Advertisement

Algebraic-Geometric Techniques for Linear Periodic Discrete-Time Systems

  • Osvaldo Maria Grasselli
  • Sauro Longhi
Part of the Progress in Systems and Control Theory book series (PSCT, volume 3)

Abstract

The analysis and control of linear periodic discrete-time systems have been approached both through geometric and algebraic techniques. They are described here, together with the main problems solved through them. The geometric techniques are based on the periodic notions of (A,B)-invariant and (C,A)-invariant subspaces and some special types of such subspaces; e.g. inner (outer) controllable or reachable subspaces. Among the algebraic techniques, notions of pole and zeros of several types are emphasized for such systems. A time-invariant matrix mechanism allows to deal both with the geometric and the algebraic approaches.

Keywords

Invariant Subspace Structural Index Parallel Connection Bilinear System Invariant Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M. Araki and K. Yamamoto (1986). Multivariate multirate sampled-data systems: state space description, transfer characteristics, and Nyquist criterion, IEEE Trans. Aut. Control. 31, 145–154.CrossRefGoogle Scholar
  2. [2]
    S. Bittanti (1986). Deterministic and stochastic linear periodic systems, in Bittanti, S. (Ed.), Time Series and Linear Systems. Springer-Verlag, Berlin, pp. 141–182.CrossRefGoogle Scholar
  3. [3]
    P. Bolzern, P. Colaneri and R. Scattolini (1986). Zeros of discrete-time linear periodic systems, IEEE Trans. Autom. Control. 30, 1057–1058.CrossRefGoogle Scholar
  4. [4]
    C.T. Chen (1984). Linear System Theory and Design. New York: Holt, Rinehart and Winston.Google Scholar
  5. [5]
    C.T. Chen and C.A. Desoer (1967). Controllability and obervability of composite systems, IEEE Trans. Aut. Control. 12, 402–409.CrossRefGoogle Scholar
  6. [6]
    D.S. Evans (1972). Finite-dimensional realizations of discrete-time weighting patterns, SIAM Appl. Math., 22, 45–67.CrossRefGoogle Scholar
  7. [7]
    B.A. Francis and T.T. Georgiou (1988). Stability theory for linear time-invariant plants with periodic digital controllers, IEEE Trans. Autom. Control. 33, 820–832.CrossRefGoogle Scholar
  8. [8]
    O.M. Grasselli (1972). Controllability and obervability of series connections of systems. Ricerche di Automatica, 3, 44–53.Google Scholar
  9. [9]
    O.M. Grasselli (1980). Conditions for controllability and reconstructibility of discrete-time linear composite systems, Int. J. Control, 31, 433–441.CrossRefGoogle Scholar
  10. [10]
    O.M. Grasselli (1984). A canonical decomposition of linear periodic discrete- time systems, Int. J. Control, 40, 201–214.CrossRefGoogle Scholar
  11. [11]
    O.M. Grasselli (1984). Dead-beat observers of reduced order for linear periodic discrete-time systems with inaccessible inputs, Int. J. Control, 40, 731–745.CrossRefGoogle Scholar
  12. [12]
    O.M. Grasselli, A. Isidori and F. Nicolo’(1979). Output regulation of a class of bilinear systems under constant disturbances, Automatica, 15, 189–195.CrossRefGoogle Scholar
  13. [13]
    O.M. Grasselli, A. Isidori and F. Nicolo’ (1980). Dead-beat control of discrete-time bilinear systems, Int. J. Control, 12, 31–39.CrossRefGoogle Scholar
  14. [14]
    O.M. Grasselli and F. Lampariello (1981). Dead-beat control of linear periodic discrete-time systems. Int. J. Control, 33, 1091–1106.CrossRefGoogle Scholar
  15. [15]
    O.M. Grasselli and S. Longhi (1983). On the stabilization of a class of bilinear systems, Int. J. Control, 37, 413–420.CrossRefGoogle Scholar
  16. [16]
    O.M. Grasselli and S. Longhi (1985). Sottospazi invarianti controllati e sottospazi di controllabilità per sistemi lineari periodici a tempo discreto. RI 2/85. Dipart. Elettr. e Autom., Univ. di Ancona.Google Scholar
  17. [17]
    O.M. Grasselli and S. Longhi (1986). Output dead-beat controllers and function dead-beat observers for linear periodic discrete-time systems, Int. J. Control. 43. 517–537.CrossRefGoogle Scholar
  18. [18]
    O.M. Grasselli and S. Longhi (1986). Disturbance localization with dead-beat control for linear periodic discrete-time systems, Int. J. Control, 44,1319–1347.CrossRefGoogle Scholar
  19. [19]
    O.M. Grasselli and S. Longhi (1986). Zeri e poli di sistemi lineari periodici a tempo discreto. RI 6/86 Dipart. di Elettronica e Automatica, Università di Ancona.Google Scholar
  20. [20]
    O.M. Grasselli and S. Longhi (1987). Linear function dead-beat observers with disturbance localization for linear periodic discrete-time systems, Int. J. Control, 45, 1603–1627.CrossRefGoogle Scholar
  21. [21]
    O.M. Grasselli and S. Longhi (1988). Disturbance localization by measurement feedback for linear periodic discrete-time systems, Automatica, 24, 375–385.CrossRefGoogle Scholar
  22. [22]
    O.M. Grasselli and S. Longhi (1988). Zeros and poles of linear periodic multivariable discrete-time systems, Circuits Systems Signal Processing, 7, 361–380.CrossRefGoogle Scholar
  23. [23]
    O.M. Grasselli and S. Longhi (1988). Transmission zeros, poles and invariant zeros of linear periodic multivariable discrete-time systems, in C.I Byrnes, C.F. Martin and R.E. Saeks (Eds), Linear Circuits. Systems and Signal Processing: Theory and Application, North-Holland, Amsterdam, pp. 293–300.Google Scholar
  24. [24]
    O.M. Grasselli and S. Longhi (1989). Eigenvalue assignment for linear periodic discrete-time non-reachable systems. Proc. IEEE Int. Conf. on Control and Appl., Jerusalem (Israel), paper no. RA-2-4.Google Scholar
  25. [25]
    O.M. Grasselli and S. Longhi (1989). Input and output decoupling zeros of linear periodic discrete-time systems. Rap. Int. n. 89–06 Dipart. di Ingegneria Elettronica, Seconda Universita’ di Roma “Tor Vergata”; Preprints of IFAC Workshop on System Structure and Control: State-Space and Polynomial Methods. Prague, 215–218.Google Scholar
  26. [26]
    O.M. Grasselli and S. Longhi (1989). Algebraic-geometric techniques for linear periodic discrete-time systems. Rap. Int. n. 89–07 Dipart. di Ingegneria Elettronica, Seconda Universita’ di Roma “Tor Vergata”.Google Scholar
  27. [27]
    O.M. Grasselli and S. Longhi (1989). Robust tracking and regulation of linear periodic discrete-time systems. Rap. Int. n. 89-08 Dipart. di Ingegneria Elettronica, Seconda Universita’ di Roma “Tor Vergata”.Google Scholar
  28. [28]
    T. Kaczorek (1985). Pole placement for linear discrete-time systems by periodic output-feedback, Syst. Control Lett., 6, 267–269.CrossRefGoogle Scholar
  29. [29]
    P.P. Khargonekar, K. Polla and A. Tannenbaum (1985). Robust control of linear time-invariant plants using periodic compensation, IEEE Trans. Autom. Control, 30, 1088–1096.CrossRefGoogle Scholar
  30. [30]
    M. Kono (1980). Eigenvalue assignment in linear periodic discrete-time systems, Int. J. Control. 32, 149–158.CrossRefGoogle Scholar
  31. [31]
    M.L. Liou and Y.L. Kuo (1979). Exact analysis of switched capacitor circuits with arbitrary inputs, IEEE Trans. Ccts Syst., 26, 213–223.CrossRefGoogle Scholar
  32. [32]
    S. Longhi, A. Perdon and G. Conte (1989). Geometric and algebraic structure at infinity of discrete-time linear periodic systems, Linear Algebra and Applications. Special Issue, (to appear).Google Scholar
  33. [33]
    R.A. Meyer and C.S. Burrus (1975). A unified analysis of multirate and periodically time-varying digital filters, IEEE Trans. Circuits Syst., 22, 162–168.CrossRefGoogle Scholar
  34. [34]
    A.V. Olbrot (1987). Robust stabilization of uncertain systems by periodic feedback, Int. J. Control. 45, 747–758.CrossRefGoogle Scholar
  35. [35]
    J.A. Richards (1983). Analysis of Periodically Time-Varying Systems. Springer.Google Scholar
  36. [36]
    H.H. Rosenbrock (1970). State-space and multivariable theory. Nelson, London.Google Scholar
  37. [37]
    C.B. Schrader and M.K. Sain (1988). Research on system zeros: a survey. Proc. 27th IEEE Confer, on Decis. and Control. Austin (Texas), 890–901.Google Scholar
  38. [38]
    J.H. Schumacher (1980). Compensator synthesis using (C,A,B)-pairs, IEEE Trans. Autom. Control, 25, 1133–1138.CrossRefGoogle Scholar
  39. [39]
    E.I. Verriest (1988). The operational transfer function and parametrization of N-periodic systems. Proc. 27th IEEE Confer. on Decision and Control, Austin (Texas), 1994–1999.Google Scholar
  40. [40]
    J.L. Willems, V. Kucèra and P. Brunovsky (1984). On the assignment of invariant factors by time-varying feedback strategies, Syst. Control Lett.. 5,75–80.CrossRefGoogle Scholar
  41. [41]
    W.M. Wonham (1979). Linear Multivariable Control: A geometric Approach. Springer-Verlag.Google Scholar

Copyright information

© Birkhäuser Boston 1990

Authors and Affiliations

  • Osvaldo Maria Grasselli
  • Sauro Longhi

There are no affiliations available

Personalised recommendations