Differential geometry of a parametric family of invertible linear systems—Riemannian metric, dual affine connections, and divergence
Copyright policy )Differential geometry of a parametric family of invertible linear systems—Riemannian metric, dual affine connections, and divergence
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.
- University of Tokyo Japan
Identification in stochastic control theory, stochastic realization, Gaussian processes, order reduction, Methods of global Riemannian geometry, including PDE methods; curvature restrictions, time-invariant, approximation problem, Discrete-time control/observation systems, Riemannian metric, Realizations from input-output data, Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds, differential-geometric statistics, linear, discrete-time control systems, Geometric methods, Hilbert manifold, Geodesics in global differential geometry, connections, Linear systems in control theory, spectral density, divergence, Connections (general theory), geodesics
Identification in stochastic control theory, stochastic realization, Gaussian processes, order reduction, Methods of global Riemannian geometry, including PDE methods; curvature restrictions, time-invariant, approximation problem, Discrete-time control/observation systems, Riemannian metric, Realizations from input-output data, Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds, differential-geometric statistics, linear, discrete-time control systems, Geometric methods, Hilbert manifold, Geodesics in global differential geometry, connections, Linear systems in control theory, spectral density, divergence, Connections (general theory), geodesics
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