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Review of linear systems over commutative rings by J.W. Brewer, J.W. Bunce, and F.S. Van Vleck
Review of Linear Systems over Commutative Rings by J. W. Brewer, J. W. Bunce, and F. S. Van Vleck* Gert NaudC National Research Institute fm Mathematical Sciences Council forScientijIc and Industrial Research, P.O. Box 395, Pretoria 0001, Republic of South Africa
Submitted by George W. Barker
During the past thirty years system theory has developed into a scientific and engineering discipline that has an important impact on many aspects of modem society; ranging from engineering to economics and biology. Since the late 195Os, linear finite-dimensional continuous-time and discrete-time systems specified by state equations of the form i(t)
= k(t)+
Gu(t),
YW = HW
0)
and x(t+l)=Fx(t)+Gu(t),
y(t)
= Wt),
where F, G, and H are respectively n X n, n X m, and k X n matrices over the real numbers, have been studied-notably, in the early years, by R. E. Kalman and R. Bellman. It soon became clear that many important results depend only on the matrices (F, G, H) and the fact that the entries of the matrices involved are elements of a field. Thus, for many purposes one can regard a linear system as a triple of matrices (F, G, H), and Kalman was able to extend several aspects of linear system theory to systems over arbitrary fields. In the particular case of systems over finite fields, this provided a connection with linear sequential circuits and coding theory. The interested reader is referred to [4] for an exposition, In the early 1970s Kalman also initiated the study of linear systems over (commutative) rings, i.e., we now consider triples (F, G, H) of matrices with
*Marcel Dekker, New York, 1988, viii + 199 pp.
LINEAR ALGEBRA AND ITS APPLICATIONS 90:241-244 0 Elsevier Science Publishing Co., Inc., 1987 52 Vanderbilt Ave., New York, NY 10017
(1987)
241
0024-3795/87/$3.50
242
GERT NAUDh
entries in a commutative ring. These systems arise in a number of different ways, of which we discuss only a few. A more complete discussion of these matters can be found in Sontag [6] or Kamen [S]. Our first example is from
F31.
Consider an example of a system described by delay-differential tions:
$1(t) = zx,(t - 1)+ xi(t)+
x2(t)+
equa-
u(t),
a&>=x,(t-l)-3x&-S)+u(t-l), y(t) =x1(t)
- .Ta(t - 1).
If we define a delay operator e by
a(r)(t) = x(t - l), the above equations can be rewritten as
I:1=[ Xl
2uo+1
-;os][:;]+[:]u.
*2
y=(l-a)
r1 [ x2
1 .
These equations resemble those in [I], except that the matrices (F, G, H) have entries in the polynomial ring R [ a]. When there are incommensurable delays in the equations, one can introduce more than one delay operator er, *. *3a, and obtain a system over the ring Iw[ ul,. . . , a,]. Systems over rings also arise in the study of parametrized families of systems: these are (real or complex) systems (F,, G,, H,) where the matrices depend on a parameter x E X. Here X can be a topological, smooth, or analytic manifold. The study of such systems leads naturally to the study of systems over rings of continuous, smooth, or analytic functions on X. When studying families of systems, algebraic geometric ideas are involved. Expositions of various interesting results in this direction can be found in books by Tannenbaum [8] and Bymes and Martin 121, and in survey articles by Hazewinkel [3] and Bymes and Falb [ 11. Although linear systems over rings received some attention in [2] and [B], prior to the publication of the book under review, the best guides to the literature for the uninitiated were two survey papers by E. D. Sontag [6, 71
BOOK REVIEW
243
and a set of notes by E. Kamen [5]. Although still useful, these are all somewhat dated, and the appearance of the book by Professors Brewer, Bunce, and Van Vleck is therefore timely. The bock is aptly described on the cover jacket: Here is the first up-to-date selfcontained work to elucidate the statements and proofs of the fundamental results of linear systems over commutative rings. Emphasizing algebraic techniques, Linear Systems over Commutative Rings synthesizesin a unified
manner literature from variousmathematical engineering and control theory journals. The authors’ approach is to extend various constructions and results known in the case of fields to classes of rings, giving particular attention to those rings that are of interest from a systems point of view. The algebraic treatment of a concept is usually preceded by an explanation of how the concept arises in a systems context. The book is written in a leisurely style, and the demands on the reader are modest. Although the emphasis is on the algebraic aspects, sophisticated techniques from commutative algebra are avoided wherever possible, and an introductory chapter on commutative algebra helps to make the rest of the material accessible to nonalgebraists. Thus, the book has a wide audience, and it can be used for a graduate course in systems and control theory (possibly supplementing or following a more traditional course in control theory) or as part of a study of commutative algebra. The topics included are those that one would expect: a chapter on reachability and observability, another on feedback and pole assignability (including a clear treatment of the important concept of dynamic feedback), and a final chapter on realization theory. Each chapter is concluded with a set of exercises, ranging from the trivial and calculational to extensions of the theory. Some interesting results are omitted; for example, a result by C. I. Bymes that states that reachable systems over k[X,, . . . , X,], k an algebraically closed field, is coefficient assignable if the local Kronecker indices of the system are constant, is not mentioned. On the whole, however, the authors have succeeded in their aim of giving a unified treatment of most of the important results, and the book is a welcome addition to the literature on algebraic system theory. REFERENCES 1 2
C. I. Bymes and P. Falb, Applications of algebraic geometry in system theory, Amer. 1. Math. 101337-363 (1979). C. I. Byrnes and C. F. Martin (Eds.), Geometrical Methods for the Theory of Linear Systems, Proceedings of a NATO Advanced Study Institute and AMS
244
3 4 5 6 7 8
GERT NAUDF
Summer Seminar in Applied Mathematics held at Harvard Univ., Cambridge, Mass. 18-29 June 1979. M. Hazewinkel, A partial survey of the uses of algebraic geometry in systems and control theory, Sympos. Math. XXfV:245-292 (1979). R. Kahnan, P. Falb, and M. Arbib, Topics in Mathematical System Theory, McGraw-Hill, New York, 1969. E. Kamen, Lectures on Algebraic System Theory: Linear Systems over Rings, NASA Contractor Report 3016, 1978. E. D. Sontag, Linear systems over commutative rings: A survey, Richerche Automat. 7:1-34 (1976). E. D. Sontag, Linear systems over commutative rings, a (partial) updated survey, in Proceedings of lFAC/81, Kyoto, 1981. A. Tannenbaum, Invariance and System Theory: Algebraic and Geometric Aspects, Lecture Notes in Mathematics, Springer, 1981. Received 4 September 1986; revised 20 September 1986
Submitted by George W. Barker
During the past thirty years system theory has developed into a scientific and engineering discipline that has an important impact on many aspects of modem society; ranging from engineering to economics and biology. Since the late 195Os, linear finite-dimensional continuous-time and discrete-time systems specified by state equations of the form i(t)
= k(t)+
Gu(t),
YW = HW
0)
and x(t+l)=Fx(t)+Gu(t),
y(t)
= Wt),
where F, G, and H are respectively n X n, n X m, and k X n matrices over the real numbers, have been studied-notably, in the early years, by R. E. Kalman and R. Bellman. It soon became clear that many important results depend only on the matrices (F, G, H) and the fact that the entries of the matrices involved are elements of a field. Thus, for many purposes one can regard a linear system as a triple of matrices (F, G, H), and Kalman was able to extend several aspects of linear system theory to systems over arbitrary fields. In the particular case of systems over finite fields, this provided a connection with linear sequential circuits and coding theory. The interested reader is referred to [4] for an exposition, In the early 1970s Kalman also initiated the study of linear systems over (commutative) rings, i.e., we now consider triples (F, G, H) of matrices with
*Marcel Dekker, New York, 1988, viii + 199 pp.
LINEAR ALGEBRA AND ITS APPLICATIONS 90:241-244 0 Elsevier Science Publishing Co., Inc., 1987 52 Vanderbilt Ave., New York, NY 10017
(1987)
241
0024-3795/87/$3.50
242
GERT NAUDh
entries in a commutative ring. These systems arise in a number of different ways, of which we discuss only a few. A more complete discussion of these matters can be found in Sontag [6] or Kamen [S]. Our first example is from
F31.
Consider an example of a system described by delay-differential tions:
$1(t) = zx,(t - 1)+ xi(t)+
x2(t)+
equa-
u(t),
a&>=x,(t-l)-3x&-S)+u(t-l), y(t) =x1(t)
- .Ta(t - 1).
If we define a delay operator e by
a(r)(t) = x(t - l), the above equations can be rewritten as
I:1=[ Xl
2uo+1
-;os][:;]+[:]u.
*2
y=(l-a)
r1 [ x2
1 .
These equations resemble those in [I], except that the matrices (F, G, H) have entries in the polynomial ring R [ a]. When there are incommensurable delays in the equations, one can introduce more than one delay operator er, *. *3a, and obtain a system over the ring Iw[ ul,. . . , a,]. Systems over rings also arise in the study of parametrized families of systems: these are (real or complex) systems (F,, G,, H,) where the matrices depend on a parameter x E X. Here X can be a topological, smooth, or analytic manifold. The study of such systems leads naturally to the study of systems over rings of continuous, smooth, or analytic functions on X. When studying families of systems, algebraic geometric ideas are involved. Expositions of various interesting results in this direction can be found in books by Tannenbaum [8] and Bymes and Martin 121, and in survey articles by Hazewinkel [3] and Bymes and Falb [ 11. Although linear systems over rings received some attention in [2] and [B], prior to the publication of the book under review, the best guides to the literature for the uninitiated were two survey papers by E. D. Sontag [6, 71
BOOK REVIEW
243
and a set of notes by E. Kamen [5]. Although still useful, these are all somewhat dated, and the appearance of the book by Professors Brewer, Bunce, and Van Vleck is therefore timely. The bock is aptly described on the cover jacket: Here is the first up-to-date selfcontained work to elucidate the statements and proofs of the fundamental results of linear systems over commutative rings. Emphasizing algebraic techniques, Linear Systems over Commutative Rings synthesizesin a unified
manner literature from variousmathematical engineering and control theory journals. The authors’ approach is to extend various constructions and results known in the case of fields to classes of rings, giving particular attention to those rings that are of interest from a systems point of view. The algebraic treatment of a concept is usually preceded by an explanation of how the concept arises in a systems context. The book is written in a leisurely style, and the demands on the reader are modest. Although the emphasis is on the algebraic aspects, sophisticated techniques from commutative algebra are avoided wherever possible, and an introductory chapter on commutative algebra helps to make the rest of the material accessible to nonalgebraists. Thus, the book has a wide audience, and it can be used for a graduate course in systems and control theory (possibly supplementing or following a more traditional course in control theory) or as part of a study of commutative algebra. The topics included are those that one would expect: a chapter on reachability and observability, another on feedback and pole assignability (including a clear treatment of the important concept of dynamic feedback), and a final chapter on realization theory. Each chapter is concluded with a set of exercises, ranging from the trivial and calculational to extensions of the theory. Some interesting results are omitted; for example, a result by C. I. Bymes that states that reachable systems over k[X,, . . . , X,], k an algebraically closed field, is coefficient assignable if the local Kronecker indices of the system are constant, is not mentioned. On the whole, however, the authors have succeeded in their aim of giving a unified treatment of most of the important results, and the book is a welcome addition to the literature on algebraic system theory. REFERENCES 1 2
C. I. Bymes and P. Falb, Applications of algebraic geometry in system theory, Amer. 1. Math. 101337-363 (1979). C. I. Byrnes and C. F. Martin (Eds.), Geometrical Methods for the Theory of Linear Systems, Proceedings of a NATO Advanced Study Institute and AMS
244
3 4 5 6 7 8
GERT NAUDF
Summer Seminar in Applied Mathematics held at Harvard Univ., Cambridge, Mass. 18-29 June 1979. M. Hazewinkel, A partial survey of the uses of algebraic geometry in systems and control theory, Sympos. Math. XXfV:245-292 (1979). R. Kahnan, P. Falb, and M. Arbib, Topics in Mathematical System Theory, McGraw-Hill, New York, 1969. E. Kamen, Lectures on Algebraic System Theory: Linear Systems over Rings, NASA Contractor Report 3016, 1978. E. D. Sontag, Linear systems over commutative rings: A survey, Richerche Automat. 7:1-34 (1976). E. D. Sontag, Linear systems over commutative rings, a (partial) updated survey, in Proceedings of lFAC/81, Kyoto, 1981. A. Tannenbaum, Invariance and System Theory: Algebraic and Geometric Aspects, Lecture Notes in Mathematics, Springer, 1981. Received 4 September 1986; revised 20 September 1986