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access icon free On robust stability of switched linear systems

In this study, the robust stability of continuous-time switched linear systems is investigated under the assumptions that the matrices of the associated linear subsystems are subjected to affine perturbations. The notion of structured stability radius of a switched linear system which is asymptotically exponentially stable w.r.t. arbitrary switchings is introduced. Some lower bounds and upper bounds for estimating this radius are established, by using the system's common quadratic Lyapunov functions and via an approach based on solutions comparison principle. When the nominal switched system is of special structures (for instance when all matrices of subsystems are normal) the obtained bounds yield easily computable formulas for calculating or estimating the system's stability radius. Several examples are provided to illustrate the authors' approach.

References

    1. 1)
      • 12. Hinrichsen, D., Pritchard, A.J.: ‘Mathematical systems theory I’ (Springer, Berlin, 2005).
    2. 2)
      • 13. Hetel, L. (2007). ‘Robust stability and control of switched linear systems’. PhD thesis, TU Eindhoven.
    3. 3)
      • 27. King, C., Nathanson, M.: ‘On the existence of a common quadratic Lyapunov function for a rank one difference’, Linear Algebra Appl., 2006, 419, pp. 400416.
    4. 4)
      • 4. Shorten, R., Wirth, F., Mason, F., et al: ‘Stability criteria for switched and hybrid systems’, SIAM Rev., 2007, 47, pp. 545592.
    5. 5)
      • 28. Laffey, T.J., Smigoc, H.: ‘Common Lyapunov solutions for two matrices whose difference has rank one’, Linear Algebra Appl., 2009, 431, pp. 228240.
    6. 6)
      • 10. Hinrichsen, D., Pritchard, A.J.: ‘Stability radii of linear systems’, Syst. Control Lett., 1986, 7, pp. 110.
    7. 7)
      • 26. Mason, O., Shorten, R.N.: ‘On the simultaneous diagonal stability of a pair of positive linear systems’, Linear Algebra Appl., 2006, 23, pp. 1323.
    8. 8)
      • 11. Qiu, L., Bernhardsson, B., Rantzer, A., et al: ‘A formula for computation of the real structured stability radius’, Automatica, 1995, 31, pp. 879890.
    9. 9)
      • 7. Mori, Y., Mori, T., Kuroe, Y.: ‘A solution to the common Lyapunov function problem for continuous-time systems’. Proc. of the 36th Conf. on Decision and Control, San Diego, California, 1997, pp. 35303531.
    10. 10)
      • 31. Thuan, D.D., Ngoc, L.V.: ‘Robust stability and robust stabilizability for periodically switched linear systems’, Appl. Math. Comput., 2019, 15, pp. 112130.
    11. 11)
      • 14. Bagherzadeh, M.A., Ghaisari, J., Askari, J.: ‘Robust exponential stability and stabilisation of parametric uncertain switched linear systems under arbitrary switching’, IET Control Theory Appl., 2016, 10, pp. 381390.
    12. 12)
      • 15. Yu, Q., Wu, B.: ‘Robust stability analysis of uncertain switched linear systems with unstable subsystems’, Int. J. Syst. Sci., 2013, 21, pp. 111.
    13. 13)
      • 8. Agrachev, A.A., Liberzon, D.: ‘Lie-algebraic stability criteria for switched systems’, SIAM J. Control Optim., 2001, 40, pp. 253269.
    14. 14)
      • 20. Hinrichsen, D., Son, N.K.: ‘Stability radii of positive discrete-time systems under parameter perturbations’, Int. J. Robust Nonlinear Control, 1998, 4, pp. 11691188.
    15. 15)
      • 17. Horn, R.A., Johnson, C.R.: ‘Matrix analysis’ (Cambridge Unviversity Press, Cambridge, 1985).
    16. 16)
      • 2. Sun, Z., Ge, S.S.: ‘Stability theory of switched dynamical systems’ (Springer-Verlag, London, 2011).
    17. 17)
      • 6. Zhai, G., Xu, X., Lin, H., et al: ‘Analysis and design of switched normal systems’, Nonlinear Anal., Theory Methods Appl., 2006, 12, pp. 22482259.
    18. 18)
      • 18. Shorten, R., Narendra, K.S.: ‘On common quadratic Lyapunov functions for pairs of stable LTI systems whose system matrices are in companion form’, IEEE Trans. Autom. Control, 2003, 48, (4), pp. 618621.
    19. 19)
      • 1. Liberzon, D.: ‘Switching in systems and control’ (Birkhauser, Boston, 2003).
    20. 20)
      • 16. Zhang, L., Shi, P., Basin, M.: ‘Robust stability and stabilisation of uncertain switched linear discrete time-delay systems’, IET Control Theory Appl ., 2007, 2, pp. 606614.
    21. 21)
      • 5. Narendra, K.S., Balakrishnan, J.: ‘A common Lyapunov function for stable LTI systems with commuting A-matrices’, IEEE Trans. Autom. Control, 1994, 39, pp. 24692471.
    22. 22)
      • 3. Lin, H., Antsaklis, P.J.: ‘Stability and stabilizability of switched linear systems: a survey of recent results’, IEEE Trans. Autom. Control, 2009, 54, pp. 308332.
    23. 23)
      • 29. Shorten, R., Narendra, K.S.: ‘Necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for a finite number of stable second order linear time invariant systems’, Inter. J. Adapt. Control Signal Process., 2002, 16, pp. 709728.
    24. 24)
      • 21. Ngoc, P.H.A., Son, N.K.: ‘Stability radii of positive linear functional differential equations under multi-perturbations’, SIAM J. Control Optim., 2005, 43, pp. 22782295.
    25. 25)
      • 25. Patel, R.V., Toda, M.: ‘Quantitative measures of robustness for multivariable systems’. Proc. of Joint Automatic Control Conf., San Francisco, CA, 1980.
    26. 26)
      • 24. Gajic, Z., Tahir, M., Qureshi, J.: ‘Lyapunov matrix equation in system stability and control’ (Academic Press, San Diego, 1995).
    27. 27)
      • 23. Berman, A., Neumann, M., Stern, R.: ‘Nonnegative matrices in dynamic systems’ (John Wiley and Sons, New York, 1989).
    28. 28)
      • 22. Ngoc, P.H.A.: ‘Novel criteria for exponential stability of functional differential equations’, Proc. Am. Math. Soc., 2013, 141, pp. 30833091.
    29. 29)
      • 9. Molchanov, A.P., Pyatnitskii, E.S.: ‘Criteria of asymptotic stability of differential and difference inclusions encountered in control theory’, Syst. Control Lett., 1989, 13, pp. 5964.
    30. 30)
      • 19. Son, N.K., Hinrichsen, D.: ‘Robust stability of positive continuous-time systems’, Numer. Funct. Anal. Optim., 1996, 17, pp. 649659.
    31. 31)
      • 30. Blanchini, F., Colaneri, P., Valcher, M.E.: ‘Switched positive linear systems’, Found. Trends Syst. Control, 2015, 2, pp. 101273.
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