Rudolf E. Kálmán

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Rudolf Kalman, 2009.

Rudolf Emil Kálmán (May 19, 1930 - July 2, 2016) was a Hungarian-American electrical engineer, mathematical system theorist, and college professor, noted for his co-invention and development of the Kalman filter, a mathematical algorithm that is widely used in signal processing, control systems, and Guidance, navigation and control.


  • Let me say quite categorically that there is no such thing as a fuzzy concept... We do talk about fuzzy things but they are not scientific concepts. Some people in the past have discovered certain interesting things, formulated their findings in a non-fuzzy way, and therefore we have progressed in science.
  • I would like to comment briefly on Professor Zadeh's presentation. His proposals could be severely, ferociously, even brutally criticized from a technical point of view. This would be out of place here. But a blunt question remains: Is professor Zadeh presenting important ideas or is he indulging in wishful thinking? No doubt Professor Zadeh's enthusiasm for fuzziness has been reinforced by the prevailing climate in the U.S.-one of unprecedented permissiveness. 'Fuzzification, is a kind of scientific permissiveness; it tends to result in socially appealing slogans unaccompanied by the discipline of hard scientific work and patient observation.
  • I have been aware from the outset (end of January 1959, 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 some 1000 related publications, at least two Citation Classics, etc. There is something to be explained.
    To look for an explanation, let me suggest a historical analogy, at the risk of further immodesty. I am thinking of Newton, and specifically his most spectacular achievement, the law of Gravitation. Newton received very ample "recognition" (as it is called today) for this work. it astounded - really floored - all his contemporaries. But I am quite sure, having studied the matter and having added something to it, that nobody then (1700) really understood what Newton's contribution was. Indeed, it seemed an absolute miracle to his contemporaries that someone, an Englishman, actually a human being, in some magic and un-understandable way, could harness mathematics, an impractical and eternal something, and so use mathematics as to discover with it something fundamental about the universe.

New results in linear filtering and prediction theory (1961)

Kalman & Richard S. Bucy (1961) "New results in linear filtering and prediction theory" in: Journal of Basic Engineering March 1961. p. 95-108
  • 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.
    • p. 95 Article summary; cited in: "Rudolf E. Kálmán", MacTutor History of Mathematics archive, 2010
  • At present, a nonspecialist might well regard the Wiener-Kolmogorov theory of filtering and prediction [1, 2] as "classical' — in short, a field where the techniques are well established and only minor improvements and generalizations can be expected.
    That this is not really so can be seen convincingly from recent results of Shinbrot [3], Stceg [4], Pugachev [5, 6], and Parzen [7]. Using a variety of time-domain methods, these investigators have solved some long-stauding problems in nonstationary filtering and prediction theory. We present here a unified account of our own independent researches during the past two years (which overlap with much of the work [3-71 just mentioned), as well as numerous new results. We, too, use time-domain methods, and obtain major improvements and generalizations of the conventional Wiener theory. In particular, our methods apply without modification to multivariate problems.
    • p. 95 Opening paragraphs
  • One should clearly distinguish between two aspects of the estimation problem:
    (1) The theoretical aspect. Here interest centers on:
    (1) The general form of the solution (see Fig. 1).
    (ii) Conditions which guarantee a priori the existence, physical realizability, and stability of the optimal filter.
    (iii) Characterization of the general results in terms of some simple quantities, such as signal-to-noise ratio, information rate, bandwidth, etc
    (2) The computational aspect. The classical (more accurately, old-fashioned) view is that a mathematical problem is solved if the solution is expressed by a formula. It is not a trivial matter, however, to substitute numbers in a formula. The current literature on the Wiener problem is full of semi-rigorously derived formulas which turn out to be unusable for practical computation when the order of the system becomes even moderately large...
    • p. 102; The conclusion

Quotes about Rudolf E. Kálmán

  • The creator of modern control and system theory, Kalman theory, which was established in the early 1960s, brought a fundamental reformation to control engineering and since then laid the foundation for the rapid progress of modern control theory.
  • 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.
    • Eduardo Sontag (2010) "Kalman Receives National Medal of Science", Notices Amer. Math. Soc. 57 (1) (2010), 56-57. cited in: "Rudolf E. Kálmán", MacTutor History of Mathematics archive, 2010
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