JULY 1987 LIDS-P-1684(Revised)SENSITIVITY REDUCTION OVER A FREQUENCY BANDbyYoshito Ohta*Gilead Tadmor**Sanjoy K. MitterABSTRACTThis paper considers the problem of reducing the sensitivity of apossibly infinite dimensional linear single-input single-output system overa finite frequency interval by feedback. Specifically the following areproven: (i) if one wants to bound the overall sensitivity, the existenceof a nontrivial inner part inhibits the reduction of the sensitivity overthe interval: (ii) in a system that is continuous and has at most countablymany zeros on the imaginary axis, one can reduce the sensitivity over theinterval arbitrarily small while the overall sensitivity is kept bounded ifand only if the system is outer and has no zeros on the interval. Theseextend results for rational transfer functions.Massachusetts Institute of TechnologyLaboratory for Information and Decision SystemsCambridge, MA 02139*On leave from Department of Electronic Engineering, Osaka University, OsakaJapan. The work of this author was supported by Murata Overseas ScholarshipFoundation, Kyoto Japan.**Current address: Center for control science, Devision of AppliedMathematics, Brown University, Providence, RI 02912.The work of these authors was supported in part by the Army Research Officeunder Grants DAAG29-84-K-0005 and DAAL03-86-K-0171 and by the Air ForceOffice of Scientific Research under grant AFOSR-85-0227B.2I. INTRODUCTIONThis paper considers the problem of reducing the sensitivity of alinear single-input single-output system over a finite frequency interval byfeedback.V V2Fig. 1The feedback system is described by Fig. 1. P is a given system and weassume PeHe (i.e., stable) and C is a feedback. We say that the feedbackstabilizes the system if the transfer functions from (vlv 2 ) to (ul,u 2) allbelong to H¶.The closed loop sensitivity S is the transfer function from v 2 to u2and is given byS(s) = (1+P(s)C(s)) 1 (1.1)The problem of sensitivity reduction over a frequency band X is statedas follows. Let X be the characteristic function of a given bounded setXC(-a, A), on the imaginary axis, i.e.,, if wia,X(jW) i s (1.2)0 otherwiseFor given s>0 and M>1, find a stabilizing feedback for which thesensitivity satisfiesIIXSII < E, I lsII < M. (1.3)Here are our main results.Theorem 1.Suppose PsHf has a nontrivial inner part and X is the characteristicfunction of a sub-set of the imaginary axis which has positive Lebesguemeasure. Theninf IIXSlI| > o,IIS IO1 and the infimum is taken over all stabilizing compensators.Theorem 2.Suppose PeH is continuous on the imaginary axis and has at mostcountably many zeros on the imaginary axis. Let X be the characterizationfunction of a compact set XC(--,-) on the imaginary axis. Then for any1>e>0 and any M>1 there exist a stabilizing compensator such that|IXS1[ < a [, ||SH < M,if and only if P is outer and has no zeros on jX.Previous discussions of this problem appear in [1] - [5]. In [1], itwas shown that if the plant P is analytic, is bounded, has no zero in Res-0, and satisfies an attenuation condition at s=-, then for any s>0 and M>1the problem has a solution. Especially the problem is solvable when P is ofminimum phase. Theorem 2 generalizes this result, and seems to illuminatemore on the structural aspects of the sensitivity reduction problem. In[3], in the framework of rational plants, it was shown that if the plant Phas a right half plane zero then there exists a positive number k such thatIlxsl 1' I lSI · kHence given M>1, there is s>0 such that the problem has no solution.Theorem 1 is a natural extension of this statement. In [5], it was shownthat if the plant is analytic and has no zero in some region containing Res20, and satisfies some intricate condition near s=-, then for any s>0 andM>2+L (L is determined by the condition) the problem has a solution.However the condition seems rather difficult to check. The difficulty wasdemonstrated by the authors' wrong conclusion that for P(s) = e-S/(s+l)(which has a nontrivial inner part), and some M>2, the problem has asolution for any E>0.5II. PRELIMINARIES AND NOTATIONS2.1 Parametrization of Stabilizing FeedbacksWe parametrize feedbacks achieving stability. The parametrization wasintroduced in [6] and modified in [7]. The following is a corollary of [8]for a stable system.Proposition 3:Assume P is stable (PeHw). Then a feedback C stabilizes the system ifand only if there exists husH, Phil such thatC = -Ph (2.1)Substituting (2.1) to (1.1), we haveS = 1-Ph. (2.2)Therefore our problem is reduced to that of finding hCHW satisfyingIIX(1-Ph)|l| < e, Il-Ph||C < M (2.3)for given s>0, M>1.2.2 HP FunctionsH2 and Ha are the Handy spaces of analytic functions on the right half6plane with L2 and LC boundary values, respectively. [9], [10] are goodsources on HP spaces, inner-outer factorizations, etc. The following isfrom [10] and is worthy of note.Proposition 4 [10].Assume PeHe and let K = H20PH2 (or K = (PH2)1). Then K = (0O if andonly if P is outer.2.3 a-Inner Product and av-NormThe Laplace transformation L defines an isometric isomrophism fromL2[0,o) to H2 . We shall use both the time domain and the frequency domainin our analysis.We denote the usual inner product and norm of H2 (respectively,L2 [0,_)) by <,'> and |1-Ii. For future use we introduce also a wholefamily of additional inner products and norms as follows. Given >0O, and f,gsH2, define the a-inner product and the a-norm by = (2n) f f(a+jw)g(a+jw)dw (2.4)o (.5)11fila = = IL (f)(t)L (g)(t)e -2tdt (2.6)Ilfl I = (I IL(f) (t) 2 e 2tdt)1/ 2 (2.7)If L-l(f) and L-l(g) are supported within the compact interval [0,T],given some T>0, thene-T IIfII < If II < I lfll (2.8)andI - Il < (l-e-2T)(llflI + I|g11)2 (2.9)For xeL 2 [0,o), define xT a L2 [0,_) to be the truncation of x at time T, T>0,i.e.x(t) tx and fT -)f as T --),in the usual topology of L2 [0,o) and H2.8III. PROOFS OF THEOREMS 1 AND 2.In proving Theorem 1 we use the following observations.Lemma 5.Let {gn} C HO be a sequence such that [[gn[[| < M. Let X be thecharacteristic function of a set X C [--, a] of positive measure on theimaginary axis. Suppose IIxgnll -30 as n -in. Then for any compact set Yin the open right half plane, Ign(s)[ -30, uniformly for seY.Proof: It seems convenient to establish the lemma in the disc. For gesH,definegD(z) g( 1-z (3.1)Then gDsHe(D) and I[g[I| = I1gDoII- where D is the unit disc on the complexplane. Let YD = {zl(l+z)/(1-z)eY), and XD = {zI(l+z)/(1-z)ejX} be theinverse image of Y and X by the Mobius transformation respectively, and XDbe the characteristic function of XD. Since Y is a compact set in the openright half plane, YD C B(O,r) (- the closed disc of radius r) for someO Xell 91 > ieO+z --- > Re >e-1T ]r (3.3)Using the inequalities (3.3), for zsYD we can find a uniform upperbound for the right hand side of (3.2).logl gD (z)- 2[ Tr log Ig(e )ide - l+-r J log +Ig(e )ie dO]< __1 r Ig(e ) ideO _- J logIge1-r ) 1 1ei d6i llgDl' - 2n l+r logD (° lXg || -+rD~- 1-r *' ~c 2t 1+ lo I o )J.(XD) (3 4)wherelog+x = log(max(l,x)} > 0, (3.5)and g is the Lebesgue measure on the unit circle.Let {gnD} be the sequence in Ha(D) obtained from {gn } by thetransformation (3.1). Note thatIlIgnDi1 < M, and IixDgnD1II -O, as n ->. (3.6)Applying (3.4) to the sequence {gnD), we see that the right hand side of10(3.4) tends to -a uniformly for zeYD. Hence Ign(s)I -*0 uniformly for seY.Q.E.D.Corollary 6. Let {gn) be as in Lemma S, feH 2 and o>0. Then IlgnfIo -0> asn -ha.Proof. Fix >0. Since feH 2, there exists 1>0 such that-1 I2 8(2X) 1f jf(fi+j)j2dw < 2 (3.7)IWI> 2MThis implies that(2T) 1 s Ignf(+jW)I 2dw < (3.8)jWn1>0 2for all n, since lg n ll I < M.Applying Lemma 5 to the sequence {gn } C HI and Y = {sjs=cf+jw, IWS<}G,and Lebesgue's dominant convergence theorem, we see that(2-n) 1J gnf(+jw) 2 dw -*0 as n --> (3.9)Thus for n large enough,I IgnfII = (2n)1 J Ign£f(O+jW)i 2 < F (3.10)This implies that I|gnf Ila -30 as n -Q". Q.E.D.Claim 7. Let fgn}C ",7 I gnllI < M and f E H2. Then for each %>0, thereexist some T>O and >0 such that I - l < X for all n.ProofI - I= I< jlfIfTl Ilgn fll< , .f-f If-fTI Ifll (3.11)Utilizing (2.9), we also haveI - CoI= ' - o6i (1-e-2T )[IlfTII + [II(gnf)TII] 2i (l-e-2"T)[Ilfll + IJgnfll] 2< (1+M)2(1_e- 26T) Ilf112 (3.12)12Recall that fT -*f as T -i. Choose T sufficiently large so that IIf-fTII 0, (3.13)IISI (1 and the infimum is taken over all stabilizing compensators.Proof of Theorem 1.On the contrary, assume that there exists a sequence {S n ) ofsensitivity functions, Sn = l-Phn, hnslH with IISnII,( = (3.14)n n= - = ;in particular, for feK, f#O, = 1lflI > o. (3.15)Hence in view of Claim 7, there exist T>O, a>O, and 6>0 such that > & (3.16)for all n.On the other hand, by corollary 6Iol < IIfT III Snfl -0 as n -3 (3.17)which is a contradiction. Q.E.D.Theorem 2.Suppose PsHe is continuous on the imaginary axis and has at mostcountably many zeros on the imaginary axis. Let X be the characterizationfunction of a compact set jX; XC(-o,o) on the imaginary axis. Then forany 1>e>0 and any M>1 there exists a stabilizing compensator such that14IlXSll, < I , IISfl, ( M (3.18)if and only if P is outer and has no zeros on jX.Proof: (Necessity) If P is not outer, then the conclusion follows fromTheorem 1. Let P have a zero on X. Suppose then that there exists helHsuch thatIIX(1-Ph)ll < £, Il-Ph|Il < M (3.19)Fix 6>0. Because P is continuous on the imaginary axis,i[Xf{wl| IP(iw)I < 8lihil[}] > 0, i.e., ,[X {WI IP(jo)h(jw) < 61}] > 0where p is the Lebesgue measure on the imaginary axis. Since 6 wasarbitrary IIX(1-Ph)ll. > 1, a contradiction.(Sufficiency). The proof is by construction of hesH such thatIIX(1-Ph)ll| < a, Ill-Phil. < M (3.20)for given 01.15LetU = (u:u=-o or P(ju) = 0}. (3.21)From the assumption U is at most countable and UnX = p. Let U = fun,n=1,2,...) be an enumeration of U.Define rn,a bya(s-jun) 1 '2n a(s-jun) + 1n (3.22)r (s) =n,an+1l U-. co,ns+a] =where a>0 is a parameter to be fixed later, and the branch of the2-(n+l)th complex root is decided in such a way that the positive real lineis mapped by rn,a into itself. Eq. (3.22) defines a analytic function onthe open right half plane, since the function a(s-jun)/[a(s-jun)+l] (ora/(s+a)) maps the open right half plane into itself. Furthermore, thefollowing properties of rn a are easily proved: (i) rna e H, I Jrn alI =1; (ii) rn, a is outer; (iii) rn,a is continuous on the imaginary axisincluding c; (iv) rn, a(un) = 0; (v) IIX(1-rn, a)I[, -0 as a -hoa (note thatX n U = 0); and (vi) larg rn,a(s)| < n 2-(n+2), for any s e (Res > 0})u o}.Given e>0 (from (3.20)), we choose q>0 such that [log z[ < q, zeC,16impliesIZ-11 < e (3.23)For each n, we choose the parameter a, according to the property (v), insuch a way thatIIX log rn, alO < n (3.24)2ais satisfied. For brevity we denote rn instead of rn, a, henceforth.The properties (iii) and (iv) imply that there exists a neighborhood Wnof un in the one point compactification of R such that Wn X = 0 andIrn(jW)I < M-1, fiWn . (Note that a neighborhood of - is {[w IwI>}) for some0>O.) U is compact since it is a closed subset of a compact set, and hencethe cover U C Un=lWn has a finite subcover, say UCW = UnCNWn , where N is afinite index set.Since P and rn are outer17P(s) = X expl J 1ogP(jw) l +js (3.25)andr (S) = X exp logIr (ji) I .s+5 d 2J (3.26)n n n W~js ~-+2for X,Xn C, cI = ,lI = 1.Given 6>0, letD. = f{l IP(ji)l < 6) (3.27)and, define hN(s) byh (s) = exp I ( )+J dw (3.28)6 cn f coo+js 1 (a.28)whereC (W) = (3.29)(6) o)$D 618C(W) = log IP(jw) - log Irn(jw) (3.30)nsNThe proof will be completed if we show that h68sH and that forsufficiently small 6>0 this function satisfies (3.19).As is known [9.P53], the right hand side of (3.29) defines an HIfunction if and only if exp(-C 5 ) s L~ and C6 is integrable with respect todwj/;(l1+2). A sufficient condition holds in particular if C6 is bounded. C6is bounded because IPI and rn|1 are bounded and bounded away from zero offD6 .In verifying (3.19) we use the following equalities.Ph6 (s) =TTh n exp V" _ n rN (W)Ij d+ c (@p) ws+j dwo n7 W+js i+21-= rn(s) exp C() w+j d1 (3.31)andN and19Irn(jW)I woD 6lPh 8 (jow) =I (3.32)IP(jw) I eD8Indeed, it is also known [9.P53] that the boundary value of hb(s) as s -ejwsatisfies Ihs(jw)I = esp(-C6 (w)), a.e., and from this (3.32) follows.Note now that if oeW then Irn(jw) < M-1 for some index neN, andrn(jwo) < 1, for all neN. So for 60 as 8 ->0. Consequently, thesecond implies (3.42). This proves the claim. Q.E.D.Remark. (i) Notice that we did not require continuity of P(jw) at= _+O. In fact, the assumptions on the continuity of P(jo) and thecompactness of the subset X, can be relaxed in various ways withoutrequiring considerable changes in the analysis. The current setup waschosen for simplicity.(ii) A major part of the proof of Theorem 2 is dedicated to theconstruction of the "roll-off" functions r n , which are needed when 2 > M >231. For M>2, the assumptions can be further relaxed; e.g., if P(jw) iscontinuous, to the requirement that P be outer and have no zeros in X.24References[1] D. Bensoussan, 'Sensitivity reduction in signle-input single-output systems," Int. J. Contr., Vol. 39, pp. 321-335, 1984.[2] G. Zames and D. Bensoussan, 'Multivariable Feedback, Sensitivity,and Decentralized Control," IEEE Trans. Automat. Contr., Vol. AC-28, pp. 1030-1035, 1983.[3] B.A. Francis and G. Zames, 'On Hr-Optimal Sensitivity Theory forSISO Feedback Systems," IEEE Trans. Automat. Control, Vol. AC-29,pp. 9-16, 1984.[41 B.A. Francis, "A Course in IH Control Theory, Lecture Notes inControl and Information Sciences 88, Springer Verlag, Berlin 1987.[5] L. Pandolfi and A.W. Olbrot, "On the Minimization of Sensitivityto Additive Disturbances for Linear-Distributed parameter MIMOfeedback systems, 'Int. J. Contr., Vol. 43, pp. 389-399, 1986.[6] D.C. Youla, H.A. Jabr and J.J. Bongiorno, "Modern Wiener-HopfDesign of Optimal Controllers - Part II: The Multivariable Case",IEEE Trans. Automatic Contr. Vol. AC-21, pp. 319-338, 1976.[7] C.A. Desoer, R.W. Liu, J. Murray and R. Saeks, "Feedback SystemDesign: the Fractional Representation Approach", IEEE Trans.Automat. Contr. Vol. AC-25, pp. 399-412, 1980.[8] F. Fagnani, D. Flamm and S.K. Mitter, "Some Min-Max OptimizationProblem in Infinite-Dimensional Control Systems', LIDS-P-1640,M.I.T., Cambridge, 1987.[9] K. Hoffman: Banach Spaces of Analytic Functions, Prentice-Hall,Englewood Cliffs, 1962.[10] R.G. Douglas: Banach Algebra Techniques in Operator Theory,Academic Press, New York, 1972.

Sensitivity reduction over a frequency band

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