Mappings of the finite and infinite zero structures and invertibility structures of general linear multivariable systems under the bilinear transformation

This paper presents a comprehensive picture of the mapping of structural properties associated with general linear multivariable systems under bilinear transformation. While the mapping of poles of linear multivariable systems under such a transformation is well known, the question of how the structural invariant properties of a given system are mapped remains unanswered. This paper addresses this question. More specifically, we investigate how the finite and infinite zero structures, as well as invertibility structures, of a general continuous-time (discrete-time) linear time-invariant multivariable system are mapped to those of its discrete-time (continuous-time) counterpart under the bilinear (inverse bilinear) transformation. We demonstrate that the structural invariant indices lists I₂ and I₃ of Morse remain invariant under the bilinear transformation, while the structural invariant indices lists I₁ and I₄ of Morse are, in general, changed