Stability of Switched Positive Descriptor Systems with Average Dwell Time Switching

doi:10.1007/s12204-015-1607-x

摘要/Abstract

摘要： In this paper, the problems of stability for a class of switched positive descriptor systems (SPDSs) with average dwell time (ADT) switching are investigated. First, based on the equivalent switched system and the properties of the projector matrix, sufficient stabilities are given for the underlying systems in both continuoustime and discrete-time contexts. Then, a sufficient stability condition for the SPDS with both stable and unstable subsystems is obtained. The stability results for the SPDSs are represented in terms of a set of linear programmings (LPs) by the multiple linear co-positive Lyapunov function (MLCLF) approach. Finally, three numerical examples are given to illustrate the effectiveness of the obtained theoretical results.

关键词: average dwell time (ADT), switched positive descriptor systems (SPDSs), linear programming (LP), stability

Abstract: In this paper, the problems of stability for a class of switched positive descriptor systems (SPDSs) with average dwell time (ADT) switching are investigated. First, based on the equivalent switched system and the properties of the projector matrix, sufficient stabilities are given for the underlying systems in both continuoustime and discrete-time contexts. Then, a sufficient stability condition for the SPDS with both stable and unstable subsystems is obtained. The stability results for the SPDSs are represented in terms of a set of linear programmings (LPs) by the multiple linear co-positive Lyapunov function (MLCLF) approach. Finally, three numerical examples are given to illustrate the effectiveness of the obtained theoretical results.

Key words: average dwell time (ADT), switched positive descriptor systems (SPDSs), linear programming (LP), stability

中图分类号:

P 13

XIA Biao (夏彪), LIAN Jie* (连捷), YUAN Xue-hai (袁学海). Stability of Switched Positive Descriptor Systems with Average Dwell Time Switching[J]. 上海交通大学学报（英文版）, 2015, 20(2): 177-184.

XIA Biao (夏彪), LIAN Jie* (连捷), YUAN Xue-hai (袁学海). Stability of Switched Positive Descriptor Systems with Average Dwell Time Switching[J]. Journal of shanghai Jiaotong University (Science), 2015, 20(2): 177-184.

参考文献 19

[1]	Lewis F L. A survey of linear singular systems [J].Circuits, Systems and Signal Processing, 1986, 5(1):3-36.
[2]	Newcomb R W. The semistate description of nonlinear time-variable circuits [J]. IEEE Transactions on Circuits and Systems, 1981, 28(1): 62-71.
[3]	Herrero A, Ram′?rez M A, Thome N. Nonnegativity,stability, and regularization of discrete-time descriptor systems [J]. Linear Algebra and its Applications,2010, 432(4): 837-846.
[4]	Cant′o B, Coll C, S′anchez E. Positive solutions of a discrete-time descriptor system [J]. International Journal of Systems Science, 2008, 39(1): 81-88.
[5]	Rami M A, Napp D. Positivity of discrete singular systems and their stability: An LP-based approach [J].Automatica, 2014, 50(1): 84-91.
[6]	Virnik E. Stability analysis of positive descriptor systems[J]. Linear Algebra and its Applications, 2008,429(10): 2640-2659.
[7]	Rami M A, Napp D. Characterization and stability of autonomous positive descriptor systems [J]. IEEE Transactions on Automatic Control, 2012, 57(10):2668-2673.
[8]	Mehrmann V, Nabben R, Virnik E. Generalisation of the Perron-Frobenius theory to matrix pencils [J].Linear Algebra and its Applications, 2008, 428(1): 20-38.
[9]	Xia Y, Boukas E K, Shi P, et al. Stability and stabilization of continuous-time singular hybrid systems[J]. Automatica, 2009, 45(6): 1504-1509.
[10]	Zhai G, Xu X. A commutation condition for stability analysis of switched linear descriptor systems [J]. Nonlinear Analysis: Hybrid Systems, 2011, 5(3): 383-393.
[11]	Liu W Q, Yan W Y, Teo K L. On initial instantaneous jumps of singular systems [J]. IEEE Transactions on Automatic Control, 1995, 40(9): 1650-1655.
[12]	Liberzon D, Trenn S. On stability of linear switched differential algebraic equations [C]//Proceedings of the 48th IEEE Conference on Decision and Control.Shanghai, China, IEEE, 2009: 2156-2161.
[13]	Liberzon D, Trenn S, Wirth F. Commutativity and asymptotic stability for linear switched DAEs[C]//2011 50th IEEE Conference on Decision and Control and European Control Conference (CDCECC).Orlando, FL, USA, IEEE, 2011: 417-422.
[14]	Zhou L, Ho D W C, Zhai G. Stability analysis of switched linear singular systems [J]. Automatica, 2013,49(5): 1481-1487.
[15]	Chen Y, Fei S, Zhang K. Stability analysis for discrete-time switched linear singular systems: Average dwell time approach [J]. IMA Journal of Mathematical Control and Information, 2012. Doi:10.1093/imamci/dns027 (published online).
[16]	Zhao X, Zhang L, Shi P, et al. Stability of switched positive linear systems with average dwell time switching[J]. Automatica, 2012, 48(6): 1132-1137.
[17]	Liberzon D. Switching in systems and control [M].Berlin: Springer-Verlag, 2003.
[18]	Zhang L, Gao H. Asynchronously switched control of switched linear systems with average dwell time [J].Automatica, 2010, 46(5): 953-958.
[19]	Lian J, Liu J. New results on stability of switched positive systems: An average dwell-time approach [J]. IET Control Theory & Applications, 2013, 7(12): 1651-1658.