# Design, parametrization, and pole placement of stabilizing output feedback compensators via injective cogenerator quotient signal modulesAcademic research paper on "Mathematics"

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Linear Algebra and its Applications
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{"Behavioral systems theory" / "Quotient signal module" / "Stabilization by output feedback" / Proper / "Pole placement" / "Control systems design" / Tracking}

## Abstract of research paper on Mathematics, author of scientific article — Ingrid Blumthaler, Ulrich Oberst

Abstract Control design belongs to the most important and difficult tasks of control engineering and has therefore been treated by many prominent researchers and in many textbooks, the systems being generally described by their transfer matrices or by Rosenbrock equations and more recently also as behaviors. Our approach to controller design uses, in addition to the ideas of our predecessors on coprime factorizations of transfer matrices and on the parametrization of stabilizing compensators, a new mathematical technique which enables simpler design and also new theorems in spite of the many outstanding results of the literature: (1) We use an injective cogenerator signal module F over the polynomial algebra D = F [ s ] (F an infinite field), a saturated multiplicatively closed set T of stable polynomials and its quotient ring D T of stable rational functions. This enables the simultaneous treatment of continuous and discrete systems and of all notions of stability, called T-stability. We investigate stabilizing control design by output feedback of input/output (IO) behaviors and study the full feedback IO behavior, especially its autonomous part and not only its transfer matrix. (2) The new technique is characterized by the permanent application of the injective cogenerator quotient signal module D T F T and of quotient behaviors B T of D F -behaviors B . (3) For the control tasks of tracking, disturbance rejection, model matching, and decoupling and not necessarily proper plants we derive necessary and sufficient conditions for the existence of proper stabilizing compensators with proper and stable closed loop behaviors, parametrize all such compensators as IO behaviors and not only their transfer matrices and give new algorithms for their construction. Moreover we solve the problem of pole placement or spectral assignability for the complete feedback behavior. The properness of the full feedback behavior ensures the absence of impulsive solutions in the continuous case, and that of the compensator enables its realization by Kalman state space equations or elementary building blocks. We note that every behavior admits an IO decomposition with proper transfer matrix, but that most of these decompositions do not have this property, and therefore we do not assume the properness of the plant. (4) The new technique can also be applied to more general control interconnections according to Willems, in particular to two-parameter feedback compensators and to the recent tracking framework of Fiaz/Takaba/Trentelman. In contrast to these authors, however, we pay special attention to the properness of all constructed transfer matrices which requires more subtle algorithms.

## Academic research paper on topic "Design, parametrization, and pole placement of stabilizing output feedback compensators via injective cogenerator quotient signal modules"

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Linear Algebra and its Applications

journal homepage: www.elsevier.com/locate/laa

Design, parametrization, and pole placement of stabilizing output feedback compensators via injective cogenerator quotient signal modules

Ingrid Blumthaler *,1) Ulrich Oberst

Institut für Mathematik, Universität Innsbruck, Technikerstraße 13, A-6020 Innsbruck, Austria

ARTICLE INFO

ABSTRACT

Article history: Received 19 August 2010 Accepted 23 May 2011 Available online 17 October 2011

Submitted by V. Mehrmann

AMS classification:

Keywords:

Behavioral systems theory Quotient signal module Stabilization by output feedback Proper

Pole placement Control systems design Tracking

Control design belongs to the most important and difficult tasks of control engineering and has therefore been treated by many prominent researchers and in many textbooks, the systems being generally described by their transfer matrices or by Rosenbrock equations and more recently also as behaviors. Our approach to controller design uses, in addition to the ideas of our predecessors on coprime factorizations of transfer matrices and on the parametrization of stabilizing compensators, a new mathematical technique which enables simpler design and also new theorems in spite of the many outstanding results of the literature: (1) We use an injective co-generator signal module F over the polynomial algebra D = f[s] (f an infinite field), a saturated multiplicatively closed set t of stable polynomials and its quotient ring DT of stable rational functions. This enables the simultaneous treatment of continuous and discrete systems and of all notions of stability, called t -stability. We investigate stabilizing control design by output feedback of input/output (IO) behaviors and study the full feedback IO behavior, especially its autonomous part and not only its transfer matrix. (2) The new technique is characterized by the permanent application of the injective cogenerator quotient signal module Dt Ft and of quotient behaviors BT of DF-behaviors B. (3) For the control tasks of tracking, disturbance rejection, model matching, and decoupling and not necessarily proper plants we derive necessary and sufficient conditions for the existence of proper stabilizing compensators with proper and stable closed loop behaviors, parametrize all such compensators as

* Corresponding author.

E-mail addresses: ingrid.blumthaler@uibk.ac.at (I. Blumthaler), ulrich.oberst@uibk.ac.at (U. Oberst). 1 Financial support from the Austrian FWF through project 22535 is gratefully acknowledged.

0024-3795/$- see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2011.05.016 IO behaviors and not only their transfer matrices and give new algorithms for their construction. Moreover we solve the problem of pole placement or spectral assignability for the complete feedback behavior. The properness of the full feedback behavior ensures the absence of impulsive solutions in the continuous case, and that of the compensator enables its realization by Kalman state space equations or elementary building blocks. We note that every behavior admits an IO decomposition with proper transfer matrix, but that most of these decompositions do not have this property, and therefore we do not assume the properness of the plant. (4) The new technique can also be applied to more general control interconnections according to Willems, in particular to two-parameter feedback compensators and to the recent tracking framework of Fiaz/Takaba/Trentelman. In contrast to these authors, however, we pay special attention to the properness of all constructed transfer matrices which requires more subtle algorithms. © 2011 Elsevier Inc. All rights reserved. 1. Introduction The present paper is an elaboration of the MTNS 2010 paper [6]. Problems of control design have always been of central interest in systems theory and have been investigated by many prominent researchers, among them Antsaklis and Michel [1, Chapter 7, Part 2, pp. 589-634], Bengtsson, Blomberg and Ylinen [3], Bourles [7], Callier and Desoer [8, Chapters 7 and 9, pp. 196-242], Chen [9, Chapter 9, pp. 458-534], Falb, Feintuch and Saeks [10], Francis, Kailath [13, Section 7.5, pp. 532-538], Khargonekar, Kucera [14], Murray, Pearson, Pernebo [19], Schneider, Vardulakis [26, Chapter 7, pp. 335-354], Vidyasagar [27, Sections 5.7 and 7.5, pp. 294-317], Wolovich [29, Chapter 8, pp. 269-323], Wonham [30], Youla, Zames, their coauthors and many other contributors. We refer to the quoted books for history, origin, and development of the decisive ideas of control design which is generally described in difficult advanced chapters of these books. Due to the large number of researchers and original papers on control design we only refer to the books where these papers are quoted, used, and elaborated and to some newer papers on behavioral stabilization. We present a new technique for controller design which enables both simpler proofs and new theorems in spite of the many outstanding results of the literature, but we also use the ideas of our predecessors on coprime factorizations of transfer matrices and parametrization of stabilizing compensators. For observer constructions the corresponding work was done in [4] after Fuhrmann's authoritative survey article [12]. Our approach to the problems of the title is distinguished by the following original features: 1. We use an injective cogenerator signal module F over a polynomial algebra D = F[s] (F an infinite field) of differential or difference operators with the action d o y, d e D, y e F, and define T-stability and T-stabilization with respect to a saturated multiplicatively closed subset or submonoid T c D \{0} of stable polynomials. This enables the simultaneous discussion of discrete and continuous systems and of different stability notions, in particular of all those discussed in [12]. An input/output (IO) behavior is T-stable if its autonomous part and its transfer matrix have this property. We investigate stabilization by output feedback and control design for DF-IO behaviors instead of Rosenbrock systems or transfer matrices which are mostly used in the literature (see item 6) and pay special attention to the autonomous part of the IO feedback behavior and not only to its transfer matrix. We note that an injective and faithful (d o F = 0 d = 0) signal module F is called regular in [3, Definition 3, p. 81]. The signal module F[s]F(s) is regular, but not a cogenerator. The duality between equation modules and behaviors is valid for injective cogenerators, but not for regular signal modules. 2. The signal module DF gives rise to its quotient module FT := {y; y e F, t e t} over the quotient ring DT := Jd; d e D, t e t} (C F(s)) of stable rational functions and to the direct sum decomposition F = FT © tT(F) where DtFt is again an injective cogenerator with its own behavioral systems theory and where tT (F) is the T-torsion submodule of T-small or T-negligible signals [18], [4]. Every behavior B c Fq admits a corresponding direct sum decomposition B = BT © tT(B) into the quotient DtFt-behavior BT and its T-small (T-negligible, T-autonomous) part tT(B). The consideration of the DtFt-behaviors BT signifies to study DF-behaviors up to T-negligible ones. A transfer matrix H e Dpxn of T-stable rational functions gives rise to the IO operator Ho : F™ ^ Fp, u ^ y := H o u, which plays an essential part in our derivations. We note that the widely used subring 5 c DT of proper and T-stable rational functions also acts on FT, but not on F. The use of quotient modules and especially of the injective cogenerator quotient signal module Dt Ft and the quotient behaviors BT enables relatively short and conceptual proofs of all results on control design. 3. Like all IO behaviors every considered plant B1 has a rational transfer matrix H1. We do not assume that B1, i.e., H1, is proper and can therefore admit arbitrary decompositions of the variables of B1 into input and output components. In contrast we only consider proper IO compensators B2 such that the output feedback IO behavior fb(B1, B2) is proper and T-stable. The properness of B2 enables its realization by Kalman equations or elementary building blocks while that of fb(B1, B2) ensures the absence of impulsive solutions in the continuous case. For the control tasks oftracking, disturbance rejection, model matching and decoupling and not necessarily proper plants we derive necessary and sufficient conditions for the existence of proper stabilizing compensators with proper closed loop behaviors, parametrize all such compensators as IO behaviors and not only their transfer matrices and give new algorithms for their construction. For a plant B1 in state space form we also obtain all possible T-stabilizing compensators and their feedback behavior in the same form. The parametrization of all not necessarily proper controllers, but with stable and proper feedback behavior is considerably simpler and derived in Theorem 3.12. 4. The generality of the monoid T also permits to solve the problem of spectral assignability or pole placement for the considered control tasks constructively: under a necessary and sufficient condition on the plant B1 and the other data a least monoid Tmin can be constructed for which a Tmin-stabilizing compensator B2 for the intended control task exists. This Tmin is finitely generated up to units. The finitely many roots of the polynomials in Tmjn are then unavoidable as possible poles of the closed loop behavior. Any finite or infinite set of complex numbers which contains these unavoidable poles can be prescribed for the location of the closed loop poles. 5. New algorithms for the construction of all proper compensators B2 as described above are presented and exhibited in an example. 6. Comparison with the behavioral control interconnection literature: more general regular interconnections of plant and controller have been discussed by several authors from the behavioral point of view, for instance in [28,2,24,21]. The latter paper [21], for instance, parametrizes the set of all regularly implementing, partially interconnected controllers for which the manifest controlled behavior is autonomous and stable. Since an autonomous behavior has no transfer matrix such matrices, their properness and use in control design as in [9,8,27] and in the present paper are, of course, not discussed in [21]. While our full feedback behavior is proper and stable as IO system which is necessary for the proper functioning of any machine realization the stability of the full interconnected behavior is not a subject of [21]. The newest paper [11 ] also treats control tasks in this framework. In Blumthaler's forthcoming thesis our new technique is also applied to other control configurations like those in [3, pp. 187-189], [27, Section 6.7] (two-parameter compensators), [20, Section 10.8], [21], [11]. In contrast to the quoted references for the behavioral framework, appropriate transfer matrices and their properness and stability still play an important part in these considerations. Multidimensional proper stabilization was already treated in [18,25]. One reviewer has pointed out the importance of robustness and in particular the internal model principle as discussed, for instance, in [30, Chapter 8], [9], [27, Section 7.5], [7, Section 9.3]. We agree, but have presently only limited insight into this problem and therefore postpone its study to the future. This has to start with the definition of a metric in the set of IO behaviors and especially in the set of compensators which realize different control tasks. The plan of the paper is the following: in Section 2 we introduce the main data and explain the connection of the standard coprime factorizations and Bezout equations with the also standard split module sequences according to [16]. In systems theory this simple connection was observed by A. Quadrat [22,23], for instance. Section 3 treats stabilization by output feedback with proper compensator and proper feedback behavior, but not necessarily proper plant and develops the new technique of injective cogenerator quotient signal module as far as needed later on. The construction of all proper compensators and the spectral assignability problem require extensive considerations. The main results of this paper on tracking, disturbance rejection, model matching, and decoupling are contained in Sections 4-6. Section 7 contains the algorithms that make the results constructive. The paper concludes with a worked-out example in Section 8. 2. Preliminaries The general situation which we consider is the same as in [5,4], and so are the mathematical techniques we apply. Let D denote the polynomial ring F[s] over some infinite field F, K := quot(D) = F(s) its quotient field, and let F be an injective cogenerator over D. Later D will be the ring of operators (differential or difference operators in the standard cases), and F the signal module. The standard choices are the following: F = R, C, F = COT(R, F) or F = D'(R, F) (continuous standard cases) or F = FN (discrete standard case). The action of the indeterminate s on a signal in F is defined as differentiation in the continuous cases and as left shift in the discrete case. Furthermore, let T be a multiplicatively closed subset or submonoid of D \ {0} which we always assume saturated. The elements of T are called T-stable polynomials. As usual DT denotes the quotient ring of D w.r.t. T (also referred to as the localization of D w.r.t. T or as the ring of T-stable rational functions), i.e., Dt = I J e F(s); d e D, t e T C F (s). (1) More generally, for any D-module M we consider the quotient module MT = {X; x e M, t e T} which is a DT-module in the natural fashion, compare [15, Section II.3], [5, p. 2424]. In particular we will need quotient modules UT of row modules U c D1xl, the quotient module FT of the signal module F (which is an injective cogenerator over DT) [4, Section 1], and quotient modules BT of F-behaviors B [4, Theorem 1.8 and Corollary 1.9]. We will subsequently use the properties of FT and BT derived in [4, Section 1]. We briefly repeat the terms T-autonomy and T-stability introduced in [5, Theorem and Definition 2.15]: Definition 2.1 (T-autonomy, T-small signals, T-stability). 1. A behavior ¡3= \w eTl; R o w = 0 where R e Vkxl is called T -autonomous if there exists t e T such that t o B = 0. This is equivalent to BT = 0 or to the existence of a left inverse matrix of R in DT~xk (cf. [4, Theorem 1.9.3]). Signals w e Fl which are annihilated by some t e T are called T-small. 2. An input/output (IO) behavior [20, Section 3.3], [17, Theorem 2.69, p. 37] B = ((u) e Fp+m; P oy = Q o u) (P, -Q) e Dpx(p+m), det(P) = 0, is called T-stable if its autonomous part 8° := {y e Fp; P o y = 0} is T-autonomous. Example 2.2. 1. Assume that F = R, A c C such that A is equal to its complex conjugate A, and T := {t e R[s] \ {0}; VC(t) c A} where VC(t) := {X e C; t(X) = 0} denotes the vanishing set of t in C. 2. In particular, if we choose A := {X e C; №(X) < 0} in the continuous standard case resp. A := {X e C; |X| < 1} in the discrete standard case then a signal is T-small if and only if it is polynomial-exponential and asymptotically zero for t ^ x>. For other examples compare, e.g., [5, Example 2.16]. In the subsequent sections the following two lemmas will be basic tools: Lemma 2.3. Let Rbea commutative ring and A1 e Rpxi, B1 e Rixm such that 0 R1xP — R1xi — R1xm 0 (2) is exact (especially i = p + m). Then the following assertions hold: 1. There are a left inverse A2 e Rmxi ofB1,A<2B1 = idm, and a right inverse e RixpofA1,A1B<2 = idp, such that 0 R1 xp R1xi R1xm ^- 0 (3) is exact too. Then oA2 resp. oB2 is called a section of oB1 resp. a retraction of oA1, and both sequences (2) and (3) are split exact. 2. There are canonical bijections (U2 c R1xi; R1xpA1 © U2 = R1xiJ U2 =$

(B2 e Rixp; A1B2 = idp) B2 = $A e Rmxi; A2B1 = idm) A2 =$

Rmxp X

with U2 = ker(oB2) = R1xmA2, B2 = b0 - B1X, A2 = A^ + XA1. Then

0 R1 xP ^_ R1xi J— R1xm ^_ 0 (4)

is (split) exact too, and

P) (B2, B1) = (idp 0 ) = idp+m.

W \ 0 id J p

Proof. Assertion 1 and the first two bijections of assertion 2 follow from [16, Propositions 1.4.1-1.4.3]. The last bijection in 2 follows from the equivalences

A2B1 = idm = A0Bi ^ (A2 - A0)Bi = 0

^ R1xm(A2 - A2) C ker(oB1) = im(oA1) = R1^ ^ 3X e Rmxp : A2 - a2 = XA1, i.e., A2 = a0 + XA1. □

The parameter X in the preceding lemma furnishes the parametrization of stabilizing compensators according to Kucera and Youla et al. The direct sum decompositions were introduced by Quadrat [22,23] in this context, but were also considered by Rocha and Wood [24] in context with regular interconnections (according to Willems) and set-controllability. Behavioral direct sum decompositions were also discussed by Bisiacco, Bourles, Fliess, Lomadze, Valcher, Zerz et al.

Lemma 2.4 (Coprime factorizations, controllable realizations). Let R denote a principal ideal domain with quotient field K := quot(R). Assume a matrix H e Kpxm.

1. There exists an essentially unique (i.e., unique up to row equivalence over R) matrix (P, -Q) e Rpx(p+m) which satisfies the following equivalent conditions with U := R1xp(P, -Q): (a) The sequence

0 _^ R1xp o(p, -Q) R1x(p+m) o(K 1xm

(b) i. PH = Q, i.e., (P, -Q) (idHm ) = 0, and

ii. (P, -Q) has a right inverse in R(p+m)xp, i.e.,

rank(P, -Q) = dimK(KU) = p and U is a direct summand ofR1x(p+m) or dimK(KU) = p and the elementary divisors ofU (or (P, -Q)) are units in R. In this case R1xpP = {f e R1xp; %H e R1xmJ, det(P) = 0, and H = P-1Q. The representation

H = P-1Q is called a left coprime factorization (l.c.f.) and (P, -Q) the controllable realization ofH over R.

2. Likewise, there is an essentially unique (i.e., unique up to column equivalence) matrix ($) e R(P+m)xm such that HD = N and (%) has a left inverse in Rmx (p+m), i.e., 0 Rm -K Rp+m Kp is exact. Then det(D) = 0 and H = ND 1 is called a right coprime factorization (r.c.f.) ofH over R. ,-q,=R-- , detP^ 0, ( N ) 3. Let (P, -Q) e Rpx(p+m), det(P) = 0,(%) e R(p+m)xm, det(D) = 0,suchthatH = P-1Q = ND-1. 0 —y R1xp -—K R1x(p+m) ___Di R1xm —> 0 is exact (and thus Lemma 2.3 is applicable to it) if and only ifH = P-1 Qisa left coprime factorization and H = ND-1 is a right coprime factorization ofH overR. 4. If (P, -Q) resp. (N) satisfies the conditions in 1 resp. 2 for the ring R this is also the case for any overring R', R c R' c k. 3. Feedback systems and stabilizing compensators We consider two input/output (IO) behaviors [20, Section 3.3], [17, Theorem 2.69, p. 37], [5, p. 2419] y 1 + "2 "2 Fig. 1. The feedback behavior fb(B1, B2 ). e Fp+m; Pi o yi = Qi o ui e Fp+m ; P2 o y2 = Q2 o U2 where (P1; —Q1) e vpx(p+m), det(P1) = 0, (-Q2, P2) e Vmx(p+m>, det(P2) = 0, with associated modules of equations U1 = D1xp(P1, -Q1), U2 = D1xm(—Q2, P2). Recall that (Kalman) state space equations give rise to IO behaviors by elimination of the state [20, Chapter 6], [17, p. 27]. Definition 3.1 (Feedback behavior). The feedback behavior (compare Fig. 1) is defined as B := fb(Bi, B2) := F (p + m) + (p + m); p o y = Q O U 2 \ p := I Pi -Qi\ Q := I 0 Qi\ e v(P+m),(P+m) {ui}' I-Q2 P2 ' l02 0 j with B0 := {y e Fp; P o y = 0} and modules of equations U = v1x(p+m)(P, —Q) and U0 = T>ix(p+m)P = U1 + U2. The feedback system is well-posed if B is an input/output behavior with input u and output y, i.e., if B° is autonomous or rank(P) = p + m = rank(Pi5 -Qi) + rank(-Q2, P2) or U0 = Ui ® U2. Theorem 3.2 (Characterization of T-stable feedback behaviors). For B = fb(Bi, B2) the following conditions are equivalent: 1. B is well-posed and T-stable or B0 is T-autonomous, i.e., B° = 0. 2. P is invertible in DT, i.e., det(P) e T. 3. (a) BT is controllable and (p+m)x(p+m) (b) B is well-posed and H := P—1Q e Vf 4. U1,t © U2,t = V}x(p+m). Note that condition 4 implies that M1,T := vj-x(p+m)/U1,T = U2,T and in particular that M1,T is free since U2,T is so. This is equivalent to right invertibility of (P1, — Q1) over VT or to controllability of B1tT, compare [20, Theorem 5.2.10], [17, Theorems 7.21, 7.52, 7.53, p. 141f p. 150ff]. Proof. The equivalence ofl, 2, and 3 has already been shown in [5, Theorem and Definition 2.15]. The sum in 4. is direct since the feedback behavior is well-posed. Moreover, since localization preserves exactness, Ui,t © U2,t = (Ui © U2)t = U0 = (v1x(p+m)P)r = v}x(p+m)P. This DT-module is equal to v1x(p+m) if and only if P is invertible in DT, i.e., if condition 2 is satisfied. □ We will primarily use the direct sum characterization from item 4, having in mind the parametriza-tion of direct summands from Lemma 2.3. Definition 3.3 (T-stabilizing compensators, T-stabilizable IO behaviors). If the equivalent conditions of Theorem 3.2 are satisfied then B2 is called a T-stabilizing compensator of B\. If fb(Bi, B2 ) is in addition proper we call B2 a properly T-stabilizing compensator of Bj.The behavior B\ is said to be T-stabilizable if there exists a T-stabilizing compensator. Remark 3.4. Assume that B2 is a T-stabilizing compensator of B\. Interconnection of B\ and B2 via u1 := y2 and u2 := y\ furnishes B1 n B2 = fb(Bt, B2)0 Ç tT(F)p+m where tT (F) denotes the set of all T-small signals in F. In Willems' language a T-small behavior, viz. Bi n B2, can be achieved from Bi by regular interconnection, compare [24]. Notice, however, that in contrast to [24] we do not specify the intersection B1 n B2,but onlyits T-smallness,and that tT (F)p+m is not a subbehavior of Fp+m. In the following we will first construct all T-stabilizing compensators with proper feedback behavior fb(B1, B2 ) and then, from Lemma 3.17 to Remark 3.28, those which are additionally themselves proper. In order to study problems related to properness, we introduce the usual rings F(s)pr := (g e F(s); deg (f) := deg(f) — deg(g) < o) resp. 5 := Dt n F(s)Pr of proper resp. of proper and T-stable rational functions, compare [8, p. 169], [26, Chapter 5], [27, Chapter 2]. We will always assume that the set T contains an element (s — a) where a e F. Otherwise (in the case F = C) the saturation of T would imply T = F \{0} and 5 = C. According to [5, Definition and Lemmas 2.14,3.11 ] we obtain a := ^^, D := F[aS = Vr with T := ¡-^; t e T) and (5) Dt = Sa := [j; ; f e S, j e N = {ßaj; ßeF\{0}, j^0}- The introduction of a and a = (s — a)-1 is due to Pernebo [19]. All these rings are principal ideal domains with the following inclusions: D = F[a ] c Dr = S c F(s)pr V = F[s] c VT = Sa c F(s) = F(a) = K. Remark 3.5 (Computation of the Smith form w.r.t. S). Note that, if R e Kkyi is a rational matrix, then its Smith form w.r.t. D is also the Smith form with respect to S = Df, w.r.t. VT = Sa, w.r.t. F(s)pr, and w.r.t. F (s). In the following we will use the inclusions V c S c VT and that these rings are quotient rings of V. We will replace the defining matrices of the behaviors by matrices with entries in V or S which are row equivalent over VT to the original matrices. Recall that T-stability depends on modules (or behaviors) over VT only. Assumption 3.6. In the sequel we assume that B1T is controllable, i.e., that H1 = P_1Q1 is a left coprime factorization of H1 over VT (compare Lemma 2.4). According to Theorem 3.2 this is a necessary condition for T-stabilizability of B1. Let H1 = V—Q = N1D_1, Q := (?1, —Q1) e Vpx(p+m), (Q1 ) e V(p+m)xm denote a left resp. right coprime factorization of H1 over V. This implies that 51 xp o(V1' ^1x(p+m) °(V1x 0 —> V1 is exact. According to Lemma 2.3 let Q = (—V0, V) e Vmx(p+m) be a left inverse of (jV^ and (V0) e V(p+m)xp a right inverse of (V1, — V1) such that 1xp V0 V1> V1xp < W j1x(p+m) o( V°'V0) is also exact. Corollary 3.7. Assumption 3.6 is in force. Then UhT = ViTxpR1 = V1TxpR1, hence B1,T = y1 j e Fp+m; P1 o y1 = Q1 o u1 y1 j e Fp+m; V1 o y1 = V1 o u1 since B1T = U^T. Recall that FT is an injective VT-cogenerator and in particular a VT-module. Proof. By assumption H1 = P—1Q1 is a left coprime factorization of H1 over VT. H1 = P—1 V1 has this property over V and hence also over VT ^ S ^ V (compare Lemma 2.4). The essential uniqueness of these factorizations implies that vT-xpR1 = V^^. □ Now assume that B2 = {) e Fp+m; P2 o y2 = Q2 o u2} is a T-stabilizing compensator of B1 where R2 := (—Q2, P2) e Vmx(p+m), det(P2) = 0, H2 := P-1Q2, and U2 := V1xmR2. Hence, U1,T © U2,T = Vlx(p+m) and H2 = P—1Q2 is a left coprime factorization of H2 over VT since U2,T = R2 is a direct summand. Let H2 = V^Ch, j2 := (—Q2, V2) e Vmx(p+m), be a left coprime factorization of H2 over V. As in Corollary 3.7 we conclude that U2,T = vTxmR2 = V^"% and 1 x mV B2,T = e Ff+m; P2 o y2 = Q2 o u2 r) e FPp+m; V2 o y2 = Q2 o u2 With the notation l := p + m, we define the following matrices: 'Pi -QM e K-Ql P2 f Pi —Qi V-Q2 Q2 , 10 Qi) e Q20 Qi) e Qlxl. IQ 0 Hence, with y := (y^and u := (Ui) we get fb(Bi, B2) = {(U) e Fl+l; P o y = Q o u} and fb(Bi, B2)0 = (y e Fl; P o y = 0). Corollary 3.8. Assume the data from (9). Then vixl(P, —Q) = v\xl Q, —Q). For the quotient behaviors this implies that fb(Bi, B2)T = fb(Bi,T, B2,T) = 1 | e ; P o y = Q o u e Flj+l; Q o y = Q o u The assumption that the behavior B2 is a T-stabilizing compensator of Bi is equivalent to fb(Bi, B2)° = 0, i.e.,P e Gll(DT) or P e Gll(DT). The transfer matrix of the feedback behavior is H = P—iQ = P—iQ. Proof. By Corollary 3.7, there are (unique) matrices Ai e Glp(DT) and A2 e Glm(DT) such that AiRi = Qi and A2R2 = Q2, hence A := (A0i A°2 ) e Gll(DT) and AP = Q, AQ = Q. We deduce that v\xl(P, —Q) = v\xl(P\ —Q). □ Theorem 3.9 (Characterization of properly T-stabilizing compensators). For the IO behavior Bi and its T-stabilizing compensator B2 and the data from above the following conditions are equivalent: 1. H e Slxl, i.e., H is also proper (recall S := DT n F(s)pr). 2. SixlP = SixpQi © SixmP2 = Sixl. 3. Q e Gh(S). Under these conditions B2 is a properly T-stabilizing compensator of Bi according to Definition 3.3. Proof. The equivalence of 2 and 3 is obvious. Remember that the sum in 2 is direct since the feedback behavior is assumed to be well-posed. Condition 3 trivially implies i since H = P—iQ. Now assume i (idp 0 0 0 \ id0p id0m id0p id0m | e Glp+m+p+m(F).Then (Qi ) M = ( —Q —Q —Q = _ p 0 idm 0 0/ (P, — Q), whence the isomorphism Sixl/Sixp (Qi, — Qi) x sixl/Sixm (—Q2, q2) = Six2l/Sixl(Q, —Q), (i0) where (f1; f2) is mapped to (f1; f2)M. But Rj = (Pj, -Ch) and p2 = (-<22, p2) are right invertible over D and thus over 5 2 D by construction and hence Slxl/SJxpQJ and S1xl/S1xmp2 are free. The preceding isomorphism implies the same property for Six21 /Sixl(P, -Q) and thus the existence of with idl = (P, -Q)Z = P ((idl, -H)Z). Since H e Slxl by 1, the matrix (ide, -H)Z is an inverse of Pin Slxl and hence P e Gli(S). □ We next construct all properly T-stabilizing compensators of B1 under the (necessary) condition of controllability of B1T. From the preceding theorem we infer that direct summands of S1xpR;i in S1xl play a part. These have been classified in Lemma 2.3. Lemma 3.10. We use the data from above, in particular from Assumption 3.6 and equations (6)-(8). 1. There are bijections (y ç S1xi; S1xpPi © V = S1xi) M e S(P+m)xp. pi /M = id IpJ \P2/ p2 = (-P2, p2) e Smx((p+m); p2 P2 = (-(22, P2) where Y = ker (o (| )) = S 1xmp2, (| ) = (§ ) - (P )X, p2 = (-<22, p2) = P + Xpi P2 = P0 - Xp1,p2 = p20 - XP1. Moreover, ixp °(p1, -p1) C1 0 <S1xp V pJ C1 o((p1 ) > Sl —> 0 and o(-P2, P2) 1x are split exact sequences and ' P1 -P1\ /P2 Pi -p2 % \P2 P1, p+m ■ 2. Almost all P2 from 1 have non-zero determinant (in the sense specified in the proof and the next remark). Proof. i. The sequences with the retraction o and the section o (—Q2\ are exact since they can be obtained from (7) and (8) by applying (—)q and localization preserves exactness. Remember that S = Dp. Application of Lemma 2.3 to these exact sequences yields the assertion. 2. Let S = (Sij)i^i^m, be a matrix of indeterminates and consider the polynomial g(S) := det (Q — SQi) . Then, forX e Smxp,g(X) = det (Q — XQ^ = det (Q).Wehavetoshowg = 0. Since S = F[a]p is an infinite integral domain this implies that also the polynomial function (g\Smxp : smxp —> S) is non-zero, indeed {X e Smxp; g(X) = 0} is an open dense subset of Smxp (in the Zariski topology). The matrix ^0, D—^ is obviously a left inverse of( QP^ in Kmx(p+m).Parti of the present lemma applied to K = F(s) = quot(S) instead of S yields the existence of X e Kmxp such that (0, Q—^ = (—q20, Q1) + X (P, —Qi). Hence g(X) = det (Q — XQi) = det(Qi)—i = 0. □ Remark 3.11. If k is an infinite field a property of vectors X e kN is called generically or almost always true if it holds on a non-empty Zariski open set or, equivalently, on a special open set where g is a non-zero polynomial. In the preceding lemma this language is extended to the infinite integral domain S = F[a ]q. Theorem 3.12 (Constructive parametrization of properly T-stabilizing compensators). 1. Assume that Bi T is controllable or, equivalently, thatRi is right invertible over DT and the ensuing data from (6) to (8)/ (a) ChooseX e SmxpsuchthatQ2 := (—Q2, Q2) := p2+XQifromLemma3.10satisfiesdet(Q2) = 0, and define H2 := P—ip2. Let R2,cont = (—Q2,ornt, P2,corn) be the controllable realization ofH2 over D, i.e., letH2 = (P2 cont)—'Q cont be a left coprime factorization ofH2 over D. Furthermore choose an arbitrary A e Dmxm with det(A) e T and define R2 := (—Q2, P2) := AR2,cont. Then B2 := {) e Fp+m; P2 o y2 = Q2 o u2} is a properly T-stabilizing compensator of Bi, and all such compensators arise in this fashion. (b) The pairs (X, A) e Smxp x Dmxm with det(P0 — XQi) = 0 and det(A) e T parametrize the set of all properly T-stabilizing compensators B2 of Bi where B2 is constructed from (X, A) according to ia. Two pairs (X, A) and (X', A') give rise to the same compensator B2 if and only ifX = X' and A is row equivalent to A' over D. 2. The following conditions are equivalent for an IO behavior Bi: (a) Bi is T-stabilizable, i.e., there exists a T-stabilizing compensator B2 of Bi. (b) There exists a properly T-stabilizing compensator B2 of Bi. (c) Bi,t is controllable. Proof. 1. (a) i. By construction S1xpRi © S1xmp2 = S1x£, hence also vlxpRi © D|xmp2 = ViTxpRi © D|xmp2 = v\xi because VT = Sa. Since S1xmQ2 = STxm (-Q2, Q2) is a direct summand of S1x£, the factorization H2 = P—T Q2 is left coprime over S and thus over VT.AlsoH2 = (P2,cont)- T Q2,cont is left coprime over V and hence over VT .The essential uniqueness of these factorizations implies dtxmR2,cont = v1 xmQ2. By assumption det(A) e T, hence A e Glm(VT) which implies 1 xmD 1 xmD 1 xmfi 1 x£ 1xpD _ 1 xmD T>t R2 = T>t R2,cont = Dt R2 and T>t = Vt R 1 © T>t R2. According to Theorem 3.2 B2isa T-stabilizing compensator of B\.NowletR2 := (—Qp2, P2) e vmx(P+m) be the controllable realization ofH2 over V and hence also over S. Therefore H2 = Q— 1 q2 = P— 1Q2 are two left coprime factorizations of H2 over S which implies row 2 2 S x mPR2 = equivalence of R2 and'R2 over S, i.e., S 1 xmP2 = S 1 xmR2, and hence S 1 xpR S 1 xi. According to Theorem 3.9 B2 is indeed a properly T-stabilizing compensator of B1. Notice that the matrix R2 from Theorem 3.9 is denoted by p2 here. The matrices p2 e Vmx(p+m) and K2 e Smx(p+m) of the present proof are row equivalent over S, but not identical. ii. Let, conversely, e Fp+m; P2 ◦ y2 = Q2 o U2 , R2 := (-q2, p2) e vmx(p+m), det(P2) = 0 be any properly T-stabilizing compensator of B1 with transfer matrix H2 := P2 1Q2. Then V^[xmR2 is a direct summand by Theorem 3.2, and consequently H2 = P—1Q2 is a left co-prime factorization of H2 over VT, compare Lemma 2.4.Let R2,cont := (—Qi,cont, P2,cont) e Vmx(p+m) be the controllable realization of H2 over V. This implies a factorization R2 = AR2,cont for some A e Vmxm with det(A) = 0. Note that H2 = P—¿ontQ2,cont is a left co-prime factorization ofH2 over V and hence also over VT. Since the left coprime factorization is unique up to row equivalence we deduce that V^[xmR2 = V1xmR2,cont and consequently that A e Glm(VT), i.e., det(A) e T. Define R2 := (—p2, p2) e Vmx(p+m) as in 1ai, i.e., H2 = p—1 p2 is a left coprime factorization of H2 over V and hence also over VT 2 V. This implies v\xmpp2 = v1xmR2. Theorem 3.9 furnishes S 1xpR1 © S 1xmp2 = S1xe From Lemma 3.10 we obtain a unique R2 = (-Q2, P2) = "R® + XR1 e Smx(p+m) with g ) = idm and S 1xmp2 = S 1xm"2, hence also v\xmp2 = v1xm1?2 and H2 = P-1""!. "j1 j = idm an^ R2 = o R2, hence also ut R2 — ^t R2 and 112 — 12 (b) From 1a we conclude that all properly T-stabilizing compensators of B1 are obtained from parameters (X, A) with the asserted properties. Assume that (X, A) and (X', A') gjve rise to the same compensator B2 with transfer matrix H2. For the corresponding R2 = (—Q2, P2) = 1 + XQ1 and Q2 = (—Q2, Q2) = "0 + X/Q1, the left coprime factorizations H2 = (Q2)—1(Qh) = (Q2)—11(1Q2) of H2 over S imply the existence of B e Glm(S) withQ2 = BQ12,hence B = B idm = BQ2 ( = Q2 (ip^ = idm, and consequently Q2 = Q2 and X = X'.The row equivalence of A and A' follows from V1xmAR2,cont = V1xmR2 = V1xmR'2 = V1xmA'R2,cont. 2. The controllability of B1,T is a necessary condition for T-stabilizability by Theorem 3.2 and sufficient - even for the existence of properly T-stabilizing compensators - due to the construction in 1. Recall that almost all P2 in Lemma 3.10 have non-zero determinant and can be chosen in the construction in 1. □ Remark 3.13. Computer calculations of the data in the preceding theorem require the following possibilities only: 1. The Smith form algorithm over the polynomial algebra D = F[s], hence also over D = F[a] = F [-] a e F. 2. A decision method for the inclusion t e T. For F = Q as in all practical examples the computations of 1 are exact, no numerical approximation is required. Note however that the Smith form transformation matrices that are also required usually get very complicated. Theorem 3.14 (Computation of the transfer matrix of fb(B1, B2)). Assume that B1 is T-stabilizable, the data from (6) to (8), and a compensator B2 constructed according to Theorem 3.12. Let ' -2 ' = X e S(p+m)xp denote the right inverse ofQ corresponding to Q2 = (-Q2, P2) = Q +XQ1. LetP, Q, Ip, Q denote the matrices from (9). Then H := P-1Q = Q-1Q ' Q1Q2 D2Q1 ID1Q2 Mi lHy\,U2 Hy1,U1 \Hy2,U2 Hy2,U1 Moreover, /Q2Q1 Q1Q2\ H + idp+m = I _ _ _ _ I • \Q2Q1 Q1Q2 J Proof. The definitions of the involved matrices in (6) to (8) and Lemma 3.10 imply that ' Q1 -Q1\ ID2 Qi ,-Q2 Q2 / \Q2 Q1, D2 Q1\ = Q-1 The assertion on H follows directly by computing H = P 1Q. The claimed form of H + idp+m is a consequence of the equation ' idp 0 1 0 idm = P-1P = (D2P1 - Q1Q2 -D2Q1 + Q1p2\ = /D2P1 Q^ I Q2Q1 - Q1Q -Q2 Q1 + Q1Q2 / IQ2Q1 Q1Q2 1 — H. □ Next we discuss the question of pole placement or spectral assignability. Consider the following data: R1 = (P1; —Q1) e Dpx(p+m), det(P1) = 0, H1 := P-1Q1, U1 := D1xpR1? M1 := D1x(p+m)/U1, (y. ^ u1 B1:= Uf = Fp+m; P1 o y 1 = Q1 o u1 and the Smith form Xi^iYi = (E, 0), X1 e Glp(D), Y1 e Glp+m(D), /ei 0 \ E = I ... I e Dpxp with e1\ ...\ep = 0. Then Dep = annD(t(M1)) = {d e D; d ■ t(M1) = 0} where t(M1) = {f e M1; 3d e D \ {0} with df = 0} is the torsion module of M1. Let P denote the representative system of primes in D = F[s] containing all monic irreducible polynomials, a e F, and define t1 := (s - a)ep, P1 := {s - a} U {q e P; q\ep} = {q e P; #1}, and j8 n qKq); P e F \ {0}, ix(q) > 0 = (t e D \ {0}; 3^ : t|tf) which is the saturated monoid generated by t1. For all D-modules M the quotient modules Mt1 and MT1 coincide, especially -p e F(s); d e D> 0 t1 = Dh C F (s). By construction t1 and thus its divisors ep and ei,1 ^ i ^ p,areinvertiblein DT1,henceE e Glp (DT1). This implies that (E, 0) (E0^ = X1R1Y^E0^ and thus (p+m)xp idp = R1Y1(E—')X1, E—')X1 e D% Therefore R is right invertible over DT1. Theorem 3.15 (Pole placement). Consider the behavior B1 and the accompanying data from (12) and the saturated monoid T1 from (13). LetT C D be any other saturated monoid with (s — a) e T. 1. By definition the monoid T1 is the least saturated one which contains (s — a) and for which R is right invertible over DT1 or, in other words, for which B1 is T1 -stabilizable. 2. The behavior B1 is T-stabilizable if and only if the monoid T1 from (13) is contained in T or, in other words, if the finitely many irreducible factors of ep belong to T. 3. If, in particular, t2 e F[s] is any multiple oft1 = (s — a)ep and T2 is the saturated monoid of all divisors ofpowers oft2, i.e., pn qm(q); q P e F \ {0}, m(q) ^ 0, q irreducible factor of t2 then this T2 contains T1, B1 is T2-stabilizable and all properly T2-stabilizing compensators can be constructed according to Theorem 3.12 applied to T2. 4. If in item 3 F = R and D = R[s], then the T2-stabilizability of B1 resp. the Testability of the feedback behavior fb(B1, B2) signify that the uncontrollable poles of B1 resp. the poles of the feedback behavior fb(B1, B2) are zeros oft2. This is the generalization of the standard pole placement result via state feedback for stabilizable state space systems. Proof. 1. Assertions 1,3, and 4 are clear. 2. The T-stabilizability of B1 is equivalent to controllability of B1T, i.e., to the existence of a right inverse of R1 with entries in DT. This signifies that the greatest elementary divisor ep of R1 (w.r.t. D) is invertible in DT, i.e., contained in T. Since T is saturated and (s — a) e T by assumption, this is equivalent to t1 = (s — a)ev e T or T1 c T. □ Remark 3.16. Computer calculations of the T2-stabilizing compensators in Theorem 2.15.3 require the Smith form algorithm over F[s] (and F[a]) only. A non-zero t e F[s] belongs to T2 if and only if t | t2deg(t) and this is trivially checked. For F = Q these computations can be executed exactly with all computer algebra systems. Our next aim is the study of T-stabilizing compensators such that both B2 and fb(B1, B2) are proper. We start with further results regarding the rings F(s)pr and 5. Lemma 3.17 (The map Vq). Let R be a principal ideal domain with quotient field K and q a prime ofR. Then q induces the saturated monoid T (q) := R\Rq,thediscretevaluationringRT(q) = { f e K; f, g e R, q / g}, i.e., a principal ideal domain with the unique prime q (up to association), and the residue field k(q) := R/Rq with the canonical map can : R —> k(q), f \—> f + Rq. The canonical map can be uniquely extended to the ring epimorphism Vq : RT(q) —> k(q), r = g i—> Vq(r) := can(g)-1 can(f) = can(g'f) where g'g = 1 mod q. Then ker(Vq) = RTq)q and hence RT(q)/RTqq = k(q) = R/Rq. Proof. Obviously Vq(q) = 0 and hence RTq)q c ker (Vq : RTq —> k(q)). If, conversely, Vq(r) = can(g)-1 can(f) = 0 in k(q), then can(f) = 0 and f e Rq, hence r = g e RT(q)q. □ Application of the preceding lemma to the ring R = D = F[a] and the prime a yields Va : Dt(a) D/Da = F, f = f- g(0)—1f(0) where T(a) := D \ Da = {g e D; g(0) = 0}. Lemma 3.18 (The ring F(s)pr as quotient ring of D). F (s)pr = Dt (a) C F (s) = F (a), a = -. s— a In particular, F(s)pr is a discrete valuation ring with the unique prime a, up to association. The prime factor decomposition of a non-zero rational function r = f,f, g e D, has the form r = uav(r) where v(r) := — deg(r) = deg(g) — deg(f) is the standard valuation ofr and where u is a unit in F(s)pr, i.e., with v(u) = 0. Proof. Let 0 = r = D e F(s) = F(a),g, g e F[a], gcdf, g) = 1,g = amam + ■■■+ a<j, am = 0, g = bnan + ■■■ + b0, bn = 0, g(0) = b0. Then, for any number N ^ m, n, ga —N = am (a —1)N—m + ■■■+ aa(a —1)N = am(s — a)N—m + ■■■ + aa(s — a)N, ga —N = bn(a —1 )N—n + ■■■ + ba (a —1)N = bn(s — a)N—n + ■■■ + bp(s — a)N, l_ = ?a —N a0(s — a)N + ■■■ + am(s — a)N—m ga —N b0(s — a)N + -^ + bn(s — a)N—n' IfD e T(a), i.e., a /gor b0 = g(0) = 0, then degs(r) ^ N — N = 0, hence r e F(s)pr. Assume, conversely, b0 = g(0) = 0.Then a0 = g(0) = 0 since gcd(g, g) = 1 and hence degs(r) ^ N—(N—1) = 1 and consequently r e F(s)pr. □ Remark 3.19. In systems theory, compare [26, Chapter 3], [27, Chapter 2], and also in one-dimensional projective algebraic geometry, Da is called the place or prime at infinity and the following notation is used: r(^) := Va(r) = D(0)—1_(0) for r = g—1g e F(s)pr = DT(a), i.e., g e D, g(0) = 0. If F = R or F = C and r = g—1f where f, g e D = F[s] and g = 0, this implies r(<x>) = lim r(t) = lim -. t^ctt t^ctt g(t) Corollary 3.20 (Direct sum decomposition of D, 5, and F(s)pr). 1. F = D/Da = 5/Sa = Dt (a)/'gT(a)a = F (s)pr /F (s)pr a. Therefore all these residue fields will be identified, especially r(x>) = Va (r) for r e F(s)pr, Va (g) = g(0) forg e D. D = F ® Da C S = F ® Sa C F(s)pr = F ® F(s)pra 9 r = Va(r) + (r — Va(r)). Proof. 1. The canonical maps F = D/Da —> S/Sa —> DT(a)/DT(a)a are field homomorphisms and thus injective. The isomorphism D/Da = DT(a)/DT(a)a from Lemma 3.17 yields the assertion. 2. The injection F —> DT(a), k ——> k, is a right inverse of Va : Dt (a) —> DT(a)/DT(a)a = F, hence DT(a) = F ® ker(Va) = F ® Dt(a)a 3 r = Va(r) + (r — Va(r)). The same argument applies to the other rings. □ Remark 3.21. The ideal F(s)spr := F(s)pra of F(s)pr is the ideal of strictly proper rational functions and F(s)pr = F © F(s)pra is a subdecomposition of the standard decomposition F(s) = F[s] © F(s)pra 3 r = rpol + rSpr of a rational function into its polynomial and strictly proper part. Corollary 3.22. For all k, t e N>0 we get the decompositions flcxe = Fkxt e Vkxta c Skxt = Fkxt ® Skxta c F(s)pxt = Fkxt ® F(s)pxta 9 X = va(X) + (X — va (X)), va (X) := {va (Xj)),, va (Y) = Y(0) ifY e V Corollary 3.23 (Invertible elements of F(s)pr). 1. Since F(s)pr = VT(a) is a discrete valuation ring with the unique prime a (up to association) an element r e F (s)pr is invertible in F (s)pr, i.e., r e U (F (s)pr) if and only if a does not divide r in F (s)pr or va (r) = 0. ForQ e V this impliesQ e U(F(s)pr) ^ va(Q) = Q(0) = 0. 2. LetP e F(s)pxp. Then P is invertible in F(s)pxp, i.e., P e U(F(s)pxp) = Glp(F(s)pr) if and only if va(P) = (va(Pj )), e Glp(F). Especially, Q e Vpxp is contained in Glp(F (s)pr) if and only if Q (0) e Glp(F). Proof. We only have to prove the second assertion. But P e Glp(F(s)pr) if and only if det(P) e U(F(s)pr), and this is the case if and only if va(det(P)) = det(va(P)) = 0, i.e., if va(P) e Glp(F). □ Lemma 3.24. Assume a T-stabilizable behavior B1 and the usual data from (6) to (8). The split exact sequences 0 Q1xp Q1xp + 1xp ◦(ff1, -fl) Q1 Q1x(p+m) Jfi ) V DlK f1 (fD0) Q1 f^ (p+m) °( Q) Q1xm from (7), (8) with Q2 , ¡0) (fj^ = idm and (P1, — Ch) ^ Q0 ) = idp induce the split exact sequences C1xp Q(P1(0), — ¡31(0)) F1 x(p+m) Jf1(0)) V ¡¡1 (0)J C1 -> F1 —> 0 F1 x p VC»/ C1> (p+m) ;(—¡gOm) F with (—f?(0), ¡2(0^ (f^ rank (¡1(0), —¡1(0)) = p. idm and (¡1(0), —f1(0))(f^)) = idp. In particular, Proof. Application of the functor (—) V/Va = (—) F to (14) furnishes (15) which is again exact since additive functors preserve split exact sequences. Recall that the tensor product is only right exact in general. □ Lemma 3.25 (Characterization of compensators with proper transfer matrix). Assume that Bi,T is controllable and the ensuing data from (6) to (8). Let (—¡2 P2) := (—if, C) + X Q, —¡¡0 e Smx(p+m) where X e Smxp. According to Theorem 3.12 (—Q2, P2) gives rise to a properly T-stabilizing compensator if and only if det(f2) = 0. The following assertions are equivalent: 1. det(P2) = 0 and H2 := P2 1Q2 is proper. 2. Va (Q2) e Glm(F), i.e, det(Va(f2)) = 0 or, equivalently, q2 e Glm(F(s)Pr). Recall that for/,/ e V and g(0) = 0 the rational function g-1/ is proper, Vo(ff-r'p) = g(0)-1/ (0) and Va(Pl) = {Va ((Plhj))^ The condition 2 thus characterizes the situation where both the compensator and the feedback behavior are proper. Proof. It is obvious that the second statement implies the first one. For the other implication, apply the functor (-) ®S S/So = (-) ®S F to the split exact sequences from (11). This furnishes the split exact sequences //1(0) \ ^p °(pm, -f1(°)} ^ If1 (0)i c1xm 0 f 1xp ~f1x1 F m 0 and J Vo (f 2 ) N V Vo (f 2 ) / C1x F1xp °yVo(tf2)) F1xi ^( Vo (f2 ),Vo(/)) F 1xm ^_ 0 and especially rank (_vo(g), vo(P2)) = m. Since H2 is proper by assumption vo(H2) e Fpxm is well-defined, and P2H2 = / implies Vo(f2)Vo(H2) = Vo(P2H2) = Vo(f2), and consequently (-Voi^^oP)) = _ VoP) (_vo(H2), idm). From the fact that rank (-Vo(f2), Vo(P2)) = m, we deduce that rank (vo(P)) = m, i.e., that det {Vo(f2)) = 0 orf2 e Glm(F(s)pr). □ Lemma 3.26. Let B1T be controllable and assume the usual data from (6) to (8). Then the polynomial g(3) := det (f2(0) - 3f1(0)) e F[3] where 3 = (3j is non-zero. Note that, since F isaninfinite field,thissignifiesthatg(X0) = 0 for almost all X0 e Fmxp. Proof. 1. Recall from Lemma 2.3 and Assumption 3.6 that f0 := ( Pf0 fM e Glp+m(V) c Glp+m(S) C Glp+m(F(s)pr) and hence \-f2 P2 / ~ I f1 (0) -p1(0)\ f0(0) = ( f/0 ' f0 ; ) e Glp+m(F) and \v-(/2l(0) /0(0) J p rank( fv) = m or F 1x(p+m) ( -f™) = F1xm 2. Consider, more generally, any matrix (B) e F(p+m)xm with rank (B) = m. By induction on rank(C) =: r we transform (B) into (c+ab) with det(C + AB) = 0 by at most m - rank(C) elementary row operations, the case r = m being obvious. For r < m we assume without loss of generality that the last row of C is linearly dependent of the preceding rows, i.e., F 1xmc = ^ FCj C F1 x (p+m) Then there exists i ^ p such that B— is linearly independent of all rows of C, i.e., . Q — F 1xmC C F 1xmC © FBi— = X FCj— © F(Cm—+ Bi—) = F1xm m— i ;=1 I C(m—1)— J 1 Cm—+Bi— Obviously the last matrix has rank r + 1, and it can be obtained from C as C + AB where A is the matrix with Amj = 1 and 0 at all other entries. □ Theorem 3.27 (Constructive parametrization of proper and properly T-stabilizing compensators). Assume that B1 = {(U1) e Fp+m; P1 o y1 = Q o u^ is T-stabilizable and the derived data from (6) to (8). Then 1. The determinant det (pQ (0) - XQCh (0)) is non-zero for almost all X0 e Fmxp. 2. The triples (X0, Y, A) e Fmxp x Smxp x Dmxm with det (PQ(0) - X0Q1(0^ = 0 and det(A) e T parametrize the set of all proper and properly T-stabilizing compensators B2 = {(U2) e Fp+m; P2 o y2 = Q2 o u2} of Bi by the following construction: (a) X := Xq+aY e Smxp, (-&, ?2) := (-Q°, +X (Pi, -Qi) e Smx(p+m).Thendet(p2) = 0 andH2 := P—''Q2 e F(s)^xp. 2 ' '2/ (b) LetH2 = P2iC1ontQ2, cont be a left coprime factorization of H2 over D and define (—Q2, P2) := A —Q2 P2 Two such triples (X0, Y, A) and (Xq , Y', A' ) give rise to the same compensator if and only ifX0 = X0, Y = Y', and A and A' are row equivalent over V. 3. The transfer matrix H C is proper if and only if Qc(0) e Glp(F ). In this case X0 := Q2q(0)Pc(0)—c satisfies the inequality of item 1 and gives rise to exactly those compensators with strictly proper H2 (compare [27, Lemma 5.2.25]). 4. If the transfer matrix HC is strictly proper and hence QC(0) = PC(0)va(HC) = 0 then X0 e Fmxp can be chosen arbitrarily and hence all properly T-stabilizing compensators are proper (compare [27, Corollary 5.2.20]). Proof. 1. This is a consequence of the previous lemma. 2. By the results derived above, any X e Smxp with det (QQ(0) — va(X)Q1(0^ = 0 gives rise to a transfer matrix H2 of a proper and properly T-stabilizing compensator B2 of B1, and all such transfer matrices are obtained in this fashion. Recall the decomposition Smxp = Fmxp © Smxpa 3 X = va (X) + (X — va (X)) =: X0 + a Y from Corollary 3.22. 3. The equivalence of the properness of H1 with P1(0) e Glp(F) follows as in Lemma 3.25 where det(P1) = 0 by assumption. The inclusion n m I Q1(0) —Q1(0)\ Glp+m (F ) 9 I p lemma3.26 \ _Q0(o) qq(0) J /Q1(0) 0 \ | Qdp °\|idp —QQ(0)—qQq1(0)q \ implies V 0 idm/ Q20(0) idm/ V 0 00(0) — Q20(0)Pc(0)—CQc(0)J det (Q2(0) — X°Q1(0)) = det (Q2(0) — Q20(0)Qc(0)—CQc(0)) = 0 and hence the condition of item 1. The equations p2H2 = /2, hence Vo (P2) VoH) = Vo (Q2) = PQ(0) -X0p1(0) and Vo(f2) e Glm(F) show that H2 is strictly proper, i.e., vo (H2) = 0, if and only if Vo (/2) = /(0) - X0p1 (0) = 0, i.e., X0 = /(0)p1 (0)-1. 4. By construction the determinant det (P^O) - X0p1(0^ = det (p2'(0)) is not zero for all X0 and otherwise does not depend on the parameterXq, hence is non-zero for all Xq. □ Remark 3.28 (Pole placement). The results on pole placement in Theorem 3.15 also hold mutatis mutandis for the T-stabilizing compensators of Theorem 3.27. Remark 3.29 (State space realizations). Let the plant B1 be proper and let B2 be a compensator constructed according to Theorem 3.27. Assume that state space (Kalman) realizations s o xi = AiXi + Biui, yi = CiXi + Dm, xi e Fni, yu u2 e Fp, y2, u1 e Fm of Bi, i = 1, 2, are given or constructed. Recall that every input/output behavior admits an essentially unique observable state space representation [29, Chapter 5.2]. On the other hand every state space system gives rise to a unique IO behavior by elimination of the state (compare, for example, [20, Chapter 6], [17, Corollary and Definition 2.41, p. 27]). In order that the assumptions ofTheorem 3.27 are satisfied we need that the plant Kalman equations are T-stabilizable and T-observable (=T-detectable), i.e., that for BS1 = {(X1) e Fn1+m; sox1 = A1x1 + B1u1} the quotient behavior BS1 T is controllable and that (1 ^ ) : BS1T = B1,T is an isomorphism. We may and do assume that the Kalman equations of the compensator B2 are the essentially unique observable ones. The constant matrix Di is the constant part of the proper transfer matrix Hi. Then the proper and T-stable IO behavior fb(B1, B2) has the T-stable and T-observable Kalman realization (compare [28, Section 10.5]) s o x = Ax + Bu, y = Cx + Du with x := (x1 J e Fn1+n2, u := P J e Fm+p, y = P1 J e Fp+m, ' idp -dA 1 = (idp +D1D-1D2 D1D-^ -D2 idmj I D-1D2 D0-1 Dq := idm -D2D1 e Glm(F), A1 0 0 A 2 B1 0 0 B2 I idp -D1 - D2 idm idp -D0 B2 0 - D2 idm idp -DA B2 0 -D2 idm i' (C1:) r1 (d 3 idp -D1 - D2 idm In particular, if B1 is strictly proper and hence D1 = 0, this yields Ai + B1D2C1 B1C2^ K B2C1 ( Ci 0 \ {D2C1 C2 fBi BiD2 ' 0 B2 ' 0 0 1 0 D2 Corollary 3.30 (State space realizations). Data of Remark 3.29. Assume a T-observable (=T-detectable) state space realization of the proper and T-stabilizable plant Bi. Then every observable T-stabilizing compensator in state space form is the observable state space realization of a compensator B2 of Bi constructed according to Theorem 3.27. The equations of the compensator are contained in the preceding Remark 3.28. The condition for T-stability of the feedback behavior is det(s idni+n2 -A) e T. The same arguments apply to the later compensators for various control tasks. Notice that the theorems in [9, Section 7.5] and [28, Section 10.5], for instance, construct some, but not all stabilizing compensators in state space form. 4. Tracking and disturbance rejection Assumption 4.1. In the remainder of this paper we always consider IO behaviors e Fp+m; Pi o yi = Qi o ui e Fp+m; P2 o y2 = Q2 o u2 (Pi; -Qi) e vpx(p+m), det(Pi) = 0, (-Q2, P2) e vmx(p+m), det(P2) = 0 such that B2 is a proper T-stabilizing compensator of the plant Bi for which the feedback behavior fb(Bi, B2) is proper and, of course, T-stable. These B2 have been parametrized in Theorem 3.27. We use the data from Assumption 3.6, Lemma 3.i0 and Theorem 3.27. Furthermore, we assume a signal generator behavior B3 := {w e Fp; R o w = 0}, R e Dkxp. The trajectories of B3 are the reference signals that shall be tracked resp. the disturbances that shall be rejected in the following. Definition 4.2 (T-tracking and T-rejecting compensators). The compensator B2 is called a T-tracking compensator resp. T-(disturbance) rejecting compensator of Biforsignals u2 in B3 if u2 e B3, ui = 0,and e fb (Bi, B2) imply that e2 := yi + u2 resp. yi is T-small, compare Fig. 2 for an interconnection Fig. 2. Tracking resp. disturbance rejection interconnection. diagram. This signifies that for zero input u1 = 0 the output y1 T-tracks any signal - u2 e B3 resp. that any disturbance input u2 e B3 has no significant effect on the output y1. Corollary 4.3. Assume that B2 is a T-disturbance rejecting compensator of B1,letu2 e B3 be a disturbance signal and let u1 e Fm be arbitrary. Furthermore, assume po(:;)-qoo and Po(!■)=Qop. i.e., (y,) is an output of fb(B1, B2) to the input ), and (y^ is an output to the disturbed input (JJ^). (y1 - yA /u2\ ) = Q o ( ) , and hence y1 - y1 is T-small. Consequently, the difference between y2 - y2j_ V 0/ the disturbed output y1 and the undisturbed output y1 is T-negligible. Example 4.4. A standard choice for the behavior B3, in particular for its use via the internal model principle, is R = 4> idp where 4> e V = F[s] is a non-zero polynomial whose roots determine the frequencies and growth of the tracking resp. disturbance signals. In the following considerations we derive necessary and sufficient conditions for the existence of T-tracking resp. T-rejecting compensators of a given T-stabilizable IO behavior B1 and parametrize all such compensators. Theorem 4.5 (Characterization of T-tracking and T-rejecting compensators). Assumption 4.1 is in force. 1. The behavior B2 is a T-tracking compensator of B1 forsignalsu2 e B3 if and only if there is a matrix Zt e Vpxk such that p1p2 + idp = p1 (pQ - Xp1) + idp = ZtR. In this case rank(R) = p and B3 is thus autonomous. 2. The behavior B2 is a T-rejecting compensator of B1 for signals u2 e B3 if and only if there is a matrix Zd e Vpxk such that p1p2 = p1(p2l - Xp1) = ZdR. Proof. We prove 1, the proof of 2 is analogous. Recall from Theorem 3.14 that Hy1,u2 = N1p2 and Hy1,u2 + idp = He2,u2 = p2p1. 1. The feedback behavior is proper and T-stable by definition and hence especially P e Glp+m(VT) andH = P-1 Q e vTp+m>x(p+m>. Recall that VT acts on FT.The following equivalences hold: B2 is a T-tracking compensator of B1 for u2 e B3 ^ y1 + u2 e Fp; y, u2 e Fp, R o u2 = 0, 3y2 e Fm : P o (y1 | = Q o (u2 1 2 1 2 2 2 y 2 0 is T-autonomous yi + U2 e Fp; yi, U2 e Fp, R o u2 = 0, 3y2 e F^" : P o yi + U2 e Fp; U2 e Fp, R o u2 = 0, ^ {y i + u2 e Fp; u2 e Fp, R o u2 = 0, yi = Hy,,^ o^J = 0 =Pi 02 {(N i°2 + idp) o u2 e fP; u2 e Fp, R o u2 = 0) = 0 ^ |u2 e Fp; R o u2 = 0j c e Fp; (Nip + idp) o u2 = 0 3Zt e vpxk such that p + idp = ZtR. The second equivalence holds by [4, Theorem i .9] and since (-)T is an exact functor on behaviors, cf. [4, Corollary i .i 0], the third one since P e Glp+m(DT) and H = P-iQ e D(/+m)x(p+m). The last equivalence is true since FT is a cogenerator over DT, compare [4, Theorem i .6]. 2. Since Ni Q2 + idp = Hyi ,u2 + idp = D2Pi is non-singular, the equation N2Ni = pi p2 + idp = ZtR implies that idp = p- ip- iZtR, hence rank(R) = p. But this signifies that B3 is autonomous. □ Corollary 4.6 (Dimension relations for tracking). Let Bi be a plant and consider the behavior B3 = {w e Fp; R o w = 0}, R e Dkxp, containing tracking signals as above. Let ai | ... | as denote the invariant factors of the module v\xp/v\xkR, i.e., the elementary divisors ofR w.r.t. D which are non-units of Dt. i . If a T-tracking compensator B2 of B i for signals u2 e B3 exists then s ^ m. 2. For R = 4> idp where 4> e D \ T is non-zero (compare Example 4.4) this signifies p ^ m [8, Theorem 31, p. 201], [26, Corollary 7.6]. Proof. From Theorem 4.5 we infer rank(R) = p. If s = 0 the behavior B3 is T-autonomous and the assertion is trivial, hence assume s > 0. Let URV = (Q) be the Smith form of R w.r.t. VT with E = diag(i,..., i, ai,..., as). Let q be a prime of VT which divides ai and hence all ai with its associated canonical map (compare Lemma 3.i7) Vq : Dt -* Dt/VTq =: k(q). By Theorem 4.5.i the equation NiQ2 + idp = ZtR holds for some Zt e Dpxk, hence p = rank(idp) = rank(Vq(idp)) = rank(Vq(Zt)Vq(R) - Vq(Ni)Vq(Q2)) < rank(vq(R)) + rank(vq(PPi)) < rank(vq(R)) + m because Ni e Dpxm. The Smith form of R implies that vq(U)vq(R)vq(V) = [Vq(QE)), vq(E) = diag(i, ..., i, 0,..., 0) since q\ai, i = i,...,s. We deduce that rank(Vq(R)) = rank(Vq(E)) = rank (diag(i'"'Qi'Q''"'Q)) = p - s since Vq(U) and Vq(V) are invertible over k(q). Substituting this in (i6), we get p ^ (p - s) + m, i.e., s ^ m as asserted. □ Remark 4.7 (Signals with left bounded support, compare [8, Definition 35, p. 113]). Consider the complex continuous standard situation with the signal module F = D'(R, C), and denote by Y : R —> C the Heaviside function. Let B2 be a T-tracking resp. T-rejecting compensator of B\ for signals u2 e B3. Assume that the input to the feedback behavior fb(Bi, B2) is of the form ) = Y (u°2 ) where u2 e B3 n COT(R, C)p, and let ($2) e Fp+m be the uniquely determined output with left bounded support corresponding to this input. Then it can be shown that y\ + u2 resp. y\ is of the form Yv for some T-small signal v in the case of tracking resp. disturbance rejection. In other words, the errors occuring by tracking resp. disturbance rejection are "truncated" T-small signals. This signifies that any T-tracking resp. T-rejecting compensator does also track resp. reject signals of the form Yu2 where u2 e COT(R, C)p is a signal in B3. Properness of the feedback behavior fb(Bj, B2) (or, more precisely, of the submatrix Hyuu2 of the transfer matrix H of fb(B1, B2)) is essential for this result.

In the following we assume that the given 1O behavior B\ is T-stabilizable. Weuse thesamenotations as in Section 3, (6)-(8) in Assumption 3.6. We first treat the existence of a T-tracking resp. a T-rejecting compensator B2 for signals u2 e B3 and then parametrize all such compensators. By Theorems 4.5 and 3.27 there exists a T-tracking resp. T-rejecting compensator if and only if the equation

M = N1XP1 + ZR

JVi 02° + idp in the case of tracking

where M :=

in the case of disturbance rejection

has a solution (X, Z) e Smxp x D>pxk such that

(-&, °2) := °0 + X (°1, -°i)

has the correct 1O structure and proper transfer matrix, i.e.,

det M°2)) = det (°°(0) - va (X)°i(°)) = 0.

This condition can be checked algorithmically due to the following considerations: Assume that (17) has a solution (X17 is an inhomogeneous linear equation in the entries of X and Z, i.e., it can be rewritten as an equation of the form (x, z) (A) = m where x e S1x(mp) resp. z e v}x(pk) contains the entries of X e Smxp resp. of Z e Dpxk etc. Consequently, Algorithms 7.1 and 7.2 allow to check solvability of (17) first in vTlxp x Dpxk, then in Smxp x Vpxk, and to compute such a matrix (XZ°). Now consider the associated homogeneous equation

°1XP1 + ZR = 0 (18)

and its solution module over D

((X, Z) e Dmxp x Dpxk; (18)) = £D(Xh'i, Zh'i). (19)

Again, since (18) is a linear equation in the entries ofX and Z, Algorithm 7.1 (over the ring R = D) can be applied in order to compute the matrices (Xh'1, Zh'1) e Dmxp x Dpxk appearing in (19).

Eq. (19) implies that

Xh := (X e Vmxp; 3Z e Vpxk : (18)) = X VXh'i

and, since localization preserves exactness, that

VTXh = jX e V^; 3Z e Vpxk : (18)) = £VTXh,i.

By means of Algorithm 7.4, again after arranging the entries of the matrices Xh'1 e Dmxp as rows xM e D1x(mp), we determine B(1), ..., B(v) e Dmxp such that

VaXh n Vmxp = 0 VB°i).

Note that the B(j) e Dmxp are computed by means of Nj, Pj, and R, i.e., the B(j) depend on B1 and B3, but not on T.

Eq. (20) and application of (-)f (where Df = 5 and (Va)j = DT) to (21) imply

(X e Smxp; 3Z e Vpxk : (18)) = VTXh n Smxp = 0

Consequently, considering again the inhomogeneous equation (17) with its solution (X0, Z°) e Smxp x Vpxk, we get the result

(X e Smxp; 3Z e Vpxk : (17)) = X0 + (VTXh n Smxp) = X0 + 0

Let now X := X0 + ZJv=1 njB(), n = (n1, ■■■,nv) e Sv arbitrarily, be any element of X0 + (DTXh n Smxp). Then the matrix (—Q2, P2) := (-Q0, f>) + X (f1, -f1) defines a T-tracking resp. T-rejecting compensator if and only if det (va (P2) = 0, i.e.,

det (va(f2)) = det (P2(0) — vff(X)Qi(0^

= det (p0(0) -

va(X°) ^ va(nj)B0)(0) j=1

Q.1 (0) I = 0.

Remember that va(f) = f (0) for Q e D, and va(r) = va Q(0) = 0. Define the polynomial

L\ - QM. = L(0)

for r e F (s) pr, Q,Q e V.

O := det (L0(0) -

va(X°) ^ SjB(j)(0) j=1

Q1(0) I e F [S ]

in the indeterminates S = (S1, Sv). Notice that the polynomial g depends on X0 and the B(j) which in turn depend on B1 and B3, but not on T. This will be important for the discussion of spectral assignability.

We summarize the preceding considerations in the following theorems:

Theorem 4.8 (Existence of T-tracking and T-rejecting compensators). For a plant B1 and a signal generator B3 with the notations from above the following conditions are equivalent:

1. There exists a T-tracking resp. T-rejecting compensator B2 of B\ for signals in B3.

2. (a) B1 is T-stabilizable, i.e., (Pi, -Q1) has a right inverse matrix in D(p+m)xm, (b) theEq. (17)

M = N1XP1 + ZR (25)

N1Q2: + idp in the case of tracking 20

where M :=

N1Q20 in the case of disturbance rejection

has a solution (XZ°) e Smxp x DpTxk, and (c) the polynomial gfrom (24) is non-zero.

Proof. The assertion follows directly from Section 3, Theorem 4.5, and the considerations from above. □

Theorem 4.9 (Constructive parametrization of T-tracking and T-rejecting compensators). Assume that the conditions of the previous theorem are satisfied. Then all T-tracking resp. T-rejecting compensators are obtained in the following fashion:

1. Let £ = (£1, ...,£v) e Fv be a non-zero of the polynomial g, i.e., g(£1, ...,£v) = 0. Note that, since g = 0 and F is an infinite field, almost all £ e Fv satisfy this condition.

2. Choose arbitrary j1,...,jv e S and define

rjj := £j + a Zj, j = 1,...,v, X := X0 + X njB(), and

—2, °2) := (-°2), °0 + X (P1, -°1). Then det (Va(°2)) = g(h, ...,£v) = 0, hence % e Glm(F (s)pr) and H2 := 0-1%2 e F(s)^.

3. LetH2 = P- l1ont Q2,cont be aleftcoprime factorization of H2 over D,chooseA e Dmxmwith det(A) e T,

and define (-Q2, P2) := A(-Q2,cont, P2,cont). Then B2 := {(£ ) e Fp+m; P2 ◦ y2 = Q2 ◦ u2} is a T-tracking resp. T-rejecting compensator of B1 for u2 e B3, and all such compensators can be obtained in this fashion.

In other terms: The T-tracking resp. T-rejecting compensators are parametrized by the triples (£, j, A) e Fv x Sv x Dmxmwithg(£) = 0 and det(A) e T.

Proof. Follows directly from the above considerations. □

Corollary 4.10. Theorem 3.27.4 implies that the requirement thatg is non-zero resp. that £ is a non-zero ofg in Theorem 4.8 resp. 4.9 is automatically satisfied if the plant B1 is strictly proper.

In the following we treat the problem of pole placement or spectral assignability for tracking and disturbance rejection.

Theorem 4.11 (Pole placement for tracking and disturbance rejection). Consider a plant B1 and the signal generator B3 with the notations from above. Choose a e F and define a := (s - a)-1 and

D := F[a] as always. Let ep e D = F[s] be the greatest elementary divisor ofR1 = (Pj, —Q1) w.r.t. D as in (13).

1. Assume that (17) has a solution in F(s)mrxp x F(s)pxk; this 7.1

and 7.2 with T = F[s] \ {0}. Then Algorithm 7.1.2 yields a "minimal" polynomial t2 e D = F[s] such that (17) has a solution in Dmxp x Dp^k. Let tmin := (s — a)ept2, Tmin the saturated monoid of all divisors of powers oftmn, i.e.

Tmin —

PYl q e F [s]; 0 = p e F, m(q) > 0, q an irreducible factor of tm

and Smin := DTmin n F(s)pr. Algorithm 7.2 furnishes a solution (X0, Z0) e S^ x D^lk of (17). Define the polynomial g from (24) with this X0. Then B1 admits a Tmin-tracking resp. Tmin-rejecting compensator if and only ifg = 0.

2. IfT c D \ {0} is any saturated monoid containing (s — a) then B1 admits a T-trackingresp. T-rejecting compensator if and only if (17) has a solution in F(s)™^ x F(s)pxk, the polynomial g from item 1 is non-zero, and Tmin c T. In particular, Tmin is the least saturated monoid T containing (s — a) with a T-tracking resp. rejecting compensator for B\. All such compensators can be constructed via Theorem 4.9 with X0 and g from item 1.

3. If (17) has a solution in F(s'^x x F(s)pxk, the polynomial g from item 1 is non-zero, t3 e F[s] is any multiple of (s — a)ept2, and hence the saturated monoid T3 of all divisors of powers oft3 contains Tmin, then there are T3-tracking resp. T3-rejecting compensators of B1 and all such compensators can be constructed via Theorem 4.9 applied to T3 with the data from item 1.

4. ForF = R and D = R[s] in item 3 the poles of all feedback behaviors with the compensators from item 3 are contained in the finite set of complex numbers

Vc(t3) 2 Vc(tmin) = {a} U Vc(ep) U Vcfe) = {a} U ch(Bt) U Vcfe).

Proof.

1. follows directly from Theorem 4.8 applied to B\, B3, and Tmin.

Assume that a T-tracking resp. rejecting compensator of Bi exists. In particular, Eq. (17) has a solution in (DT n F(s)pr)mxp x D^xk c F(s)^ x F(s)pxk. This implies that the data Tmin,

(X0, Z0) e (DTmin nF(s)pr)mxp xV^xk

, andg from item 1 can be constructed. Since the monoid Tmin is the least saturated one such that (s — a) e Tmin, B1 is Tmin-stabilizable, and (17) has a solutionin D^x x Dpxk we conclude that Tmin c T.Also(X0, Z0) e (DT nF (s)pr )mxp x DpTxk. Hence the data (X0, Z0) and g canbeusedbothforTmin and for T in Theorem 4.8. The existence of a T-tracking resp. rejecting compensator and Theorem 4.8 then imply g = 0. According to item 1 there is a Tmin-tracking resp. rejecting compensator. Because of Tmin c T this is also a T-tracking resp. rejecting compensator.

3. The assertions in item 3 and 4 follow directly from item 2. □

Finally, we study the problem of simultaneously T-tracking signals in one behavior Bt and T-rejecting

signals in another behavior Bd. We also admit disturbances at the input u1 of the plant.

Corollary 4.12 (Simultaneous tracking and disturbance rejection). Assume that B1 is T-stabilizable and

three behaviors

B1,d = u e Fm; R1,d ◦ u1 = 0},

B2,d = {u2 e R2,d ◦ u2 = 0}, and

B2,t = {u2 e R2,t ◦ u2 = 0}.

Then there is a T-stabilizing compensator which simultaneously rejects disturbances u1 e B1:d at the input and u2 e B2,d at the output and tracks signals u2 e B2,t if and only if the inhomogeneous linear system

ô2qi = N1XQ1 + Zi,dRi,d,

NQ = N1XP1 + Z2,dR2,d,

Ô0P1 = JV1XP1 + Z2,tR2,t

is solvable such thatX e Smxp and the Zk have entries in DT and the ensuing polynomial corresponding to g from (24) is non-zero. All preceding results and proofs of this section are applicable to this more general situation.

5. Model matching

Consider three IO behaviors y1

e Fp+m; P1 o y1 = Q1 o u1

e Fp+m; P2 o y2 = Q2 o u2

¡ym] u2

e Fp+p; Pm o ym = Qm o u2

Hm :— Pm Qm-

Definition 5.1 (Model matching T-compensators). Under Assumption 4.1 we call B2 a model matching

T-compensator of B1 for the model behavior Bm if y1 — ym is T-small whenever

e fb(B1, B2)

Theorem 5.2 (Characterization of model matching T-compensators). Under assumption 4.1 the compensator B2 is a model matching T-compensator of B1 for Bm if and only if Bm is T-stable andHm = Hy1}U2.

Proof.

1. Assume that B2 is such a compensator. For anyym e Bm the equations P o (0) = Q o (0) and Pm o ym = Qm o 0 imply that ym = ym — 0 is T-small. But this signifies that Bm is T-stable, i.e., Pm e g1p(Vt) and Hm = P—Q e Vv/v.

2. We now show the equivalence of the two conditions under the assumption that Bm is T-stable, i.e., Pm e Glp(VT). By definition B2 is a model matching T-compensator of B1 for Bm if and only if

|y 1 — ym e Fp ; y1, ym e Fp, 3u2 e Fp 3y2 e Fm with

P o (y1 I = Q o (2 I , Pm o ym = Qm o u2 y2 0 m m m 2

is T-autonomous. This is equivalent to

u^ „. „ ^ if =|„_ ^ if:

|y1 — ym e Fp; y1, ym e Fp, e Fp 3y2 e F? with

Since P e Glp+m(DT) and Pm e Glp(DT), we can rewrite this as

P ° (y1 I = Q ° ( u2 I ' Pm ◦ ym = Qm ◦ u2

y2 0 m m m 2

y1 — ym e Fp; 3u2 e F^ : (y1 | = H o ( u2 ) , ym = Hm o u2

or, equivalently, as

{y1 — ym e Fp; 3u2 e Fp : y1 = Hy1,u2 o u2, ym = Hm O = 0, i.e., (Hyuu2 — Hm) o u2 = 0 for u2 e Fp. Since FT is an injective cogenerator over DT this is equivalent to Hy1 ,u2 = Hm. □

Theorem 5.3 (Existence of model matching T-compensators). Forgiven IO behaviors B1 and Bm the following two conditions are equivalent:

1. There exists a model matching T-compensator B2 of B1 for the model behavior Bm.

2. (a) The matrix (P1, —Q1) has a right inverse matrix in D(T+m)xт, i.e., B1 is T-stabilizable,

(b) Bm is T-stable, i.e., Pm e Glp(DT),

(c) with the notations from Assumption 3.6 the equation

f 1f20 — Hm = f1Xf1 (26)

has a solution X0 e Smxp, and

(d) the polynomial

g(S) := det (f2(0) — (X0) + U(0)^ ^(0)) e F[S] (27)

in the indeterminates s = (Sjr,^'^ is non-zero where r := rank(N1) and U e Dmx(m—r) denotes a universal right annihilator off (cf. [5, Definition and Lemma 2.7]).

Proof. By Theorem 5.2 we assume the necessary conditions 2a and 2b and have to look for compensators B2 with Hy1 ,u2 = Hm among those parametrized in Theorem 3.27, i.e., with

(—N2 f2) := (—Q20, f2) + X (f1, —Q1), X e Smxp,

Hm = Hy1,u2 = f1Q2 = NQ0 — N1XP1, det (va(P2)) = det (f0(0) — va(X)Q_1 (0^ = 0.

These equations and inequalities are solvable if and only if conditions 2c and 2d are satisfied. Since det(P1) = 0 all solutions of (26) are of the form

X = X0 + US, S e S(m—r)xp, hence va(X) = va(X0) + U(0)va(S). □

Theorem 5.4 (Constructive parametrization of model matching T-compensators). Assume that the conditions of Theorem 5.3 are satisfied and use X0 and g from that theorem. Then all model matching T-compensators are obtained in the following fashion:

1. Let £ = (£j) ,1 jp e F(m r)xp be a non-zero of the polynomial g from (27). Note that, since g = 0 and F is an infinite field, almost all £ e F(m-r)xP satisfy this condition.

2. Choose an arbitrary matrix Z e S(m-r)xp and define

S := £ + aZ e S(m-r)xp, X := X0 + US and

(—02, $2) := (-$2°, °0) + X p1, -Ch) • Then

det M°2)) = 0, °2 e Glm(F(s)pr), andH2 := P-Q e F(s)^.

3. Let R2,cont := (—Q2,com, P2,corn) be the controllable realization of H2 over D, choose an arbitrary matrix A e Vmxm with det(A) e T, and define R2 := (-Q2, P2) := AR2,cont .Then B2 := {(U2) e FP+m; P2 o y2 = Q o u2} is a model matching T-compensator of B1 for the model behavior Bm, and all such compensators can be obtained in this fashion.

Inotherterms:ThemodelmatchingT-compensatorsareparametrizedbythetriples (£, Z, A) e F(m-r)xp x s(m-r)xp x Dmxm withg(£) = 0 and det(A) e T.

Proof. The assertions follow directly from Theorem 5.3. □

Remark 5.5. The two previous theorems are constructive: the conditions 2a and 2c in Theorem 5.3 can be checked by means of Algorithm 7.1 (by transposing the occuring equations, in the case of (26) after multiplying with P-1 from the right), the universal right annihilator U can be computed as described in [5, Definition and Lemma 2.7] (again by transposing).

Finally, we treat the question of pole placement or spectral assignability for model matching. The following result is an analog of Theorem 4.11.

Theorem 5.6 (Pole placement for model matching). Consider the plant and model

B1 = ((y ) e FP+m; P1 o yx = Q1 o ut) and Bm = ((%) e FP+P; Pm o ym = Qm o .

Choose a e F and define a := (s - a)-1 and D := F[a] as always. Let e1 be the greatest elementary divisor ofR1 = (P1, -Q1) w.r.t. D as in (13) and e^ e D the greatest elementary divisor ofPm, i.e., minimally such that Bm is (e^-stable.

1. Assume that (26) has a solution in F(s)mrxp. Then Algorithm 6.1 yields a "minimal" polynomial t2 e D = F[s] such that (26) has a solution in Dmxp. Define tmin := (s - a)ep,epmt2, Tmin as the saturated monoid of all divisors of powers of tmin and Smin := DTmin n F(s)pr. Then Algorithms 7.1 and 7.2 furnish a solution X0 e S^"^ of (26) which gives rise to the polynomial g from (27).

2. Let T be any saturated monoid containing (s - a). Then B1 admits a model matching T-compensator if and only if (26) has a solution in F (s)mrxp, the polynomial g from item 1 is non-zero, and Tmin c T. In particular, Tmin is the least saturated monoid T which contains (s - a) with a model matching T-compensator.

3. Assume that (26) has a solution in F(s)mrxp, the polynomialg from item 1 is non-zero, t3 is any multiple of tmin and T3 is the saturated monoid of all divisors of powers oft3 and hence T3 2 Tmin. Then there are T3-model matching compensators of B1 for the model Bm and all these can be constructed via Theorem 5.4 applied to T3 withX0 and g from item 1.

4. If in item 3 F = R and D = R[s], then all poles of feedback behaviors constructed with the T3-model matching compensators are zeros oft3.

Proof. The proof proceeds parallel to that of Theorem 4.11. □

6. Decoupling

We consider IO behaviors B\ and B2 as in Assumption 4.1.

Definition 6.1 (Decoupling T-compensators). The compensator B2 is called a decoupling T-compensator of B1 if HyuU2 is diagonal. Recall Hy1}U2 = N1Q2 from Theorem 3.14.

Obviously, there exists a decoupling T-compensator B2 of a given T-stabilizable behavior B1 if and only if there is a matrix X e Smxp such that nQ - N1XP1 is a diagonal matrix and (-Q2, P2) :=

(-Q0, P0) + X (P1, — Q1) has the correct IO structure and proper transfer matrix, i.e., det (va (P2) =

0. Diagonality of NQ0 — N1XP1 signifies that NQ0 — N1XP1 = diag(Z1,..., Zp) for some

(Z1,...,Zp) e v\xp.

The problem of decoupling can hence be treated completely along the lines of the theory on tracking and disturbance rejection displayed in Section 4: substitute Eq. (17) by

N1Q:20 = N1XP1 + diag(Z), (28)

where X e Smxp and Z = (Z1,..., Zp) e v\xp. Since (28) is again aninhomogeneous linear equation in the entries of X and Z1,...,Zp, the existence of solutions (X, Z) e 28

be checked and one such solution (X0, Z0) can be computed by means of Algorithms 7.1 and 7.2. Analogously to the derivations in Section 4, a parametrization of all X e Smxp satisfying (28) for some diag(Z1,...,Zp) e Dpxp can be obtained, leading to the appropriate definition of a polynomial g(S) e F [S ], S = (S1,..,SV).

The characterization of the existence ofT-tracking resp. T-rejecting compensators in Theorem 4.8 and the constructive parametrization of all such compensators in Theorem 4.9 hold mutatis mutandis for the case of decoupling T-compensators. Also the results on pole placement remain valid.

7. Algorithms Algorithm 7.1.

1. We cite from [4, Algorithm 3.1] (cf. also [27, p. 152, Lemma 4]): Let Rbea principal ideal domain with quot(R) =: K and let A e Kaxd, M e Kcxd. The following algorithm determines whether there exists a matrix X e Rcxa such thatXA = M and, if this is the case, parametrizes all such matrices. Let

If 0\ / e1 0 \

I I = UAV, F = I .. I , r = rank (A),

V00/ V0 . J

be the Smith form of A with respect to R. Then the following equivalences hold:

3 X e Rcxa : XA = M ^3X e Rcxa : XU—1 UAV = MV

=:X /F 0N =:M V00/

^3X e Rcxa : X | 0\ = M \0 0

&3X e Rcxa : Mij =

Xi,e,' for 1 < j < r, 1 j 1 < i < c.

0 for r < j ^ d, &Vi e{1,..., cjVj e{1,...,r}: MAj e R, V i e{1,...,c}Vj e{r + 1,...,d}: Mij = 0. If this is the case, define X1 e Rcxa by

ie- 1 for 1 < j < r, 1 0 for r < j ^ a,

1 J ' " 1 < i < c.

ThenX1 := X1U e Rcxa satisfies

X1A = M and {X e Rcxa; XA = m} = X 1 + Rcx(a—r)U

where U =: ( U2 j e R(r+(a~r))xa, i.e., U2 is a universal left annihilator of A. 2. In part 1 consider D = F [s] c K = F (s) and the Smith form of A w.r.t. D. Assume thatXA = M has a solution in Kcxa, i.e.,

My = 0 for 1 < i < c, r < j ^ d. For 1 ^ i ^ c, 1 ^ j ^ r write

Myef1 = — e F(s), fij, gij e F[s], gcdf, gij) = 1, and define t2 := lcm^ gj. gij

Then there is a solution ofXA = M in Dc2xa, and the saturated monoid T2 of all divisors of powers of (s — a)t2 (compare (13)) is the least saturated monoid T containing s — a for which there is a solution X e DTxa ofXA = M.

Proof. Compare Algorithm 3.1 in [4]. □

Algorithm 7.2. Consider the ring DT for a multiplicatively closed saturated setT c d \ {0}. Assume that T contains an element of the form (s — a) for some a e F, define a := (s — a)—1 and D := F[a]. Let A e F(s)axd, B e F(s)bxd, M e F(s)cxd and assume that the equation

XA + ZB = M (29)

hasasolution (X1, Z^ e DcTx(a+b). Then, by Algorithm 7.1, the set of all solutions (X, Z) e DcTx(a+b) of (29) is given by (X1, Z1) +DcTxs(C, D) where s := a + b — rank (B) and (C, D) e Dsx(a+b) denotes a universal left annihilator of (A) w.r.t. D and hence also w.r.t. DT. Hence,

{x e DcTxa; 3Z e DcTxb : (29)} = X1 + DcTxsC.

1. The existence ofY e Djxs such thatX1 + YC is proper, i.e., contained in Scxa = (DT n F(s)pr)cxa, can be checked as follows (compare [5, Corollaries 3.9-3.14], [4, Algorithm 3.2]): Let

If 0\ /e1 0 \

( I = UCV, F = I .. I , r = rank(C),

Vc . J

be the Smith form ofC with respect to D. From

U e Gls(D) c Gls(DT) and V e Gla(D) c Gla(S)

we conclude the following equivalences:

3Y e VcTxs with X1 + YC e Scxa

&3Y := YU—1 e Vc/s with X1V + YCV = X1V + Y (0g) e Scxa S + DTej for 1 < j < r

& (X1V)ije & (X1V)ij e S for 1 < i < c, r < j < a

1 < i < c S for r < j ^ a

where the last equivalence holds by Lemma 3.10 in [5] or Algorithm 7.3 below. Note that in the next to last line of item 1 in Algorithm 3.2 in [4] it is incorrectly asserted that (X 1V)ij = 0 instead of (X1 V)j e S for r < j.

If this condition is satisfied Algorithm 7.3 below yields representations

(X1 V)ij = X — Yjjej e S + DTej for 1 < i < c, 1 < j < r. Enlarge the set of entries X0 and Yjj to matrices X 0 e Scxa and Y e VcTxs by (X0)ij := (X1V)ij and Yij := 0 for 1 < i < c, r < j. Then

X° = X1V + Y (00) and X0 := X°V—1 = X1 + YC e Scxa with Y = YU e VcTxs.

2. Assume that (29) has a solution in Kcx(a+b),letT2 be constructed as in Algorithm 7.1.2,andlet(X1, Z1) be a solution of (29) in V^4^. Then (29) has also a solution (X, Z) e (VT2 n F(s)pr)cxa x V^b if and only if there exists a matrix Y in F (s)cxs such thatX1 + YC is proper, i.e., that in part 1 (X1V )y is proper for 1 ^ i ^ c,r < j ^ a. Moreover T2 is then also the least saturated monoid T containing

(\cxa c„b

VT n F(s)pJ x VT .

Algorithm 7.3. Letr e VT and 0 = e e V = F[a ]. The following algorithm yields x e S andy e VT such that r = x + ye e S + VTe.

Find representations r = Qp—1a —n e VT = Sa = (Vq)a, h e V,p e Q c V, T(0) = 0,n e N, and e = e1ae e V, e1 e V, e1(0) = 0,1 e N. Since gcd(an, e1) = 1 by construction, there exist a, b e V such that 1 = aan + be1. Hence

P—1a —n = aVt—1a —n+n + bVV—1a —ne

Definingx := apQ— 1 e Vp' = S andy := biht—1a —n—l e (Vf)a = VT yields the asserted representation ofr.

Algorithm 7.4. Assume a matrix R e Vkxe. The following algorithm determines a matrix R' e Vrxe such that paxkR n V1x£ = VixrR' and rank(R') = r. Localization w.r.t. T (with Vp = S, compare (5)) then yields

V1xkR n S1xi = S 1xrR'.

(0 0) = /RY be the Smith form of R w.r.t. V,

Iei 0 \ V

F = I .. I , r = rank(R), et e V, e1| ...\er,

V 0 . er)

r = 1 ■ Qt—1a —n = aHt—1a —n+n + bHt—1a —ne1 = aht—1 + bHt—1a —n—te e S + VTe.

ei =: e[ ■ aai, ai e N, e[ e V \ Va,

E := ^ •. j , R := (E, 0)Y~1. Then R' e VrxX and VixkR n V1xi = V1xrR'.

Proof. We study the modules V\xkR and V1xrR':

VlykR = vaxk (00) Y-1 = 0Vae(Y-1)i- = (Vaaaie[(Y-1)i- = (Vae[(Y-1)-,

i= 1 i=1 i=1

V1xrR = V1xr(e>, 0)y-1 = Ve[(Y-1)-.

It is obvious that VlxrR' ç V^xkR n V

lxrp/ c V1xkR Pi

On the other hand, let £ = YJi=1 me/(Y-1)i- e vlxkR P V1xe, ni e Va. Since £ e V1xe and Y e Gle(V),wededucethat£Y e V1xl,£Y = Zr=1 me'fii where Si e F1xe, (Si)j := 8ij.Consequently nie/ e V for 1 < i < r, and hence n e Va PV ^ C Va PVv\va = V since e/ e V\Va by construction. It follows that £ e V1xkR/. □

8. Example

We conclude this paper with an example illustrating the application of the techniques described in the preceding sections.

Example 8.1. We consider the continuous standard case, i.e., F = R, V = R[s], F = COT(R, R) with T := [t e R[s]; VC(t) c A} where A := [X e C; ft(x) < 0}, i.e., a signal in F is T-small if and only if it is polynomial exponential and converges to zero for t tending to infinity. Assume the behaviors B1 := {(l\) e F1+2; P1 oy1 = Q o u}, Bt := {w e F1; Rt o w = o}, and Bd :=

{w e F1; Rd o w = o} where

P1 := (s2 - 4) , Q1 :=(-(s + 2)2, -s2(s + 2)), Rt := (s) , Rd := (s2 + 1) .

Our goal is to construct a T-stabilizing compensator that simultaneously T-tracks signals u2 e Bt and T-rejects signals u2 e Bd. This signifies that any constant signal is admissible as tracking signal whereas any signal of the form u2 (t) = a sin(t) + b cos(t) (for arbitrary constants a and b) is considered as disturbance and shall not have any significant influence onto the output y 1 of the full feedback behavior. Of course we require both the feedback behavior and the compensator to be proper.

Note that B1 is not T-stable since the greatest elementary divisor (s2 — 4) of P1 has zeroes 2 and -2 and is consequently not contained in T. B1 is also not controllable, but T-stabilizable since the greatest elementary divisor of R1 = (P1, —Q1) is (s + 2) with zero —2 e A. Note moreover that the transfer matrix

H = p—1Q1 = (—&^ )

is not proper. It is convenient to choose a := —2 for the definition of a := (s — a)-1 and V := R[a ]. Then

H = ( _L_ (2a —1)2 )

1 V 4a — 1' a(4a — 1) / '

Now we can construct the matrices introduced in Section 3, Assumption 3.6: a left coprime factorization H1 = P—1" of H1 over V can be obtained via the controllable realization (Pj, — Ch) of H1 over V, i.e., via the computation of a universal left annihilator on ^ ), compare, e.g., [5, Res. 2.10, Definition and

Lemma 2.7]. A right coprime factorization H1 = N1D—1 over V can be computed in a similar fashion. In our case we get:

N1 = (—16a2 + 4a) , pi = (—4a, —16a2 + 16a — 4

_ / N _ /—4a2 + 4a — 1 —4a + 1\

N = (0'—10 • D1 = ( a 0 )

Aleft inverse p = (—N® N?) e V2x(1+2) of ( N| ) computed by Algorithm 7.1 is

n=(—o1 —4ao+4), no={"a+j

According to Corollary 4.12 we have to solve the equation

(N|?.+/"") = (N') XN + (Z 0 )(*) . (30)

V p1P20 / W \o w

This can be achieved by means of Algorithm 7.1 after rearranging the entries of the occurring matrices, compare the considerations after Remark 4.7. Here we obtain

(X e V2xA; 3Zt e v\xA, 3Zd e v\xA : (30)) = X0 + £VTXh,i

with X0 = ( s(278s — 29) ) = | 2 0 | e 52x1,

V 20(s + 2)4 / I® (2a — 1)(585a — 278)J

» = 2, Xhi = (M , ^ = ( s(AH) = ( 0 ^

\o/ V (s + 2?) \-(2a - 1)(5a2 - 4a + 1),

with the notations from after Remark 4.7. The smallest saturated monoid T2 such that (30) has solutions (X, Zt, Zd) with entries in DT2 according to Algorithm 7.1.2 is the saturated monoid

T2 := [p(s + 2)k; p e F \{0}, k > o)

generated by t2 := (s + 2).

It is easily seen that Algorithm 7.4 yields B(j) := Xh,i e V2xA for j = 1, 2. Hence, we get

(X e S2x1; 3Zt e V¡x\ 3Zd e v\xA : (30)) = X0 + £SB^.

For X = X0 + Zj2=1 VjB() with nj e S, the matrix (-02, p2) = (-Ô20, + X (Â, -Ô1)

gives rise to a proper T-stabilizing compensator if and only if it has the right 1O structure and proper transfer matrix, i.e., det(va(P2)) = 0. Writing nj =■ Hj + aZj, Hj € F, Zj € 5, and defining the polynomial g(3) from (24),

0 ■= det (pO(0) -

VaX) + £ 3jB^)(0) j=i

Qi(0) I = -4S2 € F[Si, 32],

this is the case iff g(Hi,H2) = 0. This signifies that H2 = 0. We could just choose ni ■= 0, n2 ■= 1. However, the resulting compensator gets simpler if we make an ansatz for ni, n2 as polynomials in F[a] = V c 5 with indetermined coefficients and assign the coefficients such that the degrees (w.r.t. a) of the entries of (—Q2, P2) get as low as possible. One possible solution is the following:

ni ■= ° n2 :=— I — §a. This yields

x = x0 + V njBj = I 0 I

=i 2, (2a — 1)(35a — 38)/ and, by (—Q2, P2) ■= (—021, °2) + X (Pi, —Qi),

Q / —i —4a + 4 \

P2 = ^ — a5(2a — i)(35a — 38) — 5(2a — i)3(35a — 38) J ,

Q _ / —4a + i \

0 = I — 5(56a2 — 58a — 5)(5a2 — 4a + iW '

It can easily be checked by means of Algorithm 7.i that there really exist Zt resp. Zd in vjx i satisfying QiO2 + idi = ZtRt resp. QiO2 = ZdRd.

Now we compute the transfer matrix H2 = P—i Q2:

' 280a3 — 5i4a2 + 283a — 58 \ / 58s3 + 65s2 + 78s + 80 1

(2a — i)(35a — 38) I = ( s(s + 2)(38s + 4i)

__5__5(s + 2)2

, (2a — i)(35a — 38) ) \ — s(38s + 4i) .

Note that H2 is by construction proper. A left coprime factorization H2 = P2 iQ2 over V = F[s] is given

P2 = ( —s — 2 4s + ^ , Q2 = ( s — 2 | -38s2 — 79s — 6 i2 / y58s2 + 7s + 74/

The matrices obtained from the algorithm described above are somewhat more complicated, some elementary row operations yield the (row equivalent) matrices stated here.

The behavior B2 = J (U ) € f'1+2; P2 o y2 = Q2 o u2} is by construction a T-stabilizing compensator of BithatT-trackssignalsu2 € Bt andT-rejectssignalsu2 € Bd.Moreover,bothB2andfb(Bi, B2) are proper, i.e., have proper transfer matrices.

We can check these properties by computing the feedback behavior fb(Bi, B2) and the error behaviors

Berr,t ■=

yi + u2 € Fp; yi € Fp, u2 € Bt, 3y2 € Fm ■ € fb(Bi, B2)

resp. Berr,d ■=

yi € Fp; 3u2 € Bd 3y2 € Fm ■ £ € fb(Bi, B2)

describing the deviation of —y\ from the tracking signal u2 e Bt resp. the error y\ caused by a disturbance input u2 e B'. These computations are not carried out in detail here, but we find that fb(Bi, B2) is really proper and T-stable and Berr,t and Berr,d are really T-autonomous. More precisely:

ch(fb(B1, B2)0) = { — 2}, ch(Berr,t) = {—2}, and ch(Berr,d) = { — 2} in accordance with Theorem 4.11.

Acknowledgements

We thank the reviewers for their efforts, questions and suggestions which have influenced the revised version of the paper.

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