access icon free Bounded real lemma for positive discrete systems

The bounded real lemma for positive discrete time-invariant systems is addressed in this study. The results about bounded real lemma for continuous time-invariant systems are extended into discrete time-invariant systems. Such systems are asymptotically stable and their H norm less than one is equivalent to that the corresponding linear matrix inequality admits a diagonal positive-definite solution. The last example in this study illustrates the result does not hold for general discrete time-invariant systems.

Inspec keywords: discrete systems; linear matrix inequalities; continuous time systems

Other keywords: continuous time invariant system; bounded real lemma; positive discrete time invariant system; linear matrix inequality

Subjects: Discrete control systems; Algebra

References

    1. 1)
      • 14. Mason, O.: ‘Diagonal Riccati stability and positive time-delay systems’, Syst. Control Lett., 2012, 61, (1), pp. 610 (doi: 10.1016/j.sysconle.2011.09.022).
    2. 2)
      • 12. Virnik, E.: ‘Stability analysis of positive descriptor systems’, Linear Algebr. Appl., 2008, 429, (10), pp. 26402659 (doi: 10.1016/j.laa.2008.03.002).
    3. 3)
      • 24. Xu, S., Lam, J.: ‘Robust Control and Filtering of Singular Systems’ (Spring-Verlag, Berlin, Heidelberg, 2006).
    4. 4)
      • 22. Xu, S., Lam, J., Zou, Y.: ‘New versions of bounded real lemmas for continuous and discrete uncertain systems’, Circuits Syst. Signal Process., 2007, 26, (6), pp. 829838 (doi: 10.1007/s00034-007-9000-0).
    5. 5)
      • 30. Hou, T., Zhang, W., Ma, H.: ‘Finite horizon H2 /H control for discrete-time stochastic systems with Markovian jumps and multiplicative noise’, IEEE Trans. Autom. Control, 2010, 55, (5), pp. 11851191 (doi: 10.1109/TAC.2010.2041987).
    6. 6)
      • 18. Li, P., Lam, J., Shu, Z.: ‘H positive filtering for positive linear discrete-time systems: an augmentation approach’, IEEE Trans. Autom. Control, 2010, 55, (10), pp. 23372342 (doi: 10.1109/TAC.2010.2053471).
    7. 7)
      • 32. Berman, A., Plemmons, R.J.: ‘Nonnegative Matrices in the Mathematical Sciences’ (SIAM, Philadephia, PA, 1994).
    8. 8)
      • 11. Haddad, W.M., Chellaboina, V.: ‘Stability theory for nonnegative and compartmental dynamical systems with time delay’, Syst. Control Lett., 2004, 51, (5), pp. 355361 (doi: 10.1016/j.sysconle.2003.09.006).
    9. 9)
      • 23. Zhang, G., Xia, Y., Shi, P.: ‘New bounded real lemma for discrete-time singular systems’, Automatica, 2008, 44, (3), pp. 886890 (doi: 10.1016/j.automatica.2007.07.017).
    10. 10)
      • 2. van den Hof, J.M.: ‘Realization of continuous-time positive linear systems’, Syst. Control Lett., 1997, 31, (4), pp. 243253 (doi: 10.1016/S0167-6911(97)00049-2).
    11. 11)
      • 29. Berman, N., Shaked, U.: ‘H control for discrete-time nonlinear stochastic systems’. IEEE Conf. Decision and Control2004, pp. 25782583.
    12. 12)
      • 5. Kaczorek, T.: ‘Reachability and controllability of 2D positive linear systems with state feedbacks’, Control Cybern., 2000, 19, (1), pp. 141151.
    13. 13)
      • 31. Tanaka, T., Langbort, C.: ‘The bounded real lemma for internally positive systems and H-Infinity structure static state feedback’, IEEE Trans. Autom. Control, 2011, 56, (9), pp. 22182223 (doi: 10.1109/TAC.2011.2157394).
    14. 14)
      • 33. Barvinok, A.: ‘A Course in Convexity’ (Cambridge University Press, New York, 2003).
    15. 15)
      • 6. Rumchev, V.G., Adeane, J.: ‘Reachability and controllability of time-variant discrete-time positive linear systems’, Control Cybern., 2004, 33, (1), pp. 8594.
    16. 16)
      • 10. Liu, X., Yu, W., Wang, L.: ‘Stability analysis of positive systems with bounded time-varying delays’, IEEE Trans. Circuits Syst. II, 2009, 56, (7), pp. 600604 (doi: 10.1109/TCSII.2009.2023305).
    17. 17)
      • 8. Valcher, M.E., Santesso, P.: ‘On the reachability of single-input positive switched systems’, IEEE Conf. Decision and Control, 2008, pp. 947952.
    18. 18)
      • 20. de Souza, C.E, Xie, L.: ‘On the discrete-time bounded real lemma with application in the characterization of static state-feedback H controllers’, Syst. Control Lett., 1992, 18, (1), pp. 6171 (doi: 10.1016/0167-6911(92)90108-5).
    19. 19)
      • 26. Fridman, E., Shaked, U.: ‘New bounded real lemma representation for time-delay systems and thier application’, IEEE Trans. Autom. Control, 2001, 46, (12), pp. 19731979 (doi: 10.1109/9.975503).
    20. 20)
      • 28. Du, C., Xie, L., Zhang, C.: H control and robust stabilization of two-dimensional systems in Roesser models’, Automatica, 2001, 37, (2), pp. 205211 (doi: 10.1016/S0005-1098(00)00155-2).
    21. 21)
      • 7. Valcher, M.E.: ‘On the reachability properties of continuous-time positive systems’, Proc. 16th Mediterranean Conf. Control and Automation Congress Centre, 2008, pp. 990995.
    22. 22)
      • 19. Boyd, S., Ghaoui, L.E., Feron, E., Balakrishnana, V.: ‘Linear matrix inequalities in system and control theory’ (SIAM, Philadephia, PA, 1994).
    23. 23)
      • 3. Kaczorek, T.: ‘Realization problem for discrete-time positive linear systems’, Appl. Math. Comput. Sci., 1997, 7, (1), pp. 117124.
    24. 24)
      • 25. Chadli, M., Darouach, M.: ‘Novel bounded real lemma for discrete-time descriptor systems: application to H control design’, Automatica, 2012, 48, (2), pp. 449453 (doi: 10.1016/j.automatica.2011.10.003).
    25. 25)
      • 1. Farina, L.: ‘On the existence of a positive realization’, Syst. Control Lett., 1996, 28, (4), pp. 219226 (doi: 10.1016/0167-6911(96)00033-3).
    26. 26)
      • 16. Liu, X.: ‘Constrained control of positive systems with delays’, IEEE Trans. Autom. Control, 2009, 54, (7), pp. 15961600 (doi: 10.1109/TAC.2009.2017961).
    27. 27)
      • 13. Liu, X., Yu, W., Wang, L.: ‘Stability analysis for continuous-time positive systems with time-varying delays’, IEEE Trans. Autom. Control, 2010, 55, (4), pp. 10241028 (doi: 10.1109/TAC.2010.2041982).
    28. 28)
      • 21. Shaked, U., Suplin, V.: ‘A new bounded real lemma representation for the continuous-time case’, IEEE Trans. Autom. Control, 2001, 46, (9), pp. 14201426 (doi: 10.1109/9.948470).
    29. 29)
      • 15. Rami, M.A., Helmke, U., Tadeo, F.: ‘Positive observation problem for linear time-delay positive systems’, Proc. 15th Mediterranean Conf. Control and Automation, 2007, pp. 16.
    30. 30)
      • 27. Shaked, U., Yaesh, I., de Souza, C.I.: ‘Bounded real criteria for linear time-delay systems’, IEEE Trans. Autom. Control, 1998, 43, (7), pp. 10161022 (doi: 10.1109/9.701117).
    31. 31)
      • 9. Farina, L., Rinaldi, S.: ‘Positive linear systems: theory and applications’ (Wiley-Interscience, New York, 2000).
    32. 32)
      • 17. Li, P., Lam, J., Wang, Z.: ‘H model reduction for positive systems’, American Control Conf., 2010, pp. 62446249.
    33. 33)
      • 4. Kaczorek, T.: ‘Realization problem for positive multivariable discrete-time linear systems with delays in the state vector and inputs’, Int. J. Appl. Math. Comput. Sci., 2006, 16, (2), pp. 169174.
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