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Differential geometry of a parametric family of invertible linear systems. Riemannian metric, dual affine connections, and divergence. (English) Zbl 0632.93017
The paper reports on a nice crossfertilization of the theory of linear, discrete-time control systems and differential-geometric statistics. A Hilbert manifold of linear, SISO, minimum phase systems is defined via the spectral density function, within which some finite-dimensional families of systems are considered. The main idea of the paper lies in introducing into the manifold differential geometric structures allowing to solve the approximation problem of a given system by a system from a finite-dimensional family, using an analog of the orthogonal projection. In doing this there have been defined on the system manifold a Riemannian metric, connections, geodesics and a divergence function, investigated basic properties of those objects, and provided a sound solution to the approximation problem. Being attractive mathematically, the paper offers a new approach to some systemic problems like approximation (order reduction), identification, stochastic realization, etc.
Reviewer: K.Tchon

MSC:
93B27 Geometric methods
93C05 Linear systems in control theory
93C55 Discrete-time control/observation systems
53C05 Connections, general theory
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C22 Geodesics in global differential geometry
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
60G15 Gaussian processes
93B15 Realizations from input-output data
93E12 Identification in stochastic control theory
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