access icon free Necessary and sufficient conditions to the transitivity in simultaneous stabilisation of time-varying systems

In this study, the authors study the transmission problem in simultaneous stabilisation of discrete time-varying systems in the frame work of nest algebra. The necessary and sufficient conditions are developed for the transitivity and strong transitivity in simultaneous stabilisation, respectively. Some criterions for these two properties are established in terms of singly strong representations of only one or two given plants but not all. Finally, the authors give parameterisations of simultaneous stabilisers when the properties of transitivity hold.

Inspec keywords: discrete systems; stability; time-varying systems

Other keywords: discrete time-varying systems; necessary and sufficient conditions; simultaneous stabiliser parameterisation; simultaneous stabilisation; nest algebra; transitivity

Subjects: Stability in control theory; Time-varying control systems; Discrete control systems

References

    1. 1)
      • 9. Vidyasagar, M., Viswanadham, N.: ‘Algebraic design techniques for reliable stabilization’, IEEE Trans. Autom. Control, 1982, 27, (5), pp. 10851095 (doi: 10.1109/TAC.1982.1103086).
    2. 2)
      • 7. Feintuch, A.: ‘On the strong stabilization of slowly time-varying linear systems’, Syst. Control Lett., 2012, 61, (1), pp. 112116 (doi: 10.1016/j.sysconle.2011.09.007).
    3. 3)
      • 4. Lu, Y.F., Xu, X.P.: ‘The stabilization problem for discrete time-varying linear systems’, Syst. Control Lett., 2008, 57, pp. 936939 (doi: 10.1016/j.sysconle.2008.05.003).
    4. 4)
      • 5. Liu, L., Lu, Y.F.: ‘Stability analysis for time-varying systems via quadratic constraints’, Syst. Control Lett., 2011, 60, (10), pp. 832839 (doi: 10.1016/j.sysconle.2011.06.007).
    5. 5)
      • 16. Feintuch, A.: ‘Robust control theory in Hilbert space’ (Springer-Verlag, 1998).
    6. 6)
      • 17. Davidson, K.R., 19, Y.Q.: ‘Topological stable rank of nest algebras’, Proc. Lond. Math. Soc., 2009, 98, issuenumber3, pp. 652678 (doi: 10.1112/plms/pdn048).
    7. 7)
      • 15. Dale, W.N., Smith, M.C.: ‘Stabilizability and existence of system representations for discrete-time time-varying systems’, SIAM J. Control Optim., 1993, 60, (1), pp. 15381557 (doi: 10.1137/0331072).
    8. 8)
      • 20. Djouadi, S.M.: ‘Commutant lifting for linear time-varying systems’. American Control Con., June, 2009, pp. 40674072.
    9. 9)
      • 13. Ghosh, B.: ‘Simutaneous partial pole-placement: a new approach to multi-mode design’, IEEE Trans Autom. Control, 1986, 31, pp. 440443 (doi: 10.1109/TAC.1986.1104297).
    10. 10)
      • 11. Lu, Y.F., Xu, X.P.: ‘Simultaneous stabilization for a family of plants’, J. Math. Res. Exposition, 2008, 28, (3), pp. 529534.
    11. 11)
      • 12. Abdallah, C.T., Dorato, P., Bredemann, M.: ‘New sufficient conditions for strong simutaneous stabilization’, Automatica, 1997, 33, (6), pp. 11931196 (doi: 10.1016/S0005-1098(97)00021-6).
    12. 12)
      • 1. Zames, G.: ‘Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms and approximate inverses’, IEEE Trans. Autom. Control, 1981, 28, pp. 301320 (doi: 10.1109/TAC.1981.1102603).
    13. 13)
      • 14. Yu, T.Q.: ‘The transitivity in simultaneous stabilization’, Syst. Control Lett., 2011, 60, (1), pp. 16 (doi: 10.1016/j.sysconle.2010.09.001).
    14. 14)
      • 19. Ji, Y.Q., Zhang, Y.H.: ‘Stable ranks of split extensions of Banach algebras’, Linear Algebra Appl., 2011, 434, (10), pp. 21492157 (doi: 10.1016/j.laa.2010.12.009).
    15. 15)
      • 3. Feintuch, A., Saeks, R.: ‘System theory, a Hilbert space approach, in: series in pure and applied mathematics’ (Academic Press, New York, London, 1982).
    16. 16)
      • 18. Feintuch, A.: ‘The stable rank of a nest algebra and strong stabilization of linear time-varying systems’, Operator Theory: Adv. Appl., 2009, 197, pp. 139148.
    17. 17)
      • 2. Doyle, G.C., Glover, K., Khargonekar, P.P., Francis, B.A.: ‘State space solutions to standard ℋ2 and ℋ control problems’, IEEE Trans. Autom. Control, 1989, 34, pp. 831847 (doi: 10.1109/9.29425).
    18. 18)
      • 8. Saeks, R., Murray, J.: ‘Fractional representation algebraic geometry, and the simultaneous stabilization problem’, IEEE Trans. Autom. Control, 1982, 27, (4), pp. 895903 (doi: 10.1109/TAC.1982.1103005).
    19. 19)
      • 6. Feintuch, A.: ‘On strong stabilization of asymptotically time-invariant linear time-varying systems’, MCSS, 2011, 22, (3), pp. 229243.
    20. 20)
      • 10. Vidyasagar, M.: ‘Control system synthesis: a factorization approach’ (MIT Press, Cambridge, MA, 1985).
http://iet.metastore.ingenta.com/content/journals/10.1049/iet-cta.2013.0011
Loading

Related content

content/journals/10.1049/iet-cta.2013.0011
pub_keyword,iet_inspecKeyword,pub_concept
6
6
Loading