  In this paper, first of all, we introduce the basic concepts of generating
function in combinatorics and some combinatorial identities. In order to
facilitate the understanding of m-order time optimal control synthesis function
of discrete system (referred as m-order synthesis function), secondly, we
introduce the derivation process and control ideas of 2nd-order synthesis
function, and then deduce in detail the m-order synthesis function by means of
generating function. By use of the m-order tracking-form synthesis function
with filter factor, the methods of signal extraction and its predictive
compensation are presented in this paper, and their immunity and effectiveness
are verified by numerical simulation.
