Limits of Generalized State Space Systems underProportional and Derivative FeedbackD. HinrichsenInstitut f�ur Dynamische SystemeUniversit�at BremenD-28344 BremenGermanydh@mathematik.uni-Bremen.deJ. O'Halloran�Dept. of Mathematical SciencesPortland State UniversityPortland, OR 97207-0751U.S.A.joyce@mth.pdx.eduKeywords: high gain feedback, singular systems, proportional andderivative feedback, feedback invariants, orbit closures

AbstractIn this paper, we study the high gain feedback classi�cation problem for gener-alized state space systems. We solve this problem for proportional and derivativefeedback transformations of regularizable systems, i.e. we give necessary and su�cientconditions for a regularizable system to be a limit of a given system under high gainproportional and derivative feedback. We also derive a new complete set of invariantsfor proportional feedback equivalence and specify a set of necessary conditions for asystem to be limit of another system under these feedback transformations. The nec-essary conditions are su�cient for arbitrary state space systems and for controllablesingular systems.�The second author's work on this paper was partially supported by a grant from the National ScienceFoundation. The support from the University of Bremen is also gratefully acknowledged.1

1 IntroductionGeneralized state space systems are described by time-invariant di�erential and algebraicequations of the formE _x(t) = Ax(t) +Bu(t); t � 0; E;A 2 K q�n ; B 2 K q�m (1.1)where K is either the �eld of complex numbers or the �eld of real numbers. If q = n andE is the identity matrix I, the equation (1.1) represents a state space system. Generalizedstate space equations of the above form are studied in the theory of singular or descriptorsystems (where often q = n is supposed), see [ZST87], [HO90], [LOMK91]. Outside thearea of singular systems, generalized state space equations of the form (1.1) have also drawnattention in the recent literature on behaviors, see [Wil91], [Kui94]. In this context it isusually assumed that q � n; m � q � n and the pencil [sE � A] has generic rank n(admissibility condition). The vector u then represents the whole set of behavioral variables,see [Kui94, 4.4].We consider the following transformations of generalized state space systems:Left multiplication: LE _x = LAx + LBu; L 2 Glq(K )State change of coordinates: ER _x = ARx +Bu; R 2 Gln(K )Input change of coordinates: E _x = Ax+BWu; W 2 Glm(K )State feedback: E _x = (A+BF )x+Bu; F 2 Km�nDerivative feedback: (E +BK) _x = Ax +Bu; K 2 Km�n (1.2)The �rst two transformations de�ne a natural equivalence relation between generalizedstate space systems:(E;A;B) � (LER;LAR;LB); L 2 Glq(K ); R 2 Gln(K ):In [HO90] necessary and su�cient conditions were derived for a controllable regular system(E;A;B) to be in the closure of the equivalence class of another such system. (Here (E,A,B)is called regular if q = n and det(sE � A) 6� 0, it is called controllable if [sE + tA B] hasfull row rank for all (s; t) 2 C 2 n f(0; 0)g). Moreover it was shown that the quotient spaceof all controllable regular systems modulo equivalence is a smooth quasiprojective variety.In the behavioral context (where p = q � n > 0; m1 = m � p > 0) it was proved by Raviand Rosenthal [RR95] that the quotient space of controllable generalized state space systems2

(1.1) modulo equivalence is isomorphic to the smooth projective variety of all m1� (m1+ p)homogeneous autoregressive systems of McMillan degree n as introduced in [RR94].In this paper we will analyze the combined e�ect of equivalence and feedback transfor-mations on generalized state space systems. Let G and D be the groups generated by the�rst four and �ve transformations in (1.2), respectively. These groups act as transformationgroups on the set Sq;n;m of all systems (E;A;B) 2 K q�n � K q�n � K q�m . The action of Gextends the action of the full feedback group [Bru70] to the space Sq;n;m. It is called thegroup of proportional feedback transformations and two systems are called feedback equivalentif they can be transformed into each other by transformations in G. D is called the group ofproportional plus derivative feedback transformations and two systems are called pd-feedbackequivalent if they belong to the same D-orbit. Proportional plus derivative feedback trans-formations were introduced into the context of singular systems by Zhou, Shayman and Tarnin [ZST87].Historically, it was one of the fundamental problems in the structure theory of linear statespace systems to �nd a canonical form for the action of the full feedback group on the set ofcontrollable linear systems. This problem was solved by Brunovsky [Bru70]. While the poleshifting theorem and Rosenbrock's theorem [Ros74] showed the extent to which the spectraand the invariant factors of a controllable state space system can be changed by state feed-back, Brunovsky's Theorem identi�ed a set of invariants which cannot be altered by feedbackand which altogether are su�cient to identify a controllable state space system modulo feed-back equivalence. Several authors contributed to an extension of Brunovsky's canonical formto di�erent classes of generalized state space systems and feedback transformations. We onlymention M. A. Shayman who constructed a canonical form for controllable singular systemswith constant-ratio proportional and derivative feedback [Sha88] and H. Gl�using-L�uer�enwho extended Brunovsky's canonical form to impulse controllable systems [Glu90]. Finally,a comprehensive solution of the feedback canonical form problem for generalized state spacesystems was given by Loiseau, �Ozcaldiran, Malabre, and Karcanias in [LOMK91]. Theirresults apply to arbitrary singular systems of the form (1.1) without any controllabilityassumptions and yield complete characterizations of the equivalence classes in Sq;n;m bothunder proportional and under proportional plus derivative feedback transformations.The present paper is built on the results in [LOMK91] and studies the question of3

which systems can be obtained in the limit by applying (pd-)feedback transformationsto a given generalized state space system. A sequence of feedback equivalent systemsGn � (E;A;B); Gn 2 G may have a �nite limit (E;A;B) 2 Sq;n;m although the sequenceof transformations itself, Gn, is unbounded in G (high gain feedback). In this case thefeedback invariants of the limit (E;A;B) may be di�erent from the invariants of (E;A;B).The high gain feedback classi�cation problem consists of determining the sets of feedbackinvariants which can be obtained from a given system in the limit by applying high gainfeedback. In other words the problem consists of determining the topological closures of theG-equivalence classes (G-orbits) in the space Sq;n;m. Note that the relationG � (E;A;B) � G � (E;A;B) (1.3)de�nes an order on the set of feedback orbits in Sq;n;m (or the corresponding set of com-plete invariants), the adherence order. To solve the high gain feedback classi�cation problemthis adherence order has to be determined. An analogous problem can be formulated forproportional plus derivative feedback transformations.For controllable state space systems it has been known since the late seventies that theadherence order of the feedback orbits is given by the dominance order of the correspondinglists of controllability indices; see [HM81] for an instructive discussion of this result in awider mathematical context. An extension of this result to controllable singular systemscan be found in [GH94]. In both cases the restriction of the analysis to controllable systemsis severe since one of the important degeneration phenomena with high gain feedback isthat controllability may be lost in the limit. Without this assumption the problem is stillunsolved although many partial results have been obtained, see [HO95] and the referencestherein. Necessary conditions can be derived from the characterization of orbit closuresof singular matrix pencils in [HO92]. These necessary conditions have been proved to besu�cient in the case of state space systems, see [HO95, Thm. 4.6]. In the present paper wesolve the high gain feedback classi�cation problem for proportional plus derivative feedbacktransformations of regularizable systems, i.e. determine the adherence order of the D-orbitsin the space of regularizable systems, without any other additional assumption. We alsoderive a set of necessary conditions concerning G-orbit closure.In order to obtain these results we use the pencil approach developed in [HO95] andproceed as follows. In Sections 3 we describe the complete set of G-invariants and the G-4

canonical form constructed by Loiseau, �Ozcaldiran, Malabre, and Karcanias [LOMK91]. InSection 4 we derive new necessary and su�cient criteria for G-equivalence. We characterizeG-equivalence via the equivalence of a pair of pencils associated with each system. Moreprecisely, we associate with (1.1) the pencil [sE�A B] and its quotient by the image of theinput operator B. The Kronecker invariants of both pencils constitute together a completeset of invariants of (1.1) with respect to G-equivalence. We determine the relationshipbetween these invariants and the invariants given in [LOMK91]. Consequences of thesecharacterizations ofD- and G -equivalence are presented in Section 5 for regularizable systems(where q = n). In Section 6 we use our characterization of G-equivalence to obtain necessaryconditions for a system to lie in the topological closure of the G-orbit of another system. Wealso give necessary conditions for a system to lie in the topological closure of a D -orbit. Forregularizable systems, we show that these necessary conditions are su�cient. It is an openproblem whether, in the regularizable case, the conditions for topological closure under theaction of G are also su�cient.2 PreliminariesProportional feedback and proportional plus derivative feedback can be described in termsof group actions on the set of q � (n +m) matrix pencils [sE � A B] associated with thesystems (1.1). This will be explained in the following.A u� v matrix pencil is a pair (M;N) of u� v matrices with entries in K , denoted sM +N .Let Pu;v = Glu(K ) �Glv(K ) (2.1)Two u� v matrix pencils are said to be equivalent if one can be transformed to the other bythe "coordinate change" action of Pu;v:(U; V ) � (sM +N) = U(sM +N)V �1 = sUMV �1 + UNV �1 (2.2)A complete set of invariants for these transformations is given by the Kronecker invariants(�nite and in�nite elementary divisors, row and column indices). We refer to [Gan59] forthe de�nition of these invariants and for a description of the associated canonical form dueto Kronecker. 5

In the following, q � (n+m) pencils of the form [sE �A B] = s[E 0] + [�A B] will becalled system pencils. That is, the setSq;n;m = f[sE � A B] : E;A 2 K q�n ; B 2 K q�mgconsists of the system pencils of dimensions q; n;m. In terms of system pencils, proportionalplus derivative feedback transformations may be described as follows. Let the groupD = f(L; 2666664 R 0 00 R 0K F W 3777775) : L 2 Glq(K ); R 2 Gln(K ); W 2 Glm(K ); F;K 2 Km�ng (2.3)act on the set Sq;n;m of q � (n+m) system pencils by(L; 2666664 R 0 00 R 0K F W 3777775�1) � [sE � A B] = [sL(ER +BK)� L(AR � BF ) LBW ] (2.4)This group action describes the combined application of the �ve transformations (1.2) tosystem (1.1). Thus a system is transformed to another via proportional plus derivativefeedback if and only if the two system pencils lie in the same D-orbit.Proportional feedback transformations may be described as the restriction of the action (2.4)to the subgroup G1 of D consisting of matrix pairs in which K = 0. It is straightforward tocheck that the group G1 is isomorphic to the following subgroup of Glq(K ) �Gln+m(K ):G = f(L; 264 R 0F W 375) : L 2 Glq(K ); R 2 Gln(K );W 2 Glm(K ); F 2 Km�ng (2.5)and via the isomorphism, the corresponding action on Sq;n;m is given by matrix multiplica-tion: (L; 264 R 0F W 375�1) � [sE � A B] = L[sE � A B] 264 R 0F W 375= [sLER � L(AR� BF ) LBW ] (2.6)Since proportional feedback transformations are usually described via the generalized feed-back group G and the action (2.6), this is the group we will consider, keeping in mind that itmay be embedded in D in such a way that the action of G is a restriction of the action of D.6

Throughout this paper the terms "proportional plus derivative feedback transformations"and "proportional feedback transformations" refer to transformations of both the systems(1.1) and their associated pencils [sE � A B].As we see from (2.6), the action of G on the set Sq;n;m of q � (n +m) system pencils is arestriction of the action (2.2) of Pq;n+m on the set of q�(n+m) matrix pencils. Summarizingthe relationships among these group actions in terms of standard orbit notation, we have,for any system pencil [sE � A B] 2 Sq;n;m:G � [sE � A B] � (D � [sE � A B]) \ (Pq;n+m � [sE � A B]) (2.7)We will see in the next section that the intersection of every pencil equivalence class of asystem pencil with Sq;n;m consists of a �nite number of G-orbits.3 The G-canonical formFor the convenience of the reader, we include a concise description of the G-canonical formwhich appears in [LOMK91] for the �eld K = C . (For the case K = R, blocks given by �niteelementary divisors are in real Jordan form.) In the canonical form, the pencil sE � A andthe matrix B are in "block diagonal" form (the blocks are not necessarily square), wherethe blocks are determined by the G-invariants listed below. While retaining the notation of[LOMK91], we introduce names for the two types of column indices and the two types ofin�nite elementary divisors in the following list of G-invariants:G-invariants which are invariant under pencil equivalence (2.2):�nite elementary divisors f�i;(kij)dij=1ggi=1, �i 2 C , 0 < kij 2 Z; ki1 � ki2 � : : : � kidirow indices �1 � : : : � �z � 0, �i 2 ZTwo types of column indices:regular column indices �1 � : : : � �v � 0, �i 2 Zsingular column indices 1 � : : : � w � 1, i 2 ZTwo types of in�nite elementary divisor blocks:7

regular nilpotency indices q1 � : : : � qx � 1, qi 2 Zsingular nilpotency indices p1 � : : : � py � 0, pi 2 ZTogether, these invariants constitute a complete set of G-invariants. Hence each G-equivalenceclass is determined by a list(f�i;(kij)dij=1ggi=1, (�1; : : : ; �z), ( 1; : : : ; w), (q1; : : : ; qx), (�1; : : : ; �v), (p1; : : : ; py)) (3.1)Here the invariants are given in the following order: �nite elementary divisors, row indices,singular column indices, regular nilpotency indices, regular column indices, singular nilpo-tency indices. Note that these G-invariants are a re�nement of the Kronecker invariants of thepencil [sE�A B]. The �nite elementary divisors of [sE�A B] are f(s��i)kij : 1 � i � g,1 � j � dig and the row indices of [sE � A B] are ( �1; : : : ; �z). If we use < a1; : : : ; ar >to denote a rearrangement (ai1; : : : ; air) of the list (a1; : : : ; ar) so that ai1 � : : : � air , thenthe column indices of [sE � A B] are(c1; : : : ; cv+w) =< �1; : : : ; �v; 1 � 1; : : : ; w � 1 > (3.2)and the in�nite elementary divisor blocks in the pencil canonical form of [sE �A B] havesizes (n1; : : : ; nx+y) =< q1; : : : ; qx; p1 + 1; : : : ; py + 1 > : (3.3)Consequently, the set of system pencils Pq;n+m-equivalent to a given one is the union of a�nite number of G-orbits.In describing the blocks determined by these invariants and the construction of the G-canonical form from these blocks, we use the following notation:M �N = 264 M 00 N 375 ; � M j N � = � M N � ; 2666664 M�N 3777775 = 264 MN 375 (3.4)Ik is the k � k-identity matrix.Nk is the nilpotent k � k-matrix with 1's above the diagonal, other entries 0.For k = 0, Ik and Nk are empty. 8

For k > 0, ck is the k� 1 matrix with last entry 1, other entries 0, and M � ck is the blockdiagonal matrix (3.4) where N = ck.For k = 0, we de�neM � c0 = 2666664 M j 2666664 0...0 3777775 3777775 ; M � cT0 = 2666664 M�� 0 � � � 0 � 3777775Up to block rearrangement, every q � (n +m) system pencil is G-equivalent to exactlyone pencil of the form [sEc � Ac Bc]:sEc � Ac = (�gi=1 �dij=1 (s� �i)Ikij +Nkij )� (�zi=1 2666664 sI�i +NT�i�cT�i3777775)�(�wi=1 � sI i�1 +N i�1 j c i�1 �)� (�xi=1Iqi + sNTqi)�(�vi=1sI�i +N�i)� (�yi=1 2666664 Ipi + sNTpi�scTpi

3777775) (3.5)Bc = 2666664 0�(�vi=1c�i)� (�yi=1cpi+1)

3777775Note that zero row indices correspond to zero rows, zero regular column indices correspondto zero columns in Bc, singular column indices equal to 1 correspond to zero columns insEc � Ac, and zero singular nilpotency indices correspond to zero rows in sEc � Ac. Abovethe line in Bc there are Pgi=1Pdij=1 kij + Pzi=1(�i + 1) + Pwi=1( i � 1) + Pxi=1 qi zero rows.From the canonical form, we have the following equations, relating the G-invariants and thesystem pencil dimensions:q = Pgi=1Pdij=1 kij +Pzi=1(�i + 1) +Pwi=1( i � 1) +Pxi=1 qi +Pvi=1 �i +Pyi=1(pi + 1)n = Pgi=1Pdij=1 kij +Pzi=1 �i +Pwi=1 i +Pxi=1 qi +Pvi=1 �i +Pyi=1 pim = v + y (3.6)9

In particular, a list of the form (3.1) is the list of G-invariants of a q� (n+m) system pencilif and only if these conditions are satis�ed.Example 3.1 A system pencil with invariants�nite elem. divisors row indices sing. column indicesf3; (2; 1)g (�i) = (1) ( i) = (2; 1)reg. nilp. indices reg. column indices sing. nilp. indices(qi) = (2) (�i) = (2; 0) (pi) = (2; 0)has G-canonical form2666666666666666666666666666666666666666666664

s� 3 1 0 0 0 0 0 0 0 0 0 0 0 j 0 0 0 00 s� 3 0 0 0 0 0 0 0 0 0 0 0 j 0 0 0 00 0 s� 3 0 0 0 0 0 0 0 0 0 0 j 0 0 0 00 0 0 s 0 0 0 0 0 0 0 0 0 j 0 0 0 00 0 0 1 0 0 0 0 0 0 0 0 0 j 0 0 0 00 0 0 0 s 1 0 0 0 0 0 0 0 j 0 0 0 00 0 0 0 0 0 0 1 0 0 0 0 0 j 0 0 0 00 0 0 0 0 0 0 s 1 0 0 0 0 j 0 0 0 00 0 0 0 0 0 0 0 0 s 1 0 0 j 0 0 0 00 0 0 0 0 0 0 0 0 0 s 0 0 j 1 0 0 00 0 0 0 0 0 0 0 0 0 0 1 0 j 0 0 0 00 0 0 0 0 0 0 0 0 0 0 s 1 j 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 s j 0 0 1 00 0 0 0 0 0 0 0 0 0 0 0 0 j 0 0 0 1

3777777777777777777777777777777777777777777775Remark 3.2 Suppose q = n. Since there are continuous invariants (the �i's of the �niteelementary divisors) there are in�nitely many G-orbits in Sn;n;m. However, we have shownin [HO95] that there exists a unique G-orbit of dimension 2n2 + nm in Sn;n;m, namely theG-orbit of any controllable state space system having the generic list of Brunovsky indices(see the proof of Prop. 5.5 in [HO95]). Every other G-orbit in Sn;n;m lies in the closure ofthis orbit and is of lower dimension.10

4 Characterizations of G-equivalenceIn this section, we derive an alternative complete set of invariants for the action (2.6) ofthe group G on the set of q � (n +m) system pencils [sE � A B]. We will see that theseinvariants are closely related to the G-invariants described in Section 3. The computation ofthese invariants requires only the computation of the Kronecker invariants of two associatedpencils.Each matrix pencil [sE � A B] associated with a system (1.1) de�nes (with respect tothe standard bases of K q , K n , and Km) a pair of linear transformations E;A : K n ! K qand a linear transformation B : Km ! K q . Composing the linear transformations E and Awith the quotient map � : K q ! K q=im B, we obtain a pair EB;AB of linear transformationsfrom K n to K q=im B. Let (e1; : : : ; eq) be the standard basis of K q and �(ei1); : : : ; �(eik) theordered subset of �(e1); : : : ; �(eq) obtained by eliminating �(ei) whenever � (ei) is linearlydependent on �(e1); : : : ; �(ei�1). �(ei1); : : : ; �(eik) is a basis of K q=im B called the standardbasis induced by B. By sEB � AB we denote the (q � rankB)� n matrix pencil de�ned bythe matrix representations of EB and AB (with respect to the standard basis on K n and thestandard basis of K q=im B induced by B).The following lemma is easily veri�ed, see [LOMK91].Lemma 4.1 If the systems E1 _x = A1x+B1u and E2 _x = A2x+B2u are G-equivalent, then:i) [sE1 � A1 B1] and [sE2 � A2 B2] are pencil equivalent;ii) sE1B1 � A1B1 and sE2B2 � A2B2 are pencil equivalent.Let ' be the map associating with every pencil [sE � A B] 2 Sq;n;m the pencil equiva-lence class of the p�n pencil sEB�AB (p = q� rankB). (We de�ne the set of 0�n pencilsto be the zero vector space.) By the previous lemma the function ' is invariant with respectto left multiplication; hence its value at a given pencil [sE � A B] may be calculated bychoosing an element L in Glq(K ) such thatL[sE � A B] = 264 sE1 � A1 0sE2 � A2 B2 375where B2 has full row rank. Then '([sE�A B]) is the pencil equivalence class of sE1�A1.More generally, if the nonzero rows of B are independent, then sEB � AB is obtained from11

sE � A by deleting the rows of sE � A corresponding to rows in which B has nonzeroentries. Hence the pencil sEB�AB is obtained after applying row operations which eliminate"redundant" rows of B.By Lemma 4.1, ' induces a function from G-equivalence classes to pairs of pencil equivalenceclasses (equivalence class of [sE � A B], equivalence class of sEB � AB). Because the setof G-equivalence classes is parameterized by the lists of G-invariants (3.1) and the set ofpencil equivalence classes is parameterized by Kronecker invariants, the function ' inducesa function ~' mapping the list of G-invariants (3.1) of any q � (n +m) system pencil [sE �A B] 2 Sq;n;m to the double index list consisting of the Kronecker invariants of [sE�A B]and the Kronecker invariants of [sEB�AB ]. In the following, we write the double index listsvertically: ~' (G-invariants of [sE � A B])= ~' �f�i;(kij)dij=1ggi=1, (�1; : : : ; �z), ( 1; : : : ; w), (q1; : : : ; qx), (�1; : : : ; �v), (p1; : : : ; py)�= 0B@ Kronecker invariants of [sE � A B]Kronecker invariants of sEB � AB 1CA (4.1)= 0BB@ f�i;(kij)dij=1ggi=1 (�1; : : : ; �z) (c1; : : : ; cv+w) (n1; : : : ; nx+y)f�i;(hij) ~dij=1g~gi=1 (r1; : : : ; r!) (~c1; : : : ; ~c�) (~n1; : : : ; ~n�) 1CCA�n. elem. divisors row indices column indices inf. elem. divisors : (4.2)The following lemma determines the relationship between the G-invariants introduced in[LOMK91] and the invariants (4.2).Lemma 4.2 (i) If [sE � A B] has G-invariants�f�i;(kij)dij=1ggi=1, (�1; : : : ; �z), ( 1; : : : ; w), (q1; : : : ; qx), (�1; : : : ; �v), (p1; : : : ; py)� (4.3)then the Kronecker invariants of [sE � A B] and [sEB � AB] are given by�f�i;(kij)dij=1ggi=1; (�1; : : : ; �z); (c1; : : : ; cv+w); (n1; : : : ; nx+y)� (4.4)and �f�i;(kij)dij=1ggi=1; (�1; : : : ; �z); (~c1; : : : ; ~c�); (~n1; : : : ; ~n�)� ; (4.5)respectively, where 12

(c1; : : : ; cv+w) = < �1; : : : ; �v; 1 � 1; : : : ; w � 1 >;(n1; : : : ; nx+y) = < q1; : : : ; qx; p1 + 1; : : : ; py + 1 >;(~c1; : : : ; ~c�) = < �1 � 1; : : : ; �v�� � 1; 1 � 1; : : : ; w � 1 >; � =j f�i : �i = 0g j(~n1; : : : ; ~n�) = < q1; : : : ; qx; p1; : : : py�� >; � =j fpi : pi = 0g j (4.6)In particular, [sE�A B] and sEB�AB have the same �nite elementary divisors andthe same row indices.(ii) Conversely, given the Kronecker invariants (4.4), (4.5) of [sE�A B] and sEB�AB ,the G-invariants (4.3) of [sE � A B] are determined as follows:(a) The �nite elementary divisors f�i;(kij)dij=1ggi=1 and the row indices (�1; : : : ; �z) of[sE � A B] are as in (4.4).(b) ( 1�1; : : : ; w�1) = (ci1 ; : : : ; ciw), where (ci1 ; : : : ; ciw) is the sublist of (c1; : : : ; cv+w)consisting of the ci satisfying ci = ~ci.(c) (�1; : : : ; �v) = (ci1 ; : : : ; civ), where (ci1; : : : ; civ) is the complement in (c1; : : : ; cv+w)of the sublist given in (b).(d) (q1; : : : ; qx) = (ni1 ; : : : ; nix), where (ni1 ; : : : ; nix) is the sublist of (n1; : : : ; nx+y)consisting of the ni satisfying ni = ~ni.(e) (p1 + 1; : : : ; py + 1) = (ni1 ; : : : ; niy), where (ni1 ; : : : ; niy) is the complement in(n1; : : : ; nx+y) of the sublist given in (d).Proof. (i) Because the Kronecker invariants of [sE�A B] and of sEB�AB are invariantunder G-transformations by Lemma 4.1, it su�ces to prove the proposition for the case that[sE�A B] is in G-canonical form (3.5). In this case, the nonzero rows of B are independentand (as we remarked above) sEB�AB is obtained from sE�A by deleting the rows of sE�Ain which B has a nonzero entry. Hence the only blocks in the G-canonical form which aremodi�ed in the transition from [sE �A B] to sEB �AB correspond to the regular columnindices and the singular nilpotency indices. From this observation, we see that [sE�A B]and sEB�AB have the same �nite elementary divisors and the same row indices. Moreover,the �rst two equalities in (4.6) hold because of (3.2), (3.3). It only remains to show howthe lists of column indices and of nilpotency indices change by the deletion process used to13

obtain sEB �AB. The column indices i� 1 of [sE�A B] are not a�ected. By inspectionof the canonical form, one sees that the row deletion in sE�A de�ned by a nonzero regularcolumn index �i produces a column index block in sEB � AB corresponding to the columnindex �i � 1. If a regular column index �i is zero, it corresponds to a zero column of B andso does not contribute to the list (~c1; : : : ; ~c�) of column indices of sEB � AB. This provesthe third equation in (4.6). Similarly, the regular nilpotency indices qi belong to the list ofnilpotency indices of both [sE � A B] and sEB � AB, and a nonzero singular nilpotencyindex pi produces a pi � pi regular in�nite elementary divisor block in sEB � AB whereasa zero singular nilpotency index pi does not contribute to the list (~n1; : : : ; ~n�). This provesthe last equation in (4.6).(ii) Given the Kronecker invariants (4.4), (4.5) of [sE �A B] and sEB �AB, respectively,the �nite elementary divisors and the row indices of [sE � A B] can be read o� directlyfrom the list (4.4), see (i). Let (4.3) denote the list of G-invariants of [sE � A B]. From(i) we know (4.6) that the chain of column indices c1 � : : : � cv+w of [sE � A B] can bewritten in the form: 1� 1 = : : : = j1 � 1 = �1 = : : : = �k1 > j1+1� 1 = : : : = j2 � 1 = �k1+1 = : : : = �k2 > : : :(This is simply a matter of labeling.) Similarly, the nilpotency indices n1 � : : : � nx+y of[sE � A B] can be written in the form:q1 = : : : = qg1 = p1+1 = : : : = ph1 +1 > qg1+1 = : : : = qg2 = ph1+1+1 = : : : = ph2 +1 > : : : :The row deletions in sE �A used to obtain sEB �AB (i.e. to obtain ~ci and ~ni) correspondto the following modi�cations of these lists:1. subtract 1 from each nonzero �i and from each pi + 1 > 12. delete all zero �i and all pi which are equal to 1Then (~c1; : : : ; ~c�) =( 1 � 1; : : : ; j1 � 1; �1 � 1; : : : ; �k1 � 1; : : : ; : : : ; �kr�1+1 � 1; : : : ; �kr � 1)where �kr > 0 and �kr+1 = 0 and (~n1; : : : ; ~n�) =14

(q1; : : : ; qg1 ; p1; : : : ; ph1; : : : ; : : : ; phs�1+1; : : : ; phs)where phs > 1 and phs+1 = 1.Using the natural injections ~ci 7! ci and ~ni 7! ni, we see that the singular column indices(respectively the regular nilpotency indices) are exactly those ci (respectively ni) which areequal to their preimages.Hence b)-e) follow.Theorem 4.3 Two systems E1 _x = A1x + B1u and E2 _x = A2x + B2u are G-equivalent ifand only if(i) [sE1 � A1 B1] and [sE2 � A2 B2] are equivalent pencils;(ii) sE1B1 � A1B1 and sE2B2 � A2B2 are equivalent pencils.Proof. Necessity of conditions i) and ii) was established by Lemma 4.1. A consequence ofLemma 4.2 is that each double list of Kronecker invariants (4.2) completely determines a listof G-invariants in the correspondence de�ned by the function ~'. Hence ~' is injective and itfollows that conditions i) and ii) are su�cient.From [LOMK91, Prop.2.1] it is known that two system pencils [sE�A B]; [s eE� eA eB] 2Sq;n;m areD-equivalent if and only if sEB�AB and s eEeB� eAeB are equivalent pencils. ThereforeLemma 4.2 yields the following complete set of D-invariants:Theorem 4.4 If [sE � A B] has G-invariants�f�i;(kij)dij=1ggi=1, (�1; : : : ; �z), ( 1; : : : ; w), (q1; : : : ; qx), (�1; : : : ; �v), (p1; : : : ; py)�then the Kronecker invariants of sEB � AB form a complete set of D-invariants and aregiven by: �f�i;(kij)dij=1ggi=1; (�1; : : : ; �z); (~c1; : : : ; ~c�); (~n1; : : : ; ~n��where(~c1; : : : ; ~c�) = < �1 � 1; : : : ; �v�� � 1; 1 � 1; : : : ; w � 1 >; � =j f�i : �i = 0g j;(~n1; : : : ; ~n�) = < q1; : : : ; qx; p1; : : : py�� >; � =j fpi : pi = 0g j :As a consequence of the two preceding results we obtain that the inclusion (2.7) is an equality:G � [sE � A B] = (Pq;n+m � [sE � A B]) \ (D � [sE � A B]) (4.7)15

Next we introduce another characterization of G-orbits which will be particularly usefulin Sections 5 and 6. For each � 2 C , de�ne �� : Sq;n;m ! Sq;n;m by�a([sE � A B]) = [sA� (�A� E) B] (4.8)Because �� has an inverse ([sE � A B] 7! [s(�E � A)� E B]), �� is bijective.Noting that �a0BBBBB@(L; 2666664 R 0 00 R 0K F W 3777775�1) � [sE � A B]1CCCCCA= 0BBBBB@L; 2666664 R 0 00 R 0F �F �K W 3777775�11CCCCCA � ��([sE � A B]) (4.9)we see that �� takes D-orbits onto D-orbits.Let H� be the subgroup of D consisting of matrix pairs in which F = �K; let H� act onthe set Sq;n;m by restriction of the action of D. Using (4.8) and (4.9) we see that��0BBBBB@(L; 2666664 R 0 00 R 00 F W 3777775�1) � [sE � A B]1CCCCCA= 0BBBBB@L; 2666664 R 0 00 R 0F �F W 3777775�11CCCCCA � ��([sE � A B]) (4.10)and it follows that��(G � [sE � A B]) = H� � ��([sE � A B]); [sE � A B] 2 Sq;n;m (4.11)From (4.9), (4.11), and the fact that �� is bijective, it follows that:Proposition 4.5 Two system pencils are G-equivalent if and only if their images under ��are H�-equivalent.Remark 4.6 The feedback part of the H� action was introduced by Zhou, Shayman, andTarn in [ZST87] and was referred to as "modi�ed proportional and derivative feedback"16

(MPD). They showed that the controllable subspace is invariant under MPD and gave con-ditions for eigenvalue relocation, impulse elimination, solvability of the decoupling problemand of the disturbance localization problem using MPD.5 Feedback of regularizable systemsFor the remainder of the paper (with the exception of Propositions 6.1 and 6.2), we restrictour attention to regularizable systems; this will set the stage for use in section 6 of resultspreviously established for state space systems in [HO95]. A system of the form (1.1) (andthe corresponding system pencil [sE � A B]) is said to be regularizable if q = n and thereis a state feedback matrix F 2 Km�n such that det(sE � (A � BF )) 6� 0, see [OL89].In [HO95], we showed that a system (1.1) with q = n is regularizable if and only if thepencil [sE � A B] has no row indices. From the equations (3.6), we see that, in this case,w = y, i.e. the number of singular column indices is the same as the number of singularnilpotency indices, and there are exactly m column indices. State space systems are clearlyregularizable, and state space system pencils (which have no nilpotency indices) have onlyregular column indices. Hence, on the set of state space systems, G-equivalence coincideswith Pn;n+m-equivalence.We denote the set of state space system pencils by Ln;m, i.e.Ln;m = f[sI � A B] : A 2 K n�n , B 2 K n�mg:A proportional feedback transformation (2.6) leaves Ln;m invariant if and only if it belongsto the following subgroup of G:F = f(R�1; 264 R 0F W 375) : R 2 Gln(K ); W 2 Glm(K ); F 2 Km�ng: (5.1)The restriction of the G-action (2.6) to F�Ln;m yields the usual action of the state feedbackgroup F on Ln;m:(R�1; 264 R 0F W 375�1) � [sI � A B] = [sI �R�1(AR� BF ) R�1BW ]: (5.2)A characterization of the topological closure of F -orbits in terms of Pn;n+m-invariants isgiven in [HO95]. 17

We de�ne the following subsets of Sn;n;m:Rn;m = f[sE � A B] 2 Sn;n;m : 9F 2 Km�n such that det(sE � A+BF ) 6� 0g;R�n;m = f[sE � A B] 2 Rn;m : det(�E � A) 6= 0g:The reader may easily verify that Rn;m is D-stable and that R�n;m is H�-stable. Note thatR�n;m is a dense subset of Rn;m (with respect to the standard topology of K q�(2n+m)) andthat (G � [sE � A B]) \R�n;m is dense in G � [sE � A B] if (G � [sE � A B]) \ R�n;m 6= ;.Hence the topological closures of G � [sE � A B] and G � [sE � A B] \ R�n;m coincide if(G � [sE � A B]) \ R�n;m 6= ;.Because ��([sI � A B]) = [sA � (�A� I) B], it follows that ��(Ln;m) � R�n;m. Nowde�ne �� : R�n;m ! Ln;m by��([sE � A B]) = [sI � (�E � A)�1E (�E � A)�1B]:For [sE � A B] 2 R�n;m, we have(�E � A)�1[sE � A B] = [s(�E � A)�1E � ((�E � A)�1�E � I) (�E � A)�1B]= ��([sI � (�E � A)�1E (�E � A)�1B]):Hence �� � ��([sE � A B]) = (�E � A)�1[sE � A B]; [sE � A B] 2 R�n;m: (5.3)In particular, every element of R�n;m is equivalent via left multiplication (and hence G-equivalent) to an element of ��(Ln;m). An easy calculation shows that�� � ��([sI � A B]) = [sI � A B]; [sI � A B] 2 Ln;mso that �� � �� is the identity map on Ln;m.Restricting the equation (4.10) to the case L = R�1 and E = I, we obtain the followingrelationship between F -orbits and H�-orbits:Proposition 5.1 ��(F � [sI � A B]) � H� � ��([sI � A B]).The following proposition establishes correspondences between F -orbits of state spacesystems and D-, G-, and H�-orbits of regularizable systems.18

Proposition 5.2 Let [sE � A B] 2 R�n;m. Then�� �(D � [sE � A B]) \ R�n;m� � F � ��([sE � A B]) (5.4)Consequently: �� �(G � [sE � A B]) \R�n;m� � F � ��([sE � A B]) (5.5)and �� (H� � [sE � A B]) � F � ��([sE � A B]) (5.6)Proof. Assume that [sE � A B] 2 R�n;m and that the triple (R;K; F ) with R 2 Gln(K ),K;F 2 Km�n satis�es (L; 2666664 R 0 00 R 0K F W 3777775�1) � [sE � A B]= [sL(ER +BK)� L(AR� BF ) LBW ] 2 (D � [sE � A B]) \R�n;m;i.e. det((�E � A)R +B(�K + F )) 6= 0. Then��0BBBBB@(L; 2666664 R 0 00 R 0K F W 3777775�1) � [sE � A B]1CCCCCA = [sI �X YW ]where X = ((�E � A)R +B(�K + F ))�1(ER +BK); (5.7)Y = ((�E � A)R +B(�K + F ))�1B: (5.8)Multiplying both sides of (5.7) by ((�E � A)R + B(�K + F )) and moving some terms tothe right-hand side, we obtain (�E � A)RX = ER +BK �B(�K + F )X; it follows thatX = R�1((�E � A)�1ER + (�E � A)�1B(K � (�K + F )X)):Similarly, we obtain Y = R�1(�E � A)�1B(I � (�K + F )Y ):If I � (�K + F )Y is invertible, then we have:��0BBBBB@(L; 2666664 R 0 00 R 0K F W 3777775�1) � [sE � A B]1CCCCCA = [sI �X YW ]19

= [sI � R�1((�E � A)�1ER + (�E � A)�1B(K � (�K + F )X))R�1(�E � A)�1B(I � (�K + F )Y )W ]= (R�1; 264 R 0K � (�K + F )X (I � (�K + F )Y )W 375�1) � ��([sE � A B]) (5.9)which is in F � ��([sE � A B]).To verify that I � (�K + F )Y is invertible, suppose v = (�K + F )Y v. ThenBv = B(�K + F )((�E � A)R +B(�K + F ))�1Bvand we have, for y = ((�E � A)R +B(�K + F ))�1Bv,Bv = B(�K + F )y = ((�E � A)R +B(�K + F ))y = (�E � A)Ry +B(�K + F )y:Hence (�E�A)Ry = 0. Since (�E�A)R is nonsingular, it follows that y is zero and thereforeBv = 0. Since v = (�K+F )Y v = (�K+F )((�E�A)R+B(�K+F ))�1Bv, we conclude thatv is zero. Therefore I�(�K+F )Y is invertible. It follows that ��(D� [sE�A B]\R�n;m) �F ���([sE�A B]). The inclusions (5.5) and (5.6) then follow from the fact that the groupsG and H� are subgroups of D and that R�n;m is H�-stable.Proposition 5.3(D � [sE � A B]) \R�n;m = H� � [sE � A B]; [sE � A B] 2 R�n;m:Proof. Suppose [sE �A B] 2 R�n;m. Since R�n;m is H�-invariant and H� is a subgroup ofD, it follows that H� � [sE � A B] � D � [sE � A B] \ R�n;m:Applying the function �� to the inclusion (5.4) and using Proposition 5.1, we have�� ��� �(D � [sE � A B]) \ R�n;m�� � �� (F � ��([sE � A B])) � H� ���(��([sE�A B]))Now apply (5.3). Since left multiplication is anH�-transformation as well as aD-transforma-tion under which R�n;m is invariant, we haveD � [sE � A B] \ R�n;m � H� � [sE � A B];hence equality. 20

6 Topological closure of D-orbits and G-orbitsThe set of q � (n + m) matrix pencils [sE � A B] with entries in K may be consideredas the topological space K q�(2n+m) (under the standard topology). Identifying the limitsof a given system under high gain feedback depends on describing the topological closureof the G-orbit of that system (see [HO95]). We denote topological closure by an overbar,e.g. the topological closure of the G-orbit of [sE � A B] is denoted G � [sE � A B].The case of a regularizable, controllable system lying in the topological closure of the G-orbit of a regularizable, controllable system was completely described in [Glu91]. Assumingneither regularizability nor controllability, we obtain necessary conditions for G-orbit closure(Proposition 6.1) and we obtain necessary conditions for D-orbit closure in the case that theranks of the input matrices of the two systems are equal (Proposition 6.2). Propositions 6.1and 6.2 provide necessary conditions for G-orbit closure and D-orbit closure in terms of Ps;t-orbit closure. Unfortunately, characterizations of Ps;t-orbit closures in terms of invariants arecurrently available only in the case of regularizable system pencils. Therefore we consideronly regularizable systems in the remainder of the section. In the case of regularizablesystems, we reformulate the necessary conditions of Proposition 6.1 in terms of Kroneckerinvariants, and we obtain | as our main result | necessary and su�cient conditions for D-orbit closure in terms of Kronecker invariants. Furthermore we obtain necessary conditionsfor G-orbit closure and conjecture that these may be su�cient. The complete characterizationof G-orbit closures remains, however, an open problem.Proposition 6.1 If [sE � A B] 2 Sq;n;m thenG � [sE � A B] � Pq;n+m � [sE � A B] \ D � [sE � A B]:Proof. Taking the topological closure of both sides of (4.7), we haveG � [sE � A B] = Pq;n+m � [sE � A B] \ D � [sE � A B]� Pq;n+m � [sE � A B] \ D � [sE � A B]Proposition 6.2 If [sE � A B] 2 D � [sE � A B] and rank B = rank B = r, thensEB � AB 2 Pq�r;n � (sEB � AB):21

Proof. The results of Section 4 allow us to assume, without loss of generality, that[sE � A B] = 264 sE1 � A1 0sE2 � A2 B2 375 and [sE � A B] = 264 sE1 � A1 0sE2 � A2 B2 375where sE1�A1 and sE1�A1 are (q� r)� n pencils and B2; B2 2 K r�m have full row rankr. By hypothesis there is a sequence of matrix pairs (Lk; Tk) 2 D such thatlimk!1(Lk; T�1k ) � [sE � A B] = [sE � A B] (6.1)By the Gram-Schmidt procedure every Lk 2 Glq(K ) can be represented in the form Lk =Uk ~Lk where Uk is unitary (resp. orthogonal) and ~Lk is a lower triangular q � q-matrix.Replacing ((Lk; Tk))k2N by a subsequence if necessary we may assume (by compactness ofthe unitary group) that (Uk) is convergent. Let U = limk!1Uk, then U; ~Lk; Tk can bewritten in the following form~Lk = 264 L11k 0L21k L22k 375 ; Tk = 2666664 Rk 0 00 Rk 0Kk Fk Wk3777775 ; U�1 = 264 U11 U12U21 U22 375 :where L22k ; U22 2 Glr(K ). (6.1) implies264 L11k 0L21k L22k 375264 sE1Rk � A1Rk 0s(E2Rk +B2Kk)� (A2Rk � B2Fk) B2Wk 375! 264 U11 U12U21 U22 375264 sE1 � A1 0sE2 � A2 B2 375as k !1. Since B2 has full row rank, it follows that U12 = 0. Hencelimk!1L11k (sE1 � A1)Rk = U11(sE1 � A1)Therefore (sE1�A1) is in the topological closure of the pencil equivalence class of (sE1�A1).Note that Propositions 6.1 and 6.2 both give necessary conditions for G-orbit closure andD-orbit closure in terms of Ps;t-orbit closure. A complete description of Ps;t-orbit closure interms of Kronecker invariants exists only for regularizable system pencils, see [HO92]. Forthe remainder of this section, we restrict our attention to this class of systems pencils, wherewe can make use of this description of Ps;t -orbit closure as well as of the results in Section5.If the pencil [sE � A B] is regularizable, then for some � 2 C , D � [sE � A B] \ R�n;m isdense in D � [sE � A B] and we conclude from Proposition 5.3 that:22

Lemma 6.3 If [sE � A B] is regularizable, thenD � [sE � A B] = H� � [sE � A B] for some � 2 CUsing this proposition in conjunction with Propositions 5.1 and 5.2, we obtain the followingcharacterization of D-orbit closure:Lemma 6.4 Suppose [sE � A B]; [sE � A B] 2 R�n;m for some � 2 C . Then[sE � A B] 2 D � [sE � A B] , ��([sE � A B]) 2 F � ��([sE � A B])Proof. If [sE � A B] 2 D � [sE � A B], then, by Lemma 6.3,[sE � A B] 2 H� � [sE � A B]:Since �� is continuous, it follows that��([sE � A B]) 2 ��(H� � [sE � A B]);and the conclusion follows from (5.6).Conversely, if ��([sE � A B]) 2 F � ��([sE � A B]), then by (5.3) and the continuity of�� (sE � A)�1[sE � A B] = ��(��([sE � A B])) 2 ��(F � ��([sE � A B])):By Proposition 5.1 and Lemma 6.3, the latter set is contained inH� � (��(��([sE � A B]))) = H� � [sE � A B] = D � [sE � A B]:Therefore (sE�A)�1[sE�A B] 2 D � [sE � A B], hence [sE�A B] 2 D � [sE � A B]:Our aim is to characterize the closure of D orbits in terms of Kronecker invariants. In orderto express the previous results in terms of Kronecker invariants, we need to introduce somenotation. We will denote the G-invariants of [sE � A B] with an overbar; e.g. the regularcolumn indices of [sE � A B] will be denoted (�1; : : : ; �m�w). For a given list of integersa = (a1; : : : ; ak), 0 � ai � n, the dual list a0 = (a1; : : : ; ak)0 = (a01; : : : ; a0n) is given bya0i =j fj : aj � ig j , i = 1; : : : ; n23

Note that a01 � a02 � : : : � a0n. Given two lists b = (b1; : : : ; bn) and c = (c1; : : : cn) of lengthn with b1 � : : : � bn and c1 � : : : � cn, we say that b � c under the generalized dominanceorder if jXi=1 bi � jXi=1 ci for all j = 1; : : : ; n(See [HO92] for further details about the generalized dominance order.)The following necessary and su�cient conditions for [sI � A B] 2 F � [sI � A B] havebeen proved in [HO95] (see Theorem 4.6):Proposition 6.5 Suppose that [sI�A B]; [sI�A B] 2 Ln;m are two state space systempencils with column index lists (c1; : : : ; cm), (c1; : : : ; cm); respectively. Then [sI � A B] 2F � [sI � A B] if and only if: (c1; : : : ; cm)0 � (c1; : : : ; cm)0; (6.2)and Dj([sI � A B]) divides Dj([sI � A B]), j = 1; : : : ; n (6.3)where Dj(sM + N) denotes the greatest common divisor of the j � j minors of any givenpencil sM +N .A consequence of condition (6.3) is that the set of roots f�iggi=1 of the �nite elementarydivisors of [sI�A B] is a subset of the set of roots f�iggi=1 of the �nite elementary divisorsof [sI � A B]. In the following, we assume that �i = �i, i = 1; : : : ; g. To write condition(6.3) in terms of �nite elementary divisor block sizes (kij)dij=1, we append zeros to each list(ki1; : : : ; kidi) to obtain a list of length n: (kij)nj=1 = (ki1; : : : ; kidi; 0; : : : ; 0). Recall that theblock sizes are ordered ki1 � ki2 � : : : � kidi . In terms of these invariants, we have:Lemma 6.6 Dj([sI � A B]) divides Dj([sI � A B]) for all j = 1; : : : ; n if and only ifnXz=j kiz � nXz=j kiz; i = 1; : : : ; g; j = 1; : : : ; n (6.4)Proof. The relationship between fDj([sI�A B])gnj=1 and fkijgi;j is described in [HO90,p. 613]. Speci�cally, ifd�i;j([sI � A B]) = the multiplicity of s� �i in Dj([sI � A B])24

then kij = d�i;n�j+1 � d�i;n�j. It follows that�jz=1kiz = d�i;n � d�i;n�j:Because d�i;n = �nz=1kiz, we haved�i;n�j = d�i;n � �jz=1kiz = �nz=j+1kiz: (6.5)The condition that Dn�j([sI � A B]) divides Dn�j([sI � A B]) is equivalent tod�i;n�j([sI � A B]) � d�i;n�j([sI � A B]) for all i = 1; : : : ; g; j = 0; : : : ; n� 1:Applying (6.5), we have�nz=j+1kiz � �nz=j+1kiz for all i = 1; : : : ; g; j = 0; : : : ; n� 1:Lemma 6.7 If [sE � A B] 2 R�n;m with � 2 C n f0g has G-invariants(f�i; (kij)dij=1ggi=1; ( 1; : : : ; w); (q1; : : : ; qx); (�1; : : : ; �m�w); (p1; : : : pw)) (6.6)�n. elem. divisors; sing. col. ind.; reg. nilp. ind.; reg. col. ind.; sing. nilp. ind.then ��([sE � A B]) has the following Kronecker invariants:(a) The �nite elementary divisors are given byf(�� �i)�1; (kij)dij=1ggi=1 [ f0; < q1; : : : ; qx; p1; : : : pw�h >g;where h =j fpi : pi = 0g j :(b) The column indices are given by < �1; : : : ; �m�w; 1; : : : ; w >.Proof. We begin by noting that, for � 6= 0, � 6= 0, the k � k pencil

sM +N = 2666666664 �s� � s� �� s�s� �3777777775k�k (6.7)

25

has pencil canonical form 2666666664 s� �� 1� �� 1s� ��3777777775k�k (6.8)This fact follows from the computation ofDj(sM+N); one easily veri�es thatDj(sM+N) =(�s� �)k when j = k and 1 otherwise. It follows that the pencil canonical form of sM +Nis that shown.Next consider the � � (� + 1) pencilsM +N = [s(�I� +N�)� I� j c�] = 2666666664 �s� 1 s j� � j� s j�s� 1 j 1

3777777775��(�+1) : (6.9)By determining solutions to (sM +N)X(s) = 0, one easily veri�es that sM +N has pencilcanonical form 2666666664 s 1� �� �s 1

3777777775��(�+1) : (6.10)Noting that ��([sE�A B]) is pencil equivalent to [s(�E�A)�E B] and so has the sameKronecker invariants, it su�ces to show that [s(�E�A)�E B] has the Kronecker invariantsdescribed in (a), (b). Without loss of generality we may assume that the pencil [sE�A B]is in G-canonical form. From (3.5) it is seen that each of the square diagonal blocks of sE�A(corresponding to the �nite elementary divisors, the regular column indices and the regularnilpotency indices of sE � A) yields a square diagonal block of [s(�E � A)� E B] whichis pencil equivalent to the corresponding block in��([sE � A B]) = (�E � A)�1[s(�E � A)� E B]:We will now determine the e�ect of the map[sE � A B] 7�! [s(�E � A)� E B] (6.11)on each of these blocks (and the associated columns of B).26

1. �nite elementary divisor blocks:(s� �i)Ikij +Nkij 7�! s((�� �i)Ikij +Nkij)� IkijNote that �� �i 6= 0 since [sE �A B] 2 R�n;m. Thus, by the preceding remarks (see(6.7), (6.8)), the pencil s((���i)Ikij +Nkij )�Ikij is equivalent to (s�(���i)�1)Ikij +Nkij . This yields a kij � kij �nite elementary divisor block of [s(�E � A) � E B]corresponding to the eigenvalue (�� �i)�1, i.e. ((�� �i)�1; kij).2. regular column index blocks:[sI�i +N�i j c�i ] 7�! [s(�I�i +N�i)� I�i j c�i]By the remarks above, [s(�I�i +N�i)� I�i j c�i] is pencil equivalent to a �i � (�i + 1)column index block and thus yields a column index �i of [s(�E � A)� E B].3. regular nilpotency index blocks:Iqi + sNTqi 7�! sIqi + (�s� 1)NTqiComputing the invariants Dj(sIqi + (�s � 1)NTqi), one proves similarly as above thatthe pencil sIqi + (�s� 1)NTqi yields a qi� qi �nite elementary divisor block of [s(�E �A)� E B] corresponding to the eigenvalue 0, i.e. (0; qi).It remains to deal with the nonsquare diagonal blocks of sE�A (and the associated columnsof B). Since [sE � A B] is regularizable, only the blocks determined either by singularcolumn indices or by singular nilpotency indices have to be considered. It makes no sense toapply the mapping (6.11) individually to these blocks, since the result would in general notbe pencil equivalent to a corresponding block in ��([sE � A B]). But in the regularizablecase the number of singular column indices is equal to the number of singular nilpotencyindices. Therefore we can arrange the blocks so that each singular column index block isfollowed by a singular nilpotency index block in sE�A. Together each of these pairs of blocksforms a square diagonal submatrix of sE�A. The associated column in B is determined bythe corresponding singular nilpotency index. The matrix below shows one of these pairs ofblocks in sE � A together with the corresponding column segment of B. The vertical lineseparates the part of the block in sE � A from the part in B, the horizontal line separates27

the rows containing the upper ( � 1)� singular column block from the rows containingthe (p+1)�p singular nilpotency block and the corresponding column subvector cpi+1 of B.For ease of computation, we write the nilpotency index block "upside down" (rearrangingrows and columns):26666666666666666666666666666664

s 1 j 0� � j �� � j �s 1 j 0� � � � � � � � � � �s j 11 � j 0� � j �� s j �1 j 0

37777777777777777777777777777775(( �1)+(p+1))�( +p+1)is G-equivalent to26666666666666666666666666666664

s 1 j 0� � j �� � j �s 1 j 0� � � � � � � � � � �1 s j 11 � j 0� � j �� s j �1 j 0

37777777777777777777777777777775(( �1)+(p+1))�( +p+1)(If = 1, then there are only p+1 rows; the upper part of these matrices is absent.If p = 0,the row consists of zeros at the �rst positions and a 1 at the position + 1.) Applying28

the map (6.11), we get26666666666666666666666666666664

�s� 1 s j 0� � j �� � j ��s� 1 s j 0� � � � � � � � � � �s �s� 1 j 1s �s� 1 j 0� � j �s �s� 1 j �s j 0

37777777777777777777777777777775(( �1)+(p+1))�( +p+1)(6.12)By similar arguments as at the beginning of this proof one shows that the lower right p� pblock in sE�A can be transformed via row and column operations to sIp+Np. To illustratethe new structure of the pencil we lower the horizontal line by one row. The row operationsa�ect only the rows beneath that line. The column operations a�ect the last p columns ofthe block in sE�A so that nonzero entries may be generated in the -th row at the positionsindicated by an asterisk. As a result we obtain26666666666666666666666666666664

�s� 1 s j j 0� � j j �� � j j ��s� 1 s j j 0s j � � � � j 1� � � � � � � � � � �j s 1 j 0j � � j �j s 1 j �j s j 0

37777777777777777777777777777775( +p)�( +p+1)Now subtract appropriate multiples of the last p rows from row so that only constant termsremain in the last p entries of row . Then subtracting suitable multiples of the last column29

from columns + 1; : : : + p, we �nd that the pencil is equivalent to26666666666666666666666666666664

�s� 1 s j j 0� � j j �� � j j ��s� 1 s j j 0s j 0 � � 0 j 1� � � � � � � � � � �j s 1 j 0j � � j �j s 1 j �j s j 0

37777777777777777777777777777775( +p)�( +p+1):

Now multiply row by � and then subtract ��1 times the last column from column toobtain an upper left corner of the form (6.9). Applying the transformation (6.9) ; (6.10),we see that the above pencil has column index and �nite elementary divisor block givenby (0; p). If p = 0, the pencil has column index and no �nite elementary divisor block.We now are able to characterize D-orbit closure in terms of D-invariants:Theorem 6.8 Let [sE � A B]; [sE � A B] 2 Sn;n;m be regularizable, with D-invariantsas follows:Kronecker invariants of sEB � AB: (f�i; (kij)dij=1ggi=1; (c1; : : : ; c�); (n1; : : : ; n�)Kronecker invariants of sEB � AB: (f�i; (kij)dij=1ggi=1; (c1; : : : ; c�); (n1; : : : ; n�)Then [sE � A B] 2 D � [sE � A B] if and only if the following three conditions are sat-is�ed:(i) The elementary divisors of sEB � AB; sEB � AB can be ordered in such a way that,for each i = 1; : : : ; g, �i = �i and Pnz=j kiz � Pnz=j kiz for all j = 1; : : : ; n.(ii) (c1 + 1; : : : ; c� + 1)0 � (c1 + 1; : : : ; c� + 1)0:(iii) If v =< n1; : : : ; n�; 0; : : : ; 0 > and v =< n1; : : : ; n�; 0; : : : ; 0 > (zeros appended so thatthe lists v and v have length n) thennXz=j vz � nXz=j vz; j = 1; : : : ; n:30

Proof. Without loss of generality, we may assume that, for some � 2 C n f0g,[sE � A B]; [sE � A B] 2 R�n;m:By Lemma 6.4[sE � A B] 2 D � [sE � A B]] , ��([sE � A B]) 2 F � ��([sE � A B]):By Proposition 6.5 the latter condition is satis�ed if and only if (6.2), (6.3) hold. Lemma6.6 translates (6.3) into a condition on �nite elementary divisors. Applying Lemma 6.7 wesee that (6.2) and (6.3) hold if and only if the G-invariants of [sE�A B] and [sE�A B](see (6.6)) | after suitable reordering of the �nite elementary divisors of both pencils |satisfy the following conditions:i) (�1; : : : ; �m�w; 1; : : : ; w)0 � (�1; : : : ; �m�w; 1; : : : ; w)0:ii) For each i = 1; : : : ; g, �i = �i and Pnz=j kiz � Pnz=j kiz, j = 1; : : : ; niii) If v =< q1; : : : ; qx; p1; : : : pw; 0; : : : ; 0 > :and v =< q1; : : : ; qx; p1; : : : pw; 0; : : : ; 0 >(zeros appended so that the lists v and v have length n:),then Pnz=j vz � Pnz=j vz, j = 1; : : : ; n: (6.13)Using Proposition 4.4 to translate these conditions into D-invariants, the result follows.If we restrict Proposition 6.2 to the case of regularizable system pencils, it may be re-formulated in terms of Kronecker invariants. The condition that rankB = rankB = rimplies � = �. It follows from Theorem 4.1 in [HO95] and Lemma 6.6 that the conclusionsEB � AB 2 Pq�r;n � (sEB � AB) is equivalent to the set of conditions i), ii), iii) of The-orem 6.8. Hence, in the case of regularizable system pencils, the necessary conditions ofProposition 6.2 are su�cient:Corollary 6.9 Suppose [sE�A B], [sE�A B] 2 Sn;n;m are regularizable and rankB =rankB = r. Then[sE � A B] 2 D � [sE � A B] , sEB � AB 2 Pq�r;n � (sEB � AB):For regularizable systems, Proposition 6.1 may be stated in terms of G-invariants asfollows: 31

Corollary 6.10 Suppose [sE � A B]; [sE � A B] 2 Sn;n;m are regularizable with G-invariants�f�i;(kij)dij=1ggi=1, ( 1; : : : ; w), (q1; : : : ; qx), (�1; : : : ; �m�w), (p1; : : : ; pw)� ;�f�i;(kij)dij=1ggi=1, ( 1; : : : ; w), (q1; : : : ; qx), (�1; : : : ; �m�w), (p1; : : : ; pw)� :If [sE � A B] 2 G � [sE � A B], then the following conditions are satis�ed:(i) The elementary divisors of [sE � A B]; [sE � A B] can be ordered in such a waythat, for each i = 1; : : : ; g, �i = �i and Pnz=j kiz � Pnz=j kiz for all j = 1; : : : ; n.(ii) (�1; : : : ; �m�w; 1; : : : ; w)0 � (�1; : : : ; �m�w; 1; : : : ; w)0.(iii) If v =< q1; : : : ; qx; p1; : : : pw; 0; : : : ; 0 > and v =< q1; : : : ; qx; p1; : : : pw; 0; : : : ; 0 > thenPnz=j vz � Pnz=j vz for all j = 1; : : : ; n.(iv) (�1; : : : ; �m�w; 1 � 1; : : : ; w � 1)0 � (�1; : : : ; �m�w; 1 � 1; : : : ; w � 1)0.(v) If u =<q1; : : : ; qx ; p1 + 1; : : : pw + 1; 0; : : : ; 0> and u =<q1; : : : ; qx ; p1+1; : : : ; pw+1;0; : : : ; 0 > then Pnz=j uz � Pnz=j uz for all j = 1; : : : ; n .Proof. If [sE�A B] 2 G � [sE � A B], then the pencils [sE�A B] and [sE�A B]satisfy both the conditions of Theorem 6.8 and the conditions of [HO95, Thm.4.1] (see alsoRemark 4.4 in [HO95]). The former are equivalent to conditions (i), (ii), and (iii), see (6.13).The latter are equivalent to conditions (i), (iv) and (v), recalling the adjustments (3.2),(3.3)to the singular indices to obtain the Kronecker invariants.Conjecture 6.11 For regularizable system pencils the inclusion in Proposition 6.1 is anequality, and the necessary conditions in Corollary 6.10 are su�cient.The conjecture is supported by the multitude of su�cient conditions for G-orbit closurepresented in [HO94].7 ConclusionsIn this paper we have studied the high gain feedback classi�cation problem for generalizedstate space systems of the form (1.1). Two groups of feedback transformations have been32

considered, the group G of proportional feedback transformations and the larger group ofproportional plus derivative feedback transformations, D. Both act on the space Sq;n;m ofsystems of the form (1.1).The feedback classi�cation problem consists in constructing a complete set of invariants forthe action of G (or D) on Sq;n;m. This problem has been solved in [LOMK91] by constructinga canonical form for these group actions. In Section 4 we have derived a new complete set ofG-invariants which can be expressed via the Kronecker invariants of the pencil [sE �A B]and of the quotient pencil [sEB � AB]. The Kronecker invariants of [sEB � AB] form acomplete set of D-invariants.The high gain proportional feedback classi�cation problem can be stated as follows: Givena system (E;A;B) 2 Sq;n;m , determine the set of (E;A;B) 2 Sq;n;m which can be obtainedin the limit by applying the feedback transformations in G to (E;A;B). Equivalently, wemust determine the orbits G � (E;A;B) which lie in the topological closure of the orbitG � (E;A;B). For D-orbits whe obtain an analogous high gain proportional plus derivativefeedback classi�cation problem.The solution strategy we pursue in this paper is based on our characterization of the orbitclosures of matrix pencils without row indices [HO92]. This restricts our analysis essentiallyto regularizable systems although some necessary conditions are obtained in the general case(Propositions 6.1 and 6.2). For regularizable systems there exists a correspondence betweenD-orbits and orbits of the full feedback group on the set Ln;m of state space systems. Thiscorrespondence is described in Section 5 and by Lemma 6.4 it is seen that the correspondencecan be extended to orbit closures. This sets the stage for applying our solution of thehigh gain feedback classi�cation problem for state space systems [HO92] to regularizablegeneralized state space systems via the \translation lemma" 6.7. As a result we obtaina complete solution of the high gain proportional plus derivative feedback classi�cationproblem (Theorem 6.8). The restriction to regularizable systems seems to be quite acceptablefor applications in the theory of singular systems although from a mathematical point of viewand perhaps for applications in a behavioral context it would be nice to dispense with therestriction. This would, however, require a new approach to the problem.With regards to proportional feedback we have only been able to derive necessary condi-tions for G � (E;A;B) � G � (E;A;B). It remains an open problem if our conditions are also33

su�cient under the assumption of regularizability.References[Bru70] P. Brunovsky. A classi�cation of linear controllable systems. Kybernetika, 3: 173{187, 1970.[Gan59] E.R. Gantmacher. The Theory of Matrices, volume 1 and 2. Chelsea, New York,1959.[Glu90] H. Gl�using-L�uer�en. A feedback canonical form for singular systems. InternationalJournal of Control, 52: 347-376, 1990.[Glu91] H. Gl�using-L�uer�en. Gruppenaktionen in der Theorie singul�arer Systeme. Ph.D.thesis, Institut f�ur Dynamische Systeme, Universit�at Bremen, 1991.[GH94] H. Gl�using-L�uer�en and D. Hinrichsen. The degeneration of reachable singular sys-tems under feedback-transformations. Kybernetika, 30: 387-391, 1994.[HM81] M. Hazewinkel and C. F. Martin. Representations of the symmetric group, thespecialization order, systems and Grassmann manifolds. L'Enseignement Math�ematique,29: 53-87, 1983.[HO90] D. Hinrichsen and J. O'Halloran. A complete characterization of orbit closures ofsingular systems under restricted system equivalence. SIAM J. Control and Optimization,28: 602-623, 1990.[HO92] D. Hinrichsen and J. O'Halloran. Orbit closure of singular matrix pencils. J. Pureand Applied Algebra, 81: 117-137, 1992.[HO94] D. Hinrichsen and J. O'Halloran. A note on the orbit closure problem for the actionof the generalized feedback group. In Systems and Networks: Mathematical Theory andApplications, Vol. II, ed. U. Helmke, R. Mennicken, J. Saurer, Akademie Verlag, pages221-224, 1994.[HO95] D. Hinrichsen and J. O'Halloran. A pencil approach to high gain feedback andgeneralized state space systems. Kybernetika, 31: 109-139, 1995.34

[Kui94] M. Kuijper. First-Order Representations of Linear Systems. Birkh�auser, 1994.[LOMK91] J.J. Loiseau, K. �Ozcaldiran, M. Malabre, N. Karcanias. Feedback canonicalforms of singular systems. Kybernetika, 27: 289-305, 1991.[OL89] K. �Oz�caldiran and F. L. Lewis. On the regularizability of singular systems. IEEETransactions on Automat. Control, 35: 1156-1160, 1990.[RR94] M. S. Ravi and J. Rosenthal. A smooth compacti�cation of the space of transferfunctions with �xed McMillan degree. Acta Appl. Math., 34:329-352, 1994.[RR95] M. S. Ravi and J. Rosenthal. A general realization theory for higher order lineardi�erential equations. Systems & Control Letters, 25: 351-360, 1995.[Ros74] H.H. Rosenbrock. Structural properties of linear dynamical systems. InternationalJournal of Control, 20: 177-189, 1974.[Sha88] M. A. Shayman. Homogeneous indices, feedback invariants, and control structuretheorem for generalized systems. SIAM J. Control and Optimization, 26: 387-400, 1988.[Wil91] J.C. Willems Paradigms and puzzles in the theory of dynamical systems. IEEETrans. Automat. Control, 36: 259-294, 1991.[ZST87] Z. Zhou, M.A. Shayman, T-J. Tarn. Singular systems: A new approach in the timedomain. IEEE Trans. Automat. Control, 32: 42-50, 1987.

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