CHAPTER 7: EUREKA MOMENT

The end of World War II had precipitated a bust to the aviation industry boom years of 24/7 factory aircraft assembly lines. Industry executives realized they would have to re-invent their companies to survive. Far-sighted executives such as those at North American Aviation understood that this new industry to become known as “aerospace” would be founded on technologies that World War II had either jump-started or promoted to a new level -- e.g., radar, propulsion systems, materials science, inertial guidance, and automatic control systems based upon servo-mechanisms. Application to their business of the war-time advances would require re-training both their current workforce and college hires, most of whose text books bore pre-war publication dates. 

Walter Evans’s wartime work experience at General Electric in Schenectady, New York and his classroom experience at Washington University in St. Louis made him an ideal new-hire for North American Aviation. He could bring to bear both his experience as a practicing engineer and as a teacher of the latest developments in servomechanisms -- a field critical to North American’s goals of achieving the dominant position in guided missiles. At General Electric, Evans had become intrigued by a graphical technique called “flux plotting” invented by a Swiss Professor Paul Profos. At Washington University he worked to extend the flux plotting idea with the encouragement of two mentors -- John R. Moore and Dr. Frank Bubb. Evans later enjoyed joking that Profos’s ideas “got me off the jw axis” -- a reference to the one-dimension focus of “frequency response” -- the reigning servomechanism analysis technique of the time. Upon his arrival at North American in June 1948, Evans shared with his new colleagues the techniques he had developed and received further encouragement from two in particular, W. D. (Bill) Mullins and R. M. Osborne.

And so, no sooner had Evans moved his family from St. Louis into a small Santa Monica apartment and its short commute to North American’s Aerophysics Laboratory on Aviation Boulevard in Inglewood, that he found himself teaching his new colleagues analysis of servo-mechanisms. His carefully hand-written and typewritten teaching notes preserve a record that nails the date of the birth of root locus to a lazy Monday -- August 23, 1948. It was his eighth or ninth class of the summer, and in Evans’s notes he lays out “several different methods for designing a servo-mechanisms, including the (1) frequency response method, (2) differential equations, (3) experimental, and (4) combinations -- itemizing the advantages and disadvantages associated with each method. His classroom goal was to achieve instruction tailored to the needs of each student. His subject that day was a question guaranteed to solicit strong re-actions. Although he himself had strong opinions, he sought to understand the reasons why others held different viewpoints. His course notes that day exemplify his sensitivity to diverse points of view.

The choice (referring to servo-mechanism design methods) depends not only on the problem at hand, but also upon the previous experience of the designer. Thus many men who are accustomed to thinking of servomechanisms in terms of examples that they have seen will probably lean heavily upon their own physical picture of their operation. For them, the output lag behind the input can be decreased by using “anticipation”. Other engineers, particularly those with extensive background in analysis of electric circuits, are apt to prefer the frequency response method. In this method the servo is described in terms of its response to a wide range of frequencies and corrective action is taken in terms of band-elimination filters. 

Finally a person who has had extensive experience with differential equations will tend to stay with this method and thus find himself devoting considerable effort to finding the roots of equations. Unfortunately, no one single method by itself is as useful as knowledge of all three. Any attempt to classify the advantages and disadvantages of each method is bound to be highly colored by opinion. 

His notes then proceed to acknowledge that he is not impartial on the subject at hand:With a warning to the reader that the writer has been involved in several rather heated discussions on this matter in on previous occasions, the following is an attempt to describe the merits of each.

The situation Evans described to his students was of a field still formalizing its methodologies. MIT’s Radiation Laboratory had been the pre-eminent center of excellence of the control-system universe during World War II. In 1948, its Instrumentation Laboratory (legendary Clark S. Draper, founder and director) and its Servo-Mechanism Laboratory (Gordon S. Brown, director) were applying so-called “frequency response” analysis techniques that use the Bode Plot and Nyquist Diagram, named after their pre-war developers --Harry Nyquist and Hendrik Bode of Bell Laboratories. MIT’s Gordon Brown, along with his associate Campbell, had just published the first seminal edition of Theory of Servo-Mechanisms(Wiley, 1948) in what would become a seminal textbook. 

Evans understood the frequency response method, but counted himself among those “who had extensive experience with differential equations (and) will tend to stay with this method and thus find himself devoting considerable effort to finding the roots of equations.” The equation Evans refers to is the “characteristic equation” of a control system. For simple feedback control systems, the characteristic equation is can be expressed as a quadratic algebraic equation (ax^2 + bx + c = 0), and as middle school algebra students are taught, its solutions or “roots” are obtained from a formula: x = [- b +/- root(b^2 - 4ac)]/2a). However, more complex feedback systems of interest to Evans and his colleagues -- those with two or more sources of delay -- are of a higher order than quadratic, and, as middle school students learn, no closed form solution exists. It may have been while Evans was explaining this point on August 23 that an unknown NAA engineer/student asked him 

“How large can the second time delay in a system be compared to the first one before the rules for a quadratic to be too much in error?” 

Evans would later refer to this case as the “slightly” cubic problem, the key word being “slightly”, for his teachers had taught him the importance of finding approximate solutions for complex problems. He later attributed his grappling with answering the questions as the trigger for his root locus technique. What is clear from the dates on his class notes is that the question inspired him, because those he prepared for classes two and three weeks later introduce root locus’s graphical approach to plotting roots for characteristic equations of any order. (A-1 through A-5). His student colleagues instantly preferred this new explanation to any they had seen before. Root locus was born. In a matter of just a few more weeks Evans prepared the draft paper of a paper and submitted it to the AIEE on November 1. North American published it as report AL-787. 

Evans’s September 1948 class notes, the earliest descriptions of root locus, strongly suggest that he had arrived at the root locus idea by using approximation techniques as his Washington University classes had emphasized. On the page of his September 7 notes that apply root locus to systems with multiple time delays (i.e., higher order than slightly cubic), he wrote

The above procedure represents a trick which is often handy: Simplify the problem down to one for which the answer is known, -- then change the answer slightly to allow for the extra complications. Another trick which is frequently handy in many problems (not just this particular graphical scheme) is to take the extreme case.

In a 1961 letter to his Washington University’s Engineering School’s former Dean, Alexander S. Langsdorf, Evans wrote

The General Electric Advanced Engineering Program put great emphasis on solving practical problems in approximate form starting from a few basic principles. ... I personally learn most effectively by starting from simple examples and working up. Washington University was excellent in that professors such as yourself, Professor Glasgow, Dr. Bubb, and Dr. Middlemiss could and did take a student all the way back to the beginning if necessary, and work up to the question at hand. 

Thus, the root locus idea was born in a classroom taught and attended by practicing engineers. The form of a question resonated with a teacher who sought to satisfy his individual student’s needs. That he developed the answer into a paper was due in considerable measure to the enthusiastic response to it of students. As practicing engineers, the students who learned a new problem solving technique were in a position to make use of it immediately. As they put Evans’s ideas to practical use, the ideas began to spread within the Aerophysics Lab at North American Aviation and beyond.