Kybernetika 58 no. 1, 1-24, 2022

Growth conditions for the stability of a class of time-varying perturbed singular systems

Faten Ezzine and Mohamed Ali HammamiDOI: 10.14736/kyb-2022-1-0001

Abstract:

In this paper, we investigate the problem of stability of linear time-varying singular systems, which are transferable into a standard canonical form. Sufficient conditions on exponential stability and practical exponential stability of solutions of linear perturbed singular systems are obtained based on generalized Gronwall inequalities and Lyapunov techniques. Moreover, we study the problem of stability and stabilization for some classes of singular systems. Finally, we present a numerical example to validate the effectiveness of the abstract results of this paper.

Keywords:

linear time-varying singular systems, standard canonical form, consistent initial conditions, Gronwall inequalities, Lyapunov techniques, practical exponential stability

Classification:

37B55, 34D20

References:

  1. B. Ben Hamed, I. Ellouze and M. A. Hammami: Practical uniform stability of nonlinear differential delay equations. Mediterranean J. Math. 8 (2011), 603-616.   DOI:10.1007/s00009-010-0083-7
  2. A. Ben Abdallah, I. Ellouze and M. A. Hammami: Practical stability of nonlinear time-varying cascade systems. J. Dynamic. Control Systems 15 (2009), 45-62.   DOI:10.1007/s10883-008-9057-5
  3. R. Bellman: The stability of solutions of linear differential equations. Duke Math. J. 10 (1943), 643-647.   CrossRef
  4. D. Bainov and P. Simenov: Integral Inequalities and Applications. Springer, Kluwer Academic Publishers, Congress, Dordrecht 1992.   CrossRef
  5. T. Berger: Bohl exponent for time-varying linear differential-algebraic equations. Int. J. Control 10 (2012), 1433-1451.   DOI:10.1080/00207179.2012.688872
  6. T. Berger and A. Ilchmann: On the standard canonical form of time-varying linear DAEs. Quarterly Appl. Math. 71 (2013), 69-87.   DOI:10.1090/S0033-569X-2012-01285-1
  7. T. Berger and A. Ilchmann: On stability of time-varying linear differential-algebraic equations. Int. J. Control 86 (2013), 1060-1076.   DOI:10.1080/00207179.2013.773087
  8. L. Campbell: Singular Systems of Differential Equations. Pitman Advanced Publishing Program, London 1980.   CrossRef
  9. S. L. Campbell: Singular Systems of Differential Equations II. Pitman Advanced Publishing Program, London 1982.   CrossRef
  10. T. Caraballo, F. Ezzine and M. A. Hammami: On the exponential stability of stochastic perturbed singular systems in mean square. Appl. Math. Optim. 84 (2021), 2923-2945.   DOI:10.1007/s00245-020-09734-8
  11. T. Caraballo, F. Ezzine, M. Hammami and L. Mchiri: Practical stability with respect to a part of variables of stochastic differential equations. Stochastics Int. J. Probab. Stoch. Process. 5 (2021), 647-664.   DOI:10.1080/17442508.2020.1773826
  12. T. Caraballo, F. Ezzine and M. Hammami: Partial stability analysis of stochastic differential equations with a general decay rate. J. Engrg. Math. 130 (2021), 1-17.   DOI:10.1007/s10665-021-10164
  13. L. Dai: Singular Control Systems. Springer-Verlag, Berlin 1989.   CrossRef
  14. D. Lj. Debeljkovic, B. Jovanovic and V. Drakulic: Singular system theory in chemical engineering theory: Stability in the sense of Lyapunov: A survey. Hemijska Industrija 6(2001), 260-272.   CrossRef
  15. S. S. Dragomir: Some Gronwall Type Inequalities and Applications. Nova Science Publishers, Hauppauge 2003.   CrossRef
  16. T. H. Gronwall: Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann. Math. 20 (1919), 293-296.   DOI:10.1007/BF02342482
  17. P. Kunkel and V. Mehrmann: Differential-Algebraic Equations Analysis and Numerical Solution. EMS Publishing House, Zurich 2006.   CrossRef
  18. D. G. Luenberger: Dynamic equation in descriptor form. IEEE Trans. Automat. Control 22 (1977), 310-319.   CrossRef
  19. D. G. Luenberger: Time-invariant descriptor systems. Automatica 14 (1978), 473-480.   DOI:10.1016/0005-1098(78)90006-7
  20. D. H. Ownes and D. Lj. Debeljkovic: Consistency and Liapunov stability of linear descriptor systems: A geometric analysis. IMA J. Math. Control Inform. 2 (1985), 139-151.   DOI:10.1093/imamci/2.2.139
  21. Q. C. Pham, N. Tabareau and J. E. Slotine: A contraction theory approach to stochastic Incremental stability. IEEE Trans. Automat. Control 54 (2009), 1285-1290.   DOI:10.1109/TAC.2009.2013055
  22. H. H. Rosenbrock: Structure properties of linear dynamical systems. Int. J. Control 20 (1974), 191-202.   DOI:10.1080/00207177408932729
  23. J. F. Ritt: Systems of algebraic differential equations. Ann. Math. 36 (1935), 293-302.   CrossRef
  24. R. Vrabel: Local null controllability of the control-affine nonlinear systems with time-varying disturbances, Direct calculation of the null controllable region. Europ. J. Control 40 (2018), 80-86.   DOI:10.1016/j.ejcon.2017.12.004