Skip to main content
SpringerLink
Log in

On the Superstability of an Interval Family of Differential-Algebraic Equations

  • LINEAR SYSTEMS
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

Abstract

We consider an interval family of differential-algebraic equations (DAE) under assumptions that guarantee the coincidence of the structure of the general solution of each of the systems in this family with the structure of the general solution of the nominal system. The analysis is based on transforming the interval family of DAE to a form in which the differential and algebraic parts are separated. This transformation includes the inversion of an interval matrix. An estimate for the stability radius is found assuming the superstability of the differential subsystem of nominal DAE. Sufficient conditions for the robust stability are obtained based on the superstability condition for the differential part of the interval family.

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. Polyak, B.T. and Shcherbakov, P.S., Robastnaya ustoichivost’ i upravlenie (Robust Stability and Control), Moscow: Nauka, 2002.

    Google Scholar 

  2. Byers, R. and Nichols, N.K., On the stability radius of a generalized state-space system, Linear Algebra Appl., 1993, no. 188–189, pp. 113–134.

  3. Qiu, L. and Davisov, E.J., The stability robustness of generalized eigenvalues, IEEE Trans. Autom. Control, 1992, no. 37, pp. 886–891.

  4. Chyan, C.J., Du, N.H., and Linh, V.H., On data-dependence of exponential stability and the stability radii for linear time-varying differential-algebraic systems, J. Differ. Equat., 2008, no. 245, pp. 2078–2102.

  5. Du, N.H. and Linh, V.H., Stability radii for linear time-varying differential-algebraic equations with respect to dynamics perturbations, J. Differ. Equat., 2006, no. 230, pp. 579–599.

  6. Fang, C.-H. and Chang, F.-R., Analysis of stability robustness for generalized state-space systems with structured perturbations, Syst. Control Lett., 1993, no. 21, pp. 109–114.

  7. Lee, L., Fang, C.-H., and Hsieh, J.-G., Exact unidirectional perturbation bounds for robustness of uncertain generalized state-space systems: continuous-time cases, Automatica, 1997, no. 33, pp. 1923–1927.

  8. De Teran, F., Dopico, F.M., and Moro, J., First order spectral perturbation theory of square singular matrix pencil, Linear Algebra Appl., 2008, no. 429, pp. 548–576.

  9. Linh, V.H. and Mehrmann, V., Lyapunov, Bohl and Sacker-Sell spectral intervals for differential-algebraic equations, J. Dyn. Differ. Equat., 2009, vol. 21, pp. 153–194.

    Article  MathSciNet  Google Scholar 

  10. Lin, Ch., Lam, J., Wang, J., and Yang, G.-H., Analysis on Robust Stability for Interval Descriptor Systems, Syst. Control Lett., 2001, no. 42, pp. 267–278.

  11. Berger, T., Robustness of stability of time-varying index-1 DAEs, Math. Control Signals Syst., 2014, no. 26, pp. 403–433.

  12. Benner, P., Sima, V., and Voigt, M., \(L_{\infty } \)-norm computation for continuous-time descriptor systems using structured matrix pencils, IEEE Trans. Autom. Control, 2010, vol. 57, no. 1, pp. 233–238.

    Article  MathSciNet  Google Scholar 

  13. Du, N.H., Linh, V.H., and Mehrmann, V., Robust Stability of Differential-Algebraic Equations. Surveys in Differential-Algebraic Equations I, Ilchmann, A. and Reis, T., Eds., Berlin–Heidelberg: Springer-Verlag, 2013.

    MATH  Google Scholar 

  14. Shcheglova, A.A. and Kononov, A.D., Robust stability of differential-algebraic equations with an arbitrary unsolvability index, Autom. Remote Control, 2017, vol. 78, pp. 798–814.

    Article  MathSciNet  Google Scholar 

  15. Shcheglova, A.A. and Kononov, A.D., Stability of differential-algebraic equations under uncertainty, Differ. Equations, 2018, vol. 54, no. 7, pp. 860–869.

    Article  MathSciNet  Google Scholar 

  16. Lancaster, P., Theory of Matrices, New York–London: Academic Press, 1969. Translated under the title: Teoriya matrits, Moscow: Nauka, 1978.

    MATH  Google Scholar 

  17. Neumaier, A., Interval Methods for Systems of Equations, Cambrige: Cambrige Univ. Press, 1990.

    MATH  Google Scholar 

  18. Polyak, B.T. and Shcherbakov, P.S., Superstable linear control systems I. Analysis, Autom. Remote Control, 2002, vol. 63, no. 8, pp. 1239–1254.

    Article  MathSciNet  Google Scholar 

  19. Polyak, B.T. and Shcherbakov, P.S., Superstable linear control systems II. Design, Autom. Remote Control, 2002, vol. 63, no. 11, pp. 1745–1763.

    Article  MathSciNet  Google Scholar 

  20. Chistyakov, V.F. and Shcheglova, A.A., Izbrannye glavy teorii algebro-differentsial’nykh sistem (Selected Chapters from the Theory of Differential-Algebraic Systems), Novosibirsk: Nauka, 2003.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Matrosov Institute for System Dynamics and Control Theory, Siberian Branch, Russian Academy of Sciences, Irkutsk, 664033, Russia

    A. A. Shcheglova

Authors
  1. A. A. Shcheglova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. A. Shcheglova.

Additional information

Translated by V. Potapchouck

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shcheglova, A.A. On the Superstability of an Interval Family of Differential-Algebraic Equations. Autom Remote Control 82, 232–244 (2021). https://doi.org/10.1134/S0005117921020041

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117921020041

Keywords

Search

Navigation