Robust Stability of Differential-Algebraic Equations under Parametric Uncertainty

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription or Fee Access

Abstract

This paper considers linear differential-algebraic equations (DAEs) representing a system of ordinary differential equations with an identically singular matrix at the derivative in the domain of its
definition. The matrix coefficients of DAEs are assumed to depend on the uncertain parameters belonging to a given admissible set. For the parametric family under consideration, structural forms with separate differential and algebraic parts are built. As is demonstrated below, the robust stability of the DAE family is equivalent to the robust stability of its differential subsystem. For the structure of perturbations, sufficient conditions are established under which the separation of DAEs into the algebraic and differential components preserves the original type of functional dependence on the uncertain parameters. Sufficient conditions for robust stability are obtained by constructing a quadratic Lyapunov function.

About the authors

A. A. Shcheglova

Matrosov Institute for System Dynamics and Control Theory, Siberian Branch, Russian Academy of Sciences

Author for correspondence.
Email: shchegl@icc.ru
Irkutsk, Russia

References

  1. Brenan K.E., Campbell S.L., Petzold L.R. Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations (Classics in Applied Mathematics; 14). Philadelphia: SIAM, 1996.
  2. Поляк Б.Т., Щербаков П.С. Робастная устойчивость и управление. М.: Наука, 2002.
  3. Чистяков В.Ф., Щеглова А.А. Избранные главы теории алгебро-дифференциальных систем. Новосибирск: Наука, 2003.
  4. Byers R., Nichols N.K. On the stability radius of a generalized state-space system // Linear Algebra Appl. 1993. No. 188-189. P. 113-134.
  5. Qiu L., Davisov E.J. The stability robustness of generalized eigenvalues // IEEE Trans. Automat. Control. 1992. No. 37. P. 886-891.
  6. Chyan C.J., Du N.H., 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. P. 2078-2102.
  7. Du N.H., Linh V.H. Stability radii for linear time-varying differential-algebraic equations with respect to dynamics perturbations // J. Differ. Equat. 2006. No. 230. P. 579-599.
  8. Fang C.-H., Chang F.-R. Analysis of stability robustness for generalized statespace systems with structured perturbations // Systems Control Lett. 1993. No. 21. P. 109-114.
  9. Lee L., Fang C.-H., Hsieh J.-G. Exact uninderectional perturbation bounds for robustness of uncertain generalized state-space systems: continuous-time cases // Automatica. 1997. No. 33. P. 1923-1927.
  10. De Teran F., Dopico F.M., Moro J. First Order Spectral Perturbation Theory of Square Singular Matrix Pencil // Linear Algebra Appl. 2008. No. 429. P. 548-576.
  11. Linh V.H., Mehrmann V. Lyapunov, Bohl and Sacker-Sell Spectral Intervals for Differential-Algebraic Equations // J. Dyn. Differ. Equat. 2009. V. 21. P. 153-194.
  12. Lin Ch., Lam J., Wang J., Yang G.-H. Analysis on Robust Stability for Interval Descriptor Systems // Syst. Control Lett. 2001. No. 42. P. 267-278.
  13. Berger T. Robustness of Stability of Time-Varying Index-1 DAEs // Math. Control Signals Syst. 2014. No. 26. P. 403-433.
  14. Du N.H., Linh V.H., Mehrmann V. Robust Stability of Differential-Algebraic Equations / Surveys in Differential-Algebraic Equations I. Ilchmann A., Reis T. (Eds.). Berlin-Heidelberg: Springer-Verlag, 2013.
  15. Barbosa K., de Sousa C., Coutinho D. Robust admissibility and H∞ perfomance of time-varying descriptor systems // Proc. 10th Int. Conf. on Control Autom., IEEE, Hangzhou, China. 2013. P. 1138-1143.
  16. Bara G.I. Robust analysis and control of parameter-dependent uncertain descriptor systems // Syst. Control Lett. 2011. V. 60. No. 5. P. 356-364.
  17. Gao L., Chen W., Sun Y. On robust admissibility condition for descriptor systems with convex polytopic uncertsinty // Proc. 2003 Amer. Control Conf., Denver, USA. 2003. V. 6. P. 5083-5088.
  18. Xing S., Zhang Q., Zhu B. Mean-Square Admissibility for Stochastic T-S Fussy Singular Systems Based on Extented Quadratic Lyapunov Function Approach // Fuzzy Sets Syst. 2017. V. 307. P. 99-114.
  19. Chen G., Zheng M., Yang Sh., Li L. Admissibility Analysis of a Sampled-Data Singular System based on the Input Delay Approach // Complexity. 2022. https://doi.org/10.1155/2022/3151620.
  20. Taniguchi T., Tanaka K., Yamafuji K., Wang H.O. Fuzzy Descriptor Systems: Stability Analysis and Design via LMis // Pro. 1999 Amer. Control Conf., IEEE, San Diego, USA. 1999. V. 3. P. 1827-1831.
  21. Dong X.-Z. Admissibility Analysis of Linear Singular Systems via a Delta Operator Method // Int. J. Syst. Sci. 2014. V. 45. P. 2366-2375.
  22. Щеглова А.А., Кононов А.Д. Робастная устойчивость дифференциально алгебраических уравнений произвольного индекса // А и Т. 2017. № 5. C. 36-55.
  23. Щеглова А.А., Кононов А.Д. Устойчивость дифференциально-алгебраических уравнений в условиях неопределенности // Дифференциальные уравнения. 2018. Т. 54. № 7. C. 881-890.
  24. Демидович Б.П. Лекции по математической теории устойчивости. М.: Наука, 1967.
  25. Campbell S.L., Petzold L.R. Canonical Forms and Solvable Singular Systems of differential Equations // SIAM J. Alg. Discrete Methods. 1983. No. 4. P. 517-521.
  26. Гантмахер Ф.Р. Теория матриц. М.: Наука, 1988 (4-е изд.).

Supplementary files

Supplementary Files
Action
1. JATS XML
Download

Copyright (c) 2023 The Russian Academy of Sciences