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
16.1.Robust Stabilization of Coprime Factors 99 where Q is the solution to the following Lyapunov equation Q(A-YCC)+(A-YCC)Q+CC=0. Moreover&if the above conditions hold then for any y>min a controller achieving is given by A-BB.Xo-YC.C-YC. K(s)= -B.Xoo 0 where x=Q(- Proof. f.Note t at te Ham lton an matrx asscaesg g ven by -YC.C -BB+寸 C.CY H= 0 Ha 0 Hg=A-Yb.c -CC -(A-YC.C) poa e x,(0g）=ImQ and tre stable nvar ant subs ace of Hoo s given by -YO X(t)= 0 X (Ha)=Im y2, Hence t ere Q>0
Robust Stabilization of Coprime Factors where Q is the solution to the fol lowing Lyapunov equation QA Y CCA Y CCQ CC Moreover if the above conditions hold then for any min a control ler achieving K I I P KM is given by Ks A BBX Y CC Y C BX where X Q I Y Q Proof Note that the Hamiltonian matrix associated with X is given by H A Y CC BB Y CCY CC A Y CC Straightforward calculation shows that H I Y I Hq I Y I where Hq A Y CC CC A Y CC It is clear that the stable invariant subspace of Hq is given by XHq Im I Q and the stable invariant subspace of H is given by XH I Y I XHq Im I Y Q Q Hence there is a nonnegative de nite stabilizing solution to the algebraic Riccati equa tion of X if and only if I YQ
300 H2 LOOP SHAPING or 1 Y>- V1-Amar YQ2 and the solution<if it exists is given by 81 Note that Y and Q are the controllability Gramian and the observability Gramian of NM respectively.Therefore<we also have that the Hankel norm of NM is VAmar YQ≥ Corollary,63 Let P=MN be a normalized left coprime factorization and PA=iM+7M21+N≥ with ix2 Then there is a robustly stabilizing controller for PA if and only if esV-A-1-【市] The solutions to the normalized left coprime factorization stabilization problem are also solutions to a related H2 problem which is shown in the following lemma. Lemma,6A Let P=M8IN be a normalized left coprime factorization.Then []+x[]+x[Pg Proof. Since -M-N>is a normalized left coprime factorization of P<we have [][m]'=1 and [ǔ]2=[]了2=1. Using these equations<we PK2
H LOOP SHAPING or p maxY Q and the solution if it exists is given by X Q I Y Q Note that Y and Q are the controllability Gramian and the observability Gramian of h N M i respectively Therefore we also have that the Hankel norm of h N M i is p maxY Q Corollary Let P M N be a normalized left coprime factorization and P M M N N with h N M i Then there is a robustly stabilizing control ler for P if and only if p maxY Q r h N M i H The solutions to the normalized left coprime factorization stabilization problem are also solutions to a related H problem which is shown in the following lemma Lemma Let P M N be a normalized left coprime factorization Then K I I P KM K I I P K h I P i Proof Since M N is a normalized left coprime factorization of P we have h M N i h M N i I and h M N i h M N i Using these equations we have K I I P KM
1>1y Robust Stabilization of Coprime Factors 301 a+P[w][立N] I K ≤ a+P[aN][立N]I K This implies fu+Pu-广f+Pw[P Combining Corollary 16.3 and Lemma 16.4,we have the following result. Corollary 12<A controller solves the normalized left coprime factor robust stabiliza- tion problem if and only if it sowves the following H control problem 广a+r and -] The solution Q can also be obtained in other ways.Let X>0 be the stabilizing solution to XA+A2X-XBB2X+C2C=0 then it is easy to verify that Q=(I+xY)1,x-
Robust Stabilization of Coprime Factors K I I P KM h M N i h M N i K I I P KM h M N i h M N i K I I P K h I P i K I I P KM h M N i K I I P KM This implies K I I P KM K I I P K h I P i Combining Corollary and Lemma we have the following result Corollary A control ler solves the normalized left coprime factor robust stabiliza tion problem if and only if it solves the fol lowing H control problem I K I P K h I P i and inf K stabilizing I K I P K h I P i p maxY Q h N M i H The solution Q can also be obtained in other ways Let X be the stabilizing solution to XA AX XBBX CC then it is easy to verify that Q I XY X
30 H2 LOOP SHAPIN G Hence 1 y min a=1(耳]月) "=V个+AmXY6 Similar results can be obtained if one starts with a normalized right coprime factoriza2 tion.In fact~a rather strong relation between the normalized left and right coprime factorization problems can be established using the follow ing matrix fact. 2o8""M-“6-6 Proof.We first show that the eigenvalues of M are either 0 or 1 and M is diagonaliz2 able.In fact assume that A is an eigen alue of M and z is a corresponding eigerv ector~ thenλx=Mz=MMx=MMz6=λMz=λE~i.e-1(A6=0.This inplies that either =0 or A =1.To show that M is diagonalizable~assume M=TJT8- whereJ is a Jordan canonical form it follows inmediately that J must be diagonal by the condition M=Ml. Next~assume that M is diagonalized by a nonsingular matrix T such that 00 M=T I 0 T8- 00 Then N:=I(M=T00r8- Define A B B*D =TT and assume02入≠1.Then A>0and det-M*M(6=0 detI 0 T-T 00 (A7"T6-0 00 dt1(6A(AB’ =0 (AB*(λD dtf(六B-6=0 dt12D+B48-B6=0 入
H LOOP SHAPING Hence min p maxY Q h N M i H p maxXY Similar results can be obtained if one starts with a normalized right coprime factoriza tion In fact a rather strong relation between the normalized left and right coprime factorization problems can be established using the following matrix fact Lemma Let M be a square matrix such that M M then iM iI M for al l i such that i M Proof We rst show that the eigenvalues of M are either or and M is diagonaliz able In fact assume that is an eigenvalue of M and x is a corresponding eigenvector then x M x MMx MM x M x x ie x This implies that either or To show that M is diagonalizable assume M TJT where J is a Jordan canonical form it follows immediately that J must be diagonal by the condition M M Next assume that M is diagonalized by a nonsingular matrix T such that M T I T Then N I M T I T De ne A B B D T T and assume Then A and detMM I det I T T I T T det A B B D detD BAB det D BAB