Differential geometry of a parametric family of invertible linear systems—Riemannian metric, dual affine connections, and divergence

Abstract

A parametric model of systems is regarded as a geometric manifold imbedded in the enveloping manifold consisting of all the linear systems. The present paper aims at establishing a new geometrical method and framework for analyzing properties of manifolds of systems. A Riemannian metric and a pair of dual affine connections are introduced to a system manifold. They are calledα-connections. The duality of connections is a new concept in differential geometry. The manifold of all the linear systems isα-flat so that it admits natural and invariantα-divergence measures. Such geometric structures are useful for treating the problems of approximation, identification, and stochastic realization of systems. By using theα-divergences, we solve the problem of approximating a given system by one included in a model. For a sequence ofα-flat nesting models such as AR models and MA models, it is shown that the approximation errors are decomposed additively corresponding to each dimension of the model.