Skip to main content
SpringerLink
Log in

Explicit solution and stability of linear time-varying differential state space systems

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

Abstract

Linear time-varying (LTV) systems naturally arise when one linearizes nonlinear systems about a trajectory. In contrast the linear time-invariant (LTI) cases which have been thoroughly understood in the analysis and synthesis technologies, many features of the LTV systems are still limited and not clear. This paper addresses the problems of solution and stability of a general unforced LTV differential state space system. Unlike most of the work based on the Lyapunov theory, numerical simulations, or specific constraint systems, the paper proposes the spectral decompositions of the LTV systems by employing extended eigenpairs and with simple mathematical derivation. The spectral decompositions reveal the mechanisms of inherent characterization in general LTV systems, rather than a particular class. Moreover, a novel set of auxiliary equations is developed for guiding and obtaining the extended eigenpairs of its system matrix which completely characterize the LTV systems. The solutions to perform the commutative systems and the second-order systems with companion form are straightforward. The proposed innovative thinking provides a novel guided way to analyze the LTV systems. These findings are easily extended to LTI cases. Examples from the literature demonstrate the effectiveness and the superiority of the proposed approaches when compared with other methods. The proposed results may be of great interest in both for scientific research and application.

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. W. J. Rugh, Linear System Theory, Prentice-Hall, New Jersey, 1993.

    Google Scholar 

  2. H. K. Khalil, Nonlinear Systems, Prentice-Hall, New Jersey, 2002.

    MATH  Google Scholar 

  3. R. H. Middleton and G. C. Goodwin, “Adaptive control of time-varying linear systems,” IEEE Trans. on Automatic Control, vol. 33, no. 2, pp. 150–155, February 1988. [click]

    Article  MathSciNet  MATH  Google Scholar 

  4. W. J. Rugh and J. S. Shamma, “Research on gain scheduling,” Automatica, vol. 36, no. 10, pp. 1401–1425, October 2000. [click]

    Article  MathSciNet  MATH  Google Scholar 

  5. H.-C. Lee and J. W. Choi, “Linear time-varying eigenstructure assignment with flight control application,” IEEE Trans. on Aerospace and Electronic Systems, vol. 40, no. 1, pp. 145–157, January 2004. [click]

    Article  Google Scholar 

  6. B. Lu, H. Choi, G. D. Buckner, and K. Tammi, “Linear parameter-varying techniques for control of a magnetic bearing system,” Control Engineering Practice, vol. 16, no. 10, pp. 1161–1172, October 2008.

    Article  Google Scholar 

  7. W. B. Blair, “Series solution to the general linear time varying system,” IEEE Trans. on Automatic Control, vol. 16, no. 2, pp. 210–211, April 1971.

    Article  MathSciNet  Google Scholar 

  8. B. Zhou, G. B. Cai, and G.-R. Duan, “Stabilization of timevarying linear systems via Lyapunov differential equations,” International Journal of Control, vol. 86, no. 2, pp. 332–347, February 2013.

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Koksal and M. E. Kosal, “Commutativity of linear time-varying differential systems with nonzero initial conditions: A review and some new extensions,” Mathematical Problems in Engineering, vol. 2011, Article ID 678575, 25 pages, doi:10.1155/2011/67857, July 2011.

    Article  MathSciNet  MATH  Google Scholar 

  10. H. W. Mosteller, “Closed-form solutions for a class of linear time-varying systems,” IEEE Trans. on Automatic Control, vol. 15, no. 3, pp. 389–390, January 1970.

    Article  MathSciNet  Google Scholar 

  11. B. Arabzadeh, M. Razzaghi, and Y. Ordokhani, “Numerical solution of linear time varying differential equations using the hybrid of block-pulse and rationalized Haar functions,” Journal of Vibration and Control, vol. 12, no. 10, pp. 1081–1092, October 2006.

    Article  MathSciNet  MATH  Google Scholar 

  12. A. V. Kamyad and M. Mazandarani, “A new approach for solving of linear time varying control systems,” Advanced Studies in Contemporary Mathematics, vol. 21, no. 3, pp. 321–328, July 2011.

    MATH  Google Scholar 

  13. S. Sedaghat and Y. Ordokhani, “Stability and numerical solution of time variant linear systems with delay in both the state and control,” Iranian Journal of Mathematical Sciences and Informatics, vol. 7, no. 1, pp. 43–57, 2012.

    MathSciNet  MATH  Google Scholar 

  14. J. K. Aggarwal and E. F. Infante, “Some remarks on the stability of time-varying systems,” IEEE Trans. on Automatic Control, vol. 13, no. 6, pp. 722–723, December 1968. [click]

    Article  MathSciNet  Google Scholar 

  15. L. Markus and H. Yamabe, “Global stability criteria for differential systems,” Osaka Mathematical Journal, vol. 12, no. 2, pp. 305–317, 1960.

    MathSciNet  MATH  Google Scholar 

  16. C. A. Desoer, “Slowly varying systems x = A(t)x,” IEEE Trans. on Automatic Control, vol. 14, no. 6, pp. 780–781, December 1969. [click]

    Article  MathSciNet  Google Scholar 

  17. J. J. DaCunha, “Stability for time varying linear dynamic systems on time scales,” Journal of Computational and Applied Mathematics, vol. 176, no. 2, pp. 381–410, April 2005. [click]

    Article  MathSciNet  MATH  Google Scholar 

  18. A. Ilchmann, D. H. Owens, and D. Pratzel-Wolters, “Sufficient conditions for stability of linear time-varying systems,” Systems & Control Letters, vol. 9, no. 2, pp. 157–163, August 1987. [click]

    Article  MathSciNet  MATH  Google Scholar 

  19. J. Sun, Q.-G. Wang, and Q.-C. Zhong, “A less conservative stability test for second-order linear time-varying vector differential equations,” International Journal of Control, vol. 80, no. 4, pp. 523–526, April 2007. [click]

    Article  MathSciNet  MATH  Google Scholar 

  20. P. Mullhaupt, D. Buccieri, and D. Bonvin, “A numerical sufficiency test for the asymptotic stability of linear timevarying systems,” Automatica, vol. 43, no. 4, pp. 631–638, April 2007. [click]

    Article  MathSciNet  MATH  Google Scholar 

  21. S.-L. Tung, Y-T Juang, W.-Y. Wu, and W.-Y. Shieh, “An improved stability test and stabilisation of linear timevarying systems governed by second-order vector differential equations,” International Journal of Systems Science, vol. 42, no. 12, pp. 1975–1980, December 2011.

    Article  MathSciNet  MATH  Google Scholar 

  22. Y. Yao, K. Liu, D. Sun, V. Balakrishnan, and J. Guo, “An integral function approach to the exponential stability of linear time-varying systems,” International Journal of Control, Automation, and Systems, vol. 10, no. 6, pp. 1096–1101, December 2012. [click]

    Article  Google Scholar 

  23. O. M. Kwon, M. J. Park, J. H. Park, and S. M. Lee, “Enhancement on stability criteria for linear systems with interval time-varying delays,” International Journal of Control, Automation, and Systems, vol. 14, no. 1, pp. 12–20, February 2016. [click]

    Article  Google Scholar 

  24. P.-L. Liu, “New results on delay-range-dependent stability analysis for interval time-varying delay systems with non-linear perturbations,” ISA Transactions, vol. 57, pp. 93–100, July 2015. [click]

    Article  Google Scholar 

  25. H. Chen and P. Hu, “New result on exponential stability for singular systems with two interval time-varying delays,” IET Control Theory and Applications, vol. 7, no. 15, pp. 1941–1949, October 2013. [click]

    Article  MathSciNet  Google Scholar 

  26. B. Zhou, “On asymptotic stability of linear time-varying systems,” Automatica, vol. 68, pp. 266–276, June 2016. [click]

    Article  MathSciNet  MATH  Google Scholar 

  27. G. Chen and Y. Yang, “New stability conditions for a class of linear time-varying systems,” Automatica, vol. 71, pp. 342–347, September 2016. [click]

    Article  MathSciNet  MATH  Google Scholar 

  28. S. S. Alaviani, “A necessary and sufficient condition for delay-independent stability of linear time-varying neutral delay systems,” Journal of the Franklin Institute, vol. 351, no. 5, pp. 2574–2581, May 2014. [click]

    Article  MathSciNet  Google Scholar 

  29. M.-Y. Wu, “A new concept of eigenvalues and eigenvectors and its applications,” IEEE Trans. on Automatic Control, vol. 25, no. 4, pp. 824–826, August 1980. [click]

    Article  MathSciNet  MATH  Google Scholar 

  30. P. M. Derusso, R. J. Roy, and C. M. Close, State Variables for Engineers, Wiley, New York, 1965.

    Google Scholar 

  31. M.-Y. Wu, “On stability of linear time-varying systems,” International Journal of System Science, vol. 15, no. 2, pp. 137–150, February 1984.

    Article  MathSciNet  MATH  Google Scholar 

  32. G. Garcia, P. L. D. Peres, and S. Tarbouriech, “Assessing asymptotically stability of linear continuous time-varying systems by computing the envelope of all trajectories,” IEEE Trans. on Automatic Control, vol. 55, no. 4, pp. 998–1003, April 2010. [click]

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, St. Johns University, No. 499, Section 4, Tam King Road, Tamsui District, New Taipei City, 25135, Taiwan

    Jing-Min Wan

Authors
  1. Jing-Min Wan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jing-Min Wan.

Additional information

Recommended by Associate Editor Jun Yoneyama under the direction of Editor Duk-Sun Shim. The author would like to thank the Editor and anonymous reviewers for their valuable comments that certainly improved the quality of this paper.

Jing-Min Wang received the M.S. degree in Electrical Engineering from National Taiwan University, Taiwan, R.O.C., in 1981. He is currently an Assistant Professor in the Department of Electrical Engineering, St. John’s University, Taiwan. His research interests include control theory and applications, linear time-varying systems, home automation, optimization, and condition-based maintenance (CBM).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wan, JM. Explicit solution and stability of linear time-varying differential state space systems. Int. J. Control Autom. Syst. 15, 1553–1560 (2017). https://doi.org/10.1007/s12555-015-0404-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12555-015-0404-5

Keywords

Search

Navigation