Your browser does not support JavaScript!
http://iet.metastore.ingenta.com
1887

access icon free Asymptotic stability and stabilisation of uncertain delta operator systems with time-varying delays

  • Author(s): Xiaochen Xie 1 ; Shen Yin 1 ; Huijun Gao 1, 2 ; Okyay Kaynak 3
    • View affiliations
    • Affiliations: 1: Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin, Heilongjiang 150001, People's Republic of China;
      2: Research Institute of Mechatronics and Automation, Bohai University, Liaoning 121013, People's Republic of China;
      3: Department of Electrical and Electronic Engineering, Bogazici University, Bebek, Istanbul 80815, Turkey
  • Source: Volume 7, Issue 8, 16 May 2013, p. 1071 – 1078
    DOI:  10.1049/iet-cta.2012.0749 , Print ISSN 1751-8644, Online ISSN 1751-8652
© The Institution of Engineering and Technology
Received 20/09/2012, Accepted 18/04/2013, Revised 17/04/2013, Published

This study focuses on the asymptotic stability and stabilisation of uncertain linear systems with time-varying delays via delta operator approach. By employing a new model formulation, the time-delayed delta operator system is transformed into an interconnected system for which the uncertainties can become easy to deal with. Based on a two-term approximation of delayed state and scaled small gain theorem, new delay-dependent sufficient conditions of robust asymptotic stability and state-feedback stabilisation of an uncertain delta operator time-delayed system are established by using a novel Lyapunov–Krasovskii functional. The criteria obtained unify some previously suggested relevant methods seen in literature for achieving asymptotic stability and stabilisation of both continuous and discrete systems into the delta operator framework. Numerical examples presented explicitly demonstrate the advantages and effectiveness of the proposed methods.

Inspec keywords: Lyapunov methods; uncertain systems; approximation theory; time-varying systems; delays; linear systems; discrete time systems; state feedback; asymptotic stability

Other keywords: uncertain delta operator system stabilisation; delay-dependent sufficient condition; uncertain linear system stabilisation; approximation theory; continuous system; state feedback stabilisation; time-varying delay; robust asymptotic stability; Lyapunov– Krasovskii functional; discrete system; interconnected system

Subjects: Stability in control theory; Discrete control systems; Distributed parameter control systems; Time-varying control systems; Interpolation and function approximation (numerical analysis)

References

    1. 1)
      • 8. Fan, H.: ‘Singular root distribution problem for delta-operator based real polynomials’, Automatica, 1999, 35, (5), pp. 791–807 (doi: 10.1016/S0005-1098(98)00218-0).
    2. 2)
      • 18. Sakthivel, R., Mathiyalagan, K., Anthoni, S.M.: ‘Robust H∞ control for uncertain discrete-time stochastic neural networks with time-varying delays’, IET Control Theory Appl., 2012, 6, (9), pp. 1220–1228 (doi: 10.1049/iet-cta.2011.0254).
    3. 3)
      • 1. Goodwin, G.C., Leal, R.L., Mayne, D.Q., Middleton, R.H.: ‘Rapprochement between continuous and discrete model reference adaptive control’, Automatica, 1986, 22, (2), pp. 199–207 (doi: 10.1016/0005-1098(86)90081-6).
    4. 4)
      • 24. Peng, C.: ‘Improved delay-dependent stabilisation criteria for discrete systems with a new finite sum inequality’, IET Control Theory Appl., 2012, 6, (3), pp. 448–453 (doi: 10.1049/iet-cta.2011.0109).
    5. 5)
      • 33. Fridman, E., Shaked, U.: ‘Input-output approach to stability and l2-gain analysis of systems with time-varying delays’, Syst. Control Lett., 2006, 55, (12), pp. 1041–1053 (doi: 10.1016/j.sysconle.2006.07.002).
    6. 6)
      • 35. Qiu, J., Xia, Y., Yang, H., Zhang, J.: ‘Robust stabilisation for a class of discrete-time systems with time-varying delays via delta operators’, IET Control Theory Appl., 2008, 2, (1), pp. 87–93 (doi: 10.1049/iet-cta:20070070).
    7. 7)
      • 9. Kadirkamanathan, V., Halauca, C., Anderson, S.R.: ‘Predictive control of fast-sampled systems using the delta-operator’, Int. J. Syst. Sci., 2009, 40, (7), pp. 745–756 (doi: 10.1080/00207720902953177).
    8. 8)
      • 5. Chen, S., Wu, J., Istepanian, R.H., Chu, J., Whidborne, J.F.: ‘Optimising stability bounds of finite-precision controller structures for sampled-data systems in the δ-operator domain’, IEE Proc. Control Theory Appl., 1999, 146, (6), pp. 517–526 (doi: 10.1049/ip-cta:19990745).
    9. 9)
      • 37. Xiang, Z., Chen, Q., Zhang, R.: ‘Robust stability analysis and control for fuzzy systems with uncertainties using the delta operator’, Control Decision, 2003, 18, (6), pp. 720–723.
    10. 10)
      • 16. Yin, S.: ‘Data-driven design of fault diagnosis systems’ (VDI-Verlag Düsseldorf, Germany, 2012).
    11. 11)
      • 4. Neuman, C.P.: ‘Properties of the delta operator model of dynamic physical systems’, IEEE Trans. Syst. Man Cybern., 1993, 23, (1), pp. 296–301 (doi: 10.1109/21.214791).
    12. 12)
      • 13. Yang, H., Xia, Y., Shi, P., Fu, M.: ‘Stability analysis for high frequency networked control systems’, IEEE Trans. Autom. Control, 2012, 57, (10), pp. 2694–2700 (doi: 10.1109/TAC.2012.2190194).
    13. 13)
      • 23. Lam, J., Shu, Z., Xu, S., Boukas, E.K.: ‘Robust H∞ control of descriptor discrete-time Markovian jump systems’, Int. J. Control, 2007, 80, (3), pp. 374–385 (doi: 10.1080/00207170600999322).
    14. 14)
      • 22. Yin, S., Ding, S.X., Haghani, A., Hao, H.: ‘Data-driven monitoring for stochastic systems and its application on batch process’. Int. J. Syst. Sci., 2013, 44, (7), pp. 1366–1376 (doi: 10.1080/00207721.2012.659708).
    15. 15)
      • 20. Chen, S.H., Chou, F.I., Chou, J.H.: ‘Robust controllability of linear systems with multiple delays in control’, IET Control Theory Appl., 2012, 6, (10), pp. 1552–1556 (doi: 10.1049/iet-cta.2011.0486).
    16. 16)
      • 30. Yang, H., Xia, Y., Shi, P., Fu, M.: ‘Stability of Markovian jump systems over networks via delta operator approach’, Circuits Syst. Signal Process., 2012, 31, (1), pp. 107–125 (doi: 10.1007/s00034-010-9263-8).
    17. 17)
      • 21. Kwon, O.M., Park, M.J., Park, J.H., Lee, S.M., Cha, E.J.: ‘Improved robust stability criteria for uncertain discrete-time systems with interval time-varying delays via new zero equalities’, IET Control Theory Appl., 2012, 6, (16), pp. 2567–2575 (doi: 10.1049/iet-cta.2012.0257).
    18. 18)
      • 27. Zhang, J., Xia, Y., Shi, P., Mahmoud, M.S.: ‘New results on stability and stabilisation of systems with interval time-varying delay’, IET Control Theory Appl., 2011, 5, (3), pp. 429–436 (doi: 10.1049/iet-cta.2009.0560).
    19. 19)
      • 3. Hersh, M.A.: ‘The zeros and poles of delta operator systems’, Int. J. Control, 1993, 57, (3), pp. 557–575 (doi: 10.1080/00207179308934407).
    20. 20)
      • 11. Guo, X.G., Yang, G.H.: ‘H∞ filter design for delta operator formulated systems with low sensitivity to filter coefficient variations’, IET Control Theory Appl., 2011, 5, (15), pp. 1677–1688 (doi: 10.1049/iet-cta.2010.0500).
    21. 21)
      • 41. Jiang, X., Han, Q., Yu, X.: ‘Stability criteria for linear discrete-time systems with interval-like time-varying delay’. Proc. American Control Conf., Portland, OR, USA, June 2005, pp. 2817–2822.
    22. 22)
      • 10. Yang, H., Xia, Y., Shi, P.: ‘Observer-based sliding mode control for a class of discrete systems via delta operator approach’, J. Franklin Inst., 2010, 347, (7), pp. 1199–1213 (doi: 10.1016/j.jfranklin.2010.05.003).
    23. 23)
      • 32. Gu, K., Zhang, Y., Xu, S.: ‘Small gain problem in coupled differential difference equations, time-varying delays, and direct Lyapunov method’, Int. J. Robust Nonlinear Control, 2011, 21, (4), pp. 429–451 (doi: 10.1002/rnc.1604).
    24. 24)
      • 19. Xie, D., Wu, Y.: ‘Stabilisation of time-delay switched systems with constrained switching signals and its applications in networked control systems’, IET Control Theory Appl., 2010, 4, (10), pp. 2120–2128 (doi: 10.1049/iet-cta.2009.0016).
    25. 25)
      • 7. Suchomski, P.: ‘A J-lossless coprime factorisation approach to H∞ control in delta domain’, Automatica, 2002, 38, (10), pp. 1807–1814 (doi: 10.1016/S0005-1098(02)00096-1).
    26. 26)
      • 28. Qiu, J., Yang, H., Shi, P., Xia, Y.: ‘Robust H∞ control for class of discrete-time Markovian jump systems with time-varying delays based on delta operator’, Circuits, Syst. Signal Process., 2008, 27, (5), pp. 627–643 (doi: 10.1007/s00034-008-9046-7).
    27. 27)
      • 14. Yang, H., Xia, Y.: ‘Low frequency positive real control for delta operator systems’, Automatica, 2012, 48, (8), pp. 1791–1795 (doi: 10.1016/j.automatica.2012.05.045).
    28. 28)
      • 25. Wang, Z., Ho, D.W.C., Liu, Y., Liu, X.: ‘Robust H∞ control for a class of nonlinear discrete time-delay stochastic systems with missing measurements’, Automatica, 2009, 45, (3), pp. 684–691 (doi: 10.1016/j.automatica.2008.10.025).
    29. 29)
      • 42. Gu, K.: ‘An integral inequality in the stability problem of time-delay systems’. Proc. 39th IEEE Conf. Decision and Control, Sydney, Australia, 2000, pp. 2805–2810.
    30. 30)
      • 31. Zhou, K., Doyle, J.C., Glover, K.: ‘Robust and optimal control’ (Prentice-Hall, New Jersey, 1996).
    31. 31)
      • 38. Li, X., Gao, H.: ‘A new model transformation of discrete-time systems with time-varying delay and its application to stability analysis’, IEEE Trans. Autom. Control, 2011, 56, (9), pp. 2172–2178 (doi: 10.1109/TAC.2011.2146850).
    32. 32)
      • 17. Du, B., Lam, J., Shu, Z., Wang, Z.: ‘A delay-partitioning projection approach to stability analysis of continuous systems with multiple delay components’, IET Control Theory Appl., 2009, 3, (4), pp. 383–390 (doi: 10.1049/iet-cta.2007.0321).
    33. 33)
      • 2. Middleton, R.H., Goodwin, G.C.: ‘Improved finite word length characteristics in digital control using delta operators’, IEEE Trans. Autom. Control, 1986, 31, (11), pp. 1015–1021 (doi: 10.1109/TAC.1986.1104162).
    34. 34)
      • 12. Yang, H., Xia, Y., Shi, P., Zhao, L.: ‘Analysis and synthesis of delta operator systems’ (Springer–Verilag, Berlin, Heidelberg, 2012).
    35. 35)
      • 29. Yang, H., Xia, Y., Qiu, J., Zhang, J.: ‘Filtering for a class of discrete-time systems with time-delays via delta operator approach’, Int. J. Syst. Sci., 2010, 41, (4), pp. 423–433 (doi: 10.1080/00207720903045833).
    36. 36)
      • 39. Won, W., Lee, K.S.: ‘Identification of a multivariable delta-operator stochastic state-space model with distributed time delays: application to a rapid thermal processor’, Comput. Chem. Eng., 2012, 40, (11), pp. 223–230 (doi: 10.1016/j.compchemeng.2012.01.019).
    37. 37)
      • 40. Yang, H., Xia, Y., Shi, P., Fu, M.: ‘A novel delta operator Kalman filter design and convergence analysis’, IEEE Trans. Circuits Syst. I, 2011, 58, (10), pp. 2458–2468 (doi: 10.1109/TCSI.2011.2131330).
    38. 38)
      • 36. Yang, H., Xia, Y., Fu, M., Shi, P.: ‘Robust adaptive sliding mode control for uncertain delta operator systems’, Int. J. Adapt. Control Signal Process., 2010, 24, (8), pp. 623–632.
    39. 39)
      • 6. Suchomski, P.: ‘Numerical conditioning of delta-domain Lyapunov and Riccati equation’, IEE Proc. Control Theory Appl., 2001, 148, (6), pp. 497–501 (doi: 10.1049/ip-cta:20010774).
    40. 40)
      • 26. Wang, Z., Liu, Y., Wei, G., Liu, X.: ‘A note on control of a class of discrete-time stochastic systems with distributed delays and nonlinear disturbances’, Automatica, 2010, 46, (3), pp. 543–548 (doi: 10.1016/j.automatica.2009.11.020).
    41. 41)
      • 34. Kao, C., Lincoln, B.: ‘Simple stability criteria for systems with time-varying delays’, Automatica, 2004, 40, (8), pp. 1429–1434 (doi: 10.1016/j.automatica.2004.03.011).
    42. 42)
      • 15. Yin, S., Ding, S.X., Haghani, A., Hao, H., Zhang, P.: ‘A comparison study of basic data-driven fault diagnosis and process monitoring methods on the benchmark Tennessee Eastman process’, J. Process Control, 2012, 22, (9), pp. 1567–1581 (doi: 10.1016/j.jprocont.2012.06.009).
http://iet.metastore.ingenta.com/content/journals/10.1049/iet-cta.2012.0749
Loading

Related content

content/journals/10.1049/iet-cta.2012.0749
pub_keyword,iet_inspecKeyword,pub_concept
6
6
Loading
Print this page
This is a required field
Please enter a valid email address