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Finite Dimensional Linear Stochastic System Identification

  • P. E. Caines

Abstract

The use of state space systems to represent dynamical systems of various types has a long history in mathematics and physics. Two fundamental examples are the well-known methods for analysing high order ordinary differential equations via the corresponding first order equations and the formulation of Hamiltonian mechanics—within which classical celestial mechanics forms a magnificent special case. Equally, in theoretical engineering, including the information and control sciences, state space methods have played and continue to play a fundamental rôle. From a mathematical viewpoint, the pure state evolution models of dynamical system theory undergo an enormous generalization in control theory to input-output systems, which then fall into equivalence classes according to their input-output behavior. Furthermore, from an engineering viewpoint, a vast array of problems are either initially posed in state space terms (aerospace engineering provides many examples) or are first presented in terms of input-output models which are then transformed into input-stateoutput form (process control being one source of examples). As a result, one of the basic subjects of study in mathematical system theory is the relationship between input-state-output systems and input–output systems. The generality of this relationship is conveyed by the Nerode Theorem which states that any non-anticipative (set-valued) input—(set valued) output system S possesses a state space realization Σ(S), that is to say, there exists an input-state-output system Σ(S) which generates the same input-output trajectories as S.

Keywords

Asymptotic Normality Hankel Matrix State Space System State Space Realization Finite Dimensional Distribution 
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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Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • P. E. Caines
    • 1
  1. 1.Canadian Institute for Advanced Research and Department of Electrical EngineeringMcGill UniversityMontréalCanada

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