**Rudolf Kalman** emigrated to the United States where he studied electrical engineering at the Massachusetts Institute of Technology. He was awarded a S.B. in 1953, and continued to undertake graduate studies there leading to a S.M. in 1954. For his doctoral work he went to Columbia University in New York City where he was advised by John Ralph Ragazzini. He was appointed as an Instructor in Control Theory at Columbia University in 1955 and promoted to Adjunct Assistant Professor in 1957. He submitted his doctoral thesis *Analysis and Synthesis of Linear Systems Operating on Randomly Sampled Data* in 1958 and was awarded a D.Sci. in that year.

Before the award of his doctorate Kalman had begun to publish influential papers. For example in *Physical and mathematical mechanisms of instability in nonlinear automatic control systems* (1959) he gives (quoting from a review by Horace Trent):-

... a lucid discussion of the stability problem for control systems containing one or more nonlinear elements. It is written in a style which appeals equally to advanced design engineers and to research workers who deal with the theory of such systems. The discussion is limited to those systems which can be treated as if they were made up of a finite number of "lumped" parts with time invariant parameters. Thus, such a system can be described by a finite number of simultaneous ordinary differential equations. The author exploits the topological approach to an analysis of non-linear systems; i.e., he examines the nature of trajectories in phase space as generated by variations of inputs, variation of initial conditions, etc., and from these results deduces quantitative information about the possible existence of nodes, foci, and saddle points. During these analyses he makes use of the method of virtual critical points. One of the main goals of the paper is to propose a classification of the types of nonlinearities which can give rise to instabilities.

After the award of his doctorate Kalman was appointed as a Research Mathematician at the Research Institute for Advanced Studies in Baltimore, Maryland. He worked there from 1958 to 1964 and during this time he produced some results of outstanding importance. For example there are three papers he wrote jointly with J E Bertram: *A unified approach to the theory of sampling systems* (1959); *Control system analysis and design via the "second method" of Lyapunov. *I*. Continuous-time systems* (1960); and *Control system analysis and design via the "second method" of Lyapunov. *II*. Discrete-time systems* (1960).

There are also the three papers *A new approach to linear filtering and prediction problems* (1960), *Mathematical description of linear dynamical systems* (1963) and (with Richard S Bucy) *New results in linear filtering and prediction theory* (1961). The authors summarise this last mention paper as follows:-

A nonlinear differential equation of the Riccati type is derived for the covariance matrix of the optimal filtering error. The solution of this 'variance equation' completely specifies the optimal filter for either finite or infinite smoothing intervals and stationary or non-stationary statistics. The variance equation is closely related to the Hamiltonian(canonical)differential equations of the calculus of variations. Analytic solutions are available in some cases. The significance of the variance equation is illustrated by examples which duplicate, simplify, or extend earlier results in this field. The duality principle relating stochastic estimation and deterministic control problems plays an important role in the proof of theoretical results. In several examples, the estimation problem and its dual are discussed side-by-side. Properties of the variance equation are of great interest in the theory of adaptive systems. Some aspects of this are considered briefly.

Now we have linked these last-mentioned three papers since they "proved to be of fundamental or lasting importance in their field" and as such were awarded the 1986 Steele Prize by the American Mathematical Society. The citation reads [2]:-

The ideas presented in these papers are a cornerstone of the modern theory and practice of systems and control. Not only have they led to eminently useful developments, such as the Kalman-Bucy filter, but they have contributed to the rapid progress of systems theory, which today draws upon mathematics ranging from differential equations to algebraic geometry.

Kalman was not present at the 93^{rd} American Mathematical Society Annual meeting in San Antonio, Texas, to receive the Prize but his response, attempting to explain the significance of the papers, was read at the meeting. His response begins as follows [2]:-

I have been aware from the outset(end of January1959, the birthdate of the second paper in the citation)that the deep analysis of something which is now called Kalman filtering were of major importance. But even with this immodesty I did not quite anticipate all the reactions to this work. Up to now there have been some1000related publications, at least two Citation Classics, etc. There is something to be explained.

Read a fuller report of Kalman's 1986 response to the award of the Steele Prize at THIS LINK.

Let us return to giving details of Kalman's career. He left the Research Institute for Advanced Studies in Baltimore, Maryland, in 1964 to take up the position of Professor at Stanford University covering engineering mechanics, electrical engineering, and mathematical system theory. In 1971 he moved to the University of Florida, Gainesville, where he was Graduate Research Professor jointly in the Departments of Mathematics, Electrical Engineering, and Industrial and Systems Engineering, and Director of the Center for Mathematical System Theory. Two years later, while continuing to hold these positions at the University of Florida, he also became a Professor at the Eidgenössische Technische Hochschule in Zürich. Kalman retired from his positions at the University of Florida in 1992 (when he became Professor Emeritus) and from the Eidgenössische Technische Hochschule in Zürich in 1997.

We have looked at some of the important papers Kalman wrote in the early part of his career. To get a fuller overview of his contributions we quote from Eduardo Sontag and Yutaka Yamamoto, both former doctoral students of Kalman at Florida University (being awarded their doctorates in 1976 and 1978 respectively) [5]:-

During the1960s, Kalman was the leader in the development of a rigorous theory of control systems. Among his many outstanding contributions were the formulation and study of most fundamental state-space notions(including controllability, observability, minimality, realizability from input/output data, matrix Riccati equations, linear-quadratic control, and the separation principle)that are today ubiquitous in control. While some of these concepts were also encountered in other contexts, such as optimal control theory, it was Kalman who recognized the central role that they play in systems analysis. The paradigms formulated by Kalman and the basic results he established have become an intrinsic part of the foundations of control and systems theory and are standard tools in research as well as in every exposition of the area, from undergraduate engineering textbooks to graduate-level mathematics research monographs. During the1970s Kalman played a major role in the introduction of algebraic and geometric techniques in the study of linear and nonlinear control systems. His work since the1980s has focused on a system-theoretic approach to the foundations of statistics, econometric modelling, and identification, as a natural complement to his earlier studies of minimality and realizability.

In addition to many research papers, Kalman has jointly authored the book *Topics in mathematical system theory* (1969) with Peter L Falb and Michael A Arbib. The authors write in the Preface:-

This book does not pretend to be a systematic treatise. Rather, it aims to present a mathematical system theory as it is today - a lively, challenging, exciting, difficult, confused, rewarding, and largely unexplored field. Each of the four sections of the book is independent. No attempt has been made to eliminate duplications; there is no standardized notation.

We mentioned above that Kalman received the Steele Prize in 1986. This was one of many prizes and awards that were made to him as a consequence of his outstanding, innovative contributions. In addition to the Steele Prize he received the Institute of Electrical and Electronics Engineers' Medal of Honor in 1974, the Institute of Electrical and Electronics Engineers' Centennial Medal in 1984, and the Inamori Foundation's Kyoto Prize in High Technology in 1985 as:-

... the creator of modern control and system theory. Kalman theory, which was established in the early1960s, brought a fundamental reformation to control engineering and since then laid the foundation for the rapid progress of modern control theory.

He also received the Richard E Bellman Control Heritage Award in 1997, and the National Academy of Engineering's Charles Stark Draper Prize in 2008:-

... for the development and dissemination of the optimal digital technique(known as the Kalman Filter)that is pervasively used to control a vast array of consumer, health, commercial, and defense products.

On 7 October 2009 U.S. President Barack Obama honoured Kalman in an awards ceremony at the White House when he presented him with the National Medal of Science, the highest honour the United States can give for scientific achievement. Eduardo Sontag explained the significance of Kalman filtering in [4]:-

Among Kalman's early work was the development of what is now called the Kalman filter for detection of signals in noise. This revolutionized the field of estimation, by providing a recursive approach to the filtering problem. Before the advent of the Kalman filter, most mathematical work was based on Norbert Wiener's ideas, but the 'Wiener filtering' had proved difficult to apply. Kalman's approach, based on the use of state space techniques and a recursive least-squares algorithm, opened up many new theoretical and practical possibilities. The impact of Kalman filtering on all areas of applied mathematics, engineering, and sciences has been tremendous. It is impossible to even begin to enumerate its practical applications. Just as examples of their diversity, one may mention the guidance of the Apollo spacecraft and of commercial airplanes, uses in seismic data processing, nuclear power plant instrumentation, and demographic models, as well as applications in econometrics.

In addition to the above mentioned honours, he has been elected to the National Academy of Sciences of the United States, the American National Academy of Engineering, the American Academy of Arts and Sciences, the Hungarian Academy of Sciences, the Académie des Sciences, and the USSR Academy of Sciences.

Kalman's ideas on statistics are thought-provoking and he has written a number of interesting articles on this topic since retiring. For example he gave an informal lecture published as *Probability and science* (1993) in which he (quoting from a review by Zeno G Swijtink):-

... laments the explosive growth of applications of probability after World War II, and questions whether probabilities exist in the real world. Randomness does, and to capture it better he proposes a new definition: random is not uniquely determined by simple classical rules. All irrational numbers are said to be random under this definition, as are most "3x +1" maps. The absence of interaction between chaos theorists and probability theorists also shows, the author concludes, how irrelevant the classical models of probability are to the real world.

In 1995 he published *Randomness and probability*, and in 2002 a discussion on *What Is a Statistical Model?* He believes that [3]:-

... the currently accepted notion of a statistical model is not scientific; rather, it is a guess at what might constitute(scientific)reality without the vital element of feedback, that is, without checking the hypothesized, postulated, wished-for, natural-looking(but in fact only guessed)model against that reality. To be blunt, as far as is known today, there is no such thing as a concrete i.i.d.(independent, identically distributed)process, not because this is not desirable, nice, or even beautiful, but because Nature does not seem to be like that.(Historical aside: recall that physicists had thought at one time that aether was such a necessary, unavoidable, appealing, clear and beautiful concept that it must perforce exist; alas, all physicists now living had to learn that such argumentation cannot lead to good science.)As Bertrand Russell put it at the end of his long life devoted to philosophy, "Roughly speaking, what we know is science and what we don't know is philosophy." In the scientific context, but perhaps not in the applied area, I fear statistical modelling today belongs to the realm of philosophy.

Finally note that Kalman married Constantina Stavrou who studied economics at Johns Hopkins University; they have two children, Andrew and Elisabeth.

**Article by:** *J J O'Connor* and *E F Robertson*