Journals A-Z
All Subjects
Free Journals
sciepub.com
Automatic Control and Information Sciences
ISSN (Print): 2375-1649 ISSN (Online): 2375-1630 Website: https://www.sciepub.com/journal/acis Editor-in-chief: Apply for this position
Open Access
Journal Browser
Go
Issue 1, Volume 2
Article Metrics
  • 17705
    Views
  • 13037
    Saves
  • 3
    Citations
  • 34
    Likes
Export Article
Automatic Control and Information Sciences. 2014, 2(1), 13-19
DOI: 10.12691/acis-2-1-3
Open AccessArticle

Further Results on Stability of Singular Time Delay Systems in the Sense of Non-Lyapunov: A New Delay Dependent Conditions

Dragutin Lj. Debeljkovic1, , Sreten B. Stojanovic2, Goran V. Simeunovic3 and Nebojsa J. Dimitrijevic1

1Department of Control Eng., University of Belgrade, School of Mechanical Engineering, Belgrade, Serbia

2Faculty of Technology, University of Nis, Serbia

3University of Belgrade, School of Mechanical Engineering, Innovation Centre, Belgrade, Serbia

Pub. Date: February 21, 2014
View Full Text Full Text PDF (192 KB) Full Text ePUB(200 KB)

Cite this paper:
Dragutin Lj. Debeljkovic, Sreten B. Stojanovic, Goran V. Simeunovic and Nebojsa J. Dimitrijevic. Further Results on Stability of Singular Time Delay Systems in the Sense of Non-Lyapunov: A New Delay Dependent Conditions. Automatic Control and Information Sciences. 2014; 2(1):13-19. doi: 10.12691/acis-2-1-3

Abstract

In this paper, we consider the problem of finite-time stability of a class of linear singular continuous time delay systems. By using Lyapunov-like functional with time-delay, new delay-dependent stability condition has been derived in terms of matrix inequality such that the system under consideration is regular, impulse free and finite time stable. In the proposed stability criterion, Drazin inverse of a singular matrix is used.

Keywords:
singular time delayed systems finite time stability delay dependent conditions

Creative CommonsThis work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]  Campbell, S. L., “Singular Systems of Differential Equations”, Pitman, Marshfield, MA, 1980.
 
[2]  Müller, P.C., “Stability of Linear Mechanical Systems with Holonomic Constraints”, Applied Mechanics Review, 46 (11). 160-164. 1993.
 
[3]  Campbell, S.L., Meyer, C.D., Rose, N.J. “Application of Drazin Inverse to Linear Systems of Differential Equations”, SIAM J. Appl. Math., 31. 411-425. 1976.
 
[4]  Luenberger, D.G., “Dynamic Equations in Descriptor Form”, IEEE Trans. Automat. Cont., 22 (3). 312-321. 1977.
 
[5]  La Salle, Lefschet, S., “Stability by Lyapunov’s Direct Method”, Academic Press, New York, 1961.
 
[6]  Weiss, L., Infante, E.F., “On the Stability of Systems Defined over a Finite Time Interval”, Proc. National Acad. Sci., 54. 44-48. 1965.
 
[7]  Weiss, L., Infante, E.F., “Finite Time Stability under Perturbing Forces and on Product Spaces”, IEEE Trans. Automat. Cont., 12. 54-59. 1967.
 
[8]  Debeljkovic, D., Owens, H., “On practical stability of singular systems”, Proc. MELECON Conf. 85, Madrid, Spain, 103-105. 1985.
 
[9]  Debeljkovic, D.Lj., Bajic, V.B., Gajic, Z., Petrovic, B., “Boundedness and existence of solutions of regular and irregular singular systems", Publications of the Faculty of Electrical Eng, Belgrade, Automatic Control, 1. 69-78. 1993.
 
[10]  Debeljkovic, D.Lj., Lazarevic, M.P., Koruga, Đ., Tomasevic, S., “Finite time stability of singular systems operating under perturbing forces: Matrix measure approach”, Proc. AMSE Conference, Melbourne Australia, 447-450. October 29-31. 1997.
 
[11]  Kablar, N.A., Debeljkovic, D.Lj., “Non - Lyapunov stability of linear singular systems: Matrix measure approach”, MNTS - Mathematical Theory of Networks and Systems, Presented lecture, also in Proc. of Ext. Abstracts, TM7, Padova, Italy, July 6-10, 1998.
 
[12]  Kablar, N.A., Debeljkovic, D.Lj., “Non - Lyapunov stability of linear singular systems: Matrix measure approach”, Preprints 5th IFAC Symposium on Low Cost Automation, Shenyang, China, September 8-10, TS13. 16-20, 1998.
 
[13]  Kablar, N.A., Debeljkovic, D.Lj., “Finite time stability of time varying singular systems”, Proc. IEEE CDC 98, Florida, USA, pp.3831-3836, December 10-12, 1998.
 
[14]  Debeljkovic, D.Lj., Kablar, N.A., “On necessary and sufficient conditions of linear singular systems stability operating on finite time interval”, Proc. XII CBA, Uberlandia, Brazil, IV. 1241-1246. September 14 -18. 1998.
 
[15]  Kablar, N.A., Debeljkovic, D.Lj., “Finite time instability of time varying linear singular systems”, Proc. IEEE ACC 99, San Diego, USA, 1796-1800. June 2-4. 1999.
 
[16]  Debeljkovic, D.Lj., Kablar, N.A., “Finite time stability of linear singular systems: Bellman-Gronwall approach”, Proc. ACC 99, San Diego, USA, 1803-1806. June 2-4. 1999.
 
[17]  Yang, C.Y., Zhang, Q.L., Lin, Y.P., “Practical Stability of Descriptor Systems“, Dynamics of Continuous, Discrete and Impulsive Systems, Series B: 12.b. 44-57. 2005.
 
[18]  Nie, Y.Y., Debeljkovic, D.Lj., “Non–Lyapunov Stability of Linear Singular Systems: A Quite new Approach in Time Domain”, Dynamics of Continuous, Discrete and Impulsive Systems, Vol.11, Series A: Math. Analysis, No. 5-6, pp.751-760., 2004.
 
[19]  Yang, C.Y., Zhang, Q.L., Zhou, L., “Practical Stabilization and Controllability of Descriptor Systems“, International Journal of Information and System Science, 1 (3-4). 455-466. 2005.
 
[20]  Jun-E, F., Zhen, W., Jia-Bing, S., “Finite-Time Control of Linear Singular Systems with Parametric Uncertainties and Disturbances”, Acta Automatica Sinica, 31(4). 634-637. 2005
 
[21]  Feng, H.H., Hunsarg, J., “Stabilization of Nonlinear Singularly Perturbed Multiple Time Delay Systems by Dither”, Trans. ASME J. of Dynamic Systems, Measurement and Control, 118 (3). 177-181. 1996.
 
[22]  Debeljkovic, D.Lj., Nenadic, Z..Lj., Milinkovic, S.A., Jovanovic, M.B., “On Practical and Finite-Time Stability of Time-Delay Systems”, Proc. ECC 97, Brussels, Belgium, 307-311. July 2-6. 1997.
 
[23]  Debeljkovic, D.Lj., Nenadic, Z.Lj., Milinkovic, S.A., Jovanovic, M.B. “On the Stability of Linear Systems with Delayed State Defined over Finite Time Interval”, Proc. IEEE CDC 97, San Diego, California, USA, 2771-2772. December 21-23. 1997.
 
[24]  Nenadic, Z.Lj., Debeljkovic, D.Lj., Milinkovic, S.A., “On Practical Stability of Time Delay Systems”, Proc. IEEE American Control Conference, Albuquerque, USA, 3235-3235. June 4-6. 1997.
 
[25]  Debeljkovic, D.Lj., Koruga, Đ., Milinkovic, S.A., Jovanovic, M. B., Jacic, Lj. “Further Results on Non-Lyapunov Stability of Time Delay Systems”, Proc. MELECON 98, Tel-Aviv, Israel, 1. 509-512. May 18-20. 1998.
 
[26]  Debeljkovic, D.Lj., Lazarevic, M.P., Milinkovic, S.A., Jovanovic, M.B., “Finite Time Stability Analysis of Linear Time Delay Systems: Bellman-Gronwall Approach”, Proc. 1st IFAC Workshop on Linear Time Delay Systems, Grenoble, France, 171-175. July 6-7. 1998.
 
[27]  Debeljkovic, D.Lj., Lazarevic, M.P., Nenadic, Z.Lj., Milinkovic, S.A., “Finite Time Stability of Time Delay Systems” IMA J. Math. Control and Info., 16 (3). 101-109. 1999.
 
[28]  Debeljkovic, D.Lj., Visnjic, N.S., Pjescic, M., “The Stability of Linear Continuous Singular Systems over the Finite Time Interval: An Overview”, International Journal of Information & System Science, 4 (4). 560-584. 2008.
 
[29]  Debeljkovic, D.Lj., Lazarevic, M.P., Koruga, Đ., Milinkovic, S.A., Jovanovic, M.B., “Further results on the stability of linear nonautonomous systems with delayed state defined over finite time interval”, Proc. IEEE ACC 2000, Chicago, Illinois, USA, 1450-1451. June 28-30. 2000.
 
[30]  Yang, C., Zhang, Q., Lin, Y., Zhou, L., “Practical stability of descriptor systems with time-delay in terms of two measurements”, Journal of the Franklin Institute, 343 (6). 635-646. 2006.
 
[31]  Debeljkovic, D.Lj, Stojanovic, S.B., Aleksendric, M.S., “Stability of Singular Time Delay Systems in the Sense of Non-Lyapunov: Classical and Modern Approach”, Hemijska Industrija, 61 (5). 193-202. 2013.
 
[32]  Owens, D.H., Debeljkovic, D.Lj., “Consistency and Lyapunov Stability of Linear Descriptor Systems: a Geometric Analysis”, IMA Journal of Math. Control and Information, 2. 139-151. 1985.
 
[33]  Su, J.H., “Further results on the robust stability of linear systems with single time delay”, Systems & Control Letters 23 (5). 375-379. 1994.
 
[34]  Debeljkovic, D.Lj., Buzurovic, I.M., Nestorovic, T., Popov, D., “A New Approach to Stability of Singular Time Delay Systems in the sense of Non-Lyapunov: Delay Independent Conditions”, Proc. 2011 IEEE ICCA (International Conference on Control and Automation, Santiago (Chile), December 19-21. 312-317. 2011.
 
[35]  Xu, S. Dooren, P.V., Stefan, R., Lam, J., “Robust Stability and Stabilization for Singular Systems with State Delay and Parameter Uncertainty”, IEEE Trans. Automat. Control, 47 (7). 1122-1128. 2002.
 
[36]  Debeljkovic, D.Lj., Buzurovic, I.M., Nestorovic, T., Popov, D.,“On Finite and Practical Stability of Time Delayed Systems: Lyapunov-Krassovski Approach: Delay Dependent Criteria”, Proc. The 23 rd, Chinese Control and Decision Conference CCDC 2011, Mianyang, (China), 23-25. 331-337. 2011.
 
Manuscript template