ing (3):. ACKNOWLEDGMENT. Manuscript received January 12, 1979. .... dently] afflicted with. .. a Death Wish, fatally smothering a beautiful, classical subject.
Correction to “Steady-State Invertibility and F d o w x d cont,.ol of rinear The-hvhant sy&& E. J. DAVISON
and assume.” 2) The following clarification shouldbe made in the definition following (3): After “(which dependson qy, qp add “i.e., u is the output of a stable,
ACKNOWLEDGMENT Manuscript received January 12, 1979. The author is with the Department of Electrical Engineering. University of Toronto, Toronto, Out, Canada. ‘E. J. Davison, IEEE Trans. Automot. Corm., vol. AC-21, pp. 529-534, Aug. 1976.
The author is gratefulto Y. Inouye for bringing the above points to his attention.
Book Reviews In this section, the IEEE Control SystemsSocietypublishesreviews of books in the control field and related areas. Readers are invited to send comments on these reviews for possible publicationin the Technical Notes and Correspondence The SCS does not necessarily endorse the opinions of the reviewers. section of this TRANSACTIONS. Brian F. D o o h Associate Editor-Book Reviews Flight Dynamics and Contr. Branch, 21 1-1 NASA Ames Res. e n . Moffett Field, CA 94035 a finite Algeb~Geometricand Lie-Theoretic Techniques in Systems Xheofy, where,” namely“except on a surface that isthezero-setof Part A: Interdisciplinary Matbematics, vol. 13-Robert Hermann number of polynomials, or is a finite union of such zero-sets.” Generally speaking, most of the formal structural synthesis problems Reviewed by W. M. (Brookhe, MA: Math. Sci.Press,1977,$17.00). that have been studied in linear multivariable control could be posed as wonham. questions about the rational map or suitably extended versions of it. W. M. Wonham (W&4-SM’76-FJ77) was born in Montreal, P.Q., So it isaltogetherreasonable to askwhatalgebraicgeometrymight Canada. He receiwd the B. Eng. degree from McGill Uniwrsiv, Montreal, contribute. The answer at the moment seemsto be, disappointingiy,“not P.Q.,C a d and the P h D. degree in control engineeringfrom the Unicer- much.” si@ of Cambridge, Cambridge, England in I956 and 1961, respectiwb. The author very clearly presentsone reason why this is true (of course, He has held teaching and research positions with Purdue Uniwrsiv, the there may be other reasons, too). Suppose you takea rational map hke rl, Research Institute for Advanced Studies (MAS), Brown Unimrsiv, and (for brevity write 4: X+Y) and compute its derivative D+. For each NASA. Currentb, he is Professor of Electrical Engineering in the System fixed x E X where is defined, &(x) will be a well defined linear map Control G r o q at the Unimrsity of Toronto,Toronto, Ont., Canadz His from X to Y (keepinmind that X and Y are vectorspaces). & is research interests are centered on vnthesis problems in multivariable con- representable, as usual, by a matrix of array size dim( Y )X dim (X),and trol. its entries are rational functions of x. Suppose W ( x ) is onto at some x : namely, dim(X)>dim(Y) and rank&(x)=dim(Y). Since & point In this book,theauthor’sobjective is to displaythepotential of algebraic geometryas a tool in systems (especially linear systems) theory. is rational, it must be onto almost everywhere. Now comes the remarkable algebro-geometricfact: if the field 3c is C, you can conclude that in, it is not extensivelyexplored as such. While Lietheorycomes of rl, includes almost every Algebraic geometry is “the only area of mathematics which deals sys- itself is “almostonto,”namely,theimage of Y. So you can either hit, or get as near as you please to, every point tematically with the theory of rational mappings between vector spaces” @. 115). On the other hand, linear systems theory “generates a multi- point of the codomain! In our example,it turns out that & is onto just when ( A , B ) is tude’’ of such mappings. Here is a standard example, from multivariable control. Let A : n X n controllable. Then (X = C), for almost any A , E ex“there will exist a and B : n X m be fixed matrices with elementsin a number field 3C (later feedback map F E C? x ” such that A + BF is similar to A,. The catch is ~ w i l l b e s p e c i a I i z e d e i t h e r t o ’ ? R . o r t o C ) . L e t F : m X n a n d T : n ~ n b e that you may need F to be complex even whenA , B, and A, are all real; matrices over 3c representing, respectively,state feedback and change of and complex F may not have any systemic interpretation, and so may basis (so T is nonsingular). Thus the set of all matrices we can hit, up to not be admissible, in the original problem context. at home), the In the case ;Jc =2‘% (where system theorists feel more similarity, by means of feedback, is just the image of the map +( = implication given by
+
+
+
+: x”x”@vx”+%?’x”: (F,T)HT-’(A+BF)T.
& onto*+ almost onto
Notice the minor abuse of notation: as a rational map, X)isactually defined only on a subset of its indicated domain, in this case where det TzO. In algebraic geometry is said to be defined “almost every-
is no longer true. The foregoingapproach, of using systemic propertiesto infer onto, to infer almost onto.. . , therefore breaks down.
+
0018-9286/79/0600-0519$00.75 01979 IEEE
+
(*>
520 Now there may be other ways, as yet undiscovered, in which algebraic geometry might help. Hermann is very enthusiastic about this,which makes his essay (it is not a textbook, or research monograph) fun to read. The scenario is dramatic. The bad guys are the “modern algebraic geometers,[who are notonly]uninterested in applications-oriented inflict on the unsuspecting public their material” @. 105) but “gobbledy-gook and Bourbaki-style generalized nonsense.. . [being evidently] afflicted with. .. a Death Wish, fatally smothering a beautiful, classical subject.. .” @. 15). The good guys are the nineteenth century who are founders of the subject (Kroneckerfiguresprominently), asserted to be “much closer in spirit to the need [sic] of modern applied mathematics... [because]theywere often muchmoreknowledgeable and concerned about progress in generalscience than are today’s mathematicians!” @. 74). Even if unwillingtotakesides, one m o t helpbutrecognize an authentic cri de coeur .., The book consists of eight chapters. The first four are on algebraic
IEEE TRANSACTIONS ON
AUTOMATIC CONTROL, VOL.
AC-24,NO. 3, JUNE 1979
geometry. Amplifying one of the author‘s suggestions, I recommend that you first read van der Waerden [I, ch. 161. The last four chapters are on linear systems theory. While thereare no really new results, the develop ment I sketched above is done in some detail, and the point of view on other problems is sometimesintriguing. Unfortunately, the centrally interesting and deepimplication (a), above, is not proved or even “explained,” although a reference is cited. This book was evidently written quickly.The organization is loose; the proofreadmg was inadequate; and there are toomany instances of references cited in the text that fail to appear in the bibliography. I think it is too early to predict whether or not algebraic geometry is going to be productive of genuinely new systemic insights. Butfor those interested in the possibilities, Hermanu’s essay, despite its literary shortcomings, is an engaging and pioneering introduction. REFERENCES
[I]
B. L. van der Waerden, A/gebra. New York: Ungar, 1970.
