30- Ho LOOP SHAPIN G and output as follows. Q+I01 (I+PsKo6 g= W (I+PK6 4 W-K =Ps W· WpI+KP6fw.Kw This shows how all the closed2loop objectiv es in (16.16 and (16.6 are incorporated.As an example it is easy to see that the signal relationship in figure 16.5 is giv en by WKU+PG{EPw小 Figure 16.5:An equi alent H formlation 16.3 Ju stification for Hoo Loop Shaping The objective of this section is to provide justification for the use of parameter e as a design indicator.We will show that e is a measure of both closed2loop robust stability and the success of the design in meeting the loop shaping sp ecifications. We first examine the possibility of loop shape det erioration at frequencies of high loop gain (ty pically low frequency6 At low frequency (in particular.w (0,w166.the de2 terioration in loop shape at plant output can b e obtained by comparing a(PW KW6
H LOOP SHAPING and output as follows I K I PsKM s I K I PsK h I Ps i W W K I P K h W PW i I Ps I KPs h I K i W WP I KP h W KW i This shows how all the closedloop ob jectives in and are incorporated As an example it is easy to see that the signal relationship in gure is given by z z W W K I P K h W PW i w w e K e e W W P W W z w w z Figure An equivalent H formulation Justication for H Loop Shaping The ob jective of this section is to provide justi cation for the use of parameter as a design indicator We will show that is a measure of both closedloop robust stability and the success of the design in meeting the loop shaping speci cations We rst examine the possibility of loop shape deterioration at frequencies of high loop gain typically low frequency At low frequency in particular l the de terioration in loop shape at plant output can be obtained by comparing PWKW
1)...JustiOcation for Ho Loop Shaping 309 tog(P)e¤(WPWw6 that a(PK)e a(PW Koow)e a(WA-WPW KooW))a(W PW a(Koo)/k(W) (163) w(0noos condition numow Similarly<for loop shap orioration at plant input<hab a(KP)e a(W KooW P)e a(W KooWPW W-))a(WPW )a(K)/K(W). (164) Inoach cao(K)is oquid to obtain a bound on th rioration in th loop shap at low foqonc Noo that th condition numlors (WI and k(W)ao ocod by tho osigrorw xt<ocalling that P.noos th shapd plant<and that Koo robustly stabilios th normaliod coprin factorization of P.with stability margin e<ton ha (I+P.K)厂M。t≤ey (165) wb(N.-M.)is a normaliod oft coprin factorization of P.<and th paranory is firod to simplify th notation to followw Th following sult shows that a(K) isoxplicitly boundd by functions of e and a(P.)<t minimum singular val of th shapd plant<and bno by (163)and (164)Koo will only ha a limiodofct on th spcifod loop shap at low fogoncyw Theorem 1).1+Any controuer Koo satsfyng (16 als o sat sfies a(K(Gjw)）i Gw(八立 y(1a(P.(jw))+1 (P.0w)>”(i. Furterm ore fa(P)≥ D」 Proofw First noo that a(P.)>(1 implos that I+卫P2>I. Furtbr sin (N.-M.)is a normaliod oft coprino factorization of P.<ha M M2 I(N.N2e I(M.P.P2M2 Tlon 2.c(1+P.P3<yI
Justication for H Loop Shaping to Ps WPW Note that P K PWKW W WPWKW WPW KW where denotes condition number Similarly for loop shape deterioration at plant input we have KP WKWP WKWPWW WPW KW In each case K is required to obtain a bound on the deterioration in the loop shape at low frequency Note that the condition numbers W and W are selected by the designer Next recalling that Ps denotes the shaped plant and that K robustly stabilizes the normalized coprime factorization of Ps with stability margin then we have I K I PsKM s where N s M s is a normalized left coprime factorization of Ps and the parameter is de ned to simplify the notation to follow The following result shows that K is explicitly bounded by functions of and Ps the minimum singular value of the shaped plant and hence by and K will only have a limited e ect on the speci ed loop shape at low frequency Theorem Any control ler K satisfying where Ps is assumed square also satises Kj Psj p p Psj for al l such that Psj p Furthermore if Ps p then Kj p where denotes asymptotical ly greater than or equal to as Ps Proof First note that Ps p implies that I PsP s I Further since N s M s is a normalized left coprime factorization of Ps we have M sM s I N sN s I M sPsP s M s Then M s M s I PsP s I
310 H2 LOOP SHAPING Now 4+P21-片≤y K2 can be rewritten as +K2K2)<y+K2 P)-M M,)+P.K2). -166) We will next show that K2 i.invertibleooSuppose that there exit.an x uch that K2x=02 then x×-l6.6)×x give. y1xx≤x证Mx which implie.that x=0 inceMM<12 and hence K2 i.invertibleooEqua- tion -16c6)can now be written as K-K+1)sK+P)MK+P.). -16c7) Define W such that wWw)1-=I(y,i。=I(4+PF)1- and completing the .quare in-16c7)with re.pect toK-yield. Kd+N)Www)1-d-+N)≤→(1)RR where N=-1(1+PP)1- RR =+P.P.)-1(+P.)1- Hence we have Rl-K-+N)Hww)1-K-+NRl-≤→1I and (w1-K-+N)Rl≤”五. Ue元(w1-K}-+N)Rl))aw1-万K}-+N)zRl-)to get 元K}-+N)≤'1元w)厅R) and u.e a+N))a-K2)(-)to get 2的)”1）w万刊+7y1- -16a)
H LOOP SHAPING Now I K I PsKM s can be rewritten as I KK I KP s M s M sI PsK We will next show that K is invertible Suppose that there exists an x such that Kx then x x gives xx xM s M sx which implies that x since M s M s I and hence K is invertible Equa tion can now be written as K K I K P s M s M sK Ps De ne W such that WW I M s M s I I PsP s and completing the square in with respect to K yields K N WW K N RR where N Ps I P s Ps RR I P s Ps I P s Ps Hence we have RK N WW K NR I and W K NR p Use W K NR W K N R to get K N p W R and use K N K N to get K n W R No
1).Justiocation for Ho Loop Shaping 311 Next,note that the eigenvalues of WW.,N.N,and R'R can be computed as follows 1+:(PP;) :(WwW)=1（4:PP) :(NN)= y:(P:P:) (1(y+:PP:)尺 1+:(PP:) :(RR)=1（+:PP: therefore w-西-（品》 1+≤P) (和 P <N)=":m(N.N)= Y:mn(P.Pa) P.) 14mP,P=（+P ☒R)=:ma(R·风)= 1+:mn(P.P:) 1+s(P.) 1(+:mn(P.Pi) (1(Y+≤P Substituting these formulas into(16.1),we have ≤K)6 +》+'-(” ≤P)(((1) y(1☒Ps)+1 The main implication of Theorem 16.10is that the bound on Ko)depends only on the selected loop shape,and the stability margin of the shaped plant.The value of (=e,)directly determines the frequency range over which this result is valid-a small (large e)is desirable,as we would expect.Further,Ps has a sufficiently large loop gain,then so also will P Koo provided (=e,)is sufficiently small. In an analogous manner,we now examine the possi bility of deterioration in the loop shape at high frequency due to the inclusion of Koo.Note that at high frequency (in particular,.-(.,.))the deterioration in plant output loop shape can be obtained by comparing☒PW,KW)to≤Ps)=☒WPW,）.Note that,.analogously to(16.3) and (16.4)we have ≤PK)=(PW,KW≤WPW,)区(K)W寸- Similarly,the corresponding deterioration in plant input loop shape is obtained by comparing W,KoW P)to (WPW,)where ☒KP)=W,KWP)≤(WPW,)☒K)W,） Hence,in each case,K)is required to obtain a bound on the deterioration in the loop shape at high frequency.In an identical manner to Theorem 16.10,we now show that (K)is explicitly bounded by functions of y,and (Ps),the maximum singular value of the shaped plant
Justication for H Loop Shaping Next note that the eigenvalues of WW NN and RR can be computed as follows WW PsP s PsP s NN PsP s PsP s RR PsP s PsP s therefore W p maxWW minPsP s minPsP s Ps Ps N p maxNN p minPsP s minPsP s Ps Ps R p maxRR minPsP s minPsP s Ps Ps Substituting these formulas into we have K Ps Ps Ps Ps p p Ps The main implication of Theorem is that the bound on K depends only on the selected loop shape and the stability margin of the shaped plant The value of directly determines the frequency range over which this result is valid"a small large is desirable as we would expect Further Ps has a suciently large loop gain then so also will PsK provided is suciently small In an analogous manner we now examine the possibility of deterioration in the loop shape at high frequency due to the inclusion of K Note that at high frequency in particular h the deterioration in plant output loop shape can be obtained by comparing PWKW to Ps WPW Note that analogously to and we have P K PWKW WPW KW Similarly the corresponding deterioration in plant input loop shape is obtained by comparing WKWP to WPW where KP WKWP WPW KW Hence in each case K is required to obtain a bound on the deterioration in the loop shape at high frequency In an identical manner to Theorem we now show that K is explicitly bounded by functions of and Ps the maximum singular value of the shaped plant
312 H2 LOOP SHAPING Theorem-6.--Any controller K2 satisfying k.Ge)als o satisyes (K2(jw)≤ -I-→（乃s(jw》 1-v7y-17(Ps(jw)) for allw such that m》< Furtherm ore-if(Ps)1(vy -1-then (K2 (jw))Vy-1-where denotes asymptotically less than or equal to as (Ps)+O< Proof.Th proof of Thoom 16.11 is similar to that of Thoom 16.10,and is only stcbdb:As in th proof of Thooom 16.10,w ha MsMse (IPsPs)1 and (I→K2K2)≤y(I→K2Ps)(M)I→PsK2). (16.9) SiO(s)<立1 I-FI→RP1R>0 and thoxists a spctral factorization VVeI-yP(I→PsP)Ps. Now compbting th squao in(16.9)with osloct to K2 yolds (K2→M)VV(K2→M)≤(Y-1)YY wloo M e 7Ps(I-→(1-Y)PsPs)1· Y-Y e (-1)(I-PsPs)(I-(1-7)PsPs)1 巧no吃ha6 元(W(K2→M)Y1≤V-1 which implos K2)≤ -五 a(V)a(yI ,→(M). (16.10) As in th proof of Thoom 16.10,it isoasy to show that a(v)ea(ylre 高) G(M)e (Ps) 1-(-1)万(Ps)
H LOOP SHAPING Theorem Any control ler K satisfying also satises Kj p Psj p Psj for al l such that Psj p Furthermore if Ps p then Kj p where denotes asymptotical ly less than or equal to as Ps Proof The proof of Theorem is similar to that of Theorem and is only sketched here As in the proof of Theorem we have M s M s I PsP s and I KK I KP s M s M sI PsK Since Ps p I P s I PsP s Ps and there exists a spectral factorization V V I P s I PsP s Ps Now completing the square in with respect to K yields K M V V K M Y Y where M P s I PsP s Y Y I PsP s I PsP s Hence we have V K MY p which implies K p V Y M As in the proof of Theorem it is easy to show that V Y Ps Ps M Ps Ps