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exists a voluminous work, both on serious research in applied sciences and areas. Indeed, research on computatheory and computation of these prob- engineering should pay attention to tional methods in control theory is still lems. Theory is extremely rich; how- this aspect or research. Unfortunately, in its infancy. ever, one cannot say the same thing control theory is lagging behind with To address the need for research in about computation. Many of the meth- respect to other areas of science and these areas, I organized an invited special ods available in present control theory engineering. For example, in recent session on numerical linear algebra in textbooks are not suitable for com- years, much effort has been devoted to signals, systems, and control for this puter implementations. Most of these parallelizing sequential algorithms and conference. In my talk titled “Parallel methods, in fact, were developed before developing new parallel algorithms in Computations in Control Theory,” I the computer era and are not based on numerical linear algebra (for an account first summarized the state-of-the-art computationally sound techniques. For- of the recent developments in the area research in this area and then described tunately, the situation is changing very of parallel matrix computations, see in detail parallel algorithms for eigenfast. In the last few years, computation- the excellent survey by Gallivan et al. value assignment and observer matrix ally viable methods have been devel- (Ref 4)). Parallel software libraries based equation problems along with impleoped for several of the above problems, on these algorithms are being devel- mentational details of these algorithms and presently studies are being con- oped on some of the existing parallel both on shared memory and distribducted not only on the development of machines. It is only natural to take uted memory machines. I showed some computationally viable methods but also advantage of those parallel linear algebra actual results of implementations on on other important numerical analysis algorithms and the associated software a Cray X-MP/4, Cray Y-MP, Alliant aspects, such as perturbation analyses libraries and software packages suitable FX/8, and transputers. Prof. of the problems, stability analysis of for hierarchial computations (such as R. Plemmons spoke on parallel the algorithms by backward and for- the recent linear algebra package algorithms for linear prediction on ward round-off error analyses, etc. LAPACK) to develop parallel algo- inverse factorization. Unfortunately, most of those algorithms, rithms for linear control problems. Daniel Pierce of Boeing Computer however, are not satiable for large and One obvious advantage of doing that is Services spoke on sparse matrix techsparse problems. They are based on programming effort will be greatly niques for condition estimation based transformation of the system matrices reduced. Anyone who has experienced

reduced. Anyone who has experienced on a rank-revealing QR factorization. to some sort of condensed forms such parallel programming and computa- Attempts are being made by researchers as Hessenberg, triangular, Real-Schur tions will admit how hard it is to code such as myself and others to incorpoform, etc., and the methods used to and debug some of the new parallel/ rate these powerful tools from sparse achieve these forms, such as Gaussian vector computers. Admittedly, some of matrix computations into developing elimination, Householder and Givens the linear control algorithms (e.g., algorithms and analyzing robustness methods, the QR iterations, etc., are eigenvalue assignment methods via properties for feedback stabilization well known to give fill-in. On the other implicit QR iterations) are very sequen- associated with large second order hand, there are practical situations tial in nature. For these problems models. such as the design of large space novel parallel algorithms (parallel algo- The final talk of the session was structures (Ref 1 and 2), control of rithms that will perhaps never be used delivered by George Cybenko on linear power systems (Ref 3), etc. that give on sequential machines) need to be algebra aspects of Wavelet transforms. rise to very large problems and, like developed.

Another invited session on parallel most practical large problems, these In control theory, there are oppor- algorithm design was organized by problems are sparse and well structured, tunities for the development of both L. Thiere of Univ. Saarlandes, Germany. too. Most of the existing methods are types of algorithms. Activities in the The talks were given by A-A. Sayed of not designed to take advantage of the area in control theory of large-scale Stanford University on “Fast Algorithms structures exhibited by these problems. and parallel computations, unfortu- for Generalized Displacement Struc

Another aspect of control theory nately, are very limited. Only a handful tures,” based on joint work with Tom research that needs the attention of of papers have appeared so far (see the Kailalh; by P. Dewilde on “The Algecomputational scientists and practic- references quoted in Reference 5). A bra of Parallel Processors”; by L. Thiere ing engineers is parallel computations recent panel report (Ref 6) on “Future on "Multidimensional Discrete Event in control. Nowadays, when there is Directions in Control Theory” has Systems and Their Applications to a revolution going on in the area emphasized, among many other things, Parallel Program Design,” based on of parallel/vector computations, any the need for expanded research in these joint work with W. Backes and U. Scwniegelshohn of the IBM T.J. Mathematical Society (AMS) summer closed-loop system is internally stable Watson Research Center; and by J. Bu research conference on Linear Alge- and the Ho-norm of the transfer funcon “Design of Fixed-Size Systolic Arrays: bra and its Role in Systems Theory and tion matrix is minimized. Control Structure and Data," based on the two interdisciplinary SIAM con- Dr. Vidyasagar, in his plenary talk, joint work with Ed T. DePrettere offerences on Linear Algebra in Signals, remarked jokingly, “This conference Delft University Tech. Finally, as Systems, and Control, 1986 and 1990 has been hijacked by H. people.” I mentioned earlier, a special lecture on (Ref 7 and 8), chaired and organized by think that there was some truth in the supercomputer solution of the alge- the author. For example, for the cur- statement. The number of talks in this braic Riccati equation was given by rent MTNS, there were noticeable area in any category was far greater Pradip Pandey based on joint work activities in this area. There were sev- than any other area addressed by the with Alan Laub of UC Santa Barbara. eral invited sessions. One was orga- conference. One of the reasons for this

Finally, let me mention another talk nized by A.C. Ran of Virje University, was, of course, love for He control by by Prof. Brian Anderson on the finite The Netherlands, on Matrix Equations Japanese control theorists who were word length (FWL) design of state- and Applications. Two sessions on involved in organization. In fact, I was space digital systems with weighted Matrix Completion and Extension told by Prof. Kimura and his colleagues sensitivity minimization and spareness Problems were organized by L. Rodman that H. control techniques are not consideration, given at an invited spe- of the College of William and Mary. academic anymore; they are being incorcial session on Finite Precision and Several linear algebraists and system porated in industrial applications such Quantization Effects in Control Design theorists, such as Leiba Rodman, as in the design of large space strucII organized by Profs. E.I. Verriest of A. Ran, M.A. Kaashoek, Carlos tures. I am aware of similar activities at the Georgia Institute of Technology De-Sonza, etc., gave talks in these the Langley NASA Research Center in and M. Gevers of Louvain University sessions. There was also a very nice our country, however, the Japanese seem of Belgium. Anderson's talk centered session on Interpolation Problems for to be ahead. around the optimal FWL state-space Matrix Functions and Applications in design, which aims to identify those Systems Theory organized by CONCLUSIONS realizations that minimize the degra- M.A. Kaashoek of Virje University, The dation of the system performance due Netherlands. In addition to Kaashoek, All indications are that the conferto the FWL effects.

the participants included A.C. Antoulas, ence was a great success; the aim of the

Joe Ball, Leiba Rodman, and conference, to bring together matheRemarks

A. Tannenbaum. Profs. Kaashoek, Ball, maticians and control and systems

Rodman, and Tannenbaum are leading theorists, was somehow achieved. I, As expected, activities in the area of authorities in operator theory and their however, expected somewhat more computational methods for control work heavily involves applications of participants from industries, especially systems design and signal processing operator theory to systems theory and

operator theory to systems theory and since the conference had such a large were far less than the other areas. I Ho control Prof. Antoulas is well known number of corporate sponsorships. would like to see more special sessionsfor his work on interpolation and its As mentioned before, I strongly more contributed talks, and even ple- applications.

believe that the mutual interactions nary lectures in this area in future MTNS

and research collaborations between meetings. H. AND ROBUST CONTROL mathematicians, computer scientists,

and practicing engineers are important LINEAR ALGEBRA

One of the central areas of research ingredients in timely success of any AND CONTROL AND

in present-day control theory is H. interdisciplinary project. SYSTEMS THEORY

control. Suppose that AP is an unknown

perturbation to a nominal plant P and SITE VISITS IN JAPAN Linear algebra and control and sys- that the feedback is internally stable tems theory have long enjoyed a natu- for AP = 0. A very important question Tokyo Denki University (TDU) ral synergism; however, the interdisci- then is how large can |AP| be so that plinary activities blending these two the internal stability is maintained? The At TDU my hosts were Prof. Hiroshi areas were disappointingly fewer than question can be answered in terms of Inaba of the Department of Informaexpected. Fortunately, a significant the Ho-norm on a weighted closed- tion Sciences and Prof. T. Kamabayashi increase in activities in this area has loop transfer function. The goal is to of the Department of Mathematics. taken place since the 1984 American design a feedback controller so that the Inaba is a well-known control theorist in Japan and the group leader of a and Algorithms (Ref 9), following an current activities in this area (though research group working mainly on infi- invitation by the celebrated mathema- they are very limited) are in the Western nite dimensional systems theory. Several tician Richard Bellman.

world, mostly in the United States. of Inaba's joint projects are with two of Prof. Iri, like me, is very much interhis bright and very promising research ested and quite active in bridging the

ested and quite active in bridging the SITE VISITS TO HONG KONG, students and colleagues, Mr. N. Otsuka communication gap between mathe

communication gap between mathe- MACAO, SINGAPORE, and Mr. Ito. Their work makes heavy maticians and engineers. As I remarked AND INDIA use of tools from linear algebra and earlier, this gap is very much noticeoperator theory. My research interaction able in the West, but I learned that this Hong Kong with this group was mainly on linear is also true in Japan. A major differalgebraic aspects of the problems they ence here is that Japanese practicing In Hong Kong, I visited City Polyare working on. This group is becoming and research engineers are far more technic of Hong Kong and the Univerincreasingly interested in numerical motivated than their Western counter

motivated than their Western counter- sity of Hong Kong. In City Polytechnic, aspects of control and systems theory parts and more knowledgeable in the my host was Dr. Daniel Ho of the research.

mathematics they have been using in Department of Applied Mathematics. their work.

I found this department very active in University of Tokyo

Prof. Murota, a former student and applied research and several members

a current research collaborator and of the department are internationally My hosts at the University of Tokyo colleague of Prof. Iri, is mainly inter- visible. Their research interests include were Profs. Masao Iri and K. Murota, ested in matroid theory and its applica

ested in matroid theory and its applica- scientific computing applied to transboth from the Department of Mathe- tions to systems theory. Tools here are port modelling, computational fluid matical Engineering and Instrumenta- combinatorial in nature. This is a very dynamics, vibrating systems, heat transtion Physics. Iri is very well respected interesting and innovative approach to fer, computational aspects in control throughout the whole country as a solving systems theory problems; how- and systems theory, optimization theory, pioneer in mathematical engineering ever, I am not sure if it will have a long- etc. Ho is a specialist in computational education in Japan. Both the mathe- time impact in systems theory research. and applied control theory. He has matics and engineering communities I am not aware of any people in the

I am not aware of any people in the published joint papers in the area of in this country owe a lot to Iri. He West currently active in this research. H. control theory, output feedback stimulated the interest of mathemati

problems, computer-aided control cians to solve real-world problems and Conclusions

theory, etc. in collaboration with some educated practicing engineers to acquire

internationally known mathematicians appropriate mathematical knowledge From the brief interactions I had and control theorists such as Fletcher, in their areas of research and applica- with the Japanese scientists, I have the MJ. Grimble, etc. His research is funded tions. Iri also contributed profoundly feeling that Japanese mathematicians, by research grants from the Polytechnic. . in the growth of mathematical pro- control theorists, and engineers are very Ho belongs to a group working on gramming in Japan. He is also interna- active in almost all areas of theory and

active in almost all areas of theory and scientific computation and its applicationally well known for his scientific applications of control theory: H. tions led by Prof. James Caldwell. One contributions in several areas of math- control, robust control, adaptive con- of Caldwell's strengths is to attract ematics and engineering, such as net- trol, infinite dimensional systems theory, researchers from engineering and other work flow, graphs and matroids, numer- abstract control theory, robotics, etc. applied science areas to work in the ical methods, computational geometry, In fact, as far as industrial applications

In fact, as far as industrial applications area of applied mathematics. For examand mathematical programming. He of control theory techniques are con- ple, his joint work with Dr. Y.M. Ram, has delivered numerous invited lectures cerned, Japan seems to be ahead of all a mechanical engineer, has resulted in in many prestigious international con- the Western countries, including the publications in several SIAM journals ferences and serves on the editorial United States. However, surprisingly, in the area of vibrating systems. board of several distinguished journals, there are very few activities in the Caldwell's research is funded by overincluding the Japan Journal of Industrial computational aspects of control and seas agencies. and Applied Mathematics (formerly the systems theory--especially, there are My host at the University of Hong Japan Journal of Applied Mathematics). almost no activities in the area of large

almost no activities in the area of large. Kong was Dr. Raymond Chan, a young He wrote a book titled Network Flow, scale and parallel computations in numerical linear algebraist who Transportation and Scheduling: Theory control. As mentioned earlier, all the has already received international recognition by winning the prestigious year. He and his wife, who is a numerical N. Viswanadham, an internationally Fox Prize. His research interests include analyst, are trying to build research in renowned control theorist. I visited numerical linear algebra with applica- certain areas of mathematics, especially several departments in IISCsuch as the tions to differential equations and signal in the area of linear and numerical Institute for Supercomputing Educaprocessing. He has collaborated with linear algebra.

tion, headed by Prof. Rajaraman, and well-known numerical linear algebra

the Microprocessor Applications Laboists such as Robert Plemmons. Singapore

ratory, headed by Prof. Patnaik. I also The department is active in research

visited several government-funded both in pure and applied mathematics. In Singapore, I visited the Mathe- research laboratories in Bangalore, Some of the internationally recognized matics Department of the National including the Center for Development researchers of the department are Drs. University of Singapore (NUS). My of Advanced Computing, headed by Y.H. Au-Yeung (linear and multilinear host was Dr. Tara Nanda, a numerical Dr. U.S. Shukla; the Central Research algebra), M.C. Liu and K.M. Tsang analyst, who received his Ph.D. from Laboratory of Bharat Electronics (analytic number theory), KY. Chan the Courant Institute of Mathematical (Ref 10), headed by Dr. Paulraj; the (differential equations and mathemat- Sciences in New York and did post- Center for Artificial Intelligence and ical modelling), M.K. Siu (combina- doctoral work with Prof. Beresford Robotics, headed by Dr. M. Vidyasagar, torial number theory), S.C.K. Chu and Parlett at Berkeley. Nanda is currently and CMMACS at the National AeroT.G. Yung (operations research applied engaged in developing an interactive nautics Laboratory, headed by Dr. to mathematical modelling), and Dr. educational software package in K.S. Yajnik. R. Chan (numerical linear algebra, numerical analysis, which appears to My visits to these places revealed numerical differential equations, sig- be very i nteresting and innovative. that India is in the era of supercomputnal processing, etc.). The department The researchers at NUS have access to ing. India has acquired a Cray from the is headed by A.J. Ellis, who pursues his national supercomputing facilities; United States recently and this is being research in functional analysis and however, the facilities are underused. used in weather prediction research at convexity theory. Over the years, the I found, to my distress, that their the Indian Metrological Department department has produced many math- program for visitors is not well orga- in Delhi; the Supercomputing Instiematicians who now hold important nized. Occasionally, the university tute at IISC

is in the process of getting positions in business, government, and spends a huge amount of money to a Cray-XMP from the United States. education, both locally and interna- bring very distinguished visitors such Other parallel computers, especially tionally. Many of their graduates hold as some Nobel laureates, which is fine, the distributed computers, are presently professional appointments in some of but not of much help to build research. planned to be bought. Besides these, the leading universities of the worldA systematic program to bring active

A systematic program to bring active India has already built a transputersuch as Harvard, Stanford, and Berkeley. researchers from around the world to based, Hypercube type, distributed

In conclusion, Hong Kong seems to help build research in certain focused memory computer, PARAM, at the be in the forefront of research in modern areas is lacking.

Center for Development of Advanced mathematics and scientific computing.

Computing in Pune and is in the proThere are adequate research and com- India

cess of building others in Bangalore puting facilities and there is a sincere

and other places. These computers are desire to expand their research hori- I visited India upon an invitation presently being used in applied research zons and be recognized in the Western from the Government of India to be a such as speech synthesis, pattern recogworld.

scientific advisor. My visit was funded nition, power systems analysis, image

by the United Nations Development processing, robotics, radar signal proMacao

Plan. This was indeed an extremely well- cessing, artificial intelligence, space

organized program. I spent the first research, oil reservoir modelling, seisDuring my stay in Hong Kong, I 2 weeks in the city of Bangalore, one of mic data processing, etc. (Ref 10). One made a 1-day trip to Macao to visit the the most prominent centers for scien- bottleneck in using these computers University of East Asia. This university tific research and technological devel- has been the lack of suitable parallel is relatively unknown to the world. My opment in India.

mathematical software. Indian scienhost was Prof. Graciano de Oliviera, I was officially hosted by the Depart- tists have been trying to develop some, an internationally reputed linear alge- ment of Computer Science and Auto- but that will be a challenge. Besides braist from Portugal who has been a mation of the Indian Institute of Sci- these, several microprocessor-based visitor to the university for the last ence (IISC). My host was Prof. architectures and appropriate software are being developed, notably at the 2. S.M. Joshi, Control of Large Flexi- 9. M. Iri, Network Flow, TransportaMicroprocessor Applications Labora- ble Structures (Springer-Verlag, 1989). tion and Scheduling: Theory and Algotory in IISC.

rithms (Academic Press, New York, The last 2 weeks of my visit were 3. P. Kokotovic and B.H. Kvog, 1969). spent at the Indian Institute of Tech- “Feedback control of overloaded nology, Kharagpur, in West Bengal, networks,IEEE Trans. Auto. Control 10. An overview of CRL Projects India. I was hosted by Prof. Kanti Bhusan AC-29, 704-11 (1984).

(Central Research Laboratory of Bharat Datta, a well-known control theorist in

Electronics, India, April 1991). the Electrical Engineering Department. 4. KA. Gallivan, M. Heath, E. Ng, I visited the Departments of Computer J. Ortega, B. Peyton, R. Plemmons, Biswa Nath Datta received his Ph.D. Science and Mechanical and Aerospace C. Romine, A. Sameh, and R. Voigt, degree in mathematics in 1972 from Engineering. I found there was a sin- Parallel Algorithms for Matrix Compu- the University of Ottawa, Canada. cere desire among the engineers and tations (SIAM, 1990).

He has been a professor of mathecomputer scientists to develop collab

matical sciences at Northern Illinois orative research blending mathemat- 5. B.N. Datta, “Parallel algorithms University since 1981. His current ics, computer science, and engineer- in control theory,” in Proceedings of the research interests include linear and ing. I was invited by the director of the 30th IEEE Conference on Decision and numerical linear algebra and its institute, Prof. K.L. Chopra, to help Control, Brighton, England, 1700-1703 applications to control and systems them set up a center for interdisciplin- (1991).

theory including large-scale and ary research with a strong focus in sci

parallel computations in these areas. entific computing. I shall be seriously 6. Report of the panel on “Future Professor Datta is the author of more considering this.

Directions in Control Theory: A Math- than 50 interdisciplinary research

ematical Perspective," chaired by papers and is currently working on REFERENCES W.H. Fleming, SIAM, 1988.

two textbooks. He serves on the edi

torial board of the SIAM Joumal of 1. M. Balas, “Trends in large space 7. SLAM Newsletter, January 1987. Matrix Analysis and Applications and structure control theory: Fondest

the Journal of Mathematical Systems, dreams and wildest hopes,IEEE Trans. 8. SIAM Newsletter, March 1991. Estimation, and Control. He was a Auto. Control AC-2, 522-38 (1981).

nominator for the Japan Prize in 1989.

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