192 CHAPTER 9.SISO DESIGN FOR UNSTABLE SAMPLED-DATA SYSTEMS 9.2.5 Integral Squared Error(ISE)for Step Inputs to Stable Systems The Hi-optimal controller qr(z)minimizes the sum of squared errors (SSE)for a particular input.To correct intersample rippling,the IMC controller (z)is obtained through the modification discussed in Sec.8.1.2. The ISE can be computed for the closed-loop system with(=)from (7.5- 1)which describes the continuous plant output.For the spec.c case of a step setpcint or disturbance input (v=-r or d),we have ho(s)()=v(s)=s-1 and then (7.5-1),(7.5-2)yield e(s)=(1-p(s)(er)s1 (9.2-37) We have 1sE色产d=ere-e (9.2-38) where2 denotes the H2-norm defined in Sec.2.4.4. For step inputs,we find from Table 8.1-1 H(c)=(pi(=)》 (9.2-39) From S.1-11)we have B(=)=1 and therefore =9: 9.2-40) where)is defined in (8.1-5). Heuce we can write ISE=(1-pis)(pileT))g-(e 0.2-41j By following the steps used in the proof of Thm.4.1-3 we can9.2-41)into two parts: ISE=(1-pA(s))s-par(s)(pu(eT))1g-ie (9.2-42) where pa(s)p(s)are defined in (4.1-3 through 41-5) Note that the first termn in (9.2-45)is the minimum ISEhe continuous case.Hence the second term represents the additional ISEis introduced because of the use of a discrete rather than a continous er(designed according to Secs.8.1.1 and 8.1.2.)