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Differential geometric structures of stable state feedback systems with dual connections. (English) Zbl 0821.93029
Stabilizing feedbacks, for an \(n\)-dimensional linear system with \(m\) inputs, are diffeomorphic to a subbundle of \(PD(n) \times \text{Skew}(n)\) where \(PD(n)\) is the manifold of symmetric positive definite real matrices and \(\text{Skew}(n)\) is the vector space of skew-symmetric real matrices. A fiber metric on \(PD(n) \times \text{Skew}(n)\), invariant under system equivalence is introduced. Two mutually dual connections are shown to exist; the inner product (using the new metric) of two vectors of the fibers is preserved for two parallel displacements of each of these vectors. For the standard parallel displacement, the linear structure of \(PD(n) \times \text{Skew}(n)\) and the linearity of the equations defining the subbundle imply its flatness. It is shown it is also flat with respect to the dual connection. Stable closed-loop matrices are also described in terms of metrics on objects parametrizing them. This article allows to use the concept of divergence introduced by the second author in the context of statistical manifolds.

MSC:
93B29 Differential-geometric methods in systems theory (MSC2000)
93B52 Feedback control
93D15 Stabilization of systems by feedback
53B21 Methods of local Riemannian geometry
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
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