Article
Full entry |
PDF
(2.2 MB)
Feedback
PDF
(2.2 MB)
Feedback
Keywords:
balanced truncation; linear periodic system; model error; infinity norm
balanced truncation; linear periodic system; model error; infinity norm
Summary:
For linear periodic discrete-time systems the analysis of the model error introduced by a truncation on the balanced minimal realization is performed, and a bound for the infinity norm of the model error is introduced. The results represent an extension to the periodic systems of the well known results on the balanced truncation for time-invariant systems. The general case of periodically time-varying state-space dimension has been considered.
References:
[1] Al–Saggaf U. M., Franklin G. F.: An error bound for a discrete reduced order model of a linear multivariable system. IEEE Trans. Automat. Control AC–32 (1987), 9, 815–819 Zbl 0622.93021
[2] Bittanti S.: Deterministic and stochastic linear periodic systems. In: Time Series and Linear Systems (S. Bittanti, ed.), Springer–Verlag, Berlin 1986 MR 0897824
[3] Bolzern P., Colaneri P.: Existence and uniqueness conditions for the periodic solutions of the discrete–time periodic Lyapunov equations. In: Proc. of 25th Conference on Decision and Control, Athens 1986, pp. 1439–1443
[4] Bolzern P., Colaneri P., Scattolini R.: Zeros of discrete–time linear periodic systems. IEEE Trans. Automat. Control AC–31 (1986), 1057–1059 DOI 10.1109/TAC.1986.1104172 | Zbl 0606.93036
[5] Colaneri P., Longhi S.: The lifted and cyclic reformulations in the minimal realization of linear discrete–time periodic systems. In: Proc. of the 1st IFAC Workshop on New Trends in Design of Control Systems, Smolenice 1994, pp. 329–334
[6] Colaneri P., Longhi S.: The realization problem for linear periodic systems. Automatica 31 (1995), 775–779 DOI 10.1016/0005-1098(94)00155-C | MR 1335982 | Zbl 0822.93019
[7] Evans D. S.: Finite–dimensional realizations of discrete–time weighting patterns. SIAM J. Appl. Math. 22 (1972), 45–67 DOI 10.1137/0122006 | MR 0378915
[8] Flamm D. S.: A new shift–invariant representation for periodic linear systems. Systems Control Lett. 17 (1991), 9–14 DOI 10.1016/0167-6911(91)90093-T | MR 1116110 | Zbl 0729.93034
[9] Fortuna L., Nunnari G., Gallo A.: Model Order Reduction Techniques with Applications in Electrical Engineering. Springer–Verlag, Berlin 1992
[10] Glover K.: All optimal Hankel–norm approximation of linear multivariable systems and their $L^\infty $–errors bounds. Internat. J. Control 39 (1984), 6, 1115–1193 DOI 10.1080/00207178408933239 | MR 0748558
[11] Gohberg I., Kaashoek M. A., Lerer L.: Minimality and realization of discrete time–varying systems. Oper. Theory: Adv. Appl. 56 (1992), 261–296 MR 1173922 | Zbl 0242.93024
[12] Grasselli O. M., Longhi S.: Disturbance localization by measurements feedback for linear periodic discrete–time systems. Automatica 24 (1988), 375–385 DOI 10.1016/0005-1098(88)90078-7 | MR 0947377
[13] Grasselli O. M., Longhi S.: Zeros and poles of linear periodic discrete–time systems. Circuits Systems Signal Process. 7 (1988), 361–380 MR 0962108
[14] Grasselli O. M., Longhi S.: Robust tracking and regulation of linear periodic discrete–time systems. Internat. J. Control 54 (1991), 613–633 DOI 10.1080/00207179108934179 | MR 1117838 | Zbl 0728.93065
[15] Grasselli O. M., Longhi S.: The geometric approach for linear periodic discrete–time systems. Linear Algebra Appl. 158 (1991), 27–60 MR 1126434 | Zbl 0758.93044
[16] Gree M., Limebeer D. J. N.: Linear Robust Control. Prentice Hall, Englewood Cliffs, N. J. 1995
[17] Mayer R. A., Burrus C. S.: A unified analysis of multirate and periodically time–varying digital filters. Trans. Ccts Syst. CSA–22 (1975), 162–168 MR 0392090
[18] Moore B. C.: Principal component analysis in linear systems: controllability, observability and modal reduction. Trans. Automat. Control AC–26 (1981), 17–32 DOI 10.1109/TAC.1981.1102568 | MR 0609248
[19] Park B., Verriest E. I.: Canonical forms on discrete linear periodically time–varying systems and a control application. In: Proc. of the 28th IEEE Conference on Decision and Control, Tampa 1989, pp. 1220–1225 MR 1038997
[20] Pernebo L., Silverman L. M.: Model reduction via balanced state space representations. IEEE Trans. Automat. Control AC–27 (1982), 2, 382–387 DOI 10.1109/TAC.1982.1102945 | MR 0680103 | Zbl 0482.93024
[21] Salomon G., Zhou K., Wu E.: A new balanced realization and model reduction method for unstable systems. In: Proc. of 14th IFAC World Congress, Beijing 1999, vol. D, pp. 123–128
[22] Shokoohi S., Silverman L. M., Dooren P. Van: Linear time–variable systems: balancing and model reduction. Trans. Automat. Control AC-28 (1983), 8, 810–822 MR 0717839
[23] Shokoohi S., Silverman L. M., Dooren P. Van: Linear time–variable systems: stability of reduced models. Automatica 20 (1984), 1, 59–67 DOI 10.1016/0005-1098(84)90065-7 | MR 0737946
[24] Varga A.: Periodic Lyapunov equations: some applications and new algorithms. Internat. J. Control 67 (1997), 69–87 DOI 10.1080/002071797224360 | MR 1685840 | Zbl 0873.93057
[25] Xie B., Aripirala R. K. A. V., Syrmos V. L.: Model reduction of linear discrete–time periodic systems using Hankel–norm approximation. In: Proc. of IFAC 13th World Congress, San Francisco, 1996, pp. 245–250
Search
Partner of
