Stochastic Realization of Stationary Processes: State-Space, Matrix Fraction and ARMA Forms

  • L. Keviczky
  • J. Bokor
Part of the Progress in Systems and Control Theory book series (PSCT, volume 2)


This paper discusses results from the stochastic realization theory of second order stochastic process. The forward and backward stochastic state-space representation are derived and transfer relations are given to obtain their associated matrix fraction description and ARMA forms. The correspondence among these realizations are elaborated.


SIAM Journal White Noise Process State Space Representation Hankel Matrix Transfer Relation 
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.


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  1. [1]
    Akaike, H. (1974), Stochastic Theory of Minimal Realization, IEEE Transactions on Automatic Control AC-19, pp. 667–674.CrossRefGoogle Scholar
  2. [2]
    Akaike, H. (1975), Markovian Representation of Stochastic Processes by Canonical Variables, SIAM Journal of Control 13, pp. 162–173.CrossRefGoogle Scholar
  3. [3]
    Anderson, B.D.O. (1967), A System Theory Criterion for Positive Real Matrices, SIAM Journal of Control 5, pp. 171–172.Google Scholar
  4. [4]
    Anderson, B.D.O. (1967), An Algebraic Solution to the Spectral Factorization Problem, IEEE Transactions on Automatic Control AC-12, pp. 410–414.CrossRefGoogle Scholar
  5. [5]
    Anderson, B.D.O., and T. Kailath (1979), Forwards, Backwards and Dynamically Reversible Markovian Models of Second-Order Processes, IEEE Transactions on Circuits and Systems CAS-26, pp. 956–965.CrossRefGoogle Scholar
  6. [6]
    Bokor, J. and L. Keviczky (1982), Structural Properties and Structure Determination of Vector Difference Equations, International Journal of Control 36, pp.461–475.CrossRefGoogle Scholar
  7. [7]
    Bokor, J. and L. Keviczky (1987), ARMA Canonical Forms Obtained from Constructibility Invariants, International Journal of Control 45, pp.861–873.CrossRefGoogle Scholar
  8. [8]
    Bokor J., Cs. Bányász and L. Keviczky (1989) Realization of Stochastic Processes in State-Space MFD and ARMA Forms, in “Proceedings of the IEEE Conference on Control and Applications, ICCON ’89,” Jerusalem, pp. WC-5–4.Google Scholar
  9. [9]
    Caines, P.E. (1988), “Linear Stochastic Systems,” John Wiley & Sons.Google Scholar
  10. [10]
    Caines, P.E. and D. Delchamps (1980), Splitting Subspaces, Spectral Factorization and the Positive Real Equation: Structural Features of the Stochastic Realization Problem, in “Proceedings of the IEEE Conference on Decision and Control,” Albuquerque, NM, pp. 358–362.Google Scholar
  11. [11]
    Deistler, M. (1983), The Structure of ARMA Systems and its Relation to Estimation, in “Geometry and Identification,” Caines, P.E. and R. Hermann (Eds.), Mathematical Sciences Press, Brookline, MA, pp. 49–61.Google Scholar
  12. [12]
    Deistler, M. and M. Gevers (1989), Properties of the Parametrization of Monic ARMAX Systems, Automatica 25, pp.87–96.CrossRefGoogle Scholar
  13. [13]
    Desai, U.B., and Pal, D. (1984), A Transformation Approach to Stochastic Model Reduction, IEEE Transactions on Automatic Control AC-29, pp.1097–1100.CrossRefGoogle Scholar
  14. [14]
    Faurre, P.L. (1976), Stochastic Realization Algorithms, in “System Identification: Advances and Case Studies,” Mehra, R.K. and D.R. Lainiotis (Eds.), Academic Press, New York, pp. 1–25.CrossRefGoogle Scholar
  15. [15]
    Forney, G.D. (1975), Minimal Bases of Rational Vector Spaces, with Applications to Multivariable Linear Systems, SIAM Journal of Control 13, pp.493–520.CrossRefGoogle Scholar
  16. [16]
    Gevers, M. and V. Wertz (1984), Uniquely Identifiable State-Space and ARMA Parametrization for Multivariable Linear Systems, Automatica 20, pp.333–347.CrossRefGoogle Scholar
  17. [17]
    Gevers, M. (1986), ARMA Models, Their Kronecker Indices and Their McMillan Degree, International Journal of Control 43, pp.1745–1761.CrossRefGoogle Scholar
  18. [18]
    Gevers, M. and V. Wertz (1987), Techniques for the selection of identifiable parametrizations for multivariable linear systems, in “Control and Dynamic Systems, Vol. XXVI,” C.T. Leondes (Ed.), Academic Press, pp. 35–86.Google Scholar
  19. [19]
    Guidorzi, R.P. (1975), Canonical Structures in The Identification of Multivariable Systems, Automatica 11, pp.361–374.CrossRefGoogle Scholar
  20. [20]
    Guidorzi, R.P. (1981), Invariants and Canonical Forms for Systems: Structural and Parametric Identification, Automatica 17, pp.117–133.CrossRefGoogle Scholar
  21. [21]
    Hannan, E.J., and M. Deistler (1988), “The Statistical Theory of Linear Systems,” John Wiley & Sons, New York.Google Scholar
  22. [22]
    Ho, L. and R.E. Kalman (1965), Effective Construction of Linear State Variable Models from Input/Output Data, in “Proceedings of the 3rd Allerton Conference,” pp. 449–459.Google Scholar
  23. [23]
    Janssen, P. (1987), MFD Models and Time Delays; Some Consequences for identification, International Journal of Control 45, pp.1179–1196.CrossRefGoogle Scholar
  24. [24]
    Janssen, P. (1988), General Results on the McMillan Degree and Kronecker Indices of ARMA and MFD models, To appear in the International Journal of Control.Google Scholar
  25. [25]
    Kailath, T. (1980), “Linear Systems,” Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
  26. [26]
    Kalman, R.E., P. Falb and M. Arbib (1969), “Topics in Mathematical System Theory,” McGraw-Hill, New York.Google Scholar
  27. [27]
    Lindquist, A., and G. Picci (1977), On the structure of minimal splitting subspaces in Stochastic realization Theory, in “Proceedings of the Conference on Decision and Control New Orleans, La.,” pp. 42–48.Google Scholar
  28. [28]
    Lindquist, A., and G. Picci (1979), On the Stochastic Realization Problem, SIAM Journal of Control Theory 17, pp.365–389.CrossRefGoogle Scholar
  29. [29]
    Lindquist, A., and G. Picci (1981), State Space Models for Gaussian Stochastic Processes, in “Stochastic Systems: The Mathematics of Filtering and Identification and Applications,” Hazewinkel, M. and J.C. Willems (Eds.), Reidel, Dordrecht, pp. 169–204.CrossRefGoogle Scholar
  30. [30]
    Lindquist, A., and Pavon, M. (1984), On the Structure of State-Space Models for Discrete-Time Stochastic Vector Processes, IEEE Transactions on Automatic Control AC-29, pp.418–432.CrossRefGoogle Scholar
  31. [31]
    Ljung, L. and T. Kailath (1976), Backwards Markovian Models for Second-Order Stochastic Processes, IEEE Transactions on Information Theory IT-22, pp.488–491.CrossRefGoogle Scholar
  32. [32]
    Pavon, M. (1980), Stochastic Realizations and Invariant Directions of Matrix Riccati Equations, SIAM Journal of Control and Optimatization 18, pp.155–180.CrossRefGoogle Scholar
  33. [33]
    Picci, G. (1976), Stochastic Realization of Gaussian Processes, Proc. IEEE 64, pp.112–122.CrossRefGoogle Scholar
  34. [34]
    Popov, V.M. (1972), Invariant Description of Linear Time Invariant Controllable Systems, SIAM Journal of Control 10, pp.252–264.CrossRefGoogle Scholar
  35. [35]
    Rozanov, Yu. A. (1967), “Stacionary Random Processes,” Holden-Day, San-Francisco.Google Scholar
  36. [36]
    Sidhu, G.S., and Desai, U.B. (1976), New Smoothing Algorithms Based on Reversed-time Lumped Model, IEEE Transactions on Automatic Control, pp.538–541.Google Scholar
  37. [37]
    Verghese, G., and Kailath, T.(1979), A Further Note on Backwards Markovian Models, IEEE Transactions on Information Theory IT-25, 121–124; Correction to the above paper in ibid, IT-25, p. 50.CrossRefGoogle Scholar
  38. [38]
    Wiener, N. (1958), The Prediction Theory of Multivariate Stochastic Processes, Part I: The regularity condition, Acta Mathematica 98, pp.111–150;CrossRefGoogle Scholar
  39. [38a]
    Wiener, N. (1958), The Prediction Theory of Multivariate Stochastic Processes, Part II: The linear predictor, Acta Mathematica 99, pp.93–137.CrossRefGoogle Scholar
  40. [39]
    Willems, J.C. (1986), From Time Series to Linear Systems, Part I, Automatica 22, pp.561–580.CrossRefGoogle Scholar
  41. [39a]
    Willems, J.C. (1986), From Time Series to Linear Systems, Part II, Automatica 22, pp.675–694.CrossRefGoogle Scholar
  42. [39b]
    Willems, J.C. (1986), From Time Series to Linear Systems, Part III, Automatica 23, pp.87–115.CrossRefGoogle Scholar
  43. [40]
    Wolovich, W.A. (1974), “Linear Multivariable Systems,” Applied Mathematical Sciences, No. 11, Springer- Verlag, New York.CrossRefGoogle Scholar
  44. [41]
    Wolovich, W.A. and H. Elliott (1983), Discrete Models for Linear Multivariable Systems, International Journal of Control 38, pp.337–357.CrossRefGoogle Scholar
  45. [42]
    Youla, D.C. (1961), On the factorization of Rational Matrices, IEEE Transactions on Information Theory IT-7, pp.172–189.CrossRefGoogle Scholar

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© Springer Science+Business Media New York 1990

Authors and Affiliations

  • L. Keviczky
  • J. Bokor

There are no affiliations available

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