Abstract
We describe a new class of positive linear discrete-time switching systems for which the problems of stability or stabilizability can be resolved constructively. The systems constituting this class can be treated as a natural generalization of systems with the so-called independently switching state vector components. Distinctive feature of such systems is that their components can be arbitrarily “re-connected” in parallel or in series without loss of the “constructive resolvability” property for the problems of stability or stabilizability of a system. It is shown also that, for such systems, the individual positive trajectories with the greatest or the lowest rate of convergence to the zero can be built constructively.
References
A. F. Kleptsyn, V. S. Kozyakin, M. A. Krasnosel’skii, and N. A. Kuznetsov, “Stability of desynchronized systems,” Dokl. Akad. Nauk SSSR 274, 1053–1056 (1984).
Barabanov, “On the Lyapunov exponent of discrete inclusions. I,” Automat. Remote Control 49, 152–157 (1988).
V. S. Kozyakin, “On the absolute stability of systems with asynchronously operating pulse elements,” Automat. Remote Control 51, 1349–1355 (1991).
L. Gurvits, “Stability of discrete linear inclusion,” Linear Algebra Appl. 231, 47–85 (1995).
V. Kozyakin, “A short introduction to asynchronous systems,” in Proc. 6th Int. Conf. on Difference Equations, Boca Raton, FL, 2004 (CRC, Boca Raton, 2004), pp. 153–165.
R. Shorten, F. Wirth, O. Mason, K. Wulff, and C. King, “Stability criteria for switched and hybrid systems,” SIAM Rev. 49, 545–592 (2007).
H. Lin and P. J. Antsaklis, “Stability and stabilizability of switched linear systems: a survey of recent results,” IEEE Trans. Automat. Control 54, 308–322 (2009).
E. Fornasini and M. E. Valcher, “Stability and stabilizability criteria for discrete-time positive switched systems,” IEEE Trans. Automat. Control 57, 1208–1221 (2012).
G.-C. Rota and G. Strang, “A note on the joint spectral radius,” Proc. Nederl. Akad. Wetensch. A 63 = Indag. Math. 22, 379–381 (1960).
J. Theys, “Joint spectral radius: Theory and approximations,” Ph. D. Dissertation, Faculté des sciences appliquées, Département d’ingénierie mathématique, Center for Systems Engineering and Applied Mechanics, Univ. Catholique de Louvain (May 2005).
R. Jungers, “The joint spectral radius,” in Lecture Notes in Control and Information Sciences, Vol. 385: Theory and Applications (Springer-Verlag, Berlin, 2009).
J. Shen and J. Hu, “Stability of discrete-time switched homogeneous systems on cones and conewise homogeneous inclusions,” SIAM J. Control Optim. 50, 2216–2253 (2012).
J. Bochi and I. D. Morris, “Continuity properties of the lower spectral radius,” Proc. Lond. Math. Soc. 110, 477–509 (2015).
T. Bousch and J. Mairesse, “Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture,” J. Amer. Math. Soc. 15, 77–111 (2002) (electronic).
V. D. Blondel, J. Theys, and A. A. Vladimirov, “Switched systems that are periodically stable may be unstable,” in Proc. 15th Int. Symp. on Math. Theory of Networks and Systems (MTNS), Notre-Dame, USA, Aug. 12—16, 2002 (Univ. Notre-Dame, 2002).
V. Kozyakin, “A dynamical systems construction of a counterexample to the finiteness conjecture,” in Proc. 44th IEEE Conf. on Decision and Control, 2005 and 2005 Eur. Control Conf. CDCECC’05, 2005 (IEEE, New York, 2005), pp. 2338–2343.
A. Czornik and P. Jurgas, “Falseness of the finiteness property of the spectral subradius,” Int. J. Appl. Math. Comput. Sci. 17, 173–178 (2007).
V. S. Kozyakin, “Algebraic unsolvability of a problem on the absolute stability of desynchronized systems,” Automat. Remote Control 51, 754–759 (1990).
J. N. Tsitsiklis and V. D. Blondel, “The Lyapunov exponent and joint spectral radius of pairs of matrices are hard—when not impossible—to compute and to approximate,” Math. Control Signals Syst. 10, 31–40 (1997).
V. D. Blondel and Y. Nesterov, “Polynomial-time computation of the joint spectral radius for some sets of nonnegative matrices,” SIAM J. Matrix Anal. Appl. 31, 865–876 (2009).
Y. Nesterov and V. Y. Protasov, “Optimizing the spectral radius,” SIAM J. Matrix Anal. Appl. 34, 999–1013 (2013).
V. Kozyakin, “Hourglass alternative and the finiteness conjecture for the spectral characteristics of sets of nonnegative matrices,” Linear Algebra Appl. 489, 167–185 (2016).
V. Yu. Protasov, “Spectral simplex method,” Math. Program. Ser. A 156, 485–511 (2016).
R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd Edition, (Cambridge Univ. Press, Cambridge, 2013).
J. S. Golan, Semirings and Their Applications, (Kluwer, Academic, Dordrecht, 1999).
Author information
Authors and Affiliations
Corresponding author
Additional information
Published in Russian in Informatsionnye Protsessy, 2016, Vol. 16, No. 2, pp. 194–206.
Rights and permissions
About this article
Cite this article
Kozyakin, V.S. Constructive stability and stabilizability of positive linear discrete-time switching systems. J. Commun. Technol. Electron. 62, 686–693 (2017). https://doi.org/10.1134/S1064226917060110
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1064226917060110