32 GHAPTER.ST COM EPTS Plof of th sthatis ita a eisc3rtrollows fom terattratwetok coTimerat statwithtratis,Np ad Np aebopimea aetreternuma raGomincbrla. h6efaHetitt8eaf能Pweaeeaouapsuwma3 Theorem 2 The feedback system is internally stable i,the following two conditions hold: (a)The transfer function 1+PCF has no zeros in Res >0 (b)There is no pole zero cancellation in Res0 when the product PCF is formed P roof Rea tattereeabak systn is intehally stableif a ninetasrerrunctions 1 1·PF.F 1 1+PCF .CF PC P 1 ae ta3 egte86 O感惑&器” To Hoveb),whtepC-F a ltios of co imefolynomials: By THIGn 1 tecraallistic ply nomia NPNCNF+MPMCME gos in Re≥03 Ths terarne Mo)raeno common z(in R≥0，ad simily fortreorernumelard omincrTal3 2-)Assymea ad )3rabrPC-F a aove ad ietso beae of tecraatelistic plynomia tretis, NPNCNF+MPMcMF)2s0)=0. wemust kw thatRe(o:th wil pveinteastaiit by rkIGh 13suset te contiay tretRe≥03r 2MPMc MF)2s0)=0- te NPNCNF)0)=0. Butthis violate 3rhs 2 PMcMF)so)≠0- so weca divideby it ovet get 1+P3= thatis 1+2PCF)3o)=0- CHAPTER BASIC CONCEPTS Proof of this statement is left as an exercise It follows from the fact that we took coprime factors to start with that is NP and MP are coprime as are the other numeratordenominator pairs ￾ By Theorem internal stability can be determined simply by checking the zeros of a polynomial There is another test that provides additional insight Theorem The feedback system is internal ly stable i the fol lowing two conditions hold a The transfer function  PCF has no zeros in Res   b There is no polezero cancel lation in Res   when the product PCF is formed Proof Recall that the feedback system is internally stable i all nine transfer functions  PCF   P F F C CF P C P   are stable  Assume that the feedback system is internally stable Then in particular   PCF ￾￾ is stable ie it has no poles in Res   Hence  PCF has no zeros there This proves a To prove b write P C F as ratios of coprime polynomials P  NP MP C  NC MC F  NF MF By Theorem the characteristic polynomial NP NCNF  MPMCMF has no zeros in Res   Thus the pair NP MC  have no common zero in Res   and similarly for the other numeratordenominator pairs  Assume a and b Factor P C F as above and let s be a zero of the characteristic polynomial that is NP NCNF  MPMCMF s We must show that Res   this will prove internal stability by Theorem Suppose to the contrary that Res   If MPMCMF s then NP NCNF s But this violates b Thus MPMCMF s   so we can divide by it above to get  NP NCNF MPMCMF s that is PCF s