$\mathbb {R}$-Complex Finsler Information Geometry Applied to Manifolds of Systems

Abstract

In this article information geometry is applied to the models of linear discrete time-invariant systems $\mathcal {M}_S^{LTI}$ with external input and zeros outside the unit disc of the z-plane. A new information geometry of manifolds of systems based $\mathbb {R}$-Complex Finsler spaces with three metric tensors is presented. First, two metric tensors in Hermitian spaces $\mathcal {H}_{\delta }$ where $\mathcal {H}_{-1}$ corresponds to the zeros outside the unit disc (exogenous zeros) and $\mathcal {H}_{+1}$ to the zeros and poles inside the unit disc (endogenous zeros-poles). Then, one metric tensor as a mixing of exogenous zeros and endogenous zeros-poles in a non-Hermitian space $\overline{\mathcal {H}}$. Experimental results are presented from a semi-finite acoustic waves guide.