access icon free Quasi-finite-rank approximation of compression operators on L [0, h) with application to stability analysis of time-delay systems

This study discusses a new method for approximating compression operators, which play important roles in the operator-theoretic approach to sampled-data systems and time-delay systems. Stimulated by the success in the application of quasi-finite-rank approximation of compression operators defined on the Hilbert space L 2[0, h), the authors study a parallel problem for compression operators defined on the Banach space L [0, h). In spite of similarity between these problems, they are led to applying a completely different approach because of essential differences in the underlying spaces. More precisely, they apply the idea of the conventional fast-sample/fast-hold (FSFH) approximation technique, and show that the approximation problem can be transformed into such a linear programming problem that asymptotically leads to optimal approximation as the FSFH approximation parameter M tends to infinity. Finally, they demonstrate the effectiveness of the L [0, h)-based approximation technique through numerical examples, with particular application to stability analysis of time-delay systems.

Inspec keywords: stability; delay systems; Hilbert spaces; Banach spaces; linear programming; approximation theory; sampled data systems

Other keywords: Banach space L∞[0, h); quasi-finite rank approximation; time delay system; linear programming problem; compression operator; stability analysis; L∞[0, h)-based approximation technique; Hilbert space L2[0, h); FSFH approximation technique; operator theory approach; parallel problem; fast sample-fast hold; sampled data system

Subjects: Discrete control systems; Interpolation and function approximation (numerical analysis); Optimisation techniques; Distributed parameter control systems; Stability in control theory

References

    1. 1)
      • 11. Dullerud, G.E.: ‘Computing the L2-induced norm of a compression operator’, Syst. Control Lett., 1999, 7, (2), pp. 8791 (doi: 10.1016/S0167-6911(99)00009-2).
    2. 2)
      • 7. Sivashankar, N., Khargonekar, P.P.: ‘Robust stability and performance analysis of sampled-data systems’, IEEE Trans. Autom. Control, 1993, 38, (1), pp. 5869 (doi: 10.1109/9.186312).
    3. 3)
      • 18. Lindner, M.: ‘Infinite matrices and their finite sections: an introduction to the limit operator method’, (Birkhäuser, Basel, 2006).
    4. 4)
      • 12. Hagiwara, T., Suyama, M., Araki, M.: ‘Upper and lower bounds of the frequency response gain of sampled-data systems’, Automatica, 2001, 37, (9), pp. 13631370 (doi: 10.1016/S0005-1098(01)00076-0).
    5. 5)
      • 17. Brown, R.C., Tvrdý, M.: ‘Generalized boundary value problems with abstract side conditions and their adjoints. II’, Czech. Math. J., 1981, 31, (4), pp. 501509.
    6. 6)
      • 5. Dullerud, G.E., Francis, B.A.: ‘L1 analysis and design of sampled-data systems’, IEEE Trans. Autom. Control, 1992, 37, (4), pp. 436446 (doi: 10.1109/9.126577).
    7. 7)
      • 3. Toivonen, H.T.: ‘Sampled-data control of continuous-time systems with an H optimality criterion’, Automatica, 1992, 28, (1), pp. 4554 (doi: 10.1016/0005-1098(92)90006-2).
    8. 8)
      • 6. Dullerud, G.E., Glover, K.: ‘Robust stabilization of sampled-data systems to structured LTI perturbations’, IEEE Trans. Autom. Control, 1993, 38, (10), pp. 14971508 (doi: 10.1109/9.241563).
    9. 9)
      • 8. Hirata, K.: ‘On numerical computation of the spectrum of a class of convolution operators related to delay systems (in Japanese)’, Trans. Inst. Syst., Control Inf. Eng., 2008, 21, (3), pp. 8388.
    10. 10)
      • 1. Yamamoto, Y.: ‘A function space approach to sampled-data control systems and tracking problems’, IEEE Trans. Autom. Control, 1994, 39, (4), pp. 703712 (doi: 10.1109/9.286247).
    11. 11)
      • 15. Anderson, B.D.O., Keller, J.P.: ‘A new approach to the discretization of continuous-time controllers’, IEEE Trans. Autom. Control, 1992, 37, (2), pp. 214223 (doi: 10.1109/9.121622).
    12. 12)
      • 9. Hirata, K., Kokame, H.: ‘Stability analysis of retarded systems via lifting technique’. Proc. IEEE Conf. Decision and Control, 2003, pp. 55955596.
    13. 13)
      • 10. Hagiwara, T., Hirata, K.: ‘Fast-lifting approach to the computation of the spectrum of retarded time-delay systems’, Eur. J. Control, 2011, 17, (2), pp. 162171 (doi: 10.3166/ejc.17.162-171).
    14. 14)
      • 16. Brown, R.C., Tvrdý, M.: ‘Generalized boundary value problems with abstract side conditions and their adjoints. I’, Czech. Math. J., 1980, 30, (1), pp. 727.
    15. 15)
      • 4. Bamieh, B.A., Dahleh, M.A., Pearson, J.B.: ‘Minimization of the L-induced norm for sampled-data systems’, IEEE Trans. Autom. Control, 1993, 38, (5), pp. 717732 (doi: 10.1109/9.277236).
    16. 16)
      • 19. Kim, J.H., Hagiwara, T.: ‘Computing the L[0, h)-induced norm of a compression operator’. Proc. European Control Conf., 2013, pp. 36883693.
    17. 17)
      • 14. Hagiwara, T., Umeda, H.: ‘Modified fast-sample/fast-hold approximation for sampled-data system analysis’, Eur. J. Control, 2008, 14, (4), pp. 286296 (doi: 10.3166/ejc.14.286-296).
    18. 18)
      • 2. Bamieh, B.A., Pearson, J.B.: ‘A general framework for linear periodic systems with application to H sampled-data systems’, IEEE Trans. Autom. Control, 1992, 37, (4), pp. 418435 (doi: 10.1109/9.126576).
    19. 19)
      • 13. Hagiwara, T., Morioka, K., Okada, K.: ‘Quasi-finite-rank approximation of compression operators in sampled-data systems and time-delay systems’, Int. J. Control, 2010, 83, (11), pp. 23852394 (doi: 10.1080/00207179.2010.523849).
http://iet.metastore.ingenta.com/content/journals/10.1049/iet-cta.2013.0458
Loading

Related content

content/journals/10.1049/iet-cta.2013.0458
pub_keyword,iet_inspecKeyword,pub_concept
6
6
Loading