OCR of paper; see pdf file noether-realiz.pdf for viewwable/printable page Reprinted from JOURNAL OF COMPUTER AND SYSTEM SCIENCES Vol. 18, No. 1. February 1979 All Rights Reserved by Academic Press. New York 4nd London PrinUd in Be4euvv On the Existence of Minimal Realizations of Linear Dynamical Systems over Noetherian Integral Dornains* YVEs RoUCIIALEAU Centre de Alath~niatiques Appliqu&s, E.N.S. Mines de Paris, F-06560 Valbonne, France AND EDUARDO D. SONTAG Rutgers University, Department of Math enza tics, New Brunswick, New Jersey 08903 Received June 15, 1977; revised March 22, 1978 This paper studies the problem of obtaining minimal realizations of linear input/output maps defined over rings. In particular, it is shown that, contrary to the case of systems over fields, it is in general impossible to obtain realizations whose dimiension equals the rank of the Hankel matrix. A characterization is given of those (Noetherian) rings over which realizations of such dimensions can he always obtained, and the result is applied to delay- differential systems. A. INTRODUCTION A linear, discrete-time, constant, dynamical system Z over an integral domain R is defined by giving a finitely generated torsionfree R-module X (the state module) and a triplet of R-honiomorphisms (F, G, H), where F: X --* X, G: R", --)- X, H: X -->- R-. We call the free R-module R- the input module, R" the output module, and write the equations of the system xt+l = Fx, + Gut t C Z' yt+l ~ Hxt+l , where ut (the input at time t) belongs to R"', x, (the state at time t) to X and y, (the output at time 0 to RP. It follows from the linearity of these equations that the relation they induce between * This research was supported in part by U.S. Army Research Grant DAH C04-74-G-01 53 and U.S. Air Force Grant AFOSR72-2268 through the Center for Mathematical System Theory, University of Florida, Gainesville, Fla. 32611. 65 0022-OOW/79/010065-11$02.00/0 Copyright Cc) 1979 by Academic Press, Inc. All rights of reproduction in any form reserved. <> 66 ROUCHALFAU AND SONTAG inputs and outputs is completely characterized by the infinite sequence of p x m R-matrices S - (A, , A, _.) (the inputloutput sequence of the system) where A, is the matrix representation of the R-homomorphism UP IG: R111 ~ RP in the standard bases of R and RP. Conversely, given a sequence of p - "i R-matrices S - (A, , A, _.), the realization problem consists in finding a finitely generated torsion-free R-module X and three R-homomorphisms (F. G, II) as above such that A, --- HF'--'G, for all t ---- 0. Suppose that X can be generated as an R-module bv n elements; then we can represent the homomorphisms F, G, 11 (not necessarily uniquel_y) by R-matrices with respect to the standard bases of R and RP and the set of generators. We shall from now on not make any distinction between the homomorphisms and their chosen matrix representations. If r is the smallest cardinalitv for a set of generators of X, we shall call r the dimension of the svstem over R. When R is a field, the realization problem is completely solved. It is shown in Kalman, Falb, and Arbib [5] that an input/output sequence can be realized by a fi nite- dimensional svsteni Iff its associated behavior matrix (~l -42 A3 ... B '42 -4:3 ~14 has finite rank n, that n is the minimal dimension for a realization of the sequence, and that a system realizing the sequence has minimal dimension iff it is canonical, i.e. both reachable (the map (G, FG,..., Fn 1G): Rnm --->- X is onto) and observable (the map (H', FH ...... (F)1~ IIF)': X ~ R" is one-to-one). An algorithm is also given to construct such a minimal realization. When R is not a field, it was first shown in Rouchalcau, Wyman, and Kalman [10] that, under fairly general conditions on R, the criterion for the existence of a realization is exactIv the same as above, namely that the behavior matrix have finite rank (an up-to-date summary of these existence results is given in Section B of this paper). That paper did not consider the question of the minimal dimension of realizations. Such a concept of course is easv to define: a realizable input!output sequence S (which therefore has a behavior matrix of finite rank) can be realized by linear systems of finite dimension; a minimal realization of S over R will be one, the dimension of which is smaller than that of any other realization of S. It, of course, always exists but is not equivalent any more to the notion of canonical realization; its dimension may be larger than the rank of the behavior matrix, and is not in general easily determined from inputloutput data. The purpose of the present paper will be to study a more restrictive and stronger version of minimality. Instead of asking, as in the a aformentioned paper (Rouchaleau, Wyman, and Kalman [10]): "When does an R-Input~output sequence S realizable over the quotient field K of R also have a realization over R ?", we shall ask: "When does S have an R-realization which has the same dimension as a minimal realization over K;". Since R is assumed to be an integral domain, we may consider its quotient field K. To a system (X, F, G, H) over R we may associate a system (X OX R K, F OR K, G OR K, <> LINEAR DYNAMICAL SYSTEMS OVER NOETHERIAN INTEGRAL DOMAINS 67 H K) over K which has the same input Toutput sequence of R-matrices. Furthermore, It is clear that if the system over R is canonical, then so is the associated system over K. Since the R-sequence S -- (Al , A, ...) is a fortiori a K-sequence, we can find a realization for it over K-1 the system (.Y ';)R K, F (1)1? K, G xll? K, 11 (X,), K) is an example of such a realization. To determine a minimal realization for the R-sequence S over K is a solved problem. We are thus led to the follov,-mg (1.1) DEFINITION. A realization (-V, F, G, 11) of a sequence S over R is called absolutely minimal iff its dimension is the same as that of a minimal realization of S over the quotient field K. (1.2) Remark. This definition is equivalent to requesting that the system over K defined by the matrices F, G, H be canonical. An absolutclv minimal realization remains minimal under any ring extension of R. (1.3) LEMMA. The state module X of an absolutely minimal systein Z is free. Z is observable and weakly reachable (i.e., rank, (G,FG_.,1,'11 1G) ~ n, dimension of the system), and converseIV an observable and weahlY reachable system i's absolutelv minimal. Proof. The n generators of X as an R-module are also generators of X R K as a K-vector space. If tbey are not linearly independent, the dimension of X (111? K is less than n, contradicting minimality over K. If the svstern were not observable, there VMUld be a state x -/ 0 in X such that x'(J1Y1l' - - - (F') " - 1 HT - 0; but this would a fortiori mean that there is a state x / 0 in X r`~,)~ K which is unobservable for the system (X Q,,, K, F, G, H) over K, contradicting its canonicity. The proof of the converse is just as trivial. I The aim of this paper is to characterize those rings R over which any realizable mput/ output sequence can have an absolutely minimal realization. The interest of such a characterization is two-fold. First, it will tell Lis exactly when we do not lose anything (from the point of View of dimension) by realizing an input,foutput sequence over the ring R rather than over an overfield of R. Second, one of the motivations for studying SN-stems over rings is their use in modeling delay-differential systems (c.f. Kamen [6]). In this case the rings under consideration are polynomial rings; a sufficient condition for the point-,vise controllability of delay- di fferen tial systems is that the associated ring model be veakly reachable (see Sontag [12, Section 5]). Furthermore (as pointed out to the authors by E. Kamen) only for absolutely minimal realizations can internal stability be deduced from external (bounded inputlbounded Output) stability. So it is important to know the polynomial rings over which this condition is ahAavs true. After having reviewed in Section B of this paper the conditions under which a realiza- tion exists, Nve shall study the problem of ahS01LJtelV minimal realizations. We shall see in c ection C that, for single input systems, canonical realizations are absolutely minimal over S very general rings. Section D shows that for general multivariable systems this verv nice <> 68 ROUCHALFAU AND SONTAG property holds only over principal- I dea I domains; we then give an algorithill for con- structing such a canonical, absolutely minimal realization. In Section E, we shall give a condition guaranteeing the existence of absolutely minimal realizations; specializing the result to the case of polynomial rings, we shall find that only those in one or two variables sati-fv the condition. 13. SURVEY OF THE CONDITIONS UNDiR Nviucii REALIZATION-, ]EXIST OVER A RING The fact that (2.1) THEORP0. An inputloutput sequence S of inatrices over a Noetherian domain R is realizable over R iff it is realizable over the quotient field K of R. was first established by Rouchalcau, Wvman, and Kalman [10]. To make the paper self contained, we shall now giN,e a simple proof of this result. It is well-known (see, for instance, Sontag [12, Lemma (3.1)1) that an input,routput sequence S is realizable (whether over a field or over a ring) iff the (infinite) columns of its beha6or Triatrix B ~1, A:, -44 ... mav he written as linear combinations of a finite subset of columns; in other words, iff the columns of B generate a finitelv generated module X. We can then obtain a canonical realization as follows: take X as state module; consider the shift operator on X defined by s k s ending each column of B to the column occupying the same position in the next bloc col Unin ~ it extends to a well defined module endomorphism F of X because of the Hankel pattern of B-, define a linear transformation G: R- --). X bv mapping thejth standard basis vector of R into thejth clementary column of B; finally, define H: X ~ RP bv taking as the Image of any column of B the vector composed of the first p elements of that column (in other ~~ ords, the intersection of the column with the first block row). Then (X, F, G, H) is a canonical realization of Let us now assume that the R-sequence,5 is realizable over K, and assume that vj Vn arc a set of basis columns for B over K. Then anv column v of B can be written as 11 v Y v, j(v) u K. 2-1 The cocfficients Ai(v) in this linear dependence can be obtained using Cramer's formulas -&Z-) JIM J, (v), J c R. j Both these determinants, composed bv additions and multiplications from elements of R, belong to R. <> LINEAR DYNAMICAL SYSTEMS OVER NOETHERIAN INTEGRAL DOMAINS 69 Dcfine no,,v u, - vj~'J. We have, for any column v of B, n v -- y Jj(7) iii, J i(v) e R, i=1 hence the R-module generated by the columns v of B is contained in the R-module generated by ii, _., u. . Since "Noetherian" is equivalent to "every submodule of a finitely generated module is finitely generated", the theorem follows. If -,vcre lax the Noetherian assumption, then we can use the following result of Cahen and Chabert [16]: (2.2) 141sui,,r. Let R be completely integrall -y closed and K its quotient field. Then an I.itputioutput sequence ~~ of R-matrice;is realizable over R whenever S is realizable over K. Furthermore, the monic recurrence relation of S of minimal degree over K has all of its coeffi- cients in R. Proof. See Eilenberg [3, Chap. XVI, Theorem 12.2]. But it has been shown (Rouchaleau, Kalman, and Wyman [10]) that an R-sequence is realizable over R whenevei it is realizable over the integral closure R of R. Hence we have: (2.3) RESULT. If the integral closure R? of a ring R is completely integrally closed, then an R-sequence i's realizable over R iff it is realizable over the quotient field K of R. A slightly less general result was proved by Rouchaleau and Wyman [11], using a generalization of classical stability theory. Extensions to reduced rings can be found in Rouchalcau [9]. C. SINGLE-1NPuT OR SINGLE-OuTt,irr SYSTEMS Let S be such that in - 1, p being an arbitrary finite integer. It is well known that the existence of a realization is linked to that of a monic recurrence relation between the elements of .;. We have precisely. (3.1) LFm,-vlA. If an input/output sequence S with in -- I satisfies a monic recurrence relation over R of degree n, then S has a realization of dimension n over R. Proof. Assume that aAk A,Ak ~n-l '-Ik-,n 0 for all k > 0, with ii c- R for all 1'. Then the R-matrices 0 0 0 ... 0 1 0 0 ... 0 0 0 1 0 ... 0 0 F G 11 ~ (A, -4, A (0 0 0 ... I - (0) 0 0 0 ... 0 0 together with the state module Rn constitute a realization of size n. <> 70 ROUCHALEAU AND SONTAG (3.2) LEMMA' ' If the domain R i's integrally closed and the inputloutput sequence S is realizable over R, then its monic recurrence polynomial of minimal degree over the quotient field of R has all of its coefficients in R. Proof. Since S is realizable over R, it satisfies a monic recurrence relation with coefficients in R (given, for example, by the characteristic polynomial h(z) of F in one of its realizations). If we now view S as a sequence over K, the set of recurrence polynomials of S is an ideal j in K[z] (nonempty, since j contains h(z)). This ideal is principal, hence has a monic gencratorf(z), the monic recurrence polynomial of minimal degree of S over K. Thus we have: h(z) = g(z) f (z), h(z) monic in R[z], f [z] monic in K[z]. This is the exact setup of Zarlski and Samuel [Vol. 1, Chap. V, Sect. 3, Theorem 51, and it follows from the integral closure of R that f (z) is in R[z]. I Assume then that our input/output sequence S has a minimal realization over K of dimension n and that S is realizable over R. This means that the associated behavior matrix B(S) has rank n (see for example Kalman, Falb, and Arbib [5, Chap. 10]), hence that the first n + I columns of B(S) are linearly dependent. Then there is a recurrence relation of degree n over K between the elements (in the present case, vectors) of the input/output sequence. If R is integrally closed, it follows from (3.2) that, there is a monic recurrence relation of degree > LINEAR DYNAMICAL SYSTEMS OVER NOETHERIAN INTEGRAL DOMAINS 71 Then the input/output sequence A, - - (A g) - - A, -- - - - is realized minimally over R by F ~ 1, G - (a 0), H ~ 1. If we take R as a state module, then this absolutely minimal realization is not reachable since J is proper. A canonical realization would have a state module isomorphic to J (which is non principal) hence its dimension would be 2, and it would not be absolutely minimal. The dual of such a realization, on the other hand, is canonical. We have here an example of a "strongly observable" system (see Sontag [14]), as well as a breakdown of the fact that the dual of a canonical system over K field is canonical. (3.5) Remark. If we assume that R is not Just integrally closed but even completely integrally closed (for example, if it is integrally closed and Noetherian) then we need assume only in (3.3) and (3.4) that the input,~output sequence S has a minimal realization of dimension n over K. That it is R-realizable follow directly from (2.2). Let us now consider the general mulfivariable case. D. WHEN ARF CANONwAi. RFALIZATIONS ALso R-'\IININTAI,? The answer to this question for Multi-input, multi-output systems is very simple. (4.1) PROPOSITION. The canonical realization of ever -v inputioutput sequence S over a Noetherian domain R is absolutelY minimal if and on/Y if R i's a principal-ideal domain. Proof. Sufficiency. Let X be the state module of a canonical realization (X, F, G, H) of the sequence S; it is by definition finitely generated and torsion free. Since R is a princi- pal ideal domain, X is a free module. Let dim X n. Consider the associated K-system, (X K, F (~~)R K, G (~X,'~R K, H (/), K). Its state space X CX R K has the same dimension n (as a K-vector space) as X (as an R-module) since X is free. As was pointed out in the introduction, this associated K-system is canonical since the original R-system was. So n is the dimension of a minimal realization of S over K. The system (X, F, G, H) is therefore necessarily an absolutely minimal realization of S. Necessity. Let us first establish the general fact, of interest by itself, that any finitely generated torsion-free module X may be the state module of a canonical svstern. Since X is finitely generated, with, say, in generators, there is a projection R"' U ~ X -- 0. Since X is finitely generated, torsion-free, and the rings we are considering are integral domains, there is also an injection, for some p, 0 - - ~ A'-', RP. (see Rotman [8, Theorem 4.21]). So (X,F, G, H), A,Itli X as a state module and with F - identify, G ~ u, H - - v, is a canonical system. It was pointed out in the Introduction (Lernma 1.3) that the state module of an <> 72 ROUCHALEAU AND SONTAG absolutely minimal realization is alwavs free. So if the state module of any canonical system is R-minimal, then any finitely generated, torsion free-module is free. But this, together with the Noetherian assumption, implies that R is a principal ideal domain. I We shall give an algorithm for constructing such a canonical realization. First we ascertain the rank of the behavior matrix B(S) (which can be done over any field containing R, using the rank condition of Kalman, Falb, and Arbib [5, Chap. 10, Condition 11.231). Then we find a nonsingular submatrix (1) of maximal rank, sav n, and a basis over R for B,,.,, (the submatrix of B(S) consisting of the first n block rows and columns of the behavior matrix). This can be done in the following way. (i) Let 1, be the submatrix of B,,,, containing the rows of op, and a, the L~reatest common divisor of the elements in the first row of B,,,,, . Call x1 the linear combination of the columns of L having a, as its leading coefficient. (ii) Subtract from every column x of L a multiple of x, a(x)x, (x(x) in R) such that the first clernent of x - cx(x)xi be 0. This is possible by definition of a, . We get a next matrix Li with zero top row, and such that its columns together with x, still generate the colunins of L. (iii) We apply the same procedure to the second row of L, , obtaining x, and L, , etc... . At the end of the process, we shall have a basis made up of vectors (x . ..... X.) - Let F be the n x m submatrix of L having its columns in the first block columns and A the p 'X n submatrix of B,, corresponding to the columns of 0 and the first block row. Thcri NNc can write out a realization (theso-called Silverman realization): F - (X . ..... NJ (70) 0- I(x . ..... X,,), G (x1 x,J 'F, H - /l (P I (v . ..... X,J, \%here cy designates the shift operator. The matrix (X, I ... I x,,), beings lower triangular, is easy to invert. As to 0, its inverse is a byproduct of the determination of the rank of the behavior matrix. E. NVHPN CAN WE GUARANTEE THE EXISTENCE OF AN ABSOLUTELY MINIMM. REALIZATION? We have just seen that we cannot expect every canonical multivariable svstern to be also absolutely minimal, unless the ring is a principal ideal domain. If, however, we are willing to consider absolutely minimal systems which may not be canonical, then we can guarantee their existence over more general rings. These are given exactly by the following (5.1) THEOREM. EverY realizable input!" output sequence S over a Noetherian ring R has an absoluteh, minitnal realization iff everY finitelY generated reflexive module over R is free. Proof. lVecessitY. Let X be the state module of a canonical realization of ~; over R and <> LINEAR DYNAMICAL SYSTEMS OVER NOETHERIAN INTE.GRAL DOMAINS 73 M t lie state module of an absolutely minimal realization of '~. By Leninia (1.3) an absolu- tely inininial realization is always observable, hence Zelgcr's lemma (see Kalman, Falb, an I I , d Arbib [5, Chap. 10. Lemma 6.2]) Implies the existence of an injection X , W. Also, bv definition of a minimal realization, 11 '~)R K is the state space of a canonical realization of .'~ ox er K. So 11 K is isomorphic with X I , K. Thus we have, up to isomorphism, X C 31 ~- X , , K, where M is free (Leturna (1.3)). We are exactly in the situation described by Bourbaki [2, Sect. 4, No. 1, Corollaire de la Proposition 1], and both ]?-modules X and .1I are "r6seaux" of the K-vector space X R K, .1I being furthermore free and containing X. As is i )inted out in the above rcf'cr(-n((, (Proposition 3, (IN') and Remark 3), Hom,(.11, R) C lJorn,(X, R), that 1s, let A he a finitely generated, torsion-frec module over R. Its dual X- is finitely generated ,,Iiicc R is Noctherian. Ixt ~,u . ..... u,,; be generators of X'. We may construct 'I calmnical ~,-~stcin Zoo , -'Nith state module Xas follows. We choose the number of inputs 11 and the niatrl.\ G as in the proof of (4.1), take for F the identlt%- matrix and dcfine 11: X - ~ R" in such a way that Its p rows are gi~ en b~ the functions [u . ..... UJ- Suppose the system Z, has an absolutely minimal rtalization .17, - (31, P, G, 1Y). By the first paragraph of the proof, XC 11. So the inap 1/: A'- R1, extends to a inap fl: J I - - l?". Since in the system .1',, the ro,~vs of 11 generate X, the generators of X' extend to lincar inaps 11 - R. So X' C 11'. It follo-,vs that M' A". Consequentl\ if A I's -Cflcxl\cl then X M and so X is free. Nlofficlenr.i% Assume that (X, F, G, II) is a canonical system. X is therefore of finite tvpc, hence a 'Wseau" of X (1)R K (see Bourbaki [2, Sect. 4, No. 1, Proposition 11). It follo\N s that X` and X- ' are reflexive (Bourbaki [2, Sect. 4, No. 2, comments following Th~OFCTIIC 11). SA) .\'and X* * have the saine dual, and the map H: X -, RP may be viewed as a inap If- ': X*'* --, R". The map I,': X - X canonically induces a map F*: X*- - V-". N\ hich in turn canonicallv induces F* ': X- X". Hence a system (X, F, G, H) canoincall.\ induces a system .1" - (X*-, F", G, [I**) having the same inputloutput inap. Since X" is reflexive, is free hv assumption. But -V'- is a free "r6scau" of X K (13ourbaki [2, Sect. 4, No. 2, COTrllllCntS preceeding Theorem 11), hence dim, X- dlul,(A', , K) (BOUrbaki [2, Sect. 4, No. 1, ExaMple 2]). X K being the state space of a canonical realization over K, !:' ' is an absolutelv minimal realization. (5.2) Remark. Absolutcly minimal realizations are not necessarily unique. In fact, thev areasubclass of the lattice of minimal-rank realizations, studied in Sontag [13]. When R is a pi incipal-deal domain, this subclass coincides with the entire lattice. NVe have thus obtained an abstract characterization of those rings over which absolutely mininial realizations alwaNs exist. We ,,hall now show that among rings ofpolvnomials over a field onli, those in one or two indeterminates meet the condition of Theorem (5. 1). (The case of one indeterminate has already been treated in the previous section). (5.3) LEM-MA. Ever_v finitely, generated refleviTe R-module is projective ~ff the global d7mension of R is inferior or equal to 2. <> 74 ROUCHALEAV AND SONTAG Proof. This result - - - due to Bass - - - may be found in Faith [4]. (5.4) LEMMA. If R is a ring of poij,nomials in two unknowns over a field, then everv projective module of finite t3pe over R isfiree. Proof. See Bass [1, Part 11, Chap. 4, Sect. 6]. Hilbert's theorem on syzygics implies that the global dimension of a polynomial ring in n unknowns is n (see for example, Kaplanski [7, Part 111, Theorem 7]). This and the last two lemmas show that our claim is true. Observation. A counterexample for the case of polynomials in three variables over a field K (R K[x, Y, z]) is given bv the following inputloutput map with m - 3, 3. 0 0 __ (0 -0), Z -4, - ~43 0 0 0 0 X, 0 0 0) Although rank B - 2, there exists no R-realization of dimension 2. Indeed, the canonical state module X is isomorphic to the column space of .4, , and this module can be proven to be reflexive but not free. In view of (5.3), the general problern of deciding if a given R satisfies the condition of (5.1) breaks down into the subproblems: (i) determine if global dim R ~~ 2 (easy) and (ii) decide if finitely generated projectives over R are necessarily free. This latter problem is very difficult, but is currently an important research area in commutative algebra (viz. "Serre's conjecture", etc.). Returning to the case of delay- dl ffe rential systems mentioned in the Introduction, we deduce from (5.1), (5.2) and (5.4) that only up to two rationally independent delays may be in general allowed if realizations of the "right" dimensions are to exist. F. CONCLUSION lVe have shown under exactly what conditions we can realize an Input/output sequence over a ring with matrices over the same ring without losing any of the nice properties guaranteed by classical realization theorv over a field. Admittedly, the class of rings thus characterized I- rather narrow; however, it does contain, besides principal-ideal domains, polynOTIlial rings in two indeterminates (which have applications in the theorv of linear delay-differential systems studied by Kamen [6]). It is possible to give an upper bound on the increase in size due to the choice of a canonical realization (see Swan [15]). In particular, over a Dedekind ring, it can be shown that this bound is equal to I (Bourbaki [2, Chap. 7, Sect. 4, No. 9, Theorem 6]). In fact, the first example of a canonical, vet nonminimal system was given to the authors by Professor B. F. Wyman using a Dedeking ring. <> LINEAR DYNAMICAL SY~,TEMS OVER NOETHERIAN INTEGRAL DOMAINS 75 ACKNOWLEDGMENT The ;lUtbors are very grateful to Professor R. E. Kalman for his numerous suggestions and comments during the writing of this paper. REFERENCES I .11. BASS, "Algebraic K-Thcory," Benjamin, New York, 1968. 2. N. BOURBAKI, "Alg~hre Commutative," Chap. 7, Diviseurs, Herm~,nn, Paris, 1965. 3. S. "Automata, Languages, and Machines," Vol. A, Academic Press, New York, 1974. 4. C. ]"All If, "Rings, Modules, and Categories," Springer-Verlag, New York/Berlin, 1973. 5. R. E. KAL-MAN, 1'. L. FALB, AND M. A. ARBIB, "Topics in Mathematical System Theory," McG I a\% -Hill, New York, 1969. 6. E- W. KANIEN, An algebraic theory of systems defined by convolution operators, j. Math. S1,stein Theo?jy 9 (1975), 57-74. 1. KAPLANSKI, "Fields and Rings," Univ. of Chicago Press, Chicago, 1969. 8. J. J. Ro'r',IAN, "Notes on Homological Algebra," Van Nostrand-Reinhold, Princeton, N.J., 1970. 9. Y. ROUCHALEAIT, "Linear, discrete-time, finite-Llimensional, dynamical systems over some classes Of Commutative rings," Ph.D. Thebis, Stanford University, 1972. 10. Y. ROUCHALFAl-, B. F. WYNIAN, AND R. E. KAL.MAN, Algebraic structure of linear dynamical systems. 111. Realization theory over a commutative ring, Proc. Nat. Acad. Sci. USA 69 (1972), 2404-2406. 11. Y. ROU( HALEAU AND B. F. WYMAN, Linear dynamical systems over integral domains, J. Comput. System Sci. 9 (1974), 129-142. 12. E. 1). SONTAG, Linear systems over commutative rings: A survey, Ricerche Automatica 7 (1976), 1-34. 13. E. 1). SON FAG, The lattice of minimal realizations of response maps over rings, Math. Systems Theory 11 (1977), 169-175. 14. E. 1). SONTAG, On split realizations of response maps over rings, Inform. Contr. 37(1978), 23-33. 15. R. G. SWAN, The number of generators of a module, Math. Z. 102 (1967), 318-322. 16. P. J. CAHEN AND J. L. CHABFRT, 1~16rnents quasi-entiers et extensions de Fatou, J. Algebra 36 (1975), 185-192. Printed by the St Catherine Press Ltd., Tempelhof 37, Bruges, Belgium.