  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(greater than 1)-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 fsun(), 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. In the back part of this
paper, we analyze the features of various components of 2nd-order active
disturbance rejection control (ADRC) technology, and then by constructing the
new type structures of extended state observer (ESO) and feedback controller
which satisfy the separation principle, we construct a new active disturbance
rejection control theory (ADRCT) based on the m-order tracking-form synthesis
function with filter factor. Finally, the numerical simulation shows that the
new ADRCT has good immunity and robustness.
