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298 Ho LOOP SHAPING It is clear that Yoo =0is the stabilizing solution.Hence by the formulae in Chapter 14 we have [hxL2]=[0L] and 2=1,D1=0D2=1,Da1=2-. The results are summarized in the following theorem. Theorem 16.1 Let D=0 and let L be such that A+LC is stable then there erists a controller K such that iffy>1 and there erists a stabilizing solution Xo>O solving L Xoo(A- 2-)+(4- 27川+22Cc X.-x.(BB LC y2-1 =0. Moreover,if the above conditions hold a central controller is given by K A-BB*Xo+LC L -B*Xoo 0 It is dlear that the existence of a robust stabilizing controller depends upon the choice of the stabilizing matrix L,i.e.,the choice of the coprime factorization.Now let Y>0 be the stabilizing solution to AY+YA*-YC*CY+BB*=0 and let L=-YC*.Then the left coprime factorization(M,N)given by [市]= 「A-YCCB-YC 01 is a normalized left coprime factorization (see Chapter 12). Corollary 16.2 Let D=0 and L=-YC*where Y>0 is the stabilizing solution to AY+YA*-YC*CY+BB*=0. Then P=M-IN is a normalized left coprime factorization and 年w -（-)了e H LOOP SHAPING It is clear that Y is the stabilizing solution Hence by the formulae in Chapter we have h L L i h L i and Z I D D I D p I The results are summarized in the following theorem Theorem Let D and let L be such that A LC is stable then there exists a control ler K such that K I I P KM i and there exists a stabilizing solution X solving XA LC A LC X XBB LL X CC Moreover if the above conditions hold a central control ler is given by K A BBX LC L BX It is clear that the existence of a robust stabilizing controller depends upon the choice of the stabilizing matrix L ie the choice of the coprime factorization Now let Y be the stabilizing solution to AY Y A Y CCY BB and let L Y C Then the left coprime factorization M N given by h N M i A Y CC B Y C C I is a normalized left coprime factorization see Chapter Corollary Let D and L Y C where Y is the stabilizing solution to AY Y A Y CCY BB Then P M N is a normalized left coprime factorization and inf K stabilizing K I I P KM p maxY Q h N M i H min