Rudolf Kalman: "Science proceeds by mathematics,"

  Michael Kassner, chair of the USC Viterbi School's Department of Aerospace and Mechanical Engineering, welcomed one of the truly legendary figures in contemporary engineering to an overflow crowd on Feb. 1. And Rudolf E. Kalman did not disappoint, charging that a fair amount of contemporary physics and engineering fall significantly short of the standards set by Sir Isaac Newton 325 years ago.

Kalman, inventor of a key analytic technique with applications throughout engineering, offered an intriguing case study of how a seemingly transparent mathematical strategy could contain hidden flaws not revealed for a half century after its introduction.

Kalman, a professor emeritus at the Swiss Federal Institute of Technology (ETH) in Zürich spoke on "The Newtonian Revolution: Interaction of Mathematics and High Technology," sponsored by AME, the third event in Viterbi School's Centennial Lecture Series.

Kalman is the creator of the Kalman filter, which AME's Firdaus Udwadia, in his introduction, called “perhaps the single most widely used mathematical technique of the last half-century.” The filter allows engineers to make the best of noisy and incomplete measurements by systematically analyzing the readings to decide which might be consistent with a real state of affairs, and which can be discarded as bogus.

It is now used, said Udwadia, "in all fields of engineering, and we would never have been able to reach the moon without it."

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AME's Firdaus Udwadia, left, introduced the guest from Zürich

Kalman's ambitious speech was delivered in two 45-minute parts, with a break, beginning with a long and admiring portrait of the extraordinary achievement of Newton who in Kalman's view had rescued celestial mechanics from "cookbook" empiricism by creating an absolutely rigorous mathematical foundation that did not just describe but explained the movements of planets.

Kalman underlined, as part of this, that Newton's interest was understanding, not simple prediction. "He was 100 percent interested in research," but the research could only turn into knowledge by mathematics. "Science proceeds by mathematics," Kalman said.  The process, building on previous work by Johannes Kepler, ("Kepler guessed; Newton proved") required 20 years for completion.

The balance of the presentation moved forward 250 years, to the development of system theory - electronic system theory, an intriguing tale of how a seemingly straightforward system of physics failed to find mathematical explanation measuring up to Newton's, and how what seemed to be a neat fix turned out to be concealing, rather than illuminating, the real situation.

Kalman began his story in the 1920s, with a group of researchers trying to develop useful understandings of electrical circuits that would transparently give outputs, building on the insights of physics pioneer Oliver Heaviside (1850-1925). Electrical circuits are made up of components, regarded as "black boxes," with three variables, capacitance, inductance, and resistance. The three come together to create"impedance," the difference between power in and power out.

The trick is to create a perfect mathematics that can characterize the interplay of the variables so that the process is not just described in terms of what happens, allowing measurements and predictions for engineering purposes, but is understood in the deep, mathematical Newtonian sense that Kalman holds up as the grail for science.

In his presentation, Kalman discussed a long line of post-Heaviside researchers who had attacked the problem, all brilliant, all dedicated, many personally known to Kalman.

The names included the R.M. Forster, whose reactance theorem was described by Kalman as "comparable to Newton," the brilliant German Wilhelm Cauer, and perhaps most critical to Kalman's story, Otto Brune, who produced a brilliantly successful theory by introducing an additional mathematical operator, T, and "ideal transformer," that seemed to drastically simplify the problem, and make a wider range of solutions possible.

Brune and his pupil Ernst Guilleman put a stamp on the problem with a success that seemed so deep and well-ordered that it discouraged fresh looks at the problem, particularly after another great researcher, Raoul Bott proved (in his first published paper) that the operator T could be replaced with equivalents constructed from other variables — but the number of these grew so quickly in non-trivial situations that T hardly lost its usefulness.

But in the 1950s, a new set of researchers, coming together from literally the four corners of the world, reexamined a century old paper by circuit pioneer Kirchoff and began a fresh approach, "graph theory." By the time the leader of the movement, S. Seshu met an untimely end in a 1964 car crash, the hidden trap inside Brune's analysis had become clear: allowing the T-operator mathematically prevented certain possible solutions from being found.

Kalman believes that Cauer might have come upon the result had he been in a Newtonian free-research environment, and not had to follow other research "to support his family."

Kalman suggested in follow-up questions that other areas of physics, including fluid dynamics, general relativity and quantum mechanics (with its paradoxical description of bodies as both particles and waves) might be in a situation where mathematics still had a way to go to create understanding of phenomena as rigorous as Newton's.

Judging by the enthusiastic applause from the more than 150 listeners who stayed for every word of an extremely dense and detailed presentation, many USC investigators are eager to take on the challenge.