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In this paper, first of all, we introduce the basic concepts of generatingfunction in combinatorics and some combinatorial identities. In order tofacilitate the understanding of m(greater than 1)-order time optimal controlsynthesis function of discrete system (referred as m-order synthesis function),secondly, we introduce the derivation process and control ideas of 2nd-ordersynthesis function fsun(), and then deduce in detail the m-order synthesisfunction by means of generating function. By use of the m-order tracking-formsynthesis function with filter factor, the methods of signal extraction and itspredictive compensation are presented in this paper, and their immunity andeffectiveness are verified by numerical simulation. In the back part of thispaper, we analyze the features of various components of 2nd-order activedisturbance rejection control (ADRC) technology, and then by constructing thenew type structures of extended state observer (ESO) and feedback controllerwhich satisfy the separation principle, we construct a new active disturbancerejection control theory (ADRCT) based on the m-order tracking-form synthesisfunction with filter factor. Finally, the numerical simulation shows that thenew ADRCT has good immunity and robustness.

m-Order Time Optimal Control Synthesis Function of Discrete System and Its Application