Algorithms, Modeling and Estimation for Linear Systems.
Final technical rept. 1 Apr 79-31 May 83,
KENTUCKY UNIV LEXINGTON DEPT OF MATHEMATICS
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Modeling of linear stochastic systems leads to the stochastic realization problem. In this project the author develops a comprehensive theory of stochastic realization. Such a theory should be the center-piece of stochastic systems theory. First he studies the problem from a coordinate-dependent point of view. Secondly, he develops a geometric theory of Markovian representation, which also accomodates infinite-dimensional systems. In this framework a unified theory of smoothing is provided, the basic idea being to embed the given stochastic system in a class of similar systems all having the same output process and the same Kalman-Bucy filter. This approach provides stochastic interpretations of many important smoothing procedures. The factorizations of the matrix Riccati equation underlying fast non-Riccati algorithms are analized in the context of Hamiltonian systems, and certain aspects of the algebraic Riccati equation are studied, as is the concept of invariant directions of the matrix Riccati equation. A unified approach to the partial realization problem is taken, incorporating ideas from numerical linear algebra. Also studied are questions of stability of partial realizations. Finally, a statistical approach to stochastic optimization is presented, and convergence results for algorithms based on stationary data are obtained.
- Statistics and Probability