Information geometry of system spaces

For the totality of square and regular transfer function matrices, basic differential geometrical structures such as Riemannian metrics and linear connections are introduced from the viewpoint of left and right invariance. These structures are successively introduced into the outer and inner system spaces, and then into the second and higher order cumulant spectral density spaces. The dualities of the derived geometrical structures with torsions and curvatures are investigated, further the possibilities for the existences of divergences are explored for the respective spaces. As an application of the information geometrical structures, we analyze the block oriented nonlinear output feedback mechanisms, and clarify the roles of the associated sensitivity operators. We also obtain the conditions with desired output feedback rules for which the divergences attain the ultimate zero values. Illustrative examples are given to the readers for the block oriented nonlinear output feedback expressions.