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exists a voluminous work, both on theory and computation of these problems. Theory is extremely rich; however, one cannot say the same thing about computation. Many of the methods available in present control theory textbooks are not suitable for computer implementations. Most of these methods, in fact, were developed before the computer era and are not based on computationally sound techniques. Fortunately, the situation is changing very fast. In the last few years, computationally viable methods have been developed for several of the above problems, and presently studies are being conducted not only on the development of computationally viable methods but also on other important numerical analysis aspects, such as perturbation analyses of the problems, stability analysis of the algorithms by backward and forward round-off error analyses, etc. Unfortunately, most of those algorithms, however, are not satiable for large and sparse problems. They are based on transformation of the system matrices to some sort of condensed forms such as Hessenberg, triangular, Real-Schur form, etc., and the methods used to achieve these forms, such as Gaussian elimination, Householder and Givens methods, the QR iterations, etc., are well known to give fill-in. On the other hand, there are practical situations such as the design of large space structures (Ref 1 and 2), control of power systems (Ref 3), etc. that give rise to very large problems and, like most practical large problems, these problems are sparse and well structured, too. Most of the existing methods are not designed to take advantage of the structures exhibited by these problems.

Another aspect of control theory research that needs the attention of computational scientists and practicing engineers is parallel computations in control. Nowadays, when there is a revolution going on in the area of parallel/vector computations, any

serious research in applied sciences and engineering should pay attention to this aspect or research. Unfortunately, control theory is lagging behind with respect to other areas of science and engineering. For example, in recent years, much effort has been devoted to parallelizing sequential algorithms and developing new parallel algorithms in numerical linear algebra [for an account of the recent developments in the area of parallel matrix computations, see the excellent survey by Gallivan et al. (Ref 4)]. Parallel software libraries based on these algorithms are being developed on some of the existing parallel machines. It is only natural to take advantage of those parallel linear algebra algorithms and the associated software libraries and software packages suitable for hierarchial computations (such as the recent linear algebra package LAPACK) to develop parallel algorithms for linear control problems. One obvious advantage of doing that is programming effort will be greatly reduced. Anyone who has experienced parallel programming and computations will admit how hard it is to code and debug some of the new parallel/ vector computers. Admittedly, some of the linear control algorithms (e.g., eigenvalue assignment methods via implicit QR iterations) are very sequential in nature. For these problems novel parallel algorithms (parallel algorithms that will perhaps never be used on sequential machines) need to be developed.

In control theory, there are opportunities for the development of both types of algorithms. Activities in the area in control theory of large-scale and parallel computations, unfortunately, are very limited. Only a handful nately, are very limited. Only a handful of papers have appeared so far (see the of papers have appeared so far (see the references quoted in Reference 5). A references quoted in Reference 5). A recent panel report (Ref 6) on "Future Directions in Control Theory" has emphasized, among many other things, the need for expanded research in these

areas. Indeed, research on computational methods in control theory is still in its infancy.

To address the need for research in these areas, I organized an invited special session on numerical linear algebra in signals, systems, and control for this conference. In my talk titled "Parallel Computations in Control Theory," I first summarized the state-of-the-art research in this area and then described in detail parallel algorithms for eigenvalue assignment and observer matrix equation problems along with implementational details of these algorithms both on shared memory and distributed memory machines. I showed some actual results of implementations on a Cray X-MP/4, Cray Y-MP, Alliant FX/8, and FX/8, transputers. Prof.

R. Plemmons spoke on parallel algorithms for linear prediction on inverse factorization.

Daniel Pierce of Boeing Computer Services spoke on sparse matrix techniques for condition estimation based on a rank-revealing QR factorization. Attempts are being made by researchers such as myself and others to incorporate these powerful tools from sparse matrix computations into developing algorithms and analyzing robustness properties for feedback stabilization associated with large second order models.

The final talk of the session was delivered by George Cybenko on linear algebra aspects of Wavelet transforms.

Another invited session on parallel algorithm design was organized by L. Thiere of Univ. Saarlandes, Germany. The talks were given by A-A. Sayed of Stanford University on "Fast Algorithms for Generalized Displacement Structures," based on joint work with Tom Kailalh; by P. Dewilde on "The Algebra of Parallel Processors"; by L. Thiere on "Multidimensional Discrete Event Systems and Their Applications to Parallel Program Design," based on joint work with W. Backes and

U. Scwniegelshohn of the IBM T.J. Watson Research Center; and by J. Bu on "Design of Fixed-Size Systolic Arrays: Control Structure and Data," based on joint work with Ed T. DePrettere of Delft University Tech. Finally, as mentioned earlier, a special lecture on supercomputer solution of the algebraic Riccati equation was given by Pradip Pandey based on joint work with Alan Laub of UC Santa Barbara.

Finally, let me mention another talk by Prof. Brian Anderson on the finite word length (FWL) design of statespace digital systems with weighted sensitivity minimization and spareness consideration, given at an invited special session on Finite Precision and Quantization Effects in Control Design II organized by Profs. E.I. Verriest of the Georgia Institute of Technology and M. Gevers of Louvain University of Belgium. Anderson's talk centered around the optimal FWL state-space design, which aims to identify those realizations that minimize the degradation of the system performance due to the FWL effects.

Remarks

As expected, activities in the area of computational methods for control systems design and signal processing were far less than the other areas. I would like to see more special sessions, more contributed talks, and even plenary lectures in this area in future MTNS meetings.

LINEAR ALGEBRA AND CONTROL AND SYSTEMS THEORY

Linear algebra and control and systems theory have long enjoyed a natural synergism; however, the interdisciplinary activities blending these two areas were disappointingly fewer than expected. Fortunately, a significant increase in activities in this area has taken place since the 1984 American

closed-loop system is internally stable and the H-norm of the transfer function matrix is minimized.

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Dr. Vidyasagar, in his plenary talk, remarked jokingly, "This conference has been hijacked by H people." I think that there was some truth in the statement. The number of talks in this area in any category was far greater than any other area addressed by the conference. One of the reasons for this was, of course, love for H control by Japanese control theorists who were involved in organization. In fact, I was told by Prof. Kimura and his colleagues that H control techniques are not academic anymore; they are being incorporated in industrial applications such as in the design of large space structures. I am aware of similar activities at the Langley NASA Research Center in our country, however, the Japanese seem to be ahead.

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Mathematical Society (AMS) summer Mathematical Society (AMS) summer research conference on Linear Algebra and its Role in Systems Theory and the two interdisciplinary SIAM conferences on Linear Algebra in Signals, Systems, and Control, 1986 and 1990 (Ref 7 and 8), chaired and organized by the author. For example, for the current MTNS, there were noticeable activities in this area. There were several invited sessions. One was organized by A.C. Ran of Virje University, The Netherlands, on Matrix Equations and Applications. Two sessions on Matrix Completion and Extension Problems were organized by L. Rodman of the College of William and Mary. of the College of William and Mary. Several linear algebraists and system Several linear algebraists and system theorists, such as Leiba Rodman, A. Ran, M.A. Kaashoek, Carlos De-Sonza, etc., gave talks in these sessions. There was also a very nice session on Interpolation Problems for Matrix Functions and Applications in Systems Theory organized by CONCLUSIONS M.A. Kaashoek of Virje University, The Netherlands. In addition to Kaashoek, the participants included A.C. Antoulas, Joe Ball, Leiba Rodman, and A. Tannenbaum. Profs. Kaashoek, Ball, Rodman, and Tannenbaum are leading authorities in operator theory and their authorities in operator theory and their work heavily involves applications of operator theory to systems theory and H control. Prof. Antoulas is well known for his work on interpolation and its applications.

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All indications are that the conference was a great success; the aim of the conference, to bring together mathematicians and control and systems theorists, was somehow achieved. I, however, expected somewhat more participants from industries, especially since the conference had such a large number of corporate sponsorships.

As mentioned before, I strongly believe that the mutual interactions and research collaborations between mathematicians, computer scientists, and practicing engineers are important ingredients in timely success of any interdisciplinary project.

perturbation to a nominal plant P and SITE VISITS IN JAPAN that the feedback is internally stable

for AP = 0. A very important question Tokyo Denki University (TDU) then is how large can |AP| be so that the internal stability is maintained? The question can be answered in terms of the H-norm on a weighted closedloop transfer function. The goal is to design a feedback controller so that the

At TDU my hosts were Prof. Hiroshi Inaba of the Department of Information Sciences and Prof. T. Kamabayashi of the Department of Mathematics. Inaba is a well-known control theorist

in Japan and the group leader of a research group working mainly on infinite dimensional systems theory. Several of Inaba's joint projects are with two of his bright and very promising research students and colleagues, Mr. N. Otsuka and Mr. Ito. Their work makes heavy use of tools from linear algebra and operator theory. My research interaction with this group was mainly on linear algebraic aspects of the problems they are working on. This group is becoming increasingly interested in numerical aspects of control and systems theory research.

University of Tokyo

My hosts at the University of Tokyo were Profs. Masao Iri and K. Murota, both from the Department of Mathematical Engineering and Instrumentation Physics. Iri is very well respected throughout the whole country as a pioneer in mathematical engineering education in Japan. Both the mathematics and engineering communities in this country owe a lot to Iri. He stimulated the interest of mathemati

and Algorithms (Ref 9), following an
invitation by the celebrated mathema-
tician Richard Bellman.

current activities in this area (though they are very limited) are in the Western world, mostly in the United States.

SITE VISITS TO HONG KONG,
MACAO, SINGAPORE,
AND INDIA

Prof. Iri, like me, is very much inter-
ested and quite active in bridging the
communication gap between mathe-
maticians and engineers. As I remarked
earlier, this gap is very much notice-
able in the West, but I learned that this Hong Kong
is also true in Japan. A major differ-
ence here is that Japanese practicing
and research engineers are far more
motivated than their Western counter-
parts and more knowledgeable in the
mathematics they have been using in
their work.

Prof. Murota, a former student and
a current research collaborator and
colleague of Prof. Iri, is mainly inter-
ested in matroid theory and its applica-
tions to systems theory. Tools here are
combinatorial in nature. This is a very
interesting and innovative approach to
solving systems theory problems; how-
ever, I am not sure if it will have a long-
time impact in systems theory research.
I am not aware of any people in the
West currently active in this research.

cians to solve real-world problems and Conclusions
educated practicing engineers to acquire
appropriate mathematical knowledge
in their areas of research and applica-
tions. Iri also contributed profoundly
in the growth of mathematical pro-
gramming in Japan. He is also interna-
tionally well known for his scientific
contributions in several areas of math-
ematics and engineering, such as net-
work flow, graphs and matroids, numer-
ical methods, computational geometry,
and mathematical programming. He
has delivered numerous invited lectures
in many prestigious international con-
ferences and serves on the editorial
board of several distinguished journals,
including the Japan Journal of Industrial
and Applied Mathematics (formerly the
Japan Journal of Applied Mathematics).
He wrote a book titled Network Flow,
Transportation and Scheduling: Theory

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From the brief interactions I had with the Japanese scientists, I have the feeling that Japanese mathematicians, control theorists, and engineers are very active in almost all areas of theory and applications of control theory: H。 control, robust control, adaptive control, infinite dimensional systems theory, abstract control theory, robotics, etc. In fact, as far as industrial applications In fact, as far as industrial applications of control theory techniques are concerned, Japan seems to be ahead of all the Western countries, including the United States. However, surprisingly, there are very few activities in the computational aspects of control and systems theory--especially, there are almost no activities in the area of largescale and parallel computations in control. As mentioned earlier, all the

In Hong Kong, I visited City Polytechnic of Hong Kong and the University of Hong Kong. In City Polytechnic, my host was Dr. Daniel Ho of the Department of Applied Mathematics. I found this department very active in applied research and several members of the department are internationally visible. Their research interests include scientific computing applied to transport modelling, computational fluid dynamics, vibrating systems, heat transfer, computational aspects in control and systems theory, optimization theory, etc. Ho is a specialist in etc. Ho is a specialist in computational and applied control theory. He has published joint papers in the area of H. control theory, output feedback problems, computer-aided control theory, etc. in collaboration with some internationally known mathematicians and control theorists such as Fletcher, M.J. Grimble, etc. His research is funded by research grants from the Polytechnic.

Ho belongs to a group working on scientific computation and its applications led by Prof. James Caldwell. One of Caldwell's strengths is to attract researchers from engineering and other applied science areas to work in the area of applied mathematics. For example, his joint work with Dr. Y.M. Ram, a mechanical engineer, has resulted in publications in several SIAM journals in the area of vibrating systems. Caldwell's research is funded by overseas agencies.

My host at the University of Hong Kong was Dr. Raymond Chan, a young numerical linear algebraist who has already received international

recognition by winning the prestigious Fox Prize. His research interests include numerical linear algebra with applications to differential equations and signal processing. He has collaborated with well-known numerical linear algebraists such as Robert Plemmons.

The department is active in research both in pure and applied mathematics. Some of the internationally recognized researchers of the department are Drs. Y.H. Au-Yeung (linear and multilinear algebra), M.C. Liu and K.M. Tsang (analytic number theory), K.Y. Chan (differential equations and mathematical modelling), M.K. Siu (combinatorial number theory), S.C.K. Chu and T.G. Yung (operations research applied to mathematical modelling), and Dr. R. Chan (numerical linear algebra, numerical differential equations, signal processing, etc.). The department is headed by A.J. Ellis, who pursues his research in functional analysis and convexity theory. Over the years, the department has produced many mathematicians who now hold important positions in business, government, and education, both locally and internationally. Many of their graduates hold professional appointments in some of the leading universities of the world, such as Harvard, Stanford, and Berkeley.

In conclusion, Hong Kong seems to be in the forefront of research in modern mathematics and scientific computing. There are adequate research and computing facilities and there is a sincere desire to expand their research horizons and be recognized in the Western world.

Macao

During my stay in Hong Kong, I made a 1-day trip to Macao to visit the University of East Asia. This university is relatively unknown to the world. My host was Prof. Graciano de Oliviera, an internationally reputed linear algebraist from Portugal who has been a visitor to the university for the last

year. He and his wife, who is a numerical analyst, are trying to build research in certain areas of mathematics, especially in the area of linear and numerical linear algebra.

Singapore

In Singapore, I visited the Mathematics Department of the National University of Singapore (NUS). My host was Dr. Tara Nanda, a numerical analyst, who received his Ph.D. from the Courant Institute of Mathematical Sciences in New York and did postdoctoral work with Prof. Beresford Parlett at Berkeley. Nanda is currently engaged in developing an interactive educational software package in numerical analysis, which appears to be very i nteresting and innovative. The researchers at NUS have access to national supercomputing facilities; however, the facilities are underused.

I found, to my distress, that their program for visitors is not well organized. Occasionally, the university spends a huge amount of money to bring very distinguished visitors such as some Nobel laureates, which is fine, but not of much help to build research. A systematic program to bring active researchers from around the world to help build research in certain focused areas is lacking.

India

I visited India upon an invitation from the Government of India to be a scientific advisor. My visit was funded by the United Nations Development Plan. This was indeed an extremely wellorganized program. I spent the first 2 weeks in the city of Bangalore, one of the most prominent centers for scientific research and technological development in India.

I was officially hosted by the Department of Computer Science and Automation of the Indian Institute of Science (IISC). My host was Prof.

N. Viswanadham, an internationally renowned control theorist. I visited several departments in IISC such as the Institute for Supercomputing Education, headed by Prof. Rajaraman, and the Microprocessor Applications Laboratory, headed by Prof. Patnaik. I also visited several government-funded research laboratories in Bangalore, including the Center for Development of Advanced Computing, headed by Dr. U.S. Shukla; the Central Research Laboratory of Bharat Electronics (Ref 10), headed by Dr. Paulraj; the Center for Artificial Intelligence and Robotics, headed by Dr. M. Vidyasagar, and CMMACS at the National Aeronautics Laboratory, headed by Dr. K.S. Yajnik.

My visits to these places revealed that India is in the era of supercomputing. India has acquired a Cray from the United States recently and this is being used in weather prediction research at the Indian Metrological Department in Delhi; the Supercomputing Institute at IISC is in the process of getting a Cray-XMP from the United States. Other parallel computers, especially the distributed computers, are presently planned to be bought. Besides these, India has already built a transputerbased, Hypercube type, distributed memory computer, PARAM, at the Center for Development of Advanced Computing in Pune and is in the process of building others in Bangalore and other places. These computers are presently being used in applied research such as speech synthesis, pattern recognition, power systems analysis, image processing, robotics, radar signal processing, artificial intelligence, space research, oil reservoir modelling, seismic data processing, etc. (Ref 10). One bottleneck in using these computers has been the lack of suitable parallel mathematical software. Indian scientists have been trying to develop some, but that will be a challenge. Besides these, several microprocessor-based architectures and appropriate software

are being developed, notably at the Microprocessor Applications Laboratory in IISC.

The last 2 weeks of my visit were spent at the Indian Institute of Technology, Kharagpur, in West Bengal, India. I was hosted by Prof. Kanti Bhusan Datta, a well-known control theorist in the Electrical Engineering Department. I visited the Departments of Computer Science and Mechanical and Aerospace Engineering. I found there was a sincere desire among the engineers and computer scientists to develop collaborative research blending mathematics, computer science, and engineering. I was invited by the director of the institute, Prof. K.L. Chopra, to help them set up a center for interdisciplinary research with a strong focus in scientific computing. I shall be seriously considering this.

REFERENCES

1. M. Balas, "Trends in large space structure control theory: Fondest dreams and wildest hopes,” IEEE Trans. Auto. Control AC-2, 522-38 (1981).

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9. M. Iri, Network Flow, Transportation and Scheduling: Theory and Algorithms (Academic Press, New York, 1969).

10. An overview of CRL Projects (Central Research Laboratory of Bharat Electronics, India, April 1991).

Biswa Nath Datta received his Ph.D. degree in mathematics in 1972 from the University of Ottawa, Canada. He has been a professor of mathematical sciences at Northern Illinois University since 1981. His current research interests include linear and numerical linear algebra and its applications to control and systems theory including large-scale and parallel computations in these areas. Professor Datta is the author of more than 50 interdisciplinary research papers and is currently working on two textbooks. He serves on the editorial board of the SIAM Journal of Matrix Analysis and Applications and the Journal of Mathematical Systems, Estimation, and Control. He was a nominator for the Japan Prize in 1989.

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