Discrete-time state representations, a new paradigm

  • S. Monaco
  • D. Normand-Cyrot


Continuing the study of discrete-time dynamical systems, a new state representation of discrete-time dynamics is proposed, whose development has been largely inspired by pioneering works of I. D. Landau. Breaking with the traditional idea of modelling discrete-time processes as difference equations we here propose to represent discrete-time dynamics as coupled difference and differential equations. The difference equation models, through jumps, the free evolution or a nominal one associated with some constant control value while the differential equation models the influence of the control. The differential structure is at the basis of a geometric study which enables us to stress some intriguing analogies with continuous-time, in particular regarding the control dependency of the dynamics compared to the time dependency. The present paper aims at convincing the reader that such a differential modelling might be the proper way to define and represent forced discrete-time behaviours. This would completely renew the understanding and the methodologies used when discussing discrete-time dynamics in modelling or control contexts.


Vector Field Differential Equation Model Free Evolution Differential Representation Exponential Representation 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A.A. Agracev and R.V. Gramkrelidge, Exponential representations of flows and the chronological calculus, Math. Sb., 107, 4, 1978, pp. 467–532.Google Scholar
  2. 2.
    F. Albertini and E.D. Sontag, Discrete-time transitivity and accessibility: analytic systems, SIAM J. Cont. Opt., 33, 1993, pp. 1599–1622.MathSciNetCrossRefGoogle Scholar
  3. 3.
    C. Califano, S. Monaco and D. Normand-Cyrot, Nonlinear decoupling with stability via static state feedback for a class of discrete-time systems, Proc. IEEE-CDC, 1997.Google Scholar
  4. 4.
    C. Califano, S. Monaco and D. Normand-Cyrot, A note on the discrete-time normal form, IEEE Trans. on A.C., 1998.Google Scholar
  5. 5.
    M. Fliess, Automatique en temps discret et algèbre aux différences, Forum Math., 2, 1990, pp. 213–232.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    J.W. Grizzle, A linear algebraic framework for the analysis of discrete time nonlinear systems, SIAM J. on Cont. Opt., 31, 1993, pp. 1026–44.MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    B. Jakubczyk and E.D. Sontag, Controllability of nonlinear discrete time systems; a Lie algebraic approach, SIAM J. Cont. and Opt., 28, 1990.Google Scholar
  8. 8.
    S. Monaco and D. Normand Cyrot, Invariant distributions for discrete time nonlinear systems, Syst. and Cont. Lett., 5, 1985, pp. 191–196.CrossRefGoogle Scholar
  9. 9.
    S. Monaco and D. Normand-Cyrot, Functional expansions for nonlinear discrete time systems, Math. Syst. Th., 21, 1989, pp. 235–254.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    S. Monaco and D. Normand-Cyrot, A unifying representation for nonlinear discrete-time and sampled dynamics, J. of Math. Syst., Est. and Cont., pp. 101–103 (summary) 1995, vol.7, 4, 1997, 477–503.Google Scholar
  11. 11.
    S. Monaco and D. Normand-Cyrot, Geometric properties of a class of nonlinear discrete-time dynamics, Proc. ECC-97, 1, TU-E-05, Brussels, 1997.Google Scholar
  12. 12.
    S. Monaco and D. Normand-Cyrot, On the conditions of passivity and losslessness in discrete-time, Proc ECC-97, 3, WE-A-J4, Brussels, 1997.Google Scholar
  13. 13.
    S. Monaco and D. Normand-Cyrot, Differential representations with jumps for discrete-time dynamics, Tech. Rep. 39–97, Dipt. Informatica e Sistemistica, University of Rome “La Sapienza”, 1997.Google Scholar
  14. 14.
    S. Muratori and S. Rinaldi, Structural properties of controlled population models, Syst. and Cont. Lett., 10, 1996, pp. 147–153.MathSciNetGoogle Scholar
  15. 15.
    L.A.B. San Martin, On global controllability of discrete-time control systems, MCSS 8, 1995, pp. 279–297.MATHGoogle Scholar

Copyright information

© Springer-Verlag London Limited 1998

Authors and Affiliations

  • S. Monaco
    • 1
  • D. Normand-Cyrot
    • 2
  1. 1.Dipartimento di Informatica e SistemisticaUniversità di Roma “La Sapienza”RomeItaly
  2. 2.Laboratoire des Signaux et SystèmesCNRS-ESEGif sur YvetteFrance

Personalised recommendations