188 AND U-SYNTHESIS since D1=U1D and U 0.9155-0.0713j 0.2365-0.3177j 0.1029+0.3824j -0.5111-0.7629j is a unitary matrix. 10.2.3 Well Posedness and Performance for Constant LFTs Let M be a complex matrix partitioned as M1M12 M= (10.13) M21M22 and suppose there are two defined block structures,Ai and A2,which are compatible in size with Mu and M22,respectively.Define a third structure A as (10.14) Now,we may compute u with respect to three structures.The notations we use to keep track of these computations are as follows:()is with respect to A1,u2()is with respect to A2,and (is with respect to A.In view of these notations,(M), u2 (M22)and (M)all make sense,thoug h,for instance,(M)does not. This section is interested in following constant matrix problems: 。determine whether the LFTF,(M,△z)is well defined for all△2∈△2with (△2)≤3(<3)，and, if so,then determine how "large"F(M,A2)can get for this norm-bounded set of perturbations. Let△2∈△2.Recall that F(M,△z)is well defined if I-M22△2 is invertible.The first theorem is nothing more than a restatement of the definition of u. Theorem 10.5 The linear fractional transformation F(M,A2)is well defined (a)for all△2∈B△2 if and only if u2(M22)<1. (b)for all△2∈B°△2 if and only if u2(M22)≤1. As the“perturbation”△2 deviates from zero,the matrix F(M,△z)deviates from Mi.The range of values that (F(M,A2))takes on isintimately related to A(M), as shown in the following theorem:
AND SYNTHESIS since D UD and U j j j j is a unitary matrix Well Posedness and Performance for Constant LFTs Let M be a complex matrix partitioned as M M M M M and suppose there are two dened block structures and which are compatible in size with M and M respectively Dene a third structure as Now we may compute with respect to three structures The notations we use to keep track of these computations are as follows is with respect to is with respect to and is with respect to In view of these notations M M and M all make sense though for instance M does not This section is interested in following constant matrix problems determine whether the LFT F M is well dened for all with and if so then determine how large F M can get for this normbounded set of perturbations Let Recall that F M is wel l dened if I M is invertible The rst theorem is nothing more than a restatement of the denition of Theorem The linear fractional transformation F M is wel l dened a for al l B if and only if M b for al l Bo if and only if M As the perturbation deviates from zero the matrix F M deviates from M The range of values that F M takes on is intimately related to M as shown in the following theorem
10.2.Structured Singular Value 189 Theorem 10.6(MAIN LOOP THEOREM)The following are equivalent. 3 2(M22)<1.and u(M)<1 ←→ △87(Fi(MA2)<1e 3 -2(M22)≤1.and ru(M)≤1 sup (F(M△2)≤1∈ △2∈B°△2 Proof.We shall only prove the first part of the equivalence.The proof for the second part is similar. ←Let△i∈△；be given,with(△i)≤l,and define△=diag[△△zl.Obviously △∈△.Now 1 0 det (IM)=detE (10.15) HM21△1I4M22△2 By hypot hesis I u M22A2 is invertible,and hence,det(I u MA)becomes det(IHM2z△2)det(IhM1△1hM2△2(IhM2△2)-1M2△1)∈ Collecting the Ai terms leaves det(IuM△)=det(IuM22△2)det(IuF(M△2)△)∈ (10.16) But,-1(F(M△2)<1and△1∈BA1,so IuF(M-△2)△1 must be nonsingular.. Therefore,ILMA is nonsingular and,by definition,(M)<1. Basically,the argument above is reversed. Again let△1∈B△1and △2∈B△2 be given,and define△=diag[△△l.Thh△∈B△and,by hypothesis, det(I u MA)0.It is easy to verify from the definition of-that (always) -(M)≥max{-(M1)-2(M22)}∈ We can see that2(M22)<1,which givesthat IuM22A2is also nonsingular.Therefore, the expression in (10.16)is valid,giving det(IhM22△2)det(IhF(M△2)△，)=det(IuM△)≠0e Obviously,IhF(M△2)△1 is nonsingular for all△；∈B△i,which indicates that the claim is true
Structured Singular Value Theorem MAIN LOOP THEOREM The fol lowing are equivalent M M and max B F M M M and sup Bo F M Proof We shall only prove the rst part of the equivalence The proof for the second part is similar Let i i be given with i and dene diag Obviously Now det I M det I M M M I M By hypothesis I M is invertible and hence det I M becomes det I M det I M M I M M Collecting the terms leaves det I M det I M det I F M But F M and B so I F M must be nonsingular Therefore I M is nonsingular and by denition M Basically the argument above is reversed Again let B and B be given and dene diag Then B and by hypothesis det I M It is easy to verify from the denition of that always M max f M Mg We can see that M which gives that IM is also nonsingular Therefore the expression in is valid giving det I M det I F M det I M Obviously I F M is nonsingular for all i Bi which indicates that the claim is true
190 AND u.SYNTHESIS Remark 083 This theorem forms the basis for all uses of u in linear system robust ness analysis,whether from a state-space,frequency domain,or Lyapunov approach.2 The role of the block structure,cin the MAIN LOOP theorem is dlear-it is the structure that the perturbations come from;however,the role of the perturbation structure,-is often misunderstood.Note that u-()appears on the right hand side of the theorem,so that the set,-defines what parficular property of F1(MAe)is considered.As an example,consider the theorem applied with the two simple block structures considered right after Lemma 10.1.Define,-:={5 In:5 }.Hence, for A chn(A)=p(A).Likewise,define,=cAm then for D cmm E(D)=I(D).Now,let,be the diagonal augmentation of these two sets,namely = 5Hn0wm:5-1 C△M∈ CCr旺 0mn△e Let A CI△nB cmmC cmn and D mbe given,and interpret them as the state space model of a discrete time system Ek≤-=ATk+Buk CIk+Duk- And let M Cemnbe the block state space matrix of the system M=A B CD Applying the theorem with this data gives that the following are equivalent: The spectral radius of A satisfies p(A).1,and (10.17） The maximum singular value of D satisfies T(D).1,and maxp(A+B△e(I DA2-C)·1- (10.18) The structured singular value of M satisfies a(M).1- (10.19)
AND SYNTHESIS Remark This theorem forms the basis for all uses of in linear system robustness analysis whether from a statespace frequency domain or Lyapunov approach The role of the block structure in the MAIN LOOP theorem is clear it is the structure that the perturbations come from however the role of the perturbation structure is often misunderstood Note that appears on the right hand side of the theorem so that the set denes what particular property of F M is considered As an example consider the theorem applied with the two simple block structures considered right after Lemma Dene fIn C g Hence for A C nn A A Likewise dene C mm then for D C mm D D Now let be the diagonal augmentation of these two sets namely In nm mn C C mm C nmnm Let A C nn B C nm C C mn and D C mm be given and interpret them as the state space model of a discrete time system xk Axk Buk yk Cxk Duk And let M C nmnm be the block state space matrix of the system M A B C D Applying the theorem with this data gives that the following are equivalent The spectral radius of A satises A and max C j j D C I A B The maximum singular value of D satises D and max Cmm A B I D C The structured singular value of M satises M
0979 Structured Singular Value up The first condition is recognized by two things:the sy stemis stable,and theA0 normon the transfer function fromu to y is less than u(by replacing 6 with f) ‖Gl0:,max。(D3C(zI4A)2 B), The second condition implies that (Iu DA)2is well defined for all(A)0 u and that a robust stability result holds for the ur certain difference equation (A3B△(tμDA)2rCxk where△ is any dement in cmmwithA)0but otherwise unknown. This equivalence between the small gain condition,G0<M and the stability robustness of the uncertain difference equation is well known.This is the small gain theorem in its necessary and sufficient formfor linear,time invariant systems with one of the components norm bounded,but otherwise unknown.What is important to note is that both of these conditions are equivalent to a condition involving the structured singular value of the state space matrix.Already we hav eseen that special cases of-are the spectral radius and the maximumsingular value Here we see that other important linear systemproperties,namely robust stability and input-output gain,are also related to a particular case of the structured singular value Example 09+LetM,△and△ be defined as in the beginning of this section.Now suppose <u.Fin (F1(M△)∈ B△ This can be done iteratively follow s: 器F1(MA), 5 a Hence
Structured Singular Value The rst condition is recognized by two things the system is stable and the jj jj norm on the transfer function from u to y is less than by replacing with z kGk max zC jzj D C zI A B max C jj D C I A B The second condition implies that I D is well dened for all and that a robust stability result holds for the uncertain dierence equation xk A B I D C xk where is any element in C mm with but otherwise unknown This equivalence between the small gain condition kGk and the stability robustness of the uncertain dierence equation is well known This is the small gain theorem in its necessary and su cient form for linear time invariant systems with one of the components normbounded but otherwise unknown What is important to note is that both of these conditions are equivalent to a condition involving the structured singular value of the state space matrix Already we have seen that special cases of are the spectral radius and the maximum singular value Here we see that other important linear system properties namely robust stability and inputoutput gain are also related to a particular case of the structured singular value Example Let M and be dened as in the beginning of this section Now suppose M Find max B F M This can be done iteratively as follows max B F M max B F M M M M M M M M M Hence max B F M M M M M M
1.( AND U&SYN TH ESIS For example let△μ△6L-,△∈C×, Aa 0a.ca6.na Find amax△spp(A+BA(I-D△-)C). 1△.≤μ Define△△ 4. Then a bisection search can be done to find ama△ △.22 Related MATLAB Commands4 unwrapp.muunw rap.dypert.sisorat 10.3 Structured Robust Stability and Performance 10.3.1 Robust Stability The most well-known use of u as a robustness analy sis tool is in the frequency domain. Suppose G(s)is a stable,realrational,multiinput,mlti-output transfer function of a linear system For clarity,assume G has qu inputs and Pu outputs.Let be a block structure,as in equation(10.1),and assume that the dimensions are such that =/Cxp.We want to consider feedback perturbations to G which are themselves dy namical sy stems with the block-diagonal structure of the set = Let M(=)denote the set of all block diagonal and stable rational transfer functions that have block structures such as = M(=):△{△()eRH<:△(so)e=far all soCo} Theorem 09 Let 3>0.The loop shown below is well-posed and internally stahle oral△()∈M(=)with‖△ll≤<台if and only if su吧A(G(j3)≤3 ωR e_ +W- G()
AND SYNTHESIS For example let I C A B C D Find max sup A BI DC Dene I Then a bisection search can be done to nd max M A B C D Related MATLAB Commands unwrapp muunwrap dypert sisorat Structured Robust Stability and Performance Robust Stability The most wellknown use of as a robustness analysis tool is in the frequency domain Suppose Gs is a stable realrational multiinput multioutput transfer function of a linear system For clarity assume G has q inputs and p outputs Let be a block structure as in equation and assume that the dimensions are such that C q p We want to consider feedback perturbations to G which are themselves dynamical systems with the blockdiagonal structure of the set Let M denote the set of all block diagonal and stable rational transfer functions that have block structures such as M RH so for all so C Theorem Let The loop shown below is wellposed and internally stable for all M with kk if and only if sup R Gj e e e e w w Gs