Skip to main content
Top

2015 | OriginalPaper | Chapter

9. Proper Markovian Representations in Hardy Space

Authors : Anders Lindquist, Giorgio Picci

Published in: Linear Stochastic Systems

Publisher: Springer Berlin Heidelberg

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

In this chapter we reformulate the splitting geometry in terms of functional models in Hardy space. This allows us to use the power of Hardy space theory to prove several additional results and useful characterizations. We shall only deal with proper Markovian representation, and therefore we formulate several functional criteria for properness.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Literature
78.
go back to reference Douglas, R.G., Shapiro, H.S., Shields, A.L.: Cyclic vectors and invariant subspaces for the backward shift operator. Ann. Inst. Fourier (Grenoble) 20(fasc. 1), 37–76 (1970) Douglas, R.G., Shapiro, H.S., Shields, A.L.: Cyclic vectors and invariant subspaces for the backward shift operator. Ann. Inst. Fourier (Grenoble) 20(fasc. 1), 37–76 (1970)
96.
go back to reference Foiaş, C., Frazho, A.E.: A note on unitary dilation theory and state spaces. Acta Sci. Math. (Szeged) 45(1–4), 165–175 (1983) Foiaş, C., Frazho, A.E.: A note on unitary dilation theory and state spaces. Acta Sci. Math. (Szeged) 45(1–4), 165–175 (1983)
102.
104.
go back to reference Fuhrmann, P.A.: Linear Systems and Operators in Hilbert Space. McGraw-Hill, New York (1981)MATH Fuhrmann, P.A.: Linear Systems and Operators in Hilbert Space. McGraw-Hill, New York (1981)MATH
138.
go back to reference Helson, H.: Lectures on Invariant Subspaces. Academic, New York (1964)MATH Helson, H.: Lectures on Invariant Subspaces. Academic, New York (1964)MATH
145.
go back to reference Hoffman, K.: Banach Spaces of Analytic Functions. Prentice-Hall Series in Modern Analysis. Prentice-Hall Inc., Englewood Cliffs (1962)MATH Hoffman, K.: Banach Spaces of Analytic Functions. Prentice-Hall Series in Modern Analysis. Prentice-Hall Inc., Englewood Cliffs (1962)MATH
180.
go back to reference Lax, P.D., Phillips, R.S.: Scattering Theory. Volume 26 of Pure and Applied Mathematics, 2nd edn. Academic, Boston (1989). With appendices by Cathleen S. Morawetz and Georg Schmidt Lax, P.D., Phillips, R.S.: Scattering Theory. Volume 26 of Pure and Applied Mathematics, 2nd edn. Academic, Boston (1989). With appendices by Cathleen S. Morawetz and Georg Schmidt
193.
go back to reference Lindquist, A., Pavon, M.: On the structure of state-space models for discrete-time stochastic vector processes. IEEE Trans. Autom. Control 29(5), 418–432 (1984)CrossRefMATHMathSciNet Lindquist, A., Pavon, M.: On the structure of state-space models for discrete-time stochastic vector processes. IEEE Trans. Autom. Control 29(5), 418–432 (1984)CrossRefMATHMathSciNet
194.
go back to reference Lindquist, A., Pavon, M., Picci, G.: Recent trends in stochastic realization theory. In: Mandrekar, V., Salehi, H. (eds.) Prediction Theory and Harmonic Analysis, pp. 201–224. North-Holland, Amsterdam (1983) Lindquist, A., Pavon, M., Picci, G.: Recent trends in stochastic realization theory. In: Mandrekar, V., Salehi, H. (eds.) Prediction Theory and Harmonic Analysis, pp. 201–224. North-Holland, Amsterdam (1983)
196.
go back to reference Lindquist, A., Picci, G.: A Hardy space approach to the stochastic realization problem. In: Proceedings of the 1978 IEEE Conference on Decision and Control, San Diego, 1978, pp. 933–939. IEEE, New York (1978) Lindquist, A., Picci, G.: A Hardy space approach to the stochastic realization problem. In: Proceedings of the 1978 IEEE Conference on Decision and Control, San Diego, 1978, pp. 933–939. IEEE, New York (1978)
199.
go back to reference Lindquist, A., Picci, G.: Realization theory for multivariate Gaussian processes I: state space construction. In: Proceedings of the 4th International Symposium on the Mathematical Theory of Networks and Systems, Delft, 1979, pp. 140–148 (1979) Lindquist, A., Picci, G.: Realization theory for multivariate Gaussian processes I: state space construction. In: Proceedings of the 4th International Symposium on the Mathematical Theory of Networks and Systems, Delft, 1979, pp. 140–148 (1979)
200.
go back to reference Lindquist, A., Picci, G.: Realization theory for multivariate Gaussian processes: II: state space theory revisited and dynamical representations of finite-dimensional state spaces. In: Second International Conference on Information Sciences and Systems, University of Patras, Patras, 1979, vol. II, pp. 108–129. Reidel, Dordrecht (1980) Lindquist, A., Picci, G.: Realization theory for multivariate Gaussian processes: II: state space theory revisited and dynamical representations of finite-dimensional state spaces. In: Second International Conference on Information Sciences and Systems, University of Patras, Patras, 1979, vol. II, pp. 108–129. Reidel, Dordrecht (1980)
202.
go back to reference Lindquist, A., Picci, G.: On a condition for minimality of Markovian splitting subspaces. Syst. Control Lett. 1(4), 264–269 (1981/1982) Lindquist, A., Picci, G.: On a condition for minimality of Markovian splitting subspaces. Syst. Control Lett. 1(4), 264–269 (1981/1982)
205.
go back to reference Lindquist, A., Picci, G.: Realization theory for multivariate stationary Gaussian processes. SIAM J. Control Optim. 23(6), 809–857 (1985)CrossRefMATHMathSciNet Lindquist, A., Picci, G.: Realization theory for multivariate stationary Gaussian processes. SIAM J. Control Optim. 23(6), 809–857 (1985)CrossRefMATHMathSciNet
206.
go back to reference Lindquist, A., Picci, G.: A geometric approach to modelling and estimation of linear stochastic systems. J. Math. Syst. Estim. Control 1(3), 241–333 (1991)MathSciNet Lindquist, A., Picci, G.: A geometric approach to modelling and estimation of linear stochastic systems. J. Math. Syst. Estim. Control 1(3), 241–333 (1991)MathSciNet
229.
go back to reference Moore, B., III, Nordgren, E.A.: On quasi-equivalence and quasi-similarity. Acta Sci. Math. (Szeged) 34, 311–316 (1973) Moore, B., III, Nordgren, E.A.: On quasi-equivalence and quasi-similarity. Acta Sci. Math. (Szeged) 34, 311–316 (1973)
274.
go back to reference Ruckebusch, G.: Factorisations minimales de densités spectrales et représentations markoviennes. In: Proceedings of the Colloque AFCET-SMF, Palaiseau (1978) Ruckebusch, G.: Factorisations minimales de densités spectrales et représentations markoviennes. In: Proceedings of the Colloque AFCET-SMF, Palaiseau (1978)
278.
go back to reference Ruckebusch, G.: Markovian representations and spectral factorizations of stationary Gaussian processes. In: Masani, P.R., Mandrekar, V., Salehi, H. (eds.) Prediction Theory and Harmonic Analysis, pp. 275–307. North-Holland, Amsterdam (1983) Ruckebusch, G.: Markovian representations and spectral factorizations of stationary Gaussian processes. In: Masani, P.R., Mandrekar, V., Salehi, H. (eds.) Prediction Theory and Harmonic Analysis, pp. 275–307. North-Holland, Amsterdam (1983)
Metadata
Title
Proper Markovian Representations in Hardy Space
Authors
Anders Lindquist
Giorgio Picci
Copyright Year
2015
Publisher
Springer Berlin Heidelberg
DOI
https://doi.org/10.1007/978-3-662-45750-4_9