Rudolf Calman Dies

On July 2, 2016, the world-famous scientist, engineer and inventor Rudolf Emil Kalman did not become.
To many, he is known primarily as the creator of the estimation algorithm, named after him. However, the contribution of this person to science is much more significant. R. Kalman was one of the founders of modern management theory. His work has changed the way information is handled in a wide class of applications, from navigation to economics. The ideas of R. Kalman gave a powerful impetus to the development of technology and inspired thousands of researchers and engineers for new inventions.

I suggest that readers remember this wonderful person and get acquainted with the history of the algorithm, known as Kalman Filter.

The material is an article by prof. Doctor of Technical Sciences Oleg Andreevich Stepanova , dedicated to the 80th anniversary of R. Kalman. It was published in the journal Gyroscopy and Navigation in 2010, however, I believe, it remains unknown to a wide circle of network users.
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Kalman filter. History and modernity

Introduction

This year marks the eightieth anniversary of one of the founders of modern control theory, R. Kalman. His contribution to management theory is widely known and described in numerous publications and publications. So, even when Kalman’s 60th anniversary was celebrated in 1990, a special collection “Mathematical theory of systems. The influence of R. Kalman "[1]. It includes works by leading scientists describing the contribution of R. Kalman to various areas of the theory of filtration and control. In 2001, under the general editorship of T. Bazar, a special edition “Twenty-five fundamental articles in management (1932-1982)” was published [2]. This voluminous work was prepared on the initiative of the Society for Management of the Institute of Electrical and Electronics Engineers (IEEE) in order to determine the most significant results obtained in a very important period in the development of control theory. The editorial commission of this collection includes more than a dozen of the largest scientists from different countries, in particular, such as: P. Kokotovich, L. Lyung, B. Anderson, H. Quakernak, A. Isidori, K. Ostrem. The collection included articles by H. Nyquist, N. Weener, L.S. Pontryagin, V.A. Yakubovich and a number of other famous scientists. Moreover, R. Kalman was the only one from whom three articles were selected for this edition, two of which were written at the age of 30 [3-5].

R. Kalman has repeatedly visited Russia, is familiar with many of our famous scientists. Here highly appreciate his achievements. The most important articles and books of Kalman were promptly translated into Russian and are well known to specialists [6-10]. In connection with the 80th anniversary of R. Kalman in April of this year, at the Institute of Management Problems of the Russian Academy of Sciences, with the active participation and support of the international public organization Academy of Navigation and Motion Control (ANUD), a special seminar was held dedicated to the hero of the day [12, 13] . At the regular general meeting of ANUD on June 2, 2010, the author of this note presented a report also devoted to the anniversary of the famous scientist. The report addressed two issues. One of them dealt with some of the prerequisites and consequences of obtaining one of the most important results of R. Kalman, associated with the creation of a recurrent optimal estimation procedure, which was later called the Kalman filter. The second is the creative ties of R. Kalman with the scientists of our country. This article was prepared on the basis of the report at the general meeting of the ANUD.

About Kalman

Briefly basic biographical data [1, 14-16]. Rudolf Kalman was born on May 19, 1930 in Budapest. During the war in 1943, together with his parents, he emigrated to the USA through Turkey. He received a bachelor's and master's degree respectively in 1953 and 1954. at the Massachusetts Institute of Technology (MIT). He defended his doctoral degree under the direction of J. Ragazzini (JR Ra-gazzini) in 1957 at Columbia University. From 1957 to 1958 he worked as a full-time engineer in the research laboratory of the well-known company IBM, and from 1958 at the Research Institute for Advance Study-RIAS in Baltimore, headed by Solomon Levshets (1884-1972) - American mathematician of Russian origin, which Kalman considered one of his mentors. Here R. Kalman went from a mathematician-researcher to a deputy director for scientific work. It was during this period (1958-1964) that he carried out fundamental work in the field of systems analysis and control theory. In 1964, he transferred to Stanford University to the department of "Electrical Engineering, Mechanics and Operations Research." In 1971 he became director of the Mathematical Center for System Analysis and a professor at the University of Florida. Since 1973, R. Kalman has worked at the Swiss Federal Institute of Technology in Zurich.
R. Kalman - winner of many prestigious awards and awards, such as: IEEE Medal of Honor (1974), IEEE Centennial Medal (1984), Kyoto Prize Award (1985) (Japanese analogue of the Nobel Prize in the field of new technologies), Steel Prize (Bellman Prize, 1997). Among the latest awards, of course, mention should be made of the prize awarded in January 2008 by the US National Engineering Academy and the commemorative medal to them. Charles Stark Draper for the development and implementation of "optimal discrete technology (known as Kalman filter), widely used in solving various kinds of applied problems." A commemorative medal and a prize of $ 500,000 were awarded in Washington February 19, 2008. And finally, the most recent significant award is the annual US National Medal for Science, which was awarded in 2008 and presented to R. Kalman on October 7, 2009. in the White House by US President Barack Obama.

R. Kalman is a foreign member of the American, Hungarian and French academies, and since 1994 - a foreign member of the Russian Academy of Sciences (RAS) in the field of mechanical engineering, mechanics and control processes.

Prehistory

Speaking about the prehistory of the Kalman filter (FK), one can cite the words of Kalman himself: “I was as lucky as Newton, who was fortunate enough to be born at the time when Kepler's laws were ready and waited for him” [10]. There are quite extensive literature in which the historical aspects of the development and formation of the theory of filtration are discussed [14, 17-26]. Among foreign works, undoubtedly, it is necessary to highlight the review of T. Kaylatts, the largest known filtering scientist [19]. It includes 390 titles, and in contrast to many works published abroad. The review fairly objectively reflects the contribution to the theory of filtering not only foreign, but also domestic scientists.

We briefly reflect the main milestones that are directly related to the emergence of the first work of Kalman, dedicated to his famous filter [3].

As the predecessors of Kalman in the development of the theory of estimation, the creators of the least squares method (MNC) are first of all remembered: the German mathematician, astronomer, geodesist and physicist Karl Friedrich Gauss (1777-1855) and the French mathematician Adrien Marie Legendre (1752-1833).

Gauss was 18 years old (1795) when he first used, but did not publish the least squares method. Legendre independently invented a similar method and in 1806 published his results [17–19]. To understand the complexity of the relationship between the two great scientists in terms of championing palm in the invention of MNCs, let us cite several quotations from the article [17], which can also be found in [27]. In the book Theory of the Motion of Celestial Bodies, 1809 by K.F. Gauss noted the following: “Our principle, which we have been using since 1795, was recently published by Legendre in his work Nouvelles methodes for determination, or Paris, 1806, which explains some of his other properties, which we omit for brevity ". This, naturally, could not be liked by Legendre, who wrote: “Gauss, being already quite rich in discoveries, could have had the decency not to usurp the MNC”. Gauss felt himself darkened by the shadow of Legendre and complained about this: “I think this is an evil rock - to intersect almost all theoretical issues with Legendre. So it was in higher arithmetic, .., and now again the MNC, which is also used in the writings of Legendre and is really beautifully presented. ” Since then, as noted in [17], historians have found enough reason to prove Gauss’s priority in terms of the invention of MNCs, and rather Legendre was in the shadow of Gauss.

It is curious to draw attention to the fact that Gauss initially considered the problem of estimation from a probabilistic point of view. He believed that errors are independent and the joint probability distribution function of measurement residuals is equal to the product of the corresponding functions of each of the residues, the distribution law of which, in turn, was assumed to be normal. Although Gauss understood to some extent the inferiority of this law, suggesting the possibility of infinitely large measurement errors, the probabilistic interpretation of the OLS given to them largely laid the foundation for the emergence and justification of the maximum likelihood function method proposed later by R.A. Fisher ( 1912). It is important to emphasize that in the future Gauss himself preferred the substantiation of OLS from deterministic positions — minimization of a certain function of the difference between estimation and observations [17].

Speaking about the predecessors of R. Kalman, it is necessary to recall two more scientists: Andrei Nikolaevich Kolmogorov (1903-1986), an eminent Soviet mathematician, the founder of modern probability theory, and the largest American mathematician Norbert Wiener (1894-1964), whose name is usually associated with the origin of science, which is called cybernetics.

If Gauss and Legendre at the beginning of the 18th century considered the problem of estimating a time-constant vector, Kolmogorov and Wiener were already solving the problem of estimating changing parameters. At the same time, Kolmogorov was engaged in the problem of estimating a stationary Gaussian random sequence from its measurements against the background of measurement errors, which were also assumed to be values ​​of a Gaussian stationary sequence. Considering that the correlation functions for the estimated sequence and measurement errors are known, Kolmogorov, without discussing the estimation algorithm itself, obtained expressions for the error variances of the optimal in the root mean square sense estimates. First, he published these results without proof in 1939 [28], and then he cited more detailed results in 1941 [29]. Wiener's works in this area were carried out as part of the military order and in the open press were presented as a book only in 1949 [30]. Wiener considered the problem for continuous time and he obtained an algorithm for finding an estimate in the form of a convolution of the realization of measurements with a weight function, which in turn satisfied the Wiener-Hopf integral equation. The task was initially considered for steady state at infinite time, and a solution for it was obtained based on factorization of spectral densities. Note that the third chapter from this book called “The linear filter for a single time series” was also subsequently selected as an article in the collection [2].

What did not suit R. Kalman in the formulation of the problem and the solution obtained earlier? He did not quite agree with the assumption that statistical characteristics like the correlation function are the right way to describe uncertainties, and also that the system description using the transfer function is exactly the same as the representation of the system itself [10 ]. In addition, the proposed algorithms were not quite convenient for solving applied problems, including the use of computers that are widely used. The essential limitation was the used assumption about the stationary nature of the processes and the fact that the solution was obtained for an infinite time interval.

R. Kalman, while still a student, and subsequently, for about 10 years, worked hard to identify the links between the transfer functions and linear differential equations. By the early 1960s, this was expressed in the following generalization: “The linear systems described by the transfer function matrix are identical with fully observable and controllable linear vector differential equations” [10].

Kalman filter. First publication

By the end of the 50s, R. Kalman already had a number of results on using the description of systems in the state space when considering problems of control theory. The idea of ​​the possibility of applying an approach based on the description of systems in the state space for solving the Wiener filtration problem came to him at the end of November 1958 in the late evening, when the train returned from Princeton to Baltimore, when for some reason the train stood at the station. within an hour [14].

The first public report outlining the idea of ​​solving the Wiener filtering problem using an algorithm, later called the Kalman filter, took place on April 1, 1959 in Cleveland [10], and the first paper, “A new approach to linear filtering and prediction problems”, was published in 1960 in the Transactions of the ASME (American Society of Mechanical Engineers - American Society of Mechanical Engineers) [3].

It should be noted that the article appeared not in the journal traditional for the problem considered in it, published by the American Society of Electrical Engineers, but in the journal of the Society of Mechanical Engineers. The fact is that at that time the community of electrical engineers was quite skeptical about the ideas expressed in it, and the issue of publication could be dragged out significantly. This, by the way, explains the content of one of the footnotes made by the editors. It noted that the results and conclusions obtained in the published work are the personal result of the author and ASME is not responsible for them. And this was exactly the work in which for the first time an algorithm was proposed for solving an estimation problem using the state space. The peculiarity of the first work was, in particular, that the task of evaluating one sequence against the background of another was considered in the absence of an error component in the form of white noise. In fact, this was the task that was solved by Kolmogorov and Wiener. In proving the results obtained, the orthogonal projection theorem, which is known mainly to mathematicians, was used. And although the article traditionally assumed the Gaussian nature of the sequences being evaluated, the results presented in it suggested that the proposed algorithm, optimal for Gaussian character of measurement errors and disturbing noises, retains its optimality even if they are randomly distributed in the class of linear systems . This, a very important property, is often forgotten, while it is precisely this that determines the effectiveness of applying the Kalman filter in solving applied problems. In the same article, the duality or duality theorem was also proved, which establishes the connection between filtering and control problems.

According to Kalman [10]

"... Kalman's filter was a true discovery, because:
- Nobody imagined that the result will be so simple?
- No one expected the result to be so general.
“No one thought the filter would be so useful.”

And another quote from the same work

Of course, I was well aware of the importance of my discovery and even tried to explain it to my friends. But, to be honest, I had absolutely no idea that it would turn out to be so important and necessary [10]

Before we briefly discuss the consequences of the appearance of the first work, we will discuss one more footnote, which was already done by R. Kalman himself and referred to the part of the article where the linear nature of the algorithms used was discussed. On this occasion, R. Kalman wrote: “Of course, in the general case, these problems can be solved using nonlinear filters. However, to date, little or almost nothing is known about how to obtain (theoretically and practically) these nonlinear filters ”[3]. In this connection, of course, we should remember Ruslan Leontievich Stratonovich (1930-1997), who in essence had already solved the problem of optimal nonlinear filtering, based on the theory of conditional Markov processes created by him.

R.L. Stratonovich was born on May 31, 1930 in Moscow, and he, like Kalman, would have turned 80 in May. He externally finished school and received a gold medal. In 1947, he entered the physics department of Moscow State University, where he later worked as a professor all his life [31, 32].

Stratonovich created a stochastic calculus, which is an alternative to the theory of Ito integral and convenient for use in describing physical problems, introduced the stochastic integral of Stratonovich. In terms of solving filtration problems, he obtained partial differential equations for a posteriori density, which is necessary for calculating the optimal estimate. In the discrete case, their analogue is the recurrence relations for this density [33]. The Kalman linear filter is a special case corresponding to the linear Gaussian problem. Unfortunately, the contribution of R.Stratonovich to the theory of filtering is not assessed properly, despite the outstanding results in this field, no less significant than R.Kalman and N.Winer [34–36]. It may also be noted with regret that the works of R.Stratonovich are perhaps more often quoted in foreign publications than in the works of Russian authors.

Development of applied filtering algorithms

After the publication of the first article by R. Kalman devoted to the solution of the filtering problem on the basis of the state space, this direction received a rapid development. R. Kalman found fertile ground for the application of his algorithm in the Ames Research Center (Ames Research Center), which is part of NASA, as well as in the laboratory of C. Draper in MIT [14, 20, 21]. During his visit in the fall of 1960, he met with an employee at the Ames Center, S.F. Schmidt, who immediately assessed the potential of the new method in relation to the Apollo project related to the flight to the moon. It is believed that S.F. Schmidt was the first to use FC in solving practical problems. In the mid-60s thanks to the efforts of SF Schmidt FC became part of the navigation system for the C5A transport aircraft. FC was used here in the task of complex data processing from an inertial system and radar, additionally solving also the problem of rejection of measurements with large errors [14].

R. Bucy, who also worked at the RIAS at that time, proposed Kalman to establish a connection between the Wiener-Hopf equation and the Riccati equation in FC for continuous time. This was done in their joint work [7]. In particular, it was shown that the Riccati equation can have a stable solution, even if the original system is unstable, provided that it is controllable and observable. By the way, as noted in [14], the work devoted to the Kalman – Bucy filter for continuous time was initially rejected because of the allegedly existing, but then not confirmed, error in the proof found by one of the reviewers.

When using FK when solving applied problems, immediately a lot of problems arose related to the choice of models that adequately describe the error behavior of measuring systems; with the sensitivity of the algorithms to selectable models; with a decrease in the amount of computation in the development of suboptimal filtering algorithms due to a reduction in the dimension of the estimated state vector itself, a simplified description of measurement error models and generating noise; with the computational stability of the proposed procedures, etc. Numerous publications were devoted to the development of Kalman’s ideas and the solution of these problems. Considerable attention was paid to various modifications of Kalman filters, adaptive algorithms, solving nonlinear problems [1, 17-26, 37-39]. A generalized FC was widely used, which in earlier publications was called the Kalman-Schmidt filter [20]. Then, the so-called iterative filters and higher order filters were proposed, which were various modifications of Kalman type algorithms [14, 37, 38]. To solve nonlinear problems with substantial nonlinearities, algorithms based on the use of recurrent relations for the a posteriori density were actively developed. Algorithms such as the point mass method, the method based on the Poly Gaussian approximation of the a posteriori density, the separation method, the Monte Carlo method, and a number of others [18, 40–44] have been developed here.

Until the mid-seventies, the theory of filtering and its applications developed rapidly. In addition to R.L. — .., .., .., .., .., .., .., .. [12, 33, 45-48].

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For stationary filtering problems in steady state, both approaches provide the same solution. The question of using one or another approach in such tasks is rather a taste and is largely a consequence of the school to which the developer belongs.

At the same time, it should also be taken into account that when using the frequency Wiener approach, it is possible to solve stationary filtering problems not only for Markov processes.

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'Darker side'

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R. Kalman is familiar with leading domestic scientists, he repeatedly visited the USSR, and then in Russia - in 1960 (Moscow [6]), 1968 (Yerevan [69]), 1969 (Tbilisi, Kiev [70]), 1970 ( St. Petersburg), 1984 (Moscow [9]), 2006 (Moscow, St. Petersburg [16, 71]).

Kalman met many scientists in our country during his first stay in 1960 at the World Congress of the International Federation for Automatic Control (IFAC), which is described in more detail below. It was here that R. Kalman met R. Stratonovich and subsequently they had a rather lengthy correspondence [31].

In the same year he first met with the outstanding Soviet mathematician L.S.Pontryagin (1908-1988). Their closer acquaintance took place later in 1969 in Tbilisi and then on the initiative of R. Kalman, LS Potryagin was invited to the United States [72].

Kalman also met with Ya.Z. many times. Tsypkin (1919-1997) [73], starting with their acquaintance in 1956 in Heldenberg, then in 1960 in Moscow and many times at various scientific conferences. It is Ya.Z. Tsypkin edited the book Essays on the Mathematical Theory of Systems when it was translated into Russian [8].

R. Kalman corresponded with A.I. Lurje (1901-1980), which follows from the bibliography in [5]. He was well acquainted with VS Pugachev (1911-1998) [15, 46]. By the way, in the article by Kalman [3] there are three references to the works of Soviet scientists: two to Pugachev V.S. and one - on V.V. Solodnikov (1910-1992).

Currently, R. Kalman often communicates with the chairman of the national committee on automatic control, A. B. Kurzhansky, who has significant achievements in the field of filtration theory [48].

R. Kalman is also well acquainted with the professor of St. Petersburg University V.A. Yakubovich. Professionals are well aware of the Kalman – Yakubovich – Popov lemma, which establishes a relationship between frequency methods in control theory and methods of Lyapunov functions and published in 1962 [74]. In English-language literature, it is often also called the KJP lemma (Kalman. Jukubovicn. Popov). This work, by the way, is small in size - only 4 pages - also hit the special volume of the 25 best publications in control theory and served as the beginning of further significant research in this area [75]. R. Kalman obtained similar results a year later, using the concepts of controllability and observability [76] to prove this. Visiting our city, R. Kalman always finds time to meet with V.A. Yakubovich.

When asked about what event or visit to our country he remembered most of all, R. Kalman answered that, of course, this is the first visit related to participation in the First Congress of the International Federation for Automatic Control (IFAC) [77]. It really was an outstanding event for the science of the time. The congress was held in Moscow from June 27 to July 7, 1960. 1,190 participants gathered here and over 1,000 invited from 29 countries. 285 out of 410 reports were selected. B. Widrow [78], a well-known scientist, the founder of the theory of constructing adaptive filters [79], published a curious memory of this congress not so long ago. The significance of the congress for our country can be judged, in particular, by the fact that at the congress the Deputy Chairman of the Government, who was at that time A.N. Kosygin [77, Vol.1].

The reasons why the congress was held precisely in Moscow are becoming clear if we recall the achievements of the USSR at that time in only one area - the space program. So, on October 4, 1957 - the launch of the first artificial satellite of the Earth. January 2, 1960 - the first flight to the Moon; September 14, 1960 - the first landing of the Luna-2 spacecraft on the surface of the Moon; October 7, 1960 - the first flight of the Moon by the Luna-3 spacecraft and its first shooting from the back. And, finally, April 12, 1961 - the first manned flight into space. It is clear that the development of astronautics required the solution of serious problems associated with the control and processing of signals, and the significant successes of Russian scientists were generally recognized in this field.

The first congress, perhaps, was the only one at which simultaneous translation was carried out - there were four working languages: English, Russian, German and French. A special group of translators was preparing for the congress. It is curious to note that the report of R. Kalman “On the General Theory of Control Systems” [6] was translated into Russian by Anatoly Grigorievich Butkovsky, who later became an eminent scientist, the founder of the theory of control of systems with distributed parameters. Since 1975, he has headed the research laboratory in this area at the Institute of Control Problems (ICS) of the Russian Academy of Sciences. In 1996 at the 13th Congress of the IFAC in the United States in the report "The history of management since 1960", the name of A.G. Butkovsky was included in the list of five names of outstanding scientists of Russia who made the greatest contribution to the theory of control [80]. Note that the staff of the Institute of Applied Physics RAS, headed by Academician V.A. Trapeznikov made a significant contribution to the organization and holding of the first IFAC congress [80].

During the congress, lectures were also organized at the Polytechnic Museum with translation into Russian. Lectures were held in the following sequence: S. Levshets, R. Kalman; N. Wiener. As Kalman recalls, it was very honorable for him to be in a “sandwich” with such famous scientists, whom he considered to be his teachers.

Note that the first congress participants were three members of the Academy of Navigation and Motion Control: Anatoly Yefimovich Sazonov, Georgy Nikolayevich Lebedev and Efim Natanovich Rosenwasser. And E.N. Rosenwasser made a report at this congress, and he happened to communicate quite often with R. Kalman, since their hotel rooms were nearby. Here is a rather unique photograph of the congress participants.

The story of this photo is curious. As a participant in the 16th IFAC Congress in Prague in 2005, I caught a glimpse of it at Efim Natanovich, who brought it in order to show R. Kalman.

In preparation for the report, I asked for this photo from Efim Natanovich, to which he replied with regret that the photo was lost, but the organizers of the congress in Prague took it from him in order to make a copy. I asked M. Simadle, a member of the 16th Congress National Committee, a well-known scholar from the Czech Republic, to send a copy of this photo or any other photos from the first Congress, where R. Kalman may be present. Quickly enough, I were kindly sent three photos, including the one in question, but it was noted that the photo with Kalman, unfortunately, is missing. Subsequently, all the same R. Kalman in this photo was found (second from the left in the second row) and he received confirmation that he was the one.


Congress participants, Moscow 1960. (photo taken from the IFAC secretariat in 2010)

Concluding the section on the connection of R. Kalman with our country, I note the following. For people of my and older generations, well-known scientists from abroad were rather certain symbols, and their results were perceived as something existing in itself. This was largely due to the limited possibilities of direct communication with foreign colleagues. In recent years, the situation has radically changed. The impetus to realizing that Rudolf Kalman is our contemporary, for me was his article, published in the journal Aerospace Instrument Making (chief editor G.N. Lebedev) in 2004 [10]. And not even the article itself, but the fact that it was translated and edited by an ANUD colleague, professor at the Moscow Aviation University, KK Veremeenko, who, as it turned out, was actively communicating via e-mail with Professor R. Kalman, performing this translation. And in 2005, in Prague, we managed to communicate on our own and even get an autograph on the aforementioned magazine. The final transformation of R. Kalman from a kind of legendary, abstract personality into a real scientist took place during the period of his last stay in Russia today.


After receiving an autograph. Prague 2005

This visit took place in June 2006 thanks to the efforts of Academician A. B. Kurzhansky. In Moscow, R. Kalman was received by Vice-President of the Academy of Sciences N. Plate. R. Kalman visited the "Star City" and delivered a lecture at Moscow State University, and also visited the Mathematical Institute. V.M. Steklov.


R. Kalman and Academician of the Russian Academy of Sciences V.G. Peshekhonov at a lecture in the House of Scientists. St. Petersburg, 2006

In St. Petersburg, R. Kalman met in the House of Scientists with the President of ANUD, Chairman of the St. Petersburg Group of the National Committee for Automatic Control, Academician of the Russian Academy of Sciences VG Peshekhonov, and then a memorable for many scientists of our city was held entitled “The Central Problem in systems theories: history, progress and hopes ”[71].

Conclusion

During the presentation of the Kyoto Prize to R. Kalman in 1985, he, speaking to the press, cited the following statement, which he first saw in one of the pubs in Colorado Springs [14]: “Small (little) people discuss other people; ) people discuss events, and big (big) people discuss ideas. ” I don’t know exactly in what context this phrase was uttered and what meaning R. Kalman himself put into the word big, but I think that those who are familiar with his works have no doubt that, as applied to R. Kalman, this word should be translated as "outstanding."

Thanks

The author expresses his gratitude to everyone who to one degree or another helped in the formation and preparation of materials for this article, including: V.M. Zinenko (SSC RF Concern Central Research Institute Elektropribor, St. Petersburg), M.V. Pyatnitskaya (IPU RAS, Moscow), E.N. Rosenwasser (St. Petersburg State Marine Technical University), A.Ye. Sazonov (State Maritime Academy. Adm. S.O.Makarova, St. Petersburg), Yu.A. Solov'ev (FGU 30 Central Research Institute of the Ministry of Defense of Russia, Moscow), N. B. Filimonova (Institute of Mechanical Engineering named after A. Blagonravov of the Russian Academy of Sciences, Moscow), A.L. Fradkov (Institute for Problems of Mechanical Engineering, RAS, St. Petersburg), MS Yarlykov (Air Force Academy named after prof. N.. Zhukovsky and Yu.A. Gagarin, Moscow), L. Camberlin (France).

Last years

Despite his advanced age, R. Kalman did not cease to engage in scientific and educational activities. A year ago, he visited St. Petersburg, where he gave an open report "Insight from the Outside" at the IFAC conference "Modeling, Identification and Control of Nonlinear Systems". Among the students there were many employees, students and graduate students of the department of the INS of ITMO University . R. Kalman kindly agreed to sign a general photo and wished creative success to young scientists.


R. Kalman with staff, students and graduate students of the department of INS.

From myself it remains only to add words of gratitude to R. Kalman. I am sure that the ideas proposed by him will find many more wonderful applications and will become the basis for new inventions.