On the complete "discretization" of an n(th)-order linear differential equation model to obtain the exact n(th)-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 s ystems. That procedure, called "discretization", is riot complete unless th e given initial-conditions of the differential-equation model are properly incorporated into the corresponding "initial-sequence values" associated wi th the difference-equation model, The literature of discrete-time/digital c ontrol theory appears to be consistently incomplete in this latter regard. In this paper we derive the complete and exact discretization of an arbitra ry n(th)-order linear, constant-coefficient, non-homogeneous ordinary diffe rential equation model, with arbitrary initial-conditions and a stepwise-co nstant input, to obtain the corresponding exact equivalent n(th) - order, l inear, constant-coefficient, non-homogeneous difference equation model with correct initial-sequence values.