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 <4B,i.e.,the system is robustly stable.Now it is sufficient to show that sup 1△(G(s)×sup1a(G(jw). 0R. It is clear that suPμ△(G(s)×su2μ△(G(s)6sP4△(G(jw) s C Now suppose sups CA(G(s))>,then by the definition ofu,there is an so 2 C+)f+g and a complex structured△such that(△)<W3 and det(I1G(so)△)×≤ This implies that there is a≤0g0+and≤<a0 such that det(I1G(jg)a:△)× ≤This in turn implies that u△(G(jg)>3 since(a△)<W3.In other words, sups CA(G(s))0 supo(G(jw)).The proofis complete. (5 Suppose supo R 1A(G(jw))>B.Then there is a <<wo <such that 1 ((jwo))>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 u<6 so that Giuhas quinputs and ruoutputs,and so on.Let =Ca0pO be a block structure,as in equation(uB.p).Define an augmented block structure The setup is to theoretically address the robust performance questions about the loop shown below A(s Gp(s) The transfer function fromw to z is denoted by Fu(GpA). Theorem0fLet0>g.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 =&Gpk<,a stabilizing controller K and a closed loop transfer matrix Gp p Gp(s) 0 Now generate thel singular value frequency responses of Gp >w=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