198 CHAPTER 9.SISO DESIGN FOR UNSTABLE SAMPLED-DATA SYSTEAS 9.4 Robust Stability For controllers designed via the IMC design procedure (=paf)Thm.7.6-1 becomes Cor.9.4-1. Corollary 9.4-1 (Robust Stability).Assume that all plants p in the family n ={p: p(i) (9-1) have the same number of RHP poles and that these poles do not become unob- servable after sampliig.Then the system is robusily stable if and only if the IAIC filter satisfies 胡< 0≤w≤π/T (9.-2 where a is a stabilizing controller for the nominal plant p. For stable systems f is arbitrary.Therefore.there always exists a flter f which satisfes (9.4-2:regardless of the magnitude of the uncertainty For unstable systems f is constrained to be unity at the poles of p"outside the ['C. Thus,depending onthere might not exist any filter parameter for which the constraint (9.4-2:is rnet.Indeed,there might not exist any filter--however complicated which satisfies (9.4-2).A minitnum amount of information is necessary or equivaleutly a maximum amount of uncertainty is allowed to stabilize an unstable systemn.The necessary information at=0 can be characterized easily. Corollary 9.4-2.Assume that a filter f is to be designed for a system or dis- turbance pole(s)at s=0-i.e.,f(1)=1.There exisis an f such thut ihe closed loop system is robusily stable for the jamily II described by (9.4-1)only if im(0)<1. Note that contrary to Cor.8.3-2.Cor.9.4-2 is only necessary.For unstabe systems,in general,the filter has to satisfy other constraints in addition to the one at z=1. 9.5 Robust Performance The results in Sec.7.7.2 hold for unstable systems if it is assumed that all pants in the family II have the same number of RHP poles and if the controller and filter f are stabilizing for the nominal plant p