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access icon free On stability of non-linear and switched parabolic systems

This study investigates the stability of non-linear parabolic systems. Several stability conditions based on separable Lyapunov functions are provided. The new results are applied to switched non-linear parabolic systems with all stable modes or with unstable modes. An example of heat propagation control of semiconductor power chips is taken to illustrate the proposed methods.

References

    1. 1)
      • 14. Wu, L., Ho, D.W.C., Li, C.W.: ‘Sliding mode control of switched hybrid systems with stochastic perturbation’, Syst. Control Lett., 2011, 60, (8), pp. 531539 (doi: 10.1016/j.sysconle.2011.04.007).
    2. 2)
      • 16. Xiang, Z.R., Wang, R.H.: ‘Robust control for uncertain switched non-linear systems with time delay under asynchronous switching’, IET Control Theory Applic., 2009, 3, (8), pp. 10411050 (doi: 10.1049/iet-cta.2008.0150).
    3. 3)
      • 2. Vazquez, R., Krstic, M.: ‘Control of 1-D parabolic PDEs with Volterra nonlinearities ¨ Part I: Design’, Automatica, 2008, 44, (11), pp. 27782790 (doi: 10.1016/j.automatica.2008.04.013).
    4. 4)
      • 1. Smyshlyaev, A., Krstic, M.: ‘Adaptive control of parabolic PDEs’ (Princeton University Press, 2010).
    5. 5)
      • 11. Wang, M., Zhao, J., Dimirovski, G.M.: ‘Variable structure control method to the output tracking control of cascade non-linear switched systems’, IET Control Theory Applic., 2009, 3, (12), pp. 16341640 (doi: 10.1049/iet-cta.2008.0436).
    6. 6)
      • 19. Zuazua, E.: ‘Switching control’, J. Eur. Math. Soc., 2011, 13, (1), pp. 85117 (doi: 10.4171/JEMS/245).
    7. 7)
      • 4. Fitzgibbon, W.B., Hollis, S.L., Morgan, J.J.: ‘Stability and Lyapunov functions for reaction-diffusion systems’, SIAM J. Math. Anal., 1997, 28, (3), pp. 595610 (doi: 10.1137/S0036141094272241).
    8. 8)
      • 13. Wu, L., Zheng, W.X.: ‘Weighted H model reduction for linear switched systems with time-varying delay’, Automatica, 2009, 45, (1), pp. 186193 (doi: 10.1016/j.automatica.2008.06.024).
    9. 9)
      • 18. Iftime, O.V., Demetriou, M.A.: ‘Optimal control of switched distributed parameter systems with spatially scheduled actuators’, Automatica, 2009, 45, (2), pp. 312323 (doi: 10.1016/j.automatica.2008.07.012).
    10. 10)
      • 25. Mason, O., Shorten, R.: ‘Some results on the stability of positive switched linear systems’. Proc. 43rd IEEE Conf. Decision and Control, Atlantis, Bahamas, 2004, pp. 46014606.
    11. 11)
      • 30. Chung, C., Yang, C.: ‘An autocalibrated all-digital temperature sensor for on-chip thermal monitoring’, IEEE Trans. Circuits Syst. II: Express Briefs, 2011, 58, (2), pp. 105109 (doi: 10.1109/TCSII.2010.2104016).
    12. 12)
      • 27. Mancilla-Aguilar, J.L., Garcia, R.A.: ‘On converse Lyapunov theorems for ISS and iISS switched nonlinear systems’, Syst. Control Lett., 2001, 42, (1), pp. 4753 (doi: 10.1016/S0167-6911(00)00079-7).
    13. 13)
      • 8. Liberzon, D.: ‘Switching in systems and control’ (Birkhauser, Boston, MA, 2003).
    14. 14)
      • 28. Yang, H., Jiang, B., Cocquempot, V.: ‘Observer based fault tolerant control for a class of hybrid impulsive systems’, Int. J. Robust Nonlinear Control, 2010, 20, (4), pp. 448459.
    15. 15)
      • 23. Rothe, F.: ‘Global solutions of reaction – diffusion systems’ (Springer-Verlag, Berlin, Germany, 1984).
    16. 16)
      • 24. Sontag, E., Wang, Y.: ‘New characterizations of input-to-state stability’, IEEE Trans. Autom. Control, 1996, 41, (9), pp. 12831294 (doi: 10.1109/9.536498).
    17. 17)
      • 15. Wu, L., Zheng, W.X., Gao, H.: ‘Dissipativity based sliding mode control of switched stochastic systems’, IEEE Trans. Autom. Control, 2013, 58, (3), pp. 785791 (doi: 10.1109/TAC.2012.2211456).
    18. 18)
      • 17. El-Farra, N.H., Christofides, P.D.: ‘Coordinating feedback and switching for control of spatially distributed processes’, Comput. Chem. Eng., 2004, 28, (1), pp. 111128 (doi: 10.1016/S0098-1354(03)00174-1).
    19. 19)
      • 7. Jovanovic, M.R., Arcak, M., Sontag, E.D.: ‘A passivity-based approach to stability of spatially distributed systems with a cyclic interconnection structure’, IEEE Trans. Autom. Control, 2008, 53, (1), pp. 7586 (doi: 10.1109/TAC.2007.911318).
    20. 20)
      • 5. Fridman, E., Orlov, Y.: ‘An LMI approach to boundary control of semilinear parabolic and hyperbolic systems’, Automatica, 2009, 45, (9), pp. 20602066 (doi: 10.1016/j.automatica.2009.04.026).
    21. 21)
      • 21. Sasane, A.: ‘Stability of switching infinite-dimensional systems’, Automatica, 2005, 41, (1), pp. 75C78.
    22. 22)
      • 20. Maj, B., Augustin, A., Kostka, A.: ‘Heat propagation in H-?bridge smart power chips under switching conditions’, Microelectron. J., 2002, 33, (9), pp. 727731 (doi: 10.1016/S0026-2692(02)00056-3).
    23. 23)
      • 12. Wu, L., Qi, T., Feng, Z.: ‘Average dwell time approach to L2 - L control of switched delay systems via dynamic output feedback’, IET Control Theory Applic., 2009, 3, (10), pp. 14251436 (doi: 10.1049/iet-cta.2008.0315).
    24. 24)
      • 22. Ouzahra, M.: ‘Global stabilization of semilinear systems using switching controls’, Automatica, 2012, 48, (5), pp. 837843 (doi: 10.1016/j.automatica.2012.02.018).
    25. 25)
      • 26. Kaszkurewicz, E., Bhaya, A.: ‘Matrix diagonal stability in systems and computation’ (Birkhauser, 1999).
    26. 26)
      • 10. Zhao, J., Hill, D.J.: ‘On stability, L2 gain and H control for switched systems’, Automatica, 2008, 44, (5), pp. 12201232 (doi: 10.1016/j.automatica.2007.10.011).
    27. 27)
      • 29. Yang, H., Jiang, B., Cocquempot, V., Zhang, H.: ‘Stabilization of switched nonlinear systems with all unstable modes: applications to multi-agent systems’, IEEE Trans. Autom. Control, 2011, 56, (9), pp. 22302235 (doi: 10.1109/TAC.2011.2157413).
    28. 28)
      • 3. Mazenc, F., Prieur, C.: ‘Strict Lyapunov functions for semilinear parabolic partial differential equations’, Math. Control Relat. Fields, 2011, 1, (2), pp. 231250 (doi: 10.3934/mcrf.2011.1.231).
    29. 29)
      • 9. Branicky, M.S.: ‘Multiple Lyapunov functions and other analysis tools for switched and hybrid systems’, IEEE Trans. Autom. Control, 1998, 43, (1), pp. 475482 (doi: 10.1109/9.664150).
    30. 30)
      • 6. Argomedo, F.B., Prieur, C., Witrant, E., Bremond, S.: ‘A strict control Lyapunov function for a diffusion equation with time-varying distributed coefficients’, IEEE Trans. Autom. Control, 2013, 58, (2), pp. 290303 (doi: 10.1109/TAC.2012.2209260).
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