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.