Rudolf E. Kálmán Quotes

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.

(ii) Conditions which guarantee a priori the existence, physical realizability, and stability of the optimal filter.

(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...

A 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 s. Some aspects of this are considered briefly.

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.

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 , Stceg , Pugachev [5, 6], and Parzen . Using a variety of 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.

One should clearly distinguish between two aspects of the estimation problem:

(1) The theoretical aspect. Here interest centers on:

(iii) Characterization of the general results in terms of some simple quantities, such as signal-to-noise ratio, information rate, bandwidth, etc

(1) The general form of the solution (see Fig. 1).

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 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.