### Stability of Arbitrarily Switched Discrete-time Linear Singular Systems of Index-1

## Main Article Content

## Abstract

**Abstract:** In this paper, the index-1 notion for arbitrarily switched discrete-time linear singular systems (SDLS) has been introduced. Based on the Bohl exponents of SDLS as well as properties of associated positive switched systems, some necessary and sufficient conditions have been established for exponential stability.

*Keywords:* Switched system, linear discrete-time singular system, positive system, index-1 system.

References

[1] D. Liberzon, S. Trenn, On stability of linear switched differential algebraic equations, Proc. IEEE 48th Conf. Decision Control, December (2009) 2156.

[2] Z. Sun, S.S. Ge, Stability Theory of Switched Dynamical Systems, Springer, London, 2011.

[3] S. Trenn and F. Wirth, Linear switched DAEs: Lyapunov exponents, a converse Lyapunov theorem, and Barabanov norms, In Decision and Control (CDC), IEEE 51st Annual Conference, (2012) 2666.

[4] G. Zhai, X. Xu, A unified approach to stability analysis of switched linear descriptor systems under arbitrary switching, Internat. J. Appl. Math. Comput. Sci., 20(2) (2010) 249.

[5] M. Darouach and M. Chadli, Admissibility and control of switched discrete-time singular systems, Syst. Sci. Control Engrg.: An Open Access J., 1:1 (2013) 43.

[6] G. Zhai, X. Xu, D. W.C. Ho, Stability of switched linear discrete-time descriptor systems: a new commutation condition, Internat. J. Contr., 85 (2012) 1779.

[7] S.H. Friedberg, A.J. Insel and L.E. Spence, Linear Algebra, Prentice Hall, New Jersey, 1989.

[8] P.K. Anh, P.T. Linh, Stability of periodically switched discrete-time linear singular systems, J. Difference Equ. Appl., 23(10) (2017) 1680.

[9] P.K. Anh, H.T.N. Yen, Floquet theorem for linear implicit nonautonomous difference systems, J. Math. Anal. Appl., 321 (2006) 921.

[10] P.K. Anh, N.H. Du, L.C. Loi, Singular difference equations: an overview, Vietnam J.

Math., 35 (2007) 339.

[11] L.C. Loi, N.H. Du, and P.K. Anh, On linear implicit non-autonomous systems of difference equations, J. Difference Equ. Appl. 8 (2002) 1085.

[12] G. Polya and G. Szego, Problems and Theorems in Analysis I, Springer, Berlin, Heidelberg, 1998.

[13] B.Z. Vulikh, Introduction to the Theory of Cones in Normed Spaces, Izdat. Kalinin Univ., Kaliningrad, 1977.

[14] T.S. Doan , A. Kalauch , M. Klose , S. Siegmund, Stability of positive linear switched systems on ordered Banach spaces, Syst.and Contr. Lett.75 (2015) 14.

[15] H. Lin, P.J. Antsaklis, Stability and stabilizability of switched Linear Systems: a survey of recent results, IEEE Trans. Aut. Contr. 54 (2009) 308.

*Keywords:*Switched system, linear discrete-time singular system, positive system, index-1 system

## References

[1] D. Liberzon, S. Trenn, On stability of linear switched differential algebraic equations, Proc. IEEE 48th Conf. Decision Control, December (2009) 2156.

[2] Z. Sun, S.S. Ge, Stability Theory of Switched Dynamical Systems, Springer, London, 2011.

[3] S. Trenn and F. Wirth, Linear switched DAEs: Lyapunov exponents, a converse Lyapunov theorem, and Barabanov norms, In Decision and Control (CDC), IEEE 51st Annual Conference, (2012) 2666.

[4] G. Zhai, X. Xu, A unified approach to stability analysis of switched linear descriptor systems under arbitrary switching, Internat. J. Appl. Math. Comput. Sci., 20(2) (2010) 249.

[5] M. Darouach and M. Chadli, Admissibility and control of switched discrete-time singular systems, Syst. Sci. Control Engrg.: An Open Access J., 1:1 (2013) 43.

[6] G. Zhai, X. Xu, D. W.C. Ho, Stability of switched linear discrete-time descriptor systems: a new commutation condition, Internat. J. Contr., 85 (2012) 1779.

[7] S.H. Friedberg, A.J. Insel and L.E. Spence, Linear Algebra, Prentice Hall, New Jersey, 1989.

[8] P.K. Anh, P.T. Linh, Stability of periodically switched discrete-time linear singular systems, J. Difference Equ. Appl., 23(10) (2017) 1680.

[9] P.K. Anh, H.T.N. Yen, Floquet theorem for linear implicit nonautonomous difference systems, J. Math. Anal. Appl., 321 (2006) 921.

[10] P.K. Anh, N.H. Du, L.C. Loi, Singular difference equations: an overview, Vietnam J.

Math., 35 (2007) 339.

[11] L.C. Loi, N.H. Du, and P.K. Anh, On linear implicit non-autonomous systems of difference equations, J. Difference Equ. Appl. 8 (2002) 1085.

[12] G. Polya and G. Szego, Problems and Theorems in Analysis I, Springer, Berlin, Heidelberg, 1998.

[13] B.Z. Vulikh, Introduction to the Theory of Cones in Normed Spaces, Izdat. Kalinin Univ., Kaliningrad, 1977.

[14] T.S. Doan , A. Kalauch , M. Klose , S. Siegmund, Stability of positive linear switched systems on ordered Banach spaces, Syst.and Contr. Lett.75 (2015) 14.

[15] H. Lin, P.J. Antsaklis, Stability and stabilizability of switched Linear Systems: a survey of recent results, IEEE Trans. Aut. Contr. 54 (2009) 308.