Skip to main content
SpringerLink
Log in

Positive observer design for linear impulsive positive systems with interval uncertainties and time delay

  • Regular Papers
  • Control Theory and Applications
  • Published:
International Journal of Control, Automation and Systems Aims and scope Submit manuscript

Abstract

This paper is concerned with the problem of positive observer design for impulsive positive systems (IPS) with interval uncertainties and time delay. A copositive Lyapunov-Krasovskii functional with exponential term is constructed. By applying the average impulsive interval method, sufficient conditions for the existence of the positive observer are established to guarantee the exponential stability of the corresponding augmented system, which ensures the designed positive observer can estimate the system states exponentially. Combined with the linear programming (LP) technique, an algorithm is developed to design the observer gain matrices. Finally, a numerical example is provided to show the effectiveness of the theoretical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. L. Farina and S. Rinaldi, Positive Linear Systems: Theory and Applications, John Wiley and Sons, 2011.

    MATH  Google Scholar 

  2. G. Silva-Navarro and J. Alvarez-Gallegos, “On the property sign-stability of equilibria in quasimonotone positive nonlinear systems,” Proc. of the 33rd Conf. Decision and Control, pp. 4043–4048, December 1994.

    Google Scholar 

  3. J. A. Jacquez and C. P. Simon, “Qualitative theory of compartmental systems,” Siam Review, vol. 35, no. 1, pp. 43–79, February 2012. [click]

    Article  MathSciNet  MATH  Google Scholar 

  4. S. Li, Z. Xiang, and H. R. Karimi, “Positive L-1 observer design for positive switched systems,” Circuits, Systems, and Signal Processing, vol. 33, no. 7, pp. 2085–2106, January 2014. [click]

    Article  MathSciNet  Google Scholar 

  5. M. Xiang, Z. Xiang, and H. R. Karimi, “Stabilization of positive switched systems with time-varying delays under asynchronous switching,” International Journal of Control, Automation and Systems, vol. 12, no. 5, pp. 939–947, August 2014. [click]

    Article  Google Scholar 

  6. J. Liu and K. Zhang, “State feedback with memory for constrained switched positive linear systems,” Entropy, vol. 17, no. 5, pp. 2655–2676, April 2015. [click]

    Article  MathSciNet  Google Scholar 

  7. S. Li, Z. Xiang, and H. R. Karimi, “Mixed l-infinity/l-1 fault detection observer design for positive switched systems with time-varying delay via delta operator approach,” International Journal of Control, Automation and Systems, vol. 12, no. 4, pp. 709–721, July 2014. [click]

    Article  Google Scholar 

  8. J. Zhang and Z. Han, “Robust stabilization of switched positive linear systems with uncertainties,” International Journal of Control, Automation and Systems, vol. 11, no. 1, pp. 41–47, January 2013. [click]

    Article  Google Scholar 

  9. S. Zhu, Z. Li, and C. Zhang, “Exponential stability analysis for positive systems with delays,” IET Control Theory and Applications, vol. 6, no. 6, pp. 761–767, April 2012.

    Article  MathSciNet  Google Scholar 

  10. M. Xiang and Z. Xiang, “Finite-time L-1 control for positive switched linear systems with time-varying delay,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 11, pp. 3158–3166, November 2013. [click]

    Article  MathSciNet  MATH  Google Scholar 

  11. X. Liu and J. Lam, “Relationships between asymptotic stability and exponential stability of positive delay systems,” International Journal of General Systems, vol. 42, no. 2, pp. 224–238, October 2013. [click]

    Article  MathSciNet  MATH  Google Scholar 

  12. X. Liu, “Constrained control of positive systems with delays,” IEEE Trans. on Automatic Control, vol. 54, no. 7, pp. 1596–1600, July 2009.

    Article  MathSciNet  Google Scholar 

  13. X. Liu, W. Yu, and L. Wang, “Stability analysis for continuous-time positive systems with time-varying delays,” IEEE Trans. on Automatic Control, vol. 55, no. 4, pp. 1024–1028, April 2010.

    Article  MathSciNet  Google Scholar 

  14. X. Zhao, L. Zhang, and P. Shi, “Stability of a class of switched positive linear time-delay systems,” International Journal of Robust and Nonlinear Control, vol. 23, no. 5, pp. 578–589, March 2013. [click]

    Article  MathSciNet  MATH  Google Scholar 

  15. M. A. Rami and F. Tadeo, “Controller synthesis for positive linear systems with bounded controls,” IEEE Trans. on Circuits and Systems II: Express Briefs, vol. 54, no. 2, pp. 151–155, February 2007.

    Article  Google Scholar 

  16. J. E. Feng, J. Lam, P. Li, and Z. Shu, “Decay rate constrained stabilization of positive systems using static output feedback,” International Journal of Robust and Nonlinear Control, vol. 21, no. 1, pp. 44–54, January 2011. [click]

    Article  MathSciNet  MATH  Google Scholar 

  17. M. A. Rami, F. Tadeo, and U. Helmke, “Positive observers for linear positive systems, and their implications,” International Journal of Control, vol. 84, no. 4, pp. 716–725, June 2011. [click]

    Article  MathSciNet  MATH  Google Scholar 

  18. M. Xiang and Z. Xiang, “Observer design of switched positive systems with time-varying delays,” Circuits, Systems, and Signal Processing, vol. 32, no. 5, pp. 2171–2184, February 2013. [click]

    Article  MathSciNet  Google Scholar 

  19. G. Wang, B. Li, Q. Zhang, and C. Yang, “Positive observer design for discrete-time positive system with missing data in output,” Neurocomputing, vol. 168, pp. 427–434, November 2015. [click]

    Article  Google Scholar 

  20. M. Cai and Z. Xiang, “Observer design for discrete-time switched positive systems with time-varying delays,” Proc. of 1st International Conference on Systems and Computer Science (ICSCS), pp. 1–6, August 2012.

    Google Scholar 

  21. X. Zhao, T. Wu, X. Zheng, and R. Li, “Discussions on observer design of nonlinear positive systems via T-S fuzzy modeling,” Neurocomputing, vol. 157, pp. 70–75, June 2015. [click]

    Article  Google Scholar 

  22. S. Li, Z. R. Xiang, and H. R. Karimi, “Positive L-1 observer design for positive switched systems,” Circuits, Systems, and Signal Processing, vol. 33, no. 7, pp. 2085–2106, July 2014. [click]

    Article  MathSciNet  Google Scholar 

  23. Z. Shu, J. Lam, H. Gao, B. Du, and L. Wu, “Positive observers and dynamic output-feedback controllers for interval positive linear systems,” IEEE Trans. on Circuits and Systems I: Regular Papers, vol. 55, no. 10, pp. 3209–3222, November 2008.

    Article  MathSciNet  Google Scholar 

  24. I. Zaidi, M. Chaabane, F. Tadeo, and A. Benzaouia, “Static state-feedback controller and observer design for interval positive systems with time-delay,” IEEE Trans. on Circuits and Systems II: Express Briefs, vol. 62, no. 5, pp. 506–510, May 2015.

    Article  Google Scholar 

  25. I. Zaidi, F. Tadeo, and M. Chaabane, “Positive observers for continuous time interval positive systems with timedelay,” Proc. of 14th International Conference on Sciences and Techniques of Automatic Control and Computer Engineering (STA), pp. 225–230, December 2013.

    Google Scholar 

  26. C. Wang and Z. Sheng, “Positive observer for positive continuous systems with interval uncertainties and multiple time-delay,” Proc. of 24th Conference on Chinese Control and Decision (CCDC), pp. 2319–2322, May 2012.

    Google Scholar 

  27. P. Li, J. Lam, and Z. Shu, “Positive observers for positive interval linear discrete-time delay systems,” Proc. of 48th Conference on Chinese Control Conference (CDC/CCC), pp. 6107–6112, December 2009.

    Google Scholar 

  28. I. Zaidi, M. Chaabane, and F. Tadeo, “Design of robust observers for positive delayed interval systems,” Proc. of 11th International Multi-Conference on Systems, Signals Devices (SSD), pp. 1–7, February 2014.

    Google Scholar 

  29. P. Li and J. Lam, “Positive state-bounding observer for positive interval continuous-time systems with time delay,” International Journal of Robust and Nonlinear Control, vol. 22, no. 11, pp. 1244–1257, July 2012. [click]

    Article  MathSciNet  MATH  Google Scholar 

  30. J. Tan, C. Li, and T. Huang, “Stability of impulsive systems with time window via comparison method,” International Journal of Control, Automation and Systems, vol. 13, no. 6, pp. 1346–1350, September 2015. [click]

    Article  Google Scholar 

  31. G. Zong and Q. Wang, “Robust resilient control for impulsive switched systems under asynchronous switching,” International Journal of Computer Mathematics, vol. 92, no. 6, pp. 1143–1159, August 2015. [click]

    Article  MathSciNet  MATH  Google Scholar 

  32. X. Li and Z. Xiang, “Observer design of discrete-time impulsive switched nonlinear systems with time-varying delays,” Applied Mathematics and Computation, vol. 229, pp. 327–339, February 2014. [click]

    Article  MathSciNet  Google Scholar 

  33. G. Zheng, Y. Orlov, W. Perruquetti, and J.-P. Richard, “Finite-time-observer design for nonlinear impulsive systems with impact perturbation,” International Journal of Control, vol. 87, no. 10, pp. 2097–2105, April 2014.

    MathSciNet  MATH  Google Scholar 

  34. A. Mahmoudi, A. Momeni, A. G. Aghdam, and P. Gohari, “On observer design for a class of impulsive switched systems,” Proc. of American Control Conference, pp. 4633–4639, June 2008.

    Google Scholar 

  35. Y. W. Wang, J. S. Zhang, and M. Liu, “Exponential stability of impulsive positive systems with mixed time-varying delays,” IET Control Theory and Applications, vol. 8, no. 15, pp. 1537–1542, October 2014.

    Article  MathSciNet  Google Scholar 

  36. J. S. Zhang, Y. W.Wang, J. W. Xiao, and Z.-H. Guan, “Stability Analysis of Impulsive Positive Systems,” Proc. of 19th International Federation of Automatic Control(IFAC), August 2014.

    Google Scholar 

  37. D. Luenberger, Introduction to Dynamic Systems, JohnWiley and Sons, New York, 1979.

    MATH  Google Scholar 

  38. J. Zhou and Q. Wu, “Exponential stability of impulsive delayed linear differential equations,” IEEE Trans. on Circuits and Systems II: Express Briefs, vol. 56, no. 9, pp. 744–748, September 2009.

    Article  Google Scholar 

  39. J. Lu, D. W. Ho, and J. Cao, “A unified synchronization criterion for impulsive dynamical networks,” Automatica, vol. 46, no. 7, pp. 1215–1221, July 2010. [click]

    Article  MathSciNet  MATH  Google Scholar 

  40. S. Zhu, M. Meng, and C. Zhang, “Exponential stability for positive systems with bounded time-varying delays and static output feedback stabilization,” Journal of the Franklin Institute, vol. 350, no. 3, pp. 617–636, April 2013.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Automation, Huazhong University of Science and Technology, Wuhan, 430074, China

    Meng-Jie Hu, Yan-Wu Wang & Jiang-Wen Xiao

  2. Key Laboratory of Image Processing and Intelligent Control, Huazhong University of Science and Technology, Ministry of Education, Wuhan, China

    Meng-Jie Hu, Yan-Wu Wang & Jiang-Wen Xiao

  3. Hubei Provincial Collaborative Innovation Center for New Energy Microgrid, China Three Gorges University, Yichang, China

    Meng-Jie Hu, Yan-Wu Wang & Jiang-Wen Xiao

Authors
  1. Meng-Jie Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yan-Wu Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jiang-Wen Xiao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jiang-Wen Xiao.

Additional information

Recommended by Associate Editor Hamid Reza Karimi under the direction of Editor Ju Hyun Park. This work is supported by the National Natural Science Foundation of China under Grants 61374171, 61572210, and 51537003, the Fundamental Research Funds for the Central Universities (2015TS030).

Meng-Jie Hu received her B.S. degree in School of Automation from Wuhan University of Technology(WUT), Wuhan, China in 2013. Now she is pursuing a M.S. degree in School of Automation from Huazhong University of Science and Technology( HUST), Wuhan, China. Her current research interests include positive systems, hybrid systems and impulsive control.

Yan-Wu Wang received her B.S. degree in automatic control, her M.S. degree and her Ph.D. degree in control theory and control engineering from Huazhong University of Science and Technology (HUST), Wuhan, China, in 1997, 2000, and 2003, respectively. From 2003 to 2005, she worked as a Post-Doctoral Researcher in the Department of Electronics and Information Engineering, HUST, Wuhan, China. From Feb. to April, 2005, she held the visiting position with the School of Electrical and Electronic Engineering in Nanyang Technological University. From Sep. 2008 to Sep. 2009, she held the position of visiting scholar with the Department of Mechanical Engineering in Boston University. Since Nov. 2009, she has been a Professor in the Department of Control Science and Engineering, HUST. Her current research interests include complex networks and hybrid systems et al. Dr. Wang is a recipient of the 2006 Natural Science Award of Ministry of Education Nominated State Science and Technology Award, China, and the 2004 Excellent PhD Dissertation of Hubei Province, China. In 2008, she was awarded the title of "New Century Excellent Talents" by the Chinese Ministry of Education.

Jiang-Wen Xiao received his B.S. degree in electrical engineering, his M.S. degree in control theory and control engineering, and his Ph.D. degree in systems engineering from Huazhong University of Science and Technology (HUST), Wuhan, China, in 1994, 1997, and 2001 respectively. From 2001 to 2003, he worked as a Post-Doctoral Researcher in the College of Management, HUST, Wuhan, China. Since July 2003, he has been an Associate Professor in the Department of Control Science and Engineering, HUST. His current research interests include modeling and control of nonlinear and complex systems.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hu, MJ., Wang, YW. & Xiao, JW. Positive observer design for linear impulsive positive systems with interval uncertainties and time delay. Int. J. Control Autom. Syst. 15, 1032–1039 (2017). https://doi.org/10.1007/s12555-016-0102-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12555-016-0102-y

Keywords

Search

Navigation