Abstract
The theory of Markovian Representation is set forth in an abstract Hilbert space setting. This enables us to completely characterize the set of all (finite or infinite-dimensional) Markovian Representations of a given (stationary) Gaussian process. The paper discusses when Markovian Representations can be studied by means of functional models in the Hardy space. It is shown that it is the case for all the Minimal Markovian Representations of a strictly noncyclic process. Hence, new results on their structure can be derived.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Reference
Douglas,R.G., Canonical ModelsJn Topics in Operator Theory, Math. Surveys No. 13, American Math. Society, Providence, 1974, pp. 161–218.
Fuhrmann,P.A., Linear Systems and Operators in Hilbert Space, Mc Graw-Hill, 1981.
Helton,J.W., Systems with Infinite-Dimensional State Space: The Hilbert Space Approach, Proc. IEEE, Trans. Autom. Cont., vol. 64, No. 1, January 1976, pp. 145–160.
Kalman,R.E., Lectures on Controllability and Observability, C.I.M.E. Summer Course, 1968, Cremonese, Roma, 1969, pp. 1–149.
Lax,P.D., and Phillips,R.S., Scattering Theory, Academic Press, 1967.
Lindquist,A., Mitter,S., and Picci,G., Toward a Theory of Nonlinear Stochastic Realization, in Feedback and Synthesis of Linear and Nonlinear Systems, Eds. D. Hinrichsen and A. Isidori, Springer Verlag.
Lindquist,A., Pavon,M., and Picci,G., Recent Trends in Stochastic Realization Theory, in Harmonic Analysis and Prediction Theory. The Masani Volume, Eds. V. Mandrekar and H. Salehi, North Holland.
Lindquist,A., and Picci,G., On a Condition for Minimality of Markovian Splitting Subspaces, Systems and Controls Letters, to be published.
Moore,B.,III, and Nordgren,E.A., On Quasi-Equivalence and Quasi-Similarity, Acta Sci. Math., vol. 34, 1973, pp. 311–316.
Neveu,J., Processus Aleatoires Gaussiens, Presses de I’Universite de Montreal, 1968.
Nordgren,E. A., On quasi-equivalence of matrices over H ∞, Acta Sci. Math., vol. 34, 1973, pp. 301–310.
Rozanov,Yu.A., Stationary Random Processes, Holden Day, 1967.
Ruckebusch,G., Théorie Géométrique de la Représentation Markovienne, Ann. Inst. Henri Poincare, vol. 16,No. 3, 1980, pp. 225–297.
Ruckebusch G., Markovian Representation Theory and Hardy spaces, Proc. IEEE, International Symposium on Cifcuits and Systems, Rome, May 1982.
Ruckebusch,G., Markovian Representations and Spectral Factorizations of Stationary Gaussian Processes, in Harmonic Analysis and Prediction Theory The Masani Volume, Eds. V. Mandrekar and H. Salehi, North Holland.
Sz. Nagy,B., and Foias,C., Harmonic Analysis of Operators on Hilbert Space, North Holland, 1970.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1983 D. Reidel Publishing Comapany
About this chapter
Cite this chapter
Ruckebusch, G. (1983). On the Structure of Minimal Markovian Representations. In: Bucy, R.S., Moura, J.M.F. (eds) Nonlinear Stochastic Problems. NATO ASI Series, vol 104. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-7142-4_10
Download citation
DOI: https://doi.org/10.1007/978-94-009-7142-4_10
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-009-7144-8
Online ISBN: 978-94-009-7142-4
eBook Packages: Springer Book Archive