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(=):△{△()eRH0.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
10.3.Structured Robust Stability and Performance 2 Proof.(x)Suppose sups c 1a(G(s)03.Then det(I1G(s)△(s)2≤or all s2C+)f+g whenever k△k ,then by the definition ofu,there is an so 2 C+)f+g and a complex structured△such that(△)3 since(a△)B.Then there is a B.By Remark t there is a complex Ac 2 A that each full block has rank u and (Ac)<such that I1 G(jwo)Ac is sin}ular.Next,usin}the same construction used in the prooo the small ain theorem (Theorem eu),one can find a rational△(s)such that k△(s)k+×△c)<W3,△(jwo)×△c,and△(s)desta bilizes the sy stem. 口 Hence,the peak value on the u plot ofthe frequency response determines the size o perturbations that the loop is robustly stable afainst. Remar 10.4 The internal stability with closed ball of uncertainties is more compli- cated.The ollowin}example is shown in Tits and Fan Consider <1 G(s)×s+μ and△×s)I2.Then 8盟1a(G(u》×8盟w+4 ×1△(G(js)×h On the other hand,1(G(s))<ufor all s 2 ss2 C+,and the only matrices in the orm ofr x I2 with 0 ufor which det(I1G(ST)x≤ are the compler matrices +j2.Thus,clearly,(I1 G(s)(s))1 2 RH+for all real rational A(s)x (s)I2 with kok 0 u since A(s)must be feal.This shows that
Structured Robust Stability and Performance Proof Suppose sup sC Gs Then detI Gss for all s C f g whenever kk ie the system is robustly stable Now it is su cient to show that sup sC Gs sup R Gj It is clear that sup sC Gs sup sC Gs sup Gj Now suppose supsC Gs then by the denition of there is an so C f g and a complex structured such that and detI Gso This implies that there is a and such that detI Gj This in turn implies that Gj since In other words supsC Gs sup Gj The proof is complete Suppose supR Gj Then there is a o such that Gj o By Remark there is a complex c that each full block has rank and c such that I Gj oc is singular Next using the same construction used in the proof of the small gain theorem Theorem one can nd a rational s such that ksk c j oc and s destabilizes the system Hence the peak value on the plot of the frequency response determines the size of perturbations that the loop is robustly stable against Remark The internal stability with closed ball of uncertainties is more compli cated The following example is shown in Tits and Fan Consider Gs s and sI Then sup R Gj sup R jj j Gj On the other hand Gs for all s s C and the only matrices in the form of I with j j for which detI G are the complex matrices jI Thus clearly I Gss RH for all real rational s sI with kk since must be real This shows that
194 1AND 1.SYNTHESIS supo (G(+)).1 is not necessary for (I-G(s)A(s))1E RH+with the dosed ball of structured uncertainty l<1.Similar examples with no repeated blocks are generated by setting G(s)= M where M is any real matrix with-u(M)=1 for S+1 whichr there is no real△∈，with I(△)=1 such that det(I-M△)=O.For example,. let I小 53 with士A= and 0+26=1.Then it is shown in Packard and Doyle [1993 that -u(①M)=1 and all△∈，with I(△)=1 that satisfy det(I-M△)=0 must be com plex. 1 Remark 08<6 Let AE RH be a struct ured uncertainty and G11(s)G1☒s) G(s)= ∈RH+ Ga(s)Gads) then Fu(G△)∈RH+does not necessarily imply(I-Gi△hl∈RH+whether△is in an open ball or is in a closed ball.For example,consider S+I 01 G(s) 0 190 00 1 and△= 51 /A年·1.TnRG=-贡∈R+or 5A admissible△(l△l+·1)but(I-G1△)h1∈RH+is true only for‖△l+.0以.1 10.3.2 Robust Performance Often,stability is not the only property of a closed-loop sy stem that must be robust to perturbations.Ty pically,there are exogenous disturbances acting on the system(wind gusts,sensor noise)which result in tracking and regulation errors.Under perturbation, the effect that these disturbances have on error signals can greatly increase.In most cases,long before the onset of instability,the closed-loop performance will degrade to the point of unacceptability,hence the need for a "robust performance"test.Such a test will indicate the worst-case level of performance degradation associated with a given level of pert urbations. Assume Gp is a stable,real-rational,proper transfer function with q+qainputs and pI+pAout puts.Partition Gp in the obvious manner Gp(s)= G11G1△
AND SYNTHESIS supR Gj is not necessary for I Gss RH with the closed ball of structured uncertainty kk Similar examples with no repeated blocks are generated by setting Gs sM where M is any real matrix with M for which there is no real with such that detI M For example let M i C with and Then it is shown in Packard and Doyle that M and all with that satisfy detI M must be complex Remark Let RH be a structured uncertainty and Gs Gs Gs Gs Gs RH then FuG RH does not necessarily imply I G RH whether is in an open ball or is in a closed ball For example consider Gs s s and with kk Then FuG s RH for all admissible kk but I G RH is true only for kk Robust Performance Often stability is not the only property of a closedloop system that must be robust to perturbations Typically there are exogenous disturbances acting on the system wind gusts sensor noise which result in tracking and regulation errors Under perturbation the eect that these disturbances have on error signals can greatly increase In most cases long before the onset of instability the closedloop performance will degrade to the point of unacceptability hence the need for a robust performance test Such a test will indicate the worstcase level of performance degradation associated with a given level of perturbations Assume Gp is a stable realrational proper transfer function with q q inputs and p p outputs Partition Gp in the obvious manner Gps G G G G
099 Structured Robust Stability and Performance ug.For all△(s)2M(=)with k.△ks·,the loop shown above is well-posed,internally stable,and kFu(GpA)k 00 if and only if 2=p(G(|3)00- R Note that by internal stability,sup (G3))0 0,then the proof of this theoremis exactly along the lines of the earlier proof for TheoremuB.p,but also appeals to Theoremu.>This is aremarkably useful theorem It says that a robust performance problemis equivalent to a robust stability problem with augmented uncertainty A as shown in Figure u8.6. Example 09)We shall consider again the HIMAT problem see Example <u.Use the SIMULINK block diagramin Example <u and run the following commands to get an interconnection model G,a Ho stabilizing controller K and a closed-loop transfer matrix Gp(s),F(K)(Do not bother how hinfsyn works,it will be considered in detail in Chapter 4.) ABA],linmod('aircraft') G;pck(AA); KAGpA],hinfsyn(G-00);
Structured Robust Stability and Performance so that G has q inputs and p outputs and so on Let C q p be a block structure as in equation Dene an augmented block structure P f f C q p The setup is to theoretically address the robust performance questions about the loop shown below z w Gps s The transfer function from w to z is denoted by Fu Gp Theorem Let For all s M with kk the loop shown above is wellposed internally stable and kFu Gp k if and only if sup R P Gpj Note that by internal stability supR Gj then the proof of this theorem is exactly along the lines of the earlier proof for Theorem but also appeals to Theorem This is a remarkably useful theorem It says that a robust performance problem is equivalent to a robust stability problem with augmented uncertainty as shown in Figure Example We shall consider again the HIMAT problem see Example Use the SIMULINK block diagram in Example and run the following commands to get an interconnection model G a H stabilizing controller K and a closedloop transfer matrix Gps FG K Do not bother how hinfsyn works it will be considered in detail in Chapter A B C D linmod aircraft G pckA B C D K Gp hinfsynG
196 AND U-SYNTHESIS △f △ Gp(s) Figure 10.5:Robust Performance vs Robust Stability which gives 4 =1.8612 =&Gpkw=logsp ace(-3,3,300 >Grf=frsp (Gp,w);Gpf is the frequency response of Gp >[u,s,v]vsvd(Gpf); ≥p lot(a,ms) The singular value frequency responses of Gp are shown in Figure 10.6.To test the robust stability,we need to compute &Gp: >Gp11 sel(Gp:1 p,1 p); >norm_of_Gp11 hinfnorm(Gp11,0.001); which gives &GP=0.933<1.So the system is robustly stable.To check the robust performan e,we shall compute the (Gp(j3))for each frequency with ravoh
AND SYNTHESIS f Gps Figure Robust Performance vs Robust Stability which gives kGpk a stabilizing controller K and a closed loop transfer matrix Gp z z e e Gps p p d d n n Gps Gp Gp Gp Gp Now generate the singular value frequency responses of Gp wlogspace Gpf frspGp w Gpf is the frequency response of Gp u s v vsvdGpf vplot liv m s The singular value frequency responses of Gp are shown in Figure To test the robust stability we need to compute kGpk Gp selGp norm of Gp hinfnormGp which gives kGP k So the system is robustly stable To check the robust performance we shall compute the P Gpj for each frequency with P f C f C
10.3.Structured Robust Stability and Performance 1.2 maximum singular value 1.5 1 0.5 10 10 101 100 10 102 103 frequency(rad/sec) Figure 10.):Singular Values of Gpj .bll=[p,p;4,p]: [bnds,dvec,sens,pvec]=mu(Gp f,bl vr iot(iv mpnorm(Gf)pnds) tit le(Mabimm Singular Yalue and mu x label(requency(radCsec) tert(001 17 marimum singular value text(05 mu bounds The structured singular value r·n(GpG》and(Gp(j、》are shown in Figure10.2. It is clear that the robust performance is not satistied.Note that Gpl Gp10 GrO Using a bisection algorithm we can also find the worst performance 吧1aG<A15
Structured Robust Stability and Performance 10−3 10−2 10−1 100 101 102 103 0 0.5 1 1.5 2 frequency (rad/sec) maximum singular value Figure Singular Values of Gpj blk bndsdvecsenspvecmu Gpfblk vplot liv m vnormGpf bnds title Maximum Singular Value and mu xlabel frequencyrad sec text maximum singular value text mu bounds The structured singular value P Gpj and Gpj are shown in Figure It is clear that the robust performance is not satised Note that max kk kFuGp k sup P Gp Gp Gp Gp Using a bisection algorithm we can also nd the worst performance max kk kFuGp k
