access icon free Robust stabilisation for a class of stochastic two-dimensional non-linear systems with time-varying delays

This study addresses the robust stabilisation problem for the two-dimensional (2D) discrete non-linear stochastic systems with time-varying delays in states. Based on the well-known Fornasiniā€“Marchesini local state-space model, a more general 2D discrete non-linear system is considered with both stochastic disturbances and time-varying delays. Our attention is focused on the design of a delay-dependent state-feedback controller so that the 2D stochastic non-linear system is guaranteed to be globally asymptotically stabilisable in the mean square. By using the stochastic analysis and some inequality techniques, sufficient criteria ensuring the existence of such controllers are derived in terms of matrix inequalities, which can be effectively solved by resorting to some standard convex optimisation algorithms. Furthermore, such stabilisation results are extended to more general cases where the system matrices contain either polytopic or norm-bounded parameter uncertainties. Finally, a numerical example is provided to demonstrate the applicability of the proposed controller design approach.

Inspec keywords: time-varying systems; delays; asymptotic stability; control system synthesis; mean square error methods; robust control; convex programming; state feedback; nonlinear control systems; discrete systems

Other keywords: convex optimisation algorithms; 2D stochastic nonlinear system; 2D discrete nonlinear stochastic systems; matrix inequalities; stochastic two dimensional nonlinear systems; time-varying delays; mean square error methods; robust stabilisation problem; Fornasini-Marchesini local state-space model; inequality techniques; stochastic disturbances; delay dependent state feedback controller; stochastic analysis

Subjects: Stability in control theory; Time-varying control systems; Nonlinear control systems; Distributed parameter control systems; Control system analysis and synthesis methods; Discrete control systems; Interpolation and function approximation (numerical analysis); Optimisation techniques

References

    1. 1)
      • 19. Wu, L., Lam, J., Wang, C.: ‘Robust H dynamic output feedback control for 2D linear parameter-varying systems’, IMA J. Math. Control Inf., 2009, 26, (1), pp. 2344 (doi: 10.1093/imamci/dnm028).
    2. 2)
      • 2. Liu, D., Michel, A.N.: ‘Stability analysis of state-space realizations for two-dimensional filters with overflow nonlinearities’, IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., 1994, 41, (2), pp. 127137 (doi: 10.1109/81.269049).
    3. 3)
      • 16. Du, C., Xie, L., Soh, Y.: ‘H filtering of 2-D discrete systems’, IEEE Trans. Signal Process., 2000, 48, (6), pp. 17601768 (doi: 10.1109/78.845933).
    4. 4)
      • 28. El Ghaoui, L., Oustry, F., AitRami, M.: ‘A cone complementarity linearization algorithm for static output-feedback and related problems’, IEEE Trans. Autom. Control, 1997, 42, (8), pp. 11711176 (doi: 10.1109/9.618250).
    5. 5)
      • 25. Liu, D., Michel, A.N.: ‘Asymptotic stability of discrete-time systems with saturation nonlinearities to digital filters’, IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., 1992, 39, (10), pp. 798807 (doi: 10.1109/81.199861).
    6. 6)
      • 27. Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: ‘Linear matrix inequalities in system and control theory’ (SIAM, Philadelphia, 1994).
    7. 7)
      • 21. Gao, H., Lam, J., Wang, C., Xu, S.: ‘H model reduction for uncertain two-dimensional discrete systems’, Opt. Control Appl. Methods, 2005, 26, (4), pp. 199227 (doi: 10.1002/oca.760).
    8. 8)
      • 7. Gao, H., Lam, J., Xu, S., Wang, C.: ‘Stability and stabilization of uncertain 2-D discrete systems with stochastic perturbation’, Multidimens. Syst. Signal Process., 2005, 16, pp. 85106 (doi: 10.1007/s11045-004-4739-y).
    9. 9)
      • 15. Chen, S.-F., Fong, I.-K.: ‘Delay-dependent robust H filtering for uncertain 2-D state-delayed systems’, Signal Process., 2007, 87, (11), pp. 26592672 (doi: 10.1016/j.sigpro.2007.04.015).
    10. 10)
      • 29. Leibfritz, F.: ‘An LMI-based algorithm for designing suboptimal static H2/H output feedback controllers’, SIAM J. Control Opt., 2001, 39, (6), pp. 17111735 (doi: 10.1137/S0363012999349553).
    11. 11)
      • 6. 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).
    12. 12)
      • 23. Singh, V.: ‘Robust stability of 2-D digital filters employing saturation’, IEEE Signal Process. Lett., 2005, 12, (2), pp. 142145 (doi: 10.1109/LSP.2004.839704).
    13. 13)
      • 30. Gao, H., Lam, J., Wang, Z.: ‘Discrete bilinear stochastic systems with time-varying delay: stability analysis and control synthesis’, Chaos Solitons Fractals, 2007, 34, pp. 394404 (doi: 10.1016/j.chaos.2006.03.027).
    14. 14)
      • 1. Fornasini, E., Marchesini, G.: ‘Doubly-indexed dynamical systems: state-space models and structural properties’, Math. Syst. Theory, 1978, 12, pp. 5972 (doi: 10.1007/BF01776566).
    15. 15)
      • 3. Kaczorek, T.: ‘Two-dimensional linear systems’ (Springer–Verlag, Berlin, Germany, 1985).
    16. 16)
      • 4. Liu, D.: ‘Lyapunov stability of two-dimensional digital filters with overflow nonlinearities’, IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., 1998, 45, (5), pp. 574577 (doi: 10.1109/81.668870).
    17. 17)
      • 11. Dey, A., Kar, H.: ‘Robust stability of 2-D discrete systems employing generalized overflow nonlinearities: an LMI approach’, Digital Signal Process., 2011, 21, (2), pp. 262269 (doi: 10.1016/j.dsp.2010.06.010).
    18. 18)
      • 20. Li, X., Gao, H.: ‘Robust finite frequency H filtering for uncertain 2-D Roesser systems’, Automatica, 2012, 48, (6), pp. 11631170 (doi: 10.1016/j.automatica.2012.03.012).
    19. 19)
      • 14. Li, X., Gao, H., Wang, C.: ‘Generalized Kalman–Yakubovich–Popov lemma for 2-D FM LSS model’, IEEE Trans. Autom. Control, 2012, 57, (12), pp. 30903103 (doi: 10.1109/TAC.2012.2200370).
    20. 20)
      • 24. Liu, Y., Wang, Z., Liang, J., Liu, X.: ‘Synchronization and state estimation for discrete-time complex networks with distributed delays’, IEEE Trans. Syst. Man Cybern. B, Cybern., 2008, 38, (5), pp. 13141325 (doi: 10.1109/TSMCB.2008.925745).
    21. 21)
      • 22. Yao, J., Wang, W., Ye, S.: ‘Robust stabilization of 2-D state-delayed systems with stochastic perturbation’. 11th Int. Conf. on Control Automation, Robotics and Vision, 7–10 December, Singapore, pp. 195119562010.
    22. 22)
      • 10. Xu, J.-M., Yu, L.: ‘H control for 2-D discrete state delayed systems in the second FM model’, Acta Autom. Sin., 2008, 34, (7), pp. 809813 (doi: 10.3724/SP.J.1004.2008.00809).
    23. 23)
      • 9. Xie, L., Du, C., Soh, Y., Zhang, C.: ‘H and robust control of 2-D systems in FM second model’, Multidimens. Syst. Signal Process., 2002, 13, (3), pp. 265287 (doi: 10.1023/A:1015808429836).
    24. 24)
      • 18. Wu, L., Shi, P., Gao, H., Wang, C.: ‘H filtering for 2D Markovian jump systems’, Automatica, 2008, 44, (7), pp. 18491858 (doi: 10.1016/j.automatica.2007.10.027).
    25. 25)
      • 8. Paszke, W., Lam, J., Galkowski, K., Xu, S., Lin, Z.: ‘Robust stability and stabilization of 2D discrete state-delayed systems’, Syst. Control Lett., 2004, 51, (3–4), pp. 277291 (doi: 10.1016/j.sysconle.2003.09.003).
    26. 26)
      • 5. Chen, S.-F.: ‘Stability analysis for 2-D systems with interval time-varying delays and saturation nonlinearities’, Signal Process., 2010, 90, (7), pp. 22652275 (doi: 10.1016/j.sigpro.2010.02.011).
    27. 27)
      • 17. Wu, L., Wang, Z., Gao, H., Wang, C.: ‘Filtering for uncertain two-dimensional discrete systems with state delays’, Signal Process., 2007, 87, (9), pp. 22132230 (doi: 10.1016/j.sigpro.2007.03.002).
    28. 28)
      • 12. Hmamed, A., Mesquine, F., Tadeo, F., Benhayoun, M., Benzaouia, A.: ‘Stabilization of 2D saturated systems by state feedback control’, Multidimens. Syst. Signal Process., 2010, 21, (3), pp. 277292 (doi: 10.1007/s11045-010-0107-2).
    29. 29)
      • 13. Liu, X., Yu, W., Wang, L.: ‘Necessary and sufficient asymptotic stability criterion for 2-D positive systems with time-varying state delays described by Roesser model’, IET Control Theory Appl., 2011, 5, (5), pp. 663668 (doi: 10.1049/iet-cta.2010.0206).
    30. 30)
      • 26. Zhu, Q., Hu, G.-D.: ‘Stability and absolute stability of a general 2-D non-linear FM second model’, IET Control Theory Appl., 2011, 5, (1), pp. 239246 (doi: 10.1049/iet-cta.2009.0624).
http://iet.metastore.ingenta.com/content/journals/10.1049/iet-cta.2012.0915
Loading

Related content

content/journals/10.1049/iet-cta.2012.0915
pub_keyword,iet_inspecKeyword,pub_concept
6
6
Loading