Stability and L 2 – Gain analysis of linear switched singular systems by using multiple discontinuous Lyapunov function approach

Abstract

In this paper, the problem of the E-exponential stability and $l_{2} - gain$ analysis of linear switched singular systems is investigated in discrete-time case. By using a multiple discontinuous Lyapunov function approach and adopting the mode-dependent average dwell time (MDADT) switching signals, new sufficient conditions of E-exponential stability and $l_{2} - gain$ analysis for linear switched singular systems are presented. Based on the above results, we also derive the weighted $l_{2} - gain$ performance index. In addition, by utilizing our proposed method, tighter bounds on average dwell time can be obtained for our considered systems. At last, a numerical example is given to show the effectiveness of the results.

Keywords

Linear switched singular systems mode-dependent average dwell time E-exponential stability multiple discontinuous Lyapunov function approach

Get full access to this article

View all access options for this article.

References

Branicky

(1998) Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Transactions on Automatic Control 43(4): 475–482.

Dai

(1989) Singular Control Systems. Berlin: Springer.

Dolezai

(1986) Generalized solution of semistate equations and stability. Circuits Systems and Signal Processing 5(4): 391–403.

Gao

(2017) Admissibility analysis for discrete-time singular Markov jump systems with asynchronous switching. Applied Mathenatics and Computation 313: 431–441.

Gomez-Gutierrez

Celikovsky

Ramirez-Trevino

, et al. (2013) On the observer design problem for continuous time switched linear systems with unknown switchings. Journal of the Franklin Institute 352(4): 1595–1612.

Guan

Yao

Hill

(2005) Robust

H_{\infty}

control of singular impulsive systems with uncertain perturbations. IEEE Transactions on Circuits Systems.II: Express Briefs 52(6): 293–298.

Johansson

Santucci

(2005) On hybrid control problems in communication systems. In: Proceedings of the 44th IEEE Conference on Decision and Control and European Control Conference, Seville, Spain, 12–15 December 2005, pp. 5624–5629. IEEE.

Dimirovski

, et al. (2017) Dwell-time-based stabilization of switched linear singular systems with all unstable-mode subsystems. Journal of the Franklin Institute 354(7): 2712–2724.

Liu

Yin

(2017) Stability analysis for discrete-time switched nonlinear system under MDADT switching. IEEE Transactions on Automatic Control 5: 18646–18653.

10.

Liberzon

(2003) Switching in Systems and Control. Systems and Control: Foundations and Applications. Boston: BirkhÆuser.

11.

Liberzon

Morse

(1999) Basic problems in stability and design of switched systems. IEEE Control Systems 5(19): 59–70.

12.

Lin

(2015) New results on admissibility and

l_{2} - gain

property of discrete-time switched singular state-delayed systems. In: IEEE Conference on Control Applications Conference, Sydney, Australia, 21–23 September 2015, pp. 1805–1810. IEEE.

13.

Liu

Zhao

Sun

, et al. (2018) Stability and

l_{2} - gain

analysis of discrete-time switched systems with mode-dependent average dwell time. IEEE Transactions on Systems, Man, and Cybernetics: Systems (99): 1–10.

14.

Zhao

(2010) Backstepping design for global stabilization of switched nonlinear systems in lower triangular form under arbitrary switching. Automatica 46(11): 1819–1823.

15.

Margalioat

(2006) Stability analysis of switched systems using variational principles: An introduction. Automatica 42(12): 2059–2077.

16.

Mills

Golddenberg

(1989) Force and position control of manipulators during constrained motion tasks. IEEE Transactions on Robotics and Automation 5(1): 30–46.

17.

Newcomb

Dziurla

(1989) Some circuits and systems applications of semistate theory. Circuits, Systems, and Signal Processing 8(3): 235–260.

18.

Parriaux

Millrioux

(2013) Designing self-synchronizing switched linear systems: An application to communications. Nonlinear Analysis: Hybrid Systems 7(1): 68–79.

19.

(2018) Observer-based bumpless switching control for switched linear systems with sensor faults. Transactions of the Institute of Measurement and Control 40(5): 1490–1498.

20.

Saija

Corless

Zeheb

, et al. (2013) Stability of a class of switched descriptor systems. American Control Conference, 2013. In: Proceedings of the American Control Conference, Washington, USA, 17–19 June 2013, pp. 54–58. IEEE.

21.

Tong

Huang

(2012) Finite-time boundedness and

l_{2} - gain

analysis for linear time-varying singular impulsive systems. In: Proceedings of the 24th Chinese Control and Decision Conference, Taiyuan, China, 23–25 May 2012, pp. 4031–4035. IEEE.

22.

Wang

Zou

Liu

, et al. (2015) Average dwell time approach to finite-time stabilization of switched singular linear systems. Journal of the Franklin Institute 352(7): 2920–2933.

23.

Wei

(2014) A bounded real lemma for discrete-time Markovian jump singular systems. Asian Journal of Control 16(1): 303–307.

24.

Xiang

Xiao

(2014) Stabilization of switched continuous-time systems with all modes unstable via dwell time switching. Automatica 50(3): 940–945.

25.

Xie

Wang

(2009) Average dwell-time approach to

l_{2} - gain

control synthesis of switched linear systems with time delay in detection of switching signal. IET Control Theory & Applications 3(6): 763–771.

26.

Zamani

Shafiee

(2015) Stability analysis of uncertain switched singular time-delay systems with discrete and distributed delays. Optimal Control Applications and Methods 36(1): 1–28.

27.

Zamani

Shafiee

Ibeas

(2018) On singular hybrid switched and impulsive systems. International Journal of Robust and Nonlinear Control 28(2): 437–465.

28.

Zhai

Kou

Imae

, et al. (2006) Stability analysis and design for switched descriptor systems. In: Proceedings of the 2006 IEEE International Symposium on Intelligent Control, Munich, Germany, 4–6 October 2006, pp. 482–487. IEEE.

29.

Zhai

Imae

, et al. (2007) Qualitative analysis of switched discrete-time descriptor systems. In: Proceedings of the 22nd IEEE International Symposium on Intelligent Control, Singapore, 1–3 October 2007, pp. 196–201. IEEE.

30.

Zhang

Zhao

Xue

, et al. (2011) Exponential stability for discrete-time singular switched time-delay systems with average dwell time. In: Proceedings of the 30th Chinese Control Conference, Yantai, China, 22-24 July 2011, pp. 1789–1794. IEEE.

31.

Zhao

Hill

(2008) On stability,

l_{2} - gain

and

H_{\infty}

control for switched systems. Automatica 44(5): 1220–1232.

32.

Zhao

Liu

Wang

(2013) Weighted

H_{\infty}

performance analysis of switched linear systems with mode-dependent average dwell time. International Journal of Systems Science 44(11): 2130–2139.

33.

Zhao

Shi

Yin

, et al. (2017) New results on stability of slowly switched systems: A multiple discontinuous Lyapunov function approach. IEEE Transactions on Automatic Control 62(7): 3502–3509.

34.

Zhao

Zhang

Shi

, et al. (2012a) Stability of switched positive linear systems with average dwell time switching. Automatica 48(6): 1132–1137.

35.

Zhao

Zhang

Shi

, et al. (2012b) Stability and stabilization of switched linear systems with mode-dependent average dwell time. IEEE Transactions on Automatic Control 57(7): 1809–1815.

36.

Zhai

Yasuda

, et al. (2002) Qualitative analysis of discrete-time switched systems. In: Proceedings of the American Control Conference, Anchorage, USA, 8–10 May 2002, pp. 1880–1885. IEEE.

37.

Zhai

(2011) Stability and L2 gain analysis of switched linear discrete-time descriptor systems. Open access. Available at: https://www.intechopen.com/books/discrete-time-systems/stability-and-

l_{2}

-gain-analysis-of-switched-linear-discrete-time-descriptor-systems

38.

Zhang

Shi

(2009) Stability,

l_{2} - gain

and asynchronous

H_{\infty}

control of discrete-time switched systems with average dwell time. IEEE Transactions on Automatic Control 54(9): 2192–2199.

39.

Zhai

Lin

Kim

, et al. (2005)

l_{2} - gain

analysis for switched systems with continuous-time and discrete-time subsystems. International Journal of Control 78(15): 1198–1205.

40.

Zheng

Wei

Yang

, et al. (2017) Stability analysis and

l_{2} - gain

of nonlinear switched systems with unstable subsystems. In: 2017 Chinese Automation Congress, Jinan, China, 20–22 October 2017, pp. 211–216