Periodic systems largely system equivalent to periodic discrete-time processes

Osvaldo Maria Grasselli; Sauro Longhi; Antonio Tornambè

Kybernetika (2001)

  • Volume: 37, Issue: 1, page [1]-20
  • ISSN: 0023-5954

Abstract

top
In this paper, the problem of obtaining a periodic model in state-space form of a linear process that can be modeled by linear difference equations with periodic coefficients is considered. Such a problem was already studied and solved in [r71] on the basis of the notion of system equivalence, but under the assumption that the process has no null characteristic multiplier. In this paper such an assumption is removed in order to generalize the results in [r71] to linear periodic processes with possibly the null characteristic multiplier (e. g., multirate sampled-data systems). Large system equivalence between two linear periodic models of such processes is introduced and analyzed. For a given linear periodic process the necessary and sufficient conditions are found for the existence of a linear periodic system (i. e., a linear periodic model in state-space form) that is largely system equivalent to the given model of the process, together with an algorithm for deriving such a system when these conditions are satisfied. In addition, the significance of the periodic system thus obtained for describing the original periodic process that is largely system equivalent to the system, is clarified by showing that the controllability, the reconstructibility, the stabilizability, the detectability, the stacked transfer matrix, the asymptotic stability, the rate of convergence of the free motions, and even the number and the dimensions of the Jordan blocks of the monodromy matrix corresponding to each nonnull characteristic multiplier of the periodic system, are determined by the original periodic process (although the order of the periodic system is not, in general, as well as its reachability and observability properties, because of some possible additional or removed null characteristic multipliers).

How to cite

top

Grasselli, Osvaldo Maria, Longhi, Sauro, and Tornambè, Antonio. "Periodic systems largely system equivalent to periodic discrete-time processes." Kybernetika 37.1 (2001): [1]-20. <http://eudml.org/doc/33513>.

@article{Grasselli2001,
abstract = {In this paper, the problem of obtaining a periodic model in state-space form of a linear process that can be modeled by linear difference equations with periodic coefficients is considered. Such a problem was already studied and solved in [r71] on the basis of the notion of system equivalence, but under the assumption that the process has no null characteristic multiplier. In this paper such an assumption is removed in order to generalize the results in [r71] to linear periodic processes with possibly the null characteristic multiplier (e. g., multirate sampled-data systems). Large system equivalence between two linear periodic models of such processes is introduced and analyzed. For a given linear periodic process the necessary and sufficient conditions are found for the existence of a linear periodic system (i. e., a linear periodic model in state-space form) that is largely system equivalent to the given model of the process, together with an algorithm for deriving such a system when these conditions are satisfied. In addition, the significance of the periodic system thus obtained for describing the original periodic process that is largely system equivalent to the system, is clarified by showing that the controllability, the reconstructibility, the stabilizability, the detectability, the stacked transfer matrix, the asymptotic stability, the rate of convergence of the free motions, and even the number and the dimensions of the Jordan blocks of the monodromy matrix corresponding to each nonnull characteristic multiplier of the periodic system, are determined by the original periodic process (although the order of the periodic system is not, in general, as well as its reachability and observability properties, because of some possible additional or removed null characteristic multipliers).},
author = {Grasselli, Osvaldo Maria, Longhi, Sauro, Tornambè, Antonio},
journal = {Kybernetika},
keywords = {linear periodic model; state-space form; linear periodic model; state-space form},
language = {eng},
number = {1},
pages = {[1]-20},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Periodic systems largely system equivalent to periodic discrete-time processes},
url = {http://eudml.org/doc/33513},
volume = {37},
year = {2001},
}

TY - JOUR
AU - Grasselli, Osvaldo Maria
AU - Longhi, Sauro
AU - Tornambè, Antonio
TI - Periodic systems largely system equivalent to periodic discrete-time processes
JO - Kybernetika
PY - 2001
PB - Institute of Information Theory and Automation AS CR
VL - 37
IS - 1
SP - [1]
EP - 20
AB - In this paper, the problem of obtaining a periodic model in state-space form of a linear process that can be modeled by linear difference equations with periodic coefficients is considered. Such a problem was already studied and solved in [r71] on the basis of the notion of system equivalence, but under the assumption that the process has no null characteristic multiplier. In this paper such an assumption is removed in order to generalize the results in [r71] to linear periodic processes with possibly the null characteristic multiplier (e. g., multirate sampled-data systems). Large system equivalence between two linear periodic models of such processes is introduced and analyzed. For a given linear periodic process the necessary and sufficient conditions are found for the existence of a linear periodic system (i. e., a linear periodic model in state-space form) that is largely system equivalent to the given model of the process, together with an algorithm for deriving such a system when these conditions are satisfied. In addition, the significance of the periodic system thus obtained for describing the original periodic process that is largely system equivalent to the system, is clarified by showing that the controllability, the reconstructibility, the stabilizability, the detectability, the stacked transfer matrix, the asymptotic stability, the rate of convergence of the free motions, and even the number and the dimensions of the Jordan blocks of the monodromy matrix corresponding to each nonnull characteristic multiplier of the periodic system, are determined by the original periodic process (although the order of the periodic system is not, in general, as well as its reachability and observability properties, because of some possible additional or removed null characteristic multipliers).
LA - eng
KW - linear periodic model; state-space form; linear periodic model; state-space form
UR - http://eudml.org/doc/33513
ER -

References

top
  1. Berg M. C., Amit, N., Powell J. D., 10.1109/9.14436, IEEE Trans. Automat. Control AC-33 (1988), 1139–1150 (1988) Zbl0711.93041DOI10.1109/9.14436
  2. Bittanti S., Deterministic and stochastic linear periodic systems: In: Time Series and Linear Systems (S, Bittanti, ed.), Springer–Verlag, Berlin 1986, pp. 141–182 (1986) MR0897824
  3. Callier F. M., Desoer C. A., Multivariable Feedback Systems, Springer Verlag, New York 1982 
  4. Colaneri P., 10.1016/0167-6911(90)90010-R, Systems Control Lett. 15 (1990), 2, 161–167 (1990) Zbl0712.93047MR1068922DOI10.1016/0167-6911(90)90010-R
  5. Colaneri P., Hamiltonian matrices for lifted systems and periodic Riccati equations in H 2 / H ∞ analysis and control, In: Proc. 29th IEEE Conference on Decision and Control, Brighton 1991, pp. 1914–1917 (1991) 
  6. Colaneri P., Longhi S., 10.1016/0005-1098(94)00155-C, Automatica 31 (1995), 775–779 (1995) Zbl0822.93019MR1335982DOI10.1016/0005-1098(94)00155-C
  7. Chen C. T., Linear System Theory and Design, Holt Rinchart and Winston, New York 1984 
  8. Coll C., Bru R., Sanchez, E., Hernandez V., Discrete-time linear periodic realization in the frequency domain, Linear Algebra Appl. 203–204 (1994), 301–326 (1994) Zbl0802.93041MR1275515
  9. Dahleh M. A., Voulgaris P. G., Valavani L. S., 10.1109/9.109641, IEEE Trans. Automat. Control 37 (1992), 1, 90–99 (1992) Zbl0747.93028MR1139618DOI10.1109/9.109641
  10. Evans D. S., 10.1137/0122006, SIAM J. Appl. Math. 22 (1972), 45–67 (1972) Zbl0242.93024MR0378915DOI10.1137/0122006
  11. Fuhrmann P. A., 10.1080/00207177708922211, Internat. J. Control 25 (1977), 5–10 (1977) Zbl0357.93009MR0472162DOI10.1080/00207177708922211
  12. Gohberg I., Kaashoek M. A., Lerer L., Minimality and realization of discrete time-varying systems, Oper. Theory: Adv. Appl. 56 (1992), 261–296 (1992) Zbl0747.93054MR1173922
  13. Grasselli O. M., 10.1080/00207178408933268, Internat. J. Control 40 (1984), 201–214 (1984) Zbl0546.93010MR0750419DOI10.1080/00207178408933268
  14. Grasselli O. M., Lampariello F., 10.1080/00207178108922978, Internat. J. Control 33 (1981), 1091–1106 (1981) Zbl0464.93056MR0624173DOI10.1080/00207178108922978
  15. Grasselli O. M., Longhi S., 10.1016/0005-1098(88)90078-7, Automatica 24 (1988), 3, 375–385 (1988) Zbl0653.93033MR0947377DOI10.1016/0005-1098(88)90078-7
  16. Grasselli O. M., Longhi S., Pole placement for non-reachable periodic discrete-time systems, Math. Control, Signals and Systems 4 (1991), 439–455 (1991) MR1128264
  17. Grasselli O. M., Longhi S., 10.1080/00207179108934179, Internat. J. Control 54 (1991), 3, 613–633 (1991) Zbl0728.93065MR1117838DOI10.1080/00207179108934179
  18. Grasselli O. M., Longhi S., 10.1080/00207729108910751, Internat. J. Systems Science 22 (1991), 10, 1785–1806 (1991) Zbl0743.93047MR1128912DOI10.1080/00207729108910751
  19. Grasselli O. M., Longhi S., Block decoupling with stability of linear periodic systems, J. Math. Systems, Estimation and Control 3 (1993), 4, 427–458 (1993) Zbl0785.93062MR1318606
  20. Grasselli O. M., Longhi, S., Tornambè A., 10.1137/S0363012992234578, SIAM J. Control Optim. 33 (1995), 2, 544–468 (1995) Zbl0838.93042MR1318660DOI10.1137/S0363012992234578
  21. Grasselli O. M., Tornambè A., 10.1109/9.256346, IEEE Trans. Automat. Control 37 (1992), 852–856 (1992) Zbl0760.93002MR1164565DOI10.1109/9.256346
  22. Grasselli O. M., Longhi S., Tornambè, A., Valigi P., 10.1016/0005-1098(96)00046-5, Automatica 32 (1996), 1015–1019 (1996) Zbl0854.93062MR1405456DOI10.1016/0005-1098(96)00046-5
  23. Ho B. L., Kalman R. E., Effective construction of linear state- variable models from input-output functions, Regelungstechnik 14 (1966), 545–548 (1966) Zbl0145.12701
  24. Kailath T., Linear Systems, Englewood Cliffs, Prentice Hall, NJ 1980 Zbl0870.93013MR0569473
  25. Kono M., 10.1080/00207178008922850, Internat. J. Control 32 (1980), 149–158 (1980) Zbl0443.93044MR0580197DOI10.1080/00207178008922850
  26. Lin C. A., King C. W., 10.1109/9.210146, IEEE Trans. Automat. Control AC-38 (1993), 3, 462–466 (1993) Zbl0789.93089MR1214252DOI10.1109/9.210146
  27. Meyer R. A., Burrus C. S., 10.1109/TCS.1975.1084020, IEEE Trans. Circuit and Systems 22 (1975), 162–168 (1975) MR0392090DOI10.1109/TCS.1975.1084020
  28. Park B. P., Verriest E. I., Canonical forms on discrete linear periodically time-varying systems and a control application, In: Proc. 28th IEEE Conference on Decision and Control, Tampa 1989, pp. 1220–1225 (1989) MR1038997
  29. Park B. P., Verriest E. I., Time-frequency transfer function and realization algorithm for discrete periodic linear systems, In: Proc. 32th IEEE Conference on Decision and Control, San Antonio 1993, pp. 2401–2402 (1993) 
  30. Rosenbrock H. H., State-space and Multivariable Theory, Nelson, London 1970 Zbl0246.93010MR0325201
  31. Sanchez E., Hernandez, V., Bru R., Minimal realization of discrete-time periodic systems, Linear Algebra Appl. 162–164 (1992), 685–708 (1992) MR1148426
  32. Wolovich W. A., Guidorzi R., 10.1016/0005-1098(77)90056-5, Automatica 13 (1977), 295–299 (1977) Zbl0358.93008DOI10.1016/0005-1098(77)90056-5

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.