9.3.THE DISCRETE IMC FILTER 193 9.3 The Discrete IMC Filter The philosophy behind the IMC filter is the same as for stable systems (Sec.8.2). The filter structure is fixed and only a few parameters are adjustedo meet the robustness objectives.The simplest filter form is f(:)=(1-a): (8.2-1) z-O 9.3.1 Filter Form The discrete IMC filter f(a)has to satisfy the following requirements (i)Asymptotic tracking of external system inputs (setpoints and or distur- bances)-i.e.,(1-p'gf)v has to be stable. (ii)Internal stability-i.e.,af and (1-p'gf)p'have to be stable. Since d:)has been designed so that (i)and (ii)are satisfied for f(:=1.f() should satisfy 1-f:n=0.及=0m-1，i=1,,5 d 9.3-1) where i.m;were defined in Sec.9.2.2.Note that (9.3-1)implies fot=0: ff:）=1at=T、.,Tξ 9.3-2) One can now select a filter of the form f{)=(z)fj() 0.3-3) where o=)=21 9.3-4) =0 and choose the coefficients 3o....,3 so that (9.3-1)is satisfied for sore specified a.The parameter a can be used as a tuning parameter. Note that for=1,:1.m 1,we only need)=:.For the general case (9.3-1)can be transformed into a system of Af linear equations with 30,....3 as unknowns,where M is given by (9.2-23).Lemma 8.2-1 can help simplify the necessary algebra.One should select w >Mf-Iso that the system of linear equations has one or more solutions.When w >Mf the system is