On the complete "discretization" of an nth-order linear differential equation model to obtain the exact nth-order difference equation model with correct "initial-sequence values"

The conversion of a given n(th)-order ordinary differential-equation model, with a stepwise-constant input, to an "equivalent" n(th)-order difference-equation model is an important procedure in many engineering applications, particularly in discrete-time/digital control theory for linear dynamical systems. That procedure, called "discretization", is riot complete unless the given initial-conditions of the differential-equation model are properly incorporated into the corresponding "initial-sequence values" associated with the difference-equation model, The literature of discrete-time/digital control theory appears to be consistently incomplete in this latter regard. In this paper we derive the complete and exact discretization of an arbitrary n(th)-order linear, constant-coefficient, non-homogeneous ordinary differential equation model, with arbitrary initial-conditions and a stepwise-constant input, to obtain the corresponding exact equivalent n(th) - order, linear, constant-coefficient, non-homogeneous difference equation model with correct initial-sequence values.