secondly, our results reveal that the Hurwitz stability of these matrices is equivalent to the existence of a common LCLF for arbitrary finite sets composed of continuous- and discrete-time positive linear timeinvariant( LTI) systems. Finally, a simple example is provided to illustrate the implication of our results.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1063425Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1095
 A. Berman, M. Neumann, and R. Stern, Nonnegative matrices in dynamic systems, New York: Wiley, 1989.
 T. Kaczorek, Positive 1D and 2D Systems, London: Springer-Verlag, 2002.
 L. Benvenuti, A. D. Santis, and L. Farina(Eds.), Positive Systems, Berlin, Germany: Springer-Verlag, 2003.
 R. Shorten, D. Leith, J. Foy, and R. Kilduff, "Towards an analysis and design framework for congestion control in communication networks," in: Proceedings of the 12th Yale Workshop on Adaptive and Learning Systems, 2003.
 A. Jadbabaie, J. Lin, and A. Morse, "Coordination of groups of mobile autonomous agents using nearest neighbor rules," IEEE Transaction on Automatic Control, vol. 48, pp. 988-1001, 2003.
 M. A. Rami and F. Tadeo, "Controller synthesis for positive linear systems with bounded controls," IEEE Transaction on Circuits and systems II, vol. 54, no. 2, pp. 151-155, 2007.
 R. Shorten, F. Wirth, O. Mason, K. Wulff, and C. King, "Stability Theory for Switched and Hybrid Systems, SIAM Review vol. 49, no. 4, pp. 545- 592, 2007
 O. Mason and R. Shorten, "A conjecture on the existence of common quadratic Lyapunov functions for positive linear systems, in: Proceedings of the 2003 American Control Conference, New York City, USA, 2003.
 L. Gurvits, R. Shorten, and O. Mason, "On the stability of switched positive linear systems," IEEE Transaction on Automatic Control, vol. 52, no. 6, pp. 1099-1103, 2007.
 O. Mason and R. Shorten, "On the simultaneous diagonal stability of pair of positive linear systems, Linear Algebra and its Application, vol. 413 no. 1, pp. 13-23, 2006.
 L. Farina and S. Rinaldi, Positive linear systems: theory and applications, New York: Wiley, 2000.
 O. Mason and R. Shorten, "On Linear Copositive Lyapunov Functions and the Stability of Switched Positive Linear Systems," IEEE Transaction on Automatic Control, vol. 52, no. 7, pp. 1346-1349, 2007.
 Z. Chen, Y. Gao, "On common linear copositive Lyapunov functions for pairs of stable positive linear systems," Nonlinear Analysis: Hybrid Systems, vol. 3, pp. 467-474, 2009.
 X. Liu, "Stability Analysis of Switched Positive Systems: A Switched Linear Copositive Lyapunov Function Method," IEEE Transaction on Circuits and Systtems II, vol. 56, no. 5, pp. 414-418, 2009.
 X. Ding, L. Shu, and Z. Wang, "On stability for switched linear positive systems," Mathematical and Computer Modelling, vol. 53, pp. 1044- 1055, 2011.
 R. T. Rockafellar, Convex Analysis, Princeton, NJ, USA: Prinnceton University Press, 1970.
 Z. Gajic, M. Qureshi, "Lyapunov Matrix Equation in System Stability and Control," Mathematics in Science and Engineering, vol. 195, Academic Press, 1995.
 T. Kailath, Linear Systems, Englewood Cliffs, N.J.: Prentice-Hall, 1980.
 R. Horn, C. Johnson, "Topics in matrix analysis, Cambridge University Press, 1991.