Stabilization of irrationaltransfer fun tionsby ontrollers with internal loopRuth F. CurtainDept. of Mathemati sUniversity of Groningen9700 AV GroningenThe NetherlandsGeorge WeissDept. of Ele tr. & Ele troni Eng.Imperial College of S i. & Te hnol.Exhibition Rd, London SW7 2BTUnited KingdomMartin WeissPhysi s and Ele troni s LaboratoryTNO, Oude Waalsdorperweg 632509 JG The HagueThe NetherlandsAbstra t: Transfer fun tions are alled well-posed if they are bounded andanalyti on some right half-plane. Two on epts of L2-stabilization are an-alyzed for well-posed transfer fun tions: the usual one, as in Figure 3, anda more general one alled stabilization with internal loop, see Figure 4. Inboth ases we obtain a omplete parametrization of all stabilizing ontrollersin terms of a doubly oprime fa torization of the original transfer fun tion.Moreover, the onne tion between the two stabilization on epts is lari�ed.We analyze two spe ial sub lasses of stabilizing ontrollers with internal loop, alled anoni al and dual anoni al ontrollers. We show that, in a ertainsense, every stabilizing ontroller with internal loop is equivalent to a anon-i al ontroller, and also to a dual anoni al ontroller.1. An intriguing ontrol problemWe onsider the following problem: For an unstable plant with transfer fun tionP(s) = 3 s+ 1s� 1 ;design a ontroller that performs the following two tasks:(1) it stabilizes P,

(2) even if an external (unpredi table) disturban e d is added to the input u ofthe plant, the ontroller predi ts exa tly the output y of the plant one se ond ahead.In other words, at some auxiliary output of our ontroller we an at any momentof time t read o� the value y(t+ 1).This might seem to be impossible at �rst sight sin e, as we said, d is not pre-di table. The problem is illustrated in Figure 1.d(t) u(t) y(t)y(t + 1)

++?Pl

Figure 1: A seemingly impossible problemLet us onne t our plant to a system onsisting of a delay line and a summationpoint, as shown in Figure 2. This inter onne tion has two inputs and two outputs:one input is the disturban e d and one output is the output y of the plant. Theother input is denoted by �i and the other output is denoted by �o. Realizing thisinter onne tion hanges hardly anything: the plant still works in open loop, withthe input u = d+ �i. The transfer fun tion from �i to �o is H(s) = 1� e�s +P(s).yud +++

P(s)1� e�s �o�i +l

lFigure 2: The plant and the ontroller, before losing the feedba k loopNow let us see what happens if we lose a feedba k loop from �o to �i, i.e., weset �i = �o. Closing this loop is possible, be ause 1�H is invertible and moreover,its inverse is in H1. We invite the reader to verify that now �o(t) = y(t + 1) andthe overall system is well-posed and stable. By this we mean that we may inje t2

additional input signals at any point, and the transfer fun tion from any su h inputsignal to any other signal will be in H1. Thus, we have solved our original problem.We emphasize the importan e of pro eeding in stages: �rst onne ting the devi eto the plant, and only afterwards losing the feedba k loop. Pro eeding in thewrong order, i.e., trying to onne t �i to �o �rst, will not work: it is an ill-posedfeedba k. In losed loop, the transfer fun tion from y to �o is es, but realizingthis transfer fun tion in open loop (i.e., without being onne ted to the plant) is learly impossible. This is an example of a so- alled ontroller with internal loop,introdu ed in Weiss and Curtain [17℄. We study the stru ture of su h ontrollers inthe next se tions, and we return to this example at the end of Se tion 4.2. Stabilization with internal loopIn this paper, we investigate the on ept of a stabilizing ontroller, as well asthe problem of parametrizing all stabilizing ontrollers for a well-posed (possiblyirrational) transfer fun tion. Our analysis is entirely in the frequen y domain.Throughout this paper, U and Y are Hilbert spa es, alled the input spa e andthe output spa e, respe tively. An L(U; Y )-valued transfer fun tion is alled well-posed if it is bounded and analyti on some right half-plane (where Re s > �). This lass of fun tions is a natural generalization of the proper rational transfer fun tionsand moreover, it oin ides with the transfer fun tions of well-posed linear systems,a lass for whi h there exists a well developed state spa e theory (see Salamon[7, 8℄, Sta�ans [10, 11, 12℄, Weiss [15, 16℄ and Weiss and Rebarber [18℄). We donot distinguish between two transfer fun tions de�ned on two di�erent right half-planes if one fun tion is a restri tion of the other (thus, by a transfer fun tion wemean, in fa t, an equivalen e lass of analyti fun tions). We mention that matrix-valued transfer fun tions with entries in the fa tor spa e of H1, used for examplein Georgiou and Smith [6℄ and in Foias, �Ozbay and Tannenbaum [4℄, are neither ontained in, nor do they ontain the matrix-valued well-posed transfer fun tions.For any Bana h spa e Z, we denote by H1(Z) the Bana h spa e of Z-valuedbounded analyti fun tions on the usual right half-plane (where Re s > 0), with thesup norm. We write H1 instead of H1(Z), if the spa e Z is lear from the ontext.An L(U; Y )-valued transfer fun tion P is alled stable if P 2 H1(L(U; Y )). This on ept of stability is also alled L2-stability (or input-output stability), be ause itis equivalent to the property that input fun tions u 2 L2([0;1); U) are mappedinto output fun tions y 2 L2([0;1); Y ), via the formula y = Pu, where a hat isused to indi ate the Lapla e transformation.De�nition 2.1. Assume that P and C are well-posed transfer fun tions withvalues in L(U; Y ) and L(Y; U), respe tively. We say that C is an admissible feed-ba k transfer fun tion for P if I � CP (or equivalently I � PC) has a well-posedinverse. In parti ular, we say that C stabilizes P if� I �C�P I ��1 2 H1(L(U�Y )): 3

The intuitive interpretation of the last ondition an be given by the blo kdiagram in Figure 3, where the transfer fun tion from � vpv � to �upu � isTP;C = � I �C�P I ��1 = � (I �CP)�1 C(I �PC)�1P(I �CP)�1 (I �PC)�1 � : (2.1)A few words about the onne tion with state spa e theory: If P and C are thetransfer fun tions of two well-posed linear systems and their feedba k onne tion isexponentially stable, then C stabilizes P, as is easy to see. The more surprizing fa tis that the onverse is true as well, provided that the two systems satisfy ertainnatural assumptions, su h as regularity, stabilizability and dete tability, see Se tion4 of Weiss and Curtain [17℄. Re ently, these assumptions have been repla ed by theweaker onditions of optimizability and dete tability (without assuming regularity),see Se tion 6 of Weiss and Rebarber [18℄. For the state spa e theory of oprimefa torizations we refer to Sta�ans [10℄ and to our paper [3℄.j

PjC +

++

+vp up ypv u y Figure 3: The feedba k onne tion of P and CWhile extending the theory of dynami stabilization to regular linear systems(a sub lass of the well-posed linear systems) in [17℄, it be ame apparent that amore general on ept of stabilizing ontroller is needed. Indeed, it was shownin Example 6.5 of [17℄ that even the standard observer-based ontroller is not, ingeneral, a stabilizing ontroller in the usual sense; that is, it is not a well-posed linearsystem and, orrespondingly, its transfer fun tion is not well-posed. However, theobserver-based ontroller may be regarded in a natural way as a stable well-posedlinear system with two inputs and two outputs. First we onne t one input and oneoutput to the plant, obtaining a new well-posed linear system. Next we onne tthe remaining input of the ontroller to its remaining output, so obtaining a stable losed-loop system. For the details of this pro edure see [17℄.Motivated by this fa t, a new type of ontroller was introdu ed, the so- alledstabilizing ontroller with internal loop, see De�nition 4.10 in [17℄. This was laterused in [3℄, Townley et al [13℄ and [18℄. We now formulate the frequen y domain ounterpart of this on ept. In fa t, we introdu e two new on epts, whi h gener-alize the two on epts from De�nition 2.1.4

De�nition 2.2. Let U , Y and R be Hilbert spa es. Let P andK = " K11 K12K21 K22 #be well-posed transfer fun tions with values in L(U; Y ) and in L(Y �R;U�R),respe tively, and denote G = " P 00 K #. We say that K is an admissible feedba ktransfer fun tion with internal loop for P if I�FG has a well-posed inverse, whereF = 264 0 I 0I 0 00 0 I 375 ; F 2 L(Y�U�R;U�Y�R): (2.2)In parti ular, we say that K stabilizes P with internal loop if(I � FG)�1 2 H1(L(U�Y�R)):upv

KP

+++yp

+ r��i�o

ykuk++ll

l

Figure 4: The plant P onne ted to a ontroller K with internal loopThe intuitive interpretation of De�nition 2.2 is the following: P represents theplant andK is the transfer fun tion of the ontroller in Figure 4 from � yk�i � to �uk�o �,when all the onne tions are open. The onne tion from �o to �i is the so- alledinternal loop. The admissibility part of the de�nition means that it is possible to lose all the onne tions in Figure 4, and obtain well-posed transfer fun tions fromthe three external inputs (v, r and �) to all the other signals. The stabilizationpart of the de�nition means that in fa t all these losed-loop transfer fun tions arestable. To understand that this is indeed what the de�nition says, it is helpful toredraw Figure 4 by learly indi ating the role of F as a feedba k operator for G,the parallel onne tion of P and C, as is done in Figure 5.If I�K22 has a well-posed inverse, then the two on epts introdu ed in De�nition2.2 redu e to those introdu ed in De�nition 2.1. Indeed, in this ase the internal5

ypukyk

v +

(((((((( hhhhhhhh

+�o+ +++ �i K

PF

r�up

ll

l

Figure 5: The parallel onne tion of P and K with the feedba k operator F . Thisdiagram is equivalent to the one in Figure 4.loop an be losed �rst (i.e., before onne ting the ontroller to the plant), to obtaina onventional ontroller with the following transfer fun tion from yk to uk:C = K11 +K12(I �K22)�1K21 : (2.3)In many ases, the ontroller C de�ned by (2.3) will not be well-posed in the senseof our de�nition: for example, it ould be improper or even anti ausal. However,the internal loop onstru tion is not merely an arti� e for allowing improper oranti ausal ontrollers. It in ludes ontrollers for whi h the expresssion (2.3) is notde�ned at all (this an happen if I�K22 is nowhere invertible). A �nite-dimensionalsingle input-single output (SISO) and physi ally meaningful example of a stabilizing ontroller with internal loop for whi h C does not exist was given in Se tion 1 of[17℄. Another, in�nite-dimensional SISO example follows in this se tion.The onne tion of the stabilization on ept in De�nition 2.2 to the state spa etheory of stabilizing ontrollers with internal loop (see De�nition 4.10 in [17℄) issimilar to the one we explained for the stabilization on ept in De�nition 2.1: IfP is the transfer fun tion of a well-posed plant and K is the transfer fun tion ofa stabilizing ontroller with internal loop for the same plant, then K stabilizes Pwith internal loop. The onverse holds whenever the two systems satisfy ertainnatural assumptions, see Proposition 4.11 in [17℄ (where the produ t FG is denotedby L). A stronger version of this result from [17℄ is Theorem 6.4 in [18℄, where theassumptions are optimizability and estimatability (but not regularity).6

For the remainder of this paper, if P and K are as in De�nition 2.2, we shall all K a stabilizing ontroller with internal loop for P. This learly ontradi ts theterminology of [17℄, sin e K is only the transfer fun tion of a ontroller, and notthe ontroller itself. However, sin e our dis ussion here is only in terms of transferfun tions, this will (hopefully) not lead to onfusions.Example 2.3. We take U = Y = R = C andP(s) = 21 + e�2s ; K = " 0 1�1 1 # :It is easy to see that the transfer fun tion (2.3) of the ontroller is unde�ned sin eI �K22 = 0. However, it is not diÆ ult to he k that K stabilizes P with internalloop (this veri� ation an be simpli�ed onsiderably by using Theorem 4.2). Theplant is unstable: P has in�nitely many poles on the imaginary axis. All these fa ts an be understood intuitively from the following physi al interpretation.Consider a ir uit omposed of a unit resistor onne ted in series with a loss-less transmission line of unit length, with unit distributed apa itan e and unitdistributed indu tan e. One end of the transmission line is earthed, as shown inFigure 6. The input of this system is the urrent through the resistor denoted byu(t), whereas the observation is hosen to be the input voltage of the ir uit y(t).rrr CC��CC��CC��CC

0i(t; 1)v(t; 1) = 0i(t; x)x1

v(t; x)R = 1u(t) i(t; 0)v(t; 0)y(t)Figure 6: Transmission line with resistorThe lo al urrent i(t; x) through and the lo al voltage v(t; x) a ross the trans-mission line are related by the equations�i�t = � �v�x ; �v�t = � �i�x ; (2.4)with the boundary onditionsi(t; 0) = u(t);v(t; 1) = 0 (be ause of the earthing),y(t) = u(t) + v(t; 0) (u(t) is added be ause of the resistor).7

It is readily veri�ed that the transfer fun tion from u to y, the impedan e of the ir uit, is P(s) = 21+e�2s . The in�nite number of poles on the imaginary axis anbe explained by the fa t that dis onne ting the input of the ir uit, i.e., u(t) = 0, reates in�nitely many undamped os illatory modes on the line, due to the losslessnature of the line. The only dissipative element, the resistor, annot dissipate theenergy of these os illations if no urrent traverses it. A simple interpretation of theproposed ontroller is the earthing of the input of the ir uit. This auses y(t) = 0,something that is learly not a hievable by a onventional ontroller. It is learthat the system does be ome stable under the new boundary ondition, sin e byearthing the input, the resistor will now be able to dissipate the energy from the ir uit. All these intuitive on lusions will be justi�ed rigorously in Se tion 2.Another motivation for introdu ing ontrollers with internal loop is to obtaina lean and elegant Youla parametrization. In the Youla parametrization of on-ventional ontrollers, if the plant is not stri tly proper, it is diÆ ult to see a priorihow to hoose the parameter in su h a way that the resulting ontroller will bewell-posed. Even if we hoose to ignore well-posedness, as some resear hers do, westill have to ensure that the denominator in the Youla parametrization is invert-ible. Thus, the parametrization is not lean; there is always an extra onditionon the parameter. This in turn makes it awkward to use this parametrization tosolve, for example, the H1- ontrol problem for well-posed linear systems. By on-trast, we do obtain a lean parametrization for all stabilizing anoni al or dual anoni al ontrollers. A stabilizing ontroller with internal loop K is alled anon-i al if K11 = 0; K12 = I and K21;K22 2 H1. It is alled dual anoni al ifK11 = 0; K21 = I and K12;K22 2 H1. We shall prove that, in a ertain sense, itis suÆ ient to onsider anoni al or dual anoni al ontrollers (Theorem 5.2).The organization of the paper is as follows. In Se tion 3, we re all the usualYoula parameterization of all stabilizing ontrollers in terms of a doubly oprimefa torization ( f. Fran is [5℄, Vidyasagar [14℄ and Baras [1℄). However, sin e we onsider the input and output spa es to be Hilbert spa es, ertain te hni al diÆ- ulties arise and we a tually need to redo the proofs for this situation. We also givetwo examples to illustrate the invertibility problems with this parametrization.In Se tion 4 we introdu e anoni al and dual anoni al ontrollers. We showthat a plant P is stabilizable with internal loop by a anoni al (dual anoni al) ontroller if and only if P has a right- oprime (left- oprime) fa torization. Wegive a omplete parameterization of all (dual) anoni al stabilizing ontrollers withinternal loop. Moreover, the relationship between ( onventional) stabilization andstabilization with internal loop by a (dual) anoni al ontroller is lari�ed.In Se tion 5 we introdu e the on ept of equivalen e for stabilizing ontrollerswith internal loop. We show that if the plant P has a doubly oprime fa torization,then any stabilizing ontroller with internal loop for P is equivalent to a anoni alone as well as to a dual anoni al one.8

3. Coprime fa torizationThis se tion ontains more or less well-known material on doubly oprime fa -torizations, only the statements are slightly di�erent, and the proofs are a littlemore involved, be ause we onsider operator-valued fun tions. Thus, we must be areful to distinguish between right and left invertibility, to assume the existen e of oprime fa torizations, et . The reader who is not interested in su h te hni alitiesmay skip to the examples at the end of the se tion.De�nition 3.1. Let P be a well-posed transfer fun tion. A right- oprime fa tor-ization of P over H1 is an ordered set of four operator-valued fun tions M , N , ~R,~S in H1 su h that M has a well-posed inverse, P = NM�1, and ~SM � ~RN = I.A left- oprime fa torization of P over H1 is an ordered set of four operator-valued fun tions ~M , ~N , R, S in H1 su h that ~M has a well-posed inverse, P =~M�1 ~N , and ~MS � ~NR = I.A doubly oprime fa torization of P over H1 is an ordered set of eight operator-valued fun tions M; N; R; S, ~M; ~N; ~R; ~S in H1 su h that, on some right half-plane, M and ~M have well-posed inverses, P has the fa torizationsP = NM�1 = ~M�1 ~N; (3.1)and " M RN S #�1 = " ~S � ~R� ~N ~M # : (3.2)If a left- and a right- oprime fa torization of P are given, as in (3.1), then it iseasy to onstru t R; S; ~R; ~S 2 H1 su h that we have a doubly oprime fa toriza-tion, as in (3.2), see Theorem 60 in Vidyasagar [14℄ or Lemma 4.3 in Sta�ans [10℄.For a matrix-valued fun tion, (3.2) is often written in the form" ~S � ~R� ~N ~M # " M RN S # = " I 00 I # : (3.3)Indeed, for matrix-valued fun tions, (3.3) is equivalent to (3.2), but in general itis weaker than (3.2). Most arguments about doubly oprime fa torizations relyon (3.3), but o asionally we need the fa tors from (3.3) in reversed order, see forexample the proof of Theorem 3.5.Noti e that De�nition 3.1 extends easily to transfer fun tions that are not ne -essarily well-posed, but are matrix-valued with omponents in the quotient �eld ofH1, denoted by H1=H1. Smith [9℄ showed that a transfer fun tion in this lass isstabilizable if and only if it has left- and right- oprime fa torizations. For operator-valued transfer fun tions, no similar result seems to be known. If a matrix-valuedwell-posed transfer fun tion has a left- or a right- oprime fa torization, or if it isstabilizable, then obviously its entries are in H1=H1.Expli it formulas for oprime fa torizations in terms of state spa e realizationsare known for a wide lass of in�nite-dimensional systems alled regular linear9

systems, see our paper [3℄. More general but (unavoidably) somewhat less expli itformulas for well-posed linear systems were given by Sta�ans [10℄. A modest, butuseful formula in terms of transfer fun tions is the following.Proposition 3.2. Let P and C be well-posed transfer fun tions with values inL(U; Y ) and L(Y; U) respe tively. IfC 2 H1 and P(I �CP)�1 2 H1 ; (3.4)then denotingL = " CI #P(I �CP)�1[I �C℄;the following is a doubly oprime fa torization of P:" M RN S # = I + L ; " ~S � ~R� ~N ~M # = I � L : (3.5)Proof. It is easy to see that L2 = 0, so that (I+L)(I�L) = (I�L)(I+L) = I,whi h is (3.2). To he k (3.1), we note that M = (I � CP)�1 and N = P(I �CP)�1, when e P = NM�1. The identity P = ~M�1 ~N is veri�ed similarly.Lemma 3.3. Let P and C be well-posed transfer fun tions with values in L(U; Y )and L(Y; U) respe tively and suppose that P has a left oprime fa torization P =~M�1 ~N and C has a right- oprime fa torization C = WV �1. Then C stabilizes Pif and only if � = ~MV � ~NW is invertible over H1.There is an obvious dual statement in whi h right is repla ed by left.Proof. Let R; S; ~R ; ~S be su h that~MS � ~NR = I and ~S W � ~R V = I : (3.6)Noti e that � = ~MV � ~NW = ~M(I �PC)V .Suppose �rst that C stabilizes P. Then learly ��1 exists and��1 = V �1(I �PC)�1 ~M�1:It is readily veri�ed that, with the notation from (2.1),TP;C = " I +C(I �PC)�1P C(I �PC)�1(I �PC)�1P (I �PC)�1 #= " I +W��1 ~N W��1 ~MV��1 ~N V��1 ~M # (3.7)and sin e TP;C 2 H1, all omponents are in H1.10

Now post-multiply the se ond identity in (3.6) by ��1 ~M to obtain~S W��1 ~M � ~R V��1 ~M = ��1 ~M;whi h shows that ��1 ~M 2 H1. Similarly, post-multiplying the se ond identity in(3.6) by ��1 ~N shows that ��1 ~N 2 H1. Next, pre-multiplying the �rst identityin (3.6) by ��1 gives ��1 ~MS ���1 ~NR = ��1and hen e ��1 2 H1.Conversely, if ��1 2 H1, then we see from � = ~M(I �PC)V that (I �PC)�1exists, and (3.7) shows that C stabilizes P.Remark 3.4 It is possible to weaken the assumption C 2 H1 imposed on the ontroller in Proposition 3.2 to the following: C is a well-posed transfer fun tionand C = WV �1 is a (not ne essarily oprime) fa torization with W;V 2 H1. If Cstabilizes P and P has a left- oprime fa torization P = ~M�1 ~N , then (as in Se tion8.3 in Vidyasagar [14℄) we an hoose the following right- oprime fa torization forC: V = ( ~M � ~NC)�1, W = C( ~M � ~NC)�1.The following theorem is the well known Youla-Bongiorno parameterization.Theorem 3.5. Suppose that the well-posed transfer fun tion P has a doubly o-prime fa torization over H1, as in (3.1) and (3.2).(1) Assume that C is a well-posed transfer fun tion that stabilizes P. ThenC has a right- oprime fa torization over H1 if and only if C has a left- oprimefa torization over H1. In this ase, there exists a unique Q 2 H1 su h that S+NQand ~S +Q ~N have well-posed inverses andC = (R +MQ)(S +NQ)�1 = ( ~S +Q ~N)�1( ~R +Q ~M) (3.8)are right- and left- oprime fa torizations of C.(2) Conversely, if Q 2 H1 is su h that one of S +NQ or ~S +Q ~N has a well-posed inverse, then the other also has a well-posed inverse and the equality in (3.8)holds. In this ase, the transfer fun tion C de�ned in (3.8) stabilizes P.Proof. We prove part (1). Suppose that C stabilizes P and C = WV �1 isa right- oprime fa torization. We show that C an be represented as in the �rstfa torization from (3.8). By Lemma 3.3, � = ~MV � ~NW is invertible over H1, soQ = M�1(W��1 � R) (3.9)is well de�ned and, moreover,R +MQ = W��1: (3.10)11

We al ulateS +NQ = S +NM�1(W��1 �R) (from (3.9))= S + ~M�1 ~N(W��1 �R) (sin e ~M�1 ~N = NM�1)= ~M�1( ~MS + ~NW��1 � ~NR)= ~M�1(I + ~NW��1) (from (3.2))= ~M�1( ~MV � ~NW + ~NW )��1 (by the de�nition of �):Thus S +NQ = V��1 (3.11)and S +NQ has a well-posed inverse, sin e V has one and � 2 H1.Furthermore, from (3.10) and (3.11) we obtainC = WV �1 = W��1�V �1= (R +MQ)(S +NQ)�1 :We have to prove that Q 2 H1. This follows fromQ = ( ~SM � ~RN)Q (from (3.2))= ~S(W��1 � R)� ~R(V��1 � S) (from (3.10) and (3.11))= ( ~SW � ~RV )��1 (from (3.2)).Next, we show that ~S +Q ~N has a well-posed inverse and the two fa torizationsin (3.8) are equal. From (3.3), pre-multiplying by " I �Q0 I # and post-multiplyingby " I Q0 I #, we obtain264 ~S +Q ~N � ~R �Q ~M� ~N ~M 375 264 M R +MQN S +NQ 375 = 264 I 00 I 375 : (3.12)By reversing the order of the fa tors in (3.3) and using a similar devi e we anobtain a version of (3.12) with the fa tors in the reverse order. This proves thatX = 264 ~S +Q ~N � ~R �Q ~M� ~N ~M 375 has a well-posed inverse. (Note that this does notfollow from (3.12) alone, sin e we work on in�nite-dimensional spa es.)Using (3.3), (3.7), the de�nition of � and the expression Q = ( ~SW � ~RV )��1,it is readily veri�ed thatXTP;C = " ~S +Q ~N 00 ~M # : (3.13)Sin e X and TP;C have well-posed inverses, ~S + Q ~N must have one. The (1,2)blo k of (3.12) shows that the two fa torizations in (3.8) are equal. Now we have to12

prove the oprimeness of the fa torizations in (3.8). The following identity whi hfollows from (3.2) shows that R +MQ and S +NQ are right- oprime:~M(S +NQ)� ~N(R +MQ) = I: (3.14)The fa t that the other fa torization is left- oprime is proved similarly.Finally, to prove the uniqueness of Q, �rst noti e that Q from (3.9) is inde-pendent of the hoi e of the fa tors in the right- oprime fa torization C = WV �1.Assume that there is a Q1 2 H1 su h that C = (R+MQ1)(S +NQ1)�1. DenoteW1 = R+MQ1 and V1 = S +NQ1 to obtain �1 = ~MV1� ~NW1 = I. Substitutingthis into (3.9) shows that Q = Q1. Thus we have proved part (1), assuming theexisten e of a right- oprime fa torization for C. A similar argument proves (1) ifwe start from the assumption that C has a left- oprime fa torization.To prove part (2), we assume that Q 2 H1 is su h that S+NQ has a well-posedinverse, and we de�ne C = (R +MQ)(S + NQ)�1. The identity (3.14) obtainedfrom (3.2) shows that this fa torization of C is right- oprime. We apply Lemma 3.3with V = S +NQ, W = R+MQ and � = I to on lude that C stabilizes P. Theremaining statements in part (2) follow from part (1). If we start from a Q 2 H1su h that ~S +Q ~N has a well-posed inverse, the proof of part (2) is similar.Let us apply this theorem to the transfer fun tion P from Example 2.3.Example 3.6. A doubly oprime fa torization of P(s) = 21+e�2s is given byN(s) = ~N(s) = 2,M(s) = ~M(s) = 1+e�2s, S(s) = ~S(s) = 1, R(s) = ~R(s) = 12e�2s.The Youla parametrization (3.8) yields the ontrollerC(s) = �12e�2s + (1 + e�2s)Q(s)� (1 + 2Q(s))�1for any Q 2 H1, and introdu ing ~Q = 1+2Q we obtain the more transparent formC(s) = 12(1 + e�2s)� 12 ~Q�1: (3.15)The above formula yields well-posed ontrollers provided that ~Q has a well-posedinverse. On the other hand, the ontroller suggested in Example 2.3 from physi al onsiderations is not a ounted for by (3.15) (see also Example 4.7).In the above example, it was easy to see whi h parameters ~Q should be hosento obtain a well-posed ontroller. However, in general it is not a priori lear whi hQ 2 H1 will guarantee that S + NQ has a well-posed inverse. We illustrate thispoint by another example.Example 3.7. This is the frequen y domain version of Example 6.5 in [17℄. Wetake U = Y = `2, the spa e of square summable sequen es x = (x1; x2; : : :). Thetransfer fun tion of the plant is P(s) = diag (Pn(s)) ; n = 1; 2; 3; : : :, wherePn(s) = (0:64n� 1)s� ns(s+ n) :13

This has poles of multipli ity one at s = �n, n = 1; 2; 3; : : :, a pole of in�nitemultipli ity at s = 0 and P is well-posed. An observer-based ontroller for thisplant has the transfer fun tion C(s) = diag (Cn(s)), whereCn(s) = (1� 0:64n)s+ ns2 + s(2� 0:28n) + 2n ;as omputed in [17℄. At �rst glan e this seems �ne; Cn(s) is a stri tly properfun tion for every n. However, the real parts of the poles of Cn(s) onverge to +1,as is easy to see. Consequently, C(s) annot be uniformly bounded in norm on anyright half-plane and it is therefore not well-posed.One an use this observer-based ontroller to obtain a doubly oprime fa -torization M = ~M = diag (Mn), N = ~N = diag (Nn), R = ~R = diag (Rn),S = ~S = diag (Sn), with Nn(s) = ~Nn(s) = (0:64n�1)s�ns2+s(1+0:36n)+2n , Mn = ~Mn = 1 + Nn,Sn = ~Sn = 1�Nn, Rn = ~Rn = �Nn. A ording to the Youla parametrization (3.8)a set of stabilizing ontrollers for P is given by C(s) = diag (Cn(s)), whereCn = (Rn +MnQ)(Sn +NnQ)�1; (3.16)and Q 2 H1 is su h that (S+NQ)�1 is well-posed. As we have seen in the ase ofthe observer-based ontroller (Q = 0), this is not always the ase. With some are,we an hoose the parameter Q 2 H1 su h that C is well-posed. For example,Q(s) = 1 yields C(s) = I`2. However, it is not a priori lear whi h Q 2 H1 willlead to a well-posed C.4. Canoni al and dual anoni al ontrollersIn [17℄, a pro edure was developed to design stabilizing ontrollers with inter-nal loop (in the state-spa e framework, assuming regularity, stabilizability and de-te tability). The transfer fun tions of the ontrollers obtained there were of theform K = " 0 IK21 K22 # ; with K21;K22 2 H1 : (4.1)Sin e this lass plays a spe ial role, we all the ontrollers of the form (4.1) anoni al ontrollers. Analogously, ontrollers of the formK = " 0 K12I K22 # ; with K12;K22 2 H1 ; (4.2)will be alled dual anoni al ontrollers (they an be obtained by a dual designpro edure, as explained in Se tion 5 of [17℄).We analyze the properties of (dual) anoni al ontrollers in some detail. Firstwe re all Proposition 4.8 from [17℄. 14

Proposition 4.1. Using the notation of De�nition 2.2, suppose that K11 is anadmissible feedba k transfer fun tion for P. Then the following two onditions areequivalent:(1) I �F(K;P) has a well-posed inverse, whereF(K;P) = K22 +K21P(I �K11P)�1K12:(2) K is an admissible feedba k transfer fun tion with internal loop for P.In fa t, Proposition 4.8 from [17℄ only states that (1) implies (2), but the onverseis also true, with pra ti ally the same proof.Theorem 4.2. The anoni al ontroller K stabilizes P with internal loop i�� = I �K22 �K21Pis invertible on some right half-plane and ��1, P��1 2 H1.If P has a right- oprime fa torization P = NM�1, then K stabilizes P withinternal loop i� D =M �K22M �K21N is invertible over H1.Note that � and ��1 might be de�ned only on some right half-plane of the formRe s > � with � > 0, but this is not a problem if ��1 has an analyti extension toa fun tion in H1 (see our onvention at the beginning of Se tion 2).Proof. First we prove the �rst statement. A ording to Proposition 4.1, I�FGhas a well-posed inverse i� I �F(K;P) = � has a well-posed inverse. Moreover,(I � FG)�1 = 264 I +��1K21P ��1K21 ��1P(I +��1K21P) I +P��1K21 P��1��1K21P ��1K21 ��1 375and so (I � FG)�1 2 H1 i� the expressions��1; P��1; ��1K21P; P(I +��1K21P)are all in H1. We show that if the �rst two of these expressions are in H1, thenso are the other two. We haveI +��1K21P = ��1(� +K21P)= ��1(I �K22) 2 H1and P(I +��1K21P) = P��1(I �K22) 2 H1:Thus, K stabilizes P i� ��1 2 H1 and P��1 2 H1.Let us prove the se ond assertion in the theorem. If P = NM�1 and D�1 2 H1,then from the formulae ��1 = MD�1 ; P��1 = ND�1;15

we see that ��1;P��1 2 H1. Conversely, assuming that M , N , ~R and ~S are asin De�nition 3.1 and ��1;P��1 2 H1, then~S��1 � ~RP��1 = ( ~SM � ~RN)(M �K22M �K21N)�1 = D�1:Sin e ~S and ~R are in H1, we see that D�1 2 H1.We have the following interesting onne tion with right- oprime fa torizations.Corollary 4.3. P has a right- oprime fa torization if and only if P is stabilizablewith internal loop by a anoni al ontroller.Proof. If K stabilizes P, then by Theorem 4.2, ��1 and P��1 are in H1. SoM = ��1 and N = P��1 is a right- oprime fa torization, sin e(I �K22)M �K21N = I:Conversely, if P = NM�1 is a right- oprime fa torization su h that ~SM �~RN = I, then K = " 0 I~R I � ~S # stabilizes P with internal loop. Indeed, from� = I � (I � ~S)� ~RNM�1 = ( ~SM � ~RN)M�1 =M�1, we see that ��1 =M andP��1 = N , and we an apply Theorem 4.2.Proposition 4.4. Suppose that P has a doubly oprime fa torization as in (3.1)and (3.2). Then all anoni al ontrollers that stabilize P with internal loop areparameterized by K = " 0 I~E( ~R +Q ~M) I � ~E( ~S +Q ~N) # ;where Q; ~E 2 H1 and ~E is invertible over H1.Proof. For K as above we show that it stabilizes P with internal loop. We ompute, using (3.2),D = M �K22M �K21N= ~E( ~S +Q ~N)M � ~E( ~R +Q ~M)N= ~E( ~SM � ~RN +Q( ~NM � ~MN)) = ~E:Sin e ~E is invertible over H1, Theorem 4.2 shows that K stabilizes P.Conversely, suppose that K = " 0 IK21 K22 # stabilizes P with internal loop.Theorem 4.2 shows that D =M �K22M �K21N is invertible over H1. De�ningQ = (D�1K21 � ~R) ~M�116

we evaluate~S +Q ~N = ~S +D�1K21 ~M�1 ~N � ~R ~M�1 ~N= ( ~SM � ~RN)M�1 +D�1K21NM�1= D�1(K21N +D)M�1= D�1(M �K22M)M�1= D�1(I �K22)and ~R +Q ~M = ~R + (D�1K21 � ~R) = D�1K21:Thus, denoting ~E = D, we have that K21 = ~E( ~R+Q ~M) and K22 = I� ~E( ~S+Q ~N)as laimed. It remains to prove that Q 2 H1. From (3.2) and our formulae for~R +Q ~M and ~S +Q ~N , we haveQ = Q( ~MS � ~NR)= (D�1K21 � ~R)S � [D�1(I �K22)� ~S℄R= D�1K21S �D�1(I �K22)R;and it is lear that both terms are in H1.We all the fun tion Q appearing in Proposition 4.4, the Youla parameter of K.We mention that by using (one half of) Proposition 4.4, it is possible to give analternative, shorter proof of Theorem 5.4 (the main result) in [17℄.The following orollary ontains the dual statements of Theorem 4.2, Corollary4.3 and Proposition 4.4.Corollary 4.5. Let P be a well-posed transfer fun tion.a. The dual anoni al ontroller K stabilizes P with internal loop i� � = I �K22 �PK12 is invertible on some right half-plane and ��1; ��1P 2 H1.b. If P has a left- oprime fa torization P = ~M�1 ~N , then K stabilizes P withinternal loop i� D = ~M � ~MK22 � ~NK12 is invertible over H1. . P has a left- oprime fa torization if and only if P is stabilizable with internalloop by a dual anoni al ontroller.d. If P has a doubly oprime fa torization as in (3.1) and (3.2), then all dual anoni al ontrollers whi h stabilize P with internal loop are given byK = " 0 (R +MQ)EI I � (S +NQ)E # ;where Q;E 2 H1 and E is invertible over H1.17

Again, we all Q appearing in Corollary 4.5 the Youla parameter of K. Theformulae for the parameterization of all stabilizing (dual) anoni al ontrollers arereminis ent of the Youla parameterization (3.8). Consequently, we expe t a strongrelationship between stabilization with internal loop and the usual on ept of sta-bilization. This is lari�ed in the following proposition.Proposition 4.6. Let P and C be well-posed transfer fun tions with values inL(U; Y ) and L(Y; U), respe tively. Assume that C has a left- oprime fa torizationC = (I�K22)�1K21. Then P is stabilized by C i� P is stabilized with internal loopby the anoni al ontroller K = " 0 IK21 K22 #.Proof. Suppose that C = (I �K22)�1K21 stabilizes P. De�ne� = I �K22 �K21P= (I �K22)(I �CP):Then, a ording to De�nition 2.1 and formula (2.1),��1K21 = (I �CP)�1C 2 H1;��1(I �K22) = (I �CP)�1 2 H1:Sin e K21 and I � K22 are left- oprime, there exist R and S in H1 su h that(I �K22)S �K21R = I, when e��1(I �K22)S ���1K21R = ��1:Sin e both terms on the left-hand side are in H1, we see that ��1 2 H1.Similarly, P��1 2 H1 follows from oprimeness and the fa ts thatP��1(I �K22) = P(I �CP)�1 2 H1;P��1K21 = P(I �CP)�1C 2 H1:A ording to Theorem 4.2, K stabilizes P with internal loop.Conversely, suppose that K stabilizes P with internal loop. Then, from(I �CP)�1 = ��1(I �K22)P(I �CP)�1 = P��1(I �K22)(I �CP)�1C = ��1K21(I �PC)�1 = I +P(I �CP)�1C = I +P��1K21and from Theorem 4.2 we see that all entries of TP;C from (2.1) are in H1.18

Noti e that if a anoni alK stabilizesP with internal loop, thenK21 and I�K22are left- oprime, sin e (I �K22)��1 �K21P��1 = I. Naturally, Proposition 4.6has a dual statement for right- oprime fa torizations of C.An interesting feature of the last result is that it remains true without therequirement that C be well-posed, if we modify De�nitions 2.1 and 3.1 a ordingly,repla ing everywhere well-posedness by analyti ity.We omplete this se tion by returning to two earlier examples.Example 4.7. Using the results in Example 3.6 and Proposition 4.4, we obtainthat all anoni al ontrollers that stabilize P(s) = 21+e�2s areK(s) = " 0 112 ~E(s) h�1 + (1 + e�2s) ~Q(s)i 1� ~E(s) ~Q(s) # ;where ~E; ~E�1; ~Q 2 H1. (We have used, as in Example 3.6, ~Q = 1+ 2Q.) If ~Q�1 iswell-posed, then we obtain the ontrollers (3.15) from Example 3.6, and if ~E(s) = 2and ~Q = 0, then we obtain the ontroller suggested in Example 2.3.Example 4.8. The plant in the \intriguing ontrol problem" of Se tion 1 and its ontroller are determined by the following transfer fun tions:P(s) = 3 s+ 1s� 1 ; K(s) = " 0 11 1� e�s # :Note that K is anoni al (and also dual anoni al). To he k that K stabilizes Pwith internal loop, we use Theorem 4.2. We ompute � = e�s � 3 s+1s�1 , so that��1 = � 13 � s� 1(1� e�s3 )s+ (1 + e�s3 ) :First we show that ��1 has no poles in the losed right half-plane. Indeed, if su ha pole p were to exist, it would have to satisfyp = � 1 + e�p31� e�p3 :But for Re p � 0, the right-hand side above is in the open left half-plane, whi h isa ontradi tion. Thus, ��1 is analyti on an open set ontaining the losed righthalf-plane. To prove that ��1 2 H1, it only remains to show that it does notbehave badly at in�nity. It is easy to see that j��1(s)j < 1 for jsj suÆ iently large(we onsider only s with Re s > 0). The fa t that P��1 2 H1 follows similarly.Computing the transfer fun tion from d to y and from d to �o, we see that the �rstis equal to the se ond delayed by 1 (i.e., multiplied by e�s).19

5. Redu tion to a (dual) anoni al ontrollerIn this se tion, we show that the anoni al ontrollers are not as spe ial as theyseem; any stabilizing ontroller with internal loop is, in a ertain sense, equivalentto a anoni al ontroller. To do this, let us analyze the onne tion in Figure 4 inmore detail. We use the auxiliary inputs v and r from Figure 4, so that up = uk+ vand yk = yp + r, while assuming that � = 0. If K stabilizes P with internal loop,then we an de�ne the transfer fun tion TP;K 2 H1 by" upyk # = TP;K " vr # :We all TP;K the ompensation operator of the pair (P;K). This is analogous tothe transfer fun tion TP;C introdu ed in (2.1). In fa t, TP;K is the left upper 2� 2 orner of the matrix (I � FG)�1 from De�nition 2.2.De�nition 5.1. Let P be well-posed and suppose that K1 and K2 are stabi-lizing ontrollers with internal loop for P. We all K1 and K2 equivalent if the ompensation operators of (P;K1) and (P;K2) are equal.The main result of this se tion is the following.Theorem 5.2. Suppose that P has a doubly oprime fa torization. If K is astabilizing ontroller with internal loop for P, then there exists a anoni al ontroller~K su h that ~K stabilizes P with internal loop and ~K is equivalent to K.Proof. We use the notation from (3.1) and (3.2). By Proposition 4.4, for everyQ 2 H1, the following ~K stabilizes P with internal loop:~K = " 0 I~R +Q ~M I � ( ~S +Q ~N) # : (5.1)We shall �nd a Youla parameter Q 2 H1 su h that the ompensation operatorof (P; ~K) oin ides with that of (P;K). First, we explain how this Q will be onstru ted. We onsider the extended transfer fun tion~P = " �M�1R M�1~M�1 P # :We denote the inputs of ~P by " ~wp~up # and its outputs by " ~vp~yp #, as in Figure 7.We shall prove that if K stabilizes P, then the inter onne tion of ~P and Kshown in Figure 7 is stable. Then we take Q to be the H1 transfer fun tion from~wp to ~vp in the inter onne tion in Figure 7. There are two assertions that have tobe proven in order to omplete the proof of the theorem:Assertion 1: If K stabilizes P, then the system in Figure 7 is stable.20

�i~yp~up

�ouk yk~vp

K~P~wp

Figure 7: The extended system ~P onne ted to the ontroller K with internal loop.The transfer fun tion from ~wp to ~vp is the desired Youla parameter.Assertion 2: If Q is hosen to be the transfer fun tion from ~wp to ~vp in theinter onne tion in Figure 7, and ~K is given by (5.1), then TP;K = T~P; ~K.Proof of Assertion 1: Suppose that K stabilizes P with internal loop. A - ording to De�nition 2.2, this means that the parallel onne tion of P and K as in-dependent hannels, denoted by G in the de�nition, is su h that (I�FG)�1 2 H1.Sin e F is invertible, this is equivalent to the fa t that G is stabilized by the stati output feedba k264 upyk�i 375 = 264 0 I 0I 0 00 0 I 375 264 ypuk�o 375 = F 264 ypuk�o 375 :We have to prove that the parallel onne tion of ~P and K,~G = " ~P 00 K # = 26664 �M�1R M�1 0 0~M�1 P 0 00 0 K11 K120 0 K21 K22 37775 ;is stabilized by the stati output feedba k26664 ~wp~upyk�i 37775 = 26664 0 0 0 00 0 I 00 I 0 00 0 0 I 37775 26664 ~vp~ypuk�o 37775 = ~F 26664 ~vp~ypuk�o 37775 :Equivalently, we have to prove that the four transfer fun tions from (2.1), (I �~G ~F )�1, (I � ~G ~F )�1 ~G, ~F (I � ~G ~F )�1 and (I � ~F ~G)�1, are in H1. Be ause ~F is a21

onstant transfer fun tion, it is enough to prove that the �rst two are inH1, and theother two will follow immediately (for the last one, use the identity (I � ~F ~G)�1 =I + ~F (I � ~G ~F )�1 ~G). Noti e that~G = 2666664 �M�1R h M�1 0 0 i264 ~M�100 375 G 3777775and ~F = 2666664 0 h 0 0 0 i264 000 375 F 3777775 :It is easy to see that(I � ~G ~F )�1 = 2666664 I h 0 M�1 0 i (I �GF )�1264 000 375 (I �GF )�1 3777775 :Sin e F stabilizes G, (I �GF )�1 is in H1. Therefore, (I � ~G ~F )�1 is in H1, if weshow that h 0 M�1 0 i (I�GF )�1 is inH1. Let us denote by Tjk the onstituentblo ks of (I �GF )�1, i.e.,(I �GF ) 264 T11 T12 T13T21 T22 T23T13 T23 T33 375 = 264 I 0 00 I 00 0 I 375 :>From the upper row of this equality, we dedu e thatT11 � I = PT21 ; T12 = PT22 ; T13 = PT23 :Re all from (3.2) that ~SM � ~RN = I and ompute~ST21 � ~R(T11 � I) = ~ST21 � ~RNM�1T21= ( ~SM � ~RN)M�1T21= M�1T21: (5.2)Similarly,~ST22 � ~RT12 = ~ST22 � ~RNM�1T22 =M�1T22 (5.3)and ~ST23 � ~RT13 = ~ST23 � ~RNM�1T23 =M�1T23: (5.4)22

This shows thath 0 M�1 0 i (I �GF )�1 =M�1 h T21 T22 T23 i 2 H1:So we have shown that (I � ~G ~F )�1 is in H1.Let us now show that (I � ~G ~F )�1 ~G is in H1. We an ompute, using thenotation introdu ed before, that(I � ~G ~F )�1 ~G == 2666664 �M�1R +M�1T21 ~M�1 h M�1 0 0 i +M�1 h T21 T22 T23 iG(I �GF )�1 264 ~M�100 375 (I �GF )�1G 3777775 : (5.5)Sin e F stabilizes G, we have that (I �GF )�1G is in H1. We have to he k thatthe other three blo ks in (5.5) are in H1. For this, we onsider the identity264 T11 T12 T13T21 T22 T23T13 T23 T33 375 (I �GF ) = 264 I 0 00 I 00 0 I 375 :>From the se ond row and the se ond olumn of the previous identity, we dedu eT12 = T11P ; T22 � I = T21P ; T32 = T31P ; (5.6)T22K11 + T23K21 = T21 ; T22K12 + T23(K22 � I) = 0 : (5.7)We an now pro eed to show that the remaining three blo ks in (5.5) are in H1.Blo k (2,1): We have to prove that(I �GF )�1 264 ~M�100 375 = 264 T11 ~M�1T21 ~M�1T31 ~M�1 375 2 H1:Re all from (3.2) that ~MS � ~NR = I and use the identities (5.6) to omputeT11S � T12R = T11S � T11PR= T11 ~M�1( ~MS � ~NR) = T11 ~M�1;and similarlyT21S + (I � T22)R = T21S � T21PR= T21 ~M�1( ~MS � ~NR) = T21 ~M�1;T31S � T32R = T31S � T31PR = T31 ~M�1:These identities show that blo k (2,1) in (5.5) is in H1.23

Blo k (1,1): We have, using identity (5.2) and (3.2), that�M�1R +M�1T21 ~M�1 = �M�1R + ( ~ST21 � ~RT11 + ~R) ~M�1= ~ST21 ~M�1 � ~RT11 ~M�1:As we previously proved, T21 ~M�1 and T11 ~M�1 are in H1, and so blo k (1,1) in(5.5) is in H1.Blo k (1,2): A little omputation shows that blo k (2,1) is equal toh M�1 +M�1T21P M�1(T22K11 + T23K21) M�1(T22K12 + T23K22) i :Using (5.6), we haveM�1 +M�1T21P =M�1 +M�1(T22 � I) =M�1T22;whi h is in H1 by (5.3). Using (5.7), we obtainM�1(T22K11 + T23K21) =M�1T21;whi h is in H1 by (5.2), andM�1(T22K12 + T23K22) =M�1T23;whi h is in H1 by (5.4). This on ludes the proof of Assertion 1.Proof of Assertion 2: We have hosen Q to be the transfer fun tion from ~wpto ~vp in Figure 7. By Assertion 1, this is in H1. We must show that, if we hoose~K from (5.1) with this Q, then TP;K = TP; ~K.~K from (5.1) an be realized via the system KG with outputs ~uk, ~wk, ~�o, inputs~yk, ~vk, ~�i and onne tion ~vk = Q ~wk, as in Figure 8, withKG = 264 0 0 I~M 0 � ~N~R I I � ~S 375 :To see this, onsider264 ~uk~wk~�o 375 = 264 0 0 I~M 0 � ~N~R I I � ~S 375 264 ~yk~vk~�i 375 ; ~vk = Q ~wk :These yield ~uk = ~�i and ~�o = ( ~R +Q ~M)~yk + (I � ~S �Q ~N)~�i, i.e.," ~uk~�o # = ~K " ~yk~�i # :The realization of ~K as in Figure 8 is more onvenient to show that (P; ~K) has thesame ompensation operator as (P;K), i.e.," up~yk # = TP;K " vr # : 24

~�iKG~ukg P

~wk ~vkg~�o Q~yk

up ypvr++++

Figure 8: The system in Figure 4, with ~K in pla e of K, after the realization of ~Kas the feedba k inter onne tion of KG with Q (and with � = 0)Let us expand Q in Figure 8 into its omponents to obtain Figure 9.The idea is now to prove that yk = ~yk and uk = ~uk. If this is so, then yk = yp+rand up = uk + v, and this implies that the transfer fun tion from " vr # to " upyk # isTP;K. Sin e yk = ~yk, this is enough to prove our laim. So it remains only to provethat yk = ~yk and uk = ~uk in Figure 9.First we show that the onne tion between KG and ~P in Figure 9 (representedseparately in Figure 10) is well-posed, i.e., the output feedba k �F given by26666664 ~yk~vk~�i~wp~up37777775 = 26666664 0 0 0 0 00 0 0 I 00 0 I 0 00 I 0 0 00 0 0 0 0

37777775 26666664 ~uk~wk~�o~vp~yp37777775 = �F 26666664 ~uk~wk~�o~vp~yp

37777775is admissible for the parallel onne tion of KG and ~P,�G = " KG 00 ~P # :Straightforward omputations using the identity (3.2) yield(I � �G �F )�1 = 26666664 I �R M M 00 I + ~NR � ~NM � ~NM 00 �R M M 00 � ~SR �I + ~SM ~SM 00 S �N �N I37777775 :25

~�i~uk

�o �i

~wk ~wp

gg~yk~�o

+ up ypKG

uk yk~vk~vp~up ~yp

K~P

Pvr+++

Figure 9: Expanding Q in Figure 8, using that Q is obtained in Figure 7 as thetransfer fun tion from ~wp to ~vpSo we have well-posedness and, in addition, (I � �G �F )�1 2 H1. Moreover,(I � �G �F )�1 �G = 26666664 0 � � � I� � � � �� � � � �� � � � �I � � � 037777775 ;in whi h a star denotes an irrelevant entry. This shows that" ~yk~up # = " 0 II 0 # " ~uk~yp #and hen e yk = ~yk and uk = ~uk, as laimed.Example 5.3. It is interesting to al ulate the anoni al equivalent to a dual anoni al ontroller. Suppose that K = " 0 K12I K22 # stabilizes the plant P with26

~�i~uk~wk ~wp~�o~ykKG ~vk~vp~up ~yp~P

Figure 10: Conne tion between KG and ~P in Figure 9internal loop. If P admits a doubly oprime fa torization, then the pro edure inthe proof of Theorem 5.2 produ es the anoni al ontroller~K = " 0 IM�1K12 ~��1 I �M�1(I +K12 ~��1P) # ;where ~� = I �K22 �PK12, and M omes from the oprime fa torization (3.1).It is lear that the following dual to Theorem 5.2 holds.Corollary 5.4. Suppose that P has a doubly oprime fa torization. If K is astabilizing ontroller with internal loop for P, then there exists a dual anoni al ontroller ~K su h that ~K stabilizes P with internal loop and ~K is equivalent to K.Theorem 5.5. Suppose that P has a doubly oprime fa torization as in (3.1),(3.2), and K is a stabilizing ontroller with internal loop for P. Then there existsa unique Q 2 H1 su h that K is equivalent to the anoni al ontrollerKl = " 0 I~R +Q ~M I � ( ~S +Q ~N) # ;and is also equivalent to the dual anoni al ontrollerKr = " 0 R +MQI I � (S +NQ) # :The proof is the same as for Theorem 5.2.27

6. Con lusionsIn this paper, we have examined two on epts of stabilization for the lass ofwell-posed transfer fun tions. The motivation for hoosing this lass was that itsmembers have realizations as well-posed linear systems (see Salamon [8℄, Sta�ans[12℄). We have proposed a new and more general type of ontroller, ontrollerswith an internal loop. The need for this extension is demonstrated by severalexamples of su h ontrollers, whi h annot be redu ed to onventional ontrollers.We have introdu ed anoni al and dual anoni al ontrollers, whi h are ontrollerswith internal loop of a spe ial (simple) stru ture. We have found that these are losely related to (doubly) oprime fa torizations of the plant transfer fun tion.For the ase that P has a doubly oprime fa torization, we have given a ompleteparameterization of all stabilizing ontrollers with internal loop whi h are (dual) anoni al. We have proven that any stabilizing ontroller with internal loop isequivalent to a anoni al one and also to a dual anoni al one. Finally, we remarkthat, although our motivation was to develop a new theory for the well-posed lassof irrational transfer fun tions, the on ept of stabilization with internal loop is neweven for rational transfer fun tions.Referen es[1℄ J.S. Baras: Frequen y domain design of linear distributed systems. In Pro .of the 19th IEEE De ision and Control Conferen e, pp. 728{732, 1980.[2℄ F.M. Callier and C.A. Desoer: An algebra of transfer fun tions for distributedlinear time-invariant systems. IEEE Trans. Cir uits and Systems, 25:651{663,1978. (Corre tions: vol. 26, p. 320, 1979).[3℄ R.F. Curtain, G. Weiss, and M. Weiss: Coprime fa torization for regular linearsystems. Automati a, 32:1519{1531, 1996.[4℄ C. Foias, H. �Ozbay and A. Tannenbaum: Robust Control of In�nite Dimen-sional Systems, Springer-Verlag, LNCIS 209, 1996.[5℄ B.A. Fran is: A Course in H1 Control Theory. Springer-Verlag, LNCIS vol.88, Berlin, 1987.[6℄ T.T. Georgiou and M.C. Smith: Graphs, ausality and stabilizability: linear,shift-invariant systems on L2[0;1). Math. for Control, Signals and Systems,6:195{223, 1993.[7℄ D. Salamon: In�nite-dimensional systems with unbounded ontrol and ob-servation: A fun tional analyti approa h. Trans of the Amer. Math. So .,300:383{431, 1987.[8℄ D. Salamon: Realization theory in Hilbert spa e. Math. Systems Theory,21:147{164, 1989. 28

[9℄ M.C. Smith: On stabilization and the existen e of oprime fa torizations.IEEE Trans. on Automati Control, 34:1005{1007, 1989.[10℄ O.J. Sta�ans: Coprime fa torizations and well-posed linear systems. SIAM J.Control and Optim., 36:1268{1292, 1998.[11℄ O.J. Sta�ans: Quadrati optimal ontrol of well-posed linear systems. SIAMJ. Control and Optim., 37:131{164, 1998.[12℄ O.J. Sta�ans: Admissible fa torizations of Hankel operators indu e well-posedlinear systems. Systems and Control Letters, 37:301{307, 1999.[13℄ S. Townley, G. Weiss and Y. Yamamoto: Dis retizing ontinuous-time on-trollers for in�nite-dimensional linear systems. Pro . of the MTNS Symposium,Padova, Italy, July 1998, pp. 547-550.[14℄ M. Vidyasagar: Control Systems Synthesis: A Fa torization Approa h. M.I.T.Press, 1985.[15℄ G. Weiss: Transfer fun tions of regular linear systems. Part I: hara terizationsof regularity. Trans. Amer. Math. So ., 342:827{854, 1994.[16℄ G. Weiss: Regular linear systems with feedba k. Mathemati s for Control,Signals and Systems, 7:23{57, 1994.[17℄ G. Weiss and R.F. Curtain: Dynami stabilization of regular linear systems.IEEE Trans. Aut. Control, 42:1{18, 1997.[18℄ G. Weiss and R. Rebarber: Optimizability and estimatability for in�nite-di-mensional linear systems. SIAM. J. Control and Optim., to appear in 2000.

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