Skip to main content
SpringerLink
Log in

Robust stability of linear time-varying implicit dynamic equations: a general consideration

  • Original Article
  • Published:
Mathematics of Control, Signals, and Systems Aims and scope Submit manuscript

Abstract

In this paper, the problem of robust stability for linear time-varying implicit dynamic equations is generally studied. We consider the effect of uncertain structured perturbations on all coefficient matrices of equations. A formula of stability radius with respect to dynamic structured perturbations acting on the right-hand side coefficients is obtained. In case where structured perturbations affect on both derivative and the right-hand side, the lower bounds for the stability radius are derived. The results are novel and extend many previous results about robust stability for time-varying ordinary differential/difference equations, time-varying differential algebraic equations and time-varying implicit difference equations.

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. Arendt W, Batty CJK, Hieber M, Neubrander F (2011) Vector-valued Laplace transforms and Cauchy problems. Springer, Berlin

    Book  MATH  Google Scholar 

  2. Bartosiewicz Z (2013) Linear positive control systems on time scales. Math Control Signals Syst 25:327–343

    Article  MathSciNet  MATH  Google Scholar 

  3. Berger T (2014) Robustness of stability of time-varying index-1 IDEs. Math Control Signals Syst 26:403–433

    Article  MATH  Google Scholar 

  4. Bohner M, Peterson A (2001) Dynamic equations on time scales: an introduction with applications. Birkhäuser, Boston

    Book  MATH  Google Scholar 

  5. Bracke M (2000) On stability radii of parametrized linear differential-algebraic systems. Ph.D. thesis, University of Kaiserslautern

  6. DaCunha JJ, Davis JM (2011) A unified Floquet theory for discrete, continuous, and hybrid periodic linear systems. J Differ Equ 251:2987–3027

    Article  MathSciNet  MATH  Google Scholar 

  7. Doan TS, Kalauch A, Siegmund S, Wirth F (2010) Stability radii for positive linear time-invariant systems on time scales. Syst Control Lett 59:173–179

    Article  MathSciNet  MATH  Google Scholar 

  8. Du NH, Duy TK, Viet VT (2007) Degenerate cocycle with index-1 and Lyapunov exponent. Stoch Dyn 7(2):229–245

    Article  MathSciNet  MATH  Google Scholar 

  9. Du NH, Linh VH (2006) Stability radii for linear time-varying differential–algebraic equations with respect to dynamic perturbations. J Differ Equ 230:579–599

    Article  MathSciNet  MATH  Google Scholar 

  10. Du NH, Thuan DD, Liem NC (2011) Stability radius of implicit dynamic equations with constant coefficients on time scales. Syst Control Lett 60:596–603

    Article  MathSciNet  MATH  Google Scholar 

  11. Du NH, Linh VH, Mehrmann V, Thuan DD (2013) Stability and robust stability of linear time-invariant delay differential-algebraic equations. SIAM J Matrix Anal Appl 34:1631–1654

    Article  MathSciNet  MATH  Google Scholar 

  12. Griepentrog E, März R (1986) Differential-algebraic equations and their numerical treatment. Teubner-Texte zur Mathematik, Leibzig

    MATH  Google Scholar 

  13. Guseinov GSh (2003) Integration on time scales. J Math Anal Appl 285:107–127

    Article  MathSciNet  MATH  Google Scholar 

  14. Ha NT, Du NH, Thuan DD (2016) On data dependence of stability domains, exponential stability and stability radii for implicit linear dynamic equations. Math Control Signals Syst 28(2):1–28

    Article  MathSciNet  MATH  Google Scholar 

  15. Hinrichsen D, Ilchmann A, Pritchard AJ (1989) Robustness of stability of time-varying linear systems. J Differ Equ 82:219–250

    Article  MathSciNet  MATH  Google Scholar 

  16. Hinrichsen D, Ilchmann A, Pritchard AJ (1992) A stability radius for time-varying linear systems. In: Nichols NK, Owens DH (eds) The mathematics of control theory. Oxford University Press, Oxford, pp 281–290

    Google Scholar 

  17. Hinrichsen D, Pritchard AJ (1986) Stability radii of linear systems. Syst Control Lett 7:1–10

    Article  MathSciNet  MATH  Google Scholar 

  18. Hinrichsen D, Pritchard AJ (1986) Stability for structured perturbations and the algebraic Riccati equation. Syst Control Lett 8:105–113

    Article  MathSciNet  MATH  Google Scholar 

  19. Hinrichsen D, Son NK, Ngoc PHA (2003) Stability radii of higher order positive difference systems. Syst Control Lett 49:377–388

    Article  MathSciNet  MATH  Google Scholar 

  20. Jacob B (1998) A formula for the stability radius of time-varying systems. J Differ Equ 142:167–187

    Article  MathSciNet  MATH  Google Scholar 

  21. Kunkel P, Mehrmann V (2006) Differential-algebraic equations, analysis and numerical solution. EMS Publishing House, Zürich

    Book  MATH  Google Scholar 

  22. Lamour R, März R, Tischendorf C (2013) Differential-algebraic equations: a projector based analysis. Springer, Berlin

    Book  MATH  Google Scholar 

  23. Linh VH, Nga NTT, Thuan DD (2018) Exponential stability and robust stability for linear time-varying singular systems of second-order difference equations. SIAM J Matrix Anal Appl 39:204–233

    Article  MathSciNet  MATH  Google Scholar 

  24. Luenberger DG (1977) Dynamic equations in descriptor form. IEEE Trans Autom Control 22:312–322

    Article  MathSciNet  MATH  Google Scholar 

  25. Luenberger DG (1986) Control of linear dynamic market systems. J Econ Dyn Control 10:339–351

    Article  MathSciNet  MATH  Google Scholar 

  26. Marks RJ II, Gravagne IA, Davis JM (2008) A generalized Fourier transform and convolution on time scales. J Math Anal Appl 340:901–919

    Article  MathSciNet  MATH  Google Scholar 

  27. März R (1992) Numerical methods for differential-algebraic equations. Acta Numer 1:141–198

    Article  MathSciNet  MATH  Google Scholar 

  28. Mehrmann V, Thuan DD (2015) Stability analysis of implicit difference equations under restricted perturbations. SIAM J Matrix Anal Appl 36:178–202

    Article  MathSciNet  MATH  Google Scholar 

  29. Qiu L, Davison EJ (1992) The stability robustness of generalized eigenvalues. IEEE Trans Autom Control 37:886–891

    Article  MathSciNet  MATH  Google Scholar 

  30. Rodjanadid B, Sanh NV, Ha NT, Du NH (2009) Stability radii for implicit difference equations. Asian-Eur J Math 2:95–115

    Article  MathSciNet  MATH  Google Scholar 

  31. Srivastava SM (1998) A course on Borel sets. Springer, Berlin

    Book  MATH  Google Scholar 

  32. Taousser FZ, Defoort M, Djemai M (2014) Stability analysis of a class of switched linear systems on non-uniform time domains. Syst Control Lett 74:24–31

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The first and second authors were supported by NAFOSTED project 101.01-2017.302. The third and fourth authors were supported by NAFOSTED project 101.03-2017.308. A part of this work was done when D.D. Thuan was working at the Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to thank the VIASM for providing a fruitful research environment and hospitality. The authors also gratefully thank the reviewers for useful comments that led to the improvements of the paper.

Author information

Authors and Affiliations

  1. School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, 1 Dai Co Viet Str., Hanoi, Vietnam

    Do Duc Thuan

  2. Department of Basic Science, Tan Trao University, Tuyen Quang, Vietnam

    Khong Chi Nguyen

  3. Department of Mathematics, Hanoi Pedagogical University 2, Xuan Hoa, Phuc Yen, Vinh Phuc, Vietnam

    Khong Chi Nguyen

  4. Department of Basic Science, Electric Power University, 235 Hoang Quoc Viet Str., Hanoi, Vietnam

    Nguyen Thu Ha

  5. Department of Mathematics, Mechanics and Informatics, Vietnam National University, 334 Nguyen Trai Str., Hanoi, Vietnam

    Nguyen Huu Du

Authors
  1. Do Duc Thuan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Khong Chi Nguyen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nguyen Thu Ha
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Nguyen Huu Du
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Do Duc Thuan.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Thuan, D.D., Nguyen, K.C., Ha, N.T. et al. Robust stability of linear time-varying implicit dynamic equations: a general consideration. Math. Control Signals Syst. 31, 385–413 (2019). https://doi.org/10.1007/s00498-019-0242-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00498-019-0242-8

Keywords

Search

Navigation