Rudolf Kalman: The Creator of Kalman Filters and Modern Control Theory

Rudolf Emil Kalman was an American mathematician of Hungarian descent. The Budapest-born scientist fled WW2 with his family in 1943 and immigrated to the U.S. He lived a long life dedicated to research mathematics until he passed away in 2016, leaving a rich legacy of applied mathematics in signal processing, control systems, and navigation.

Kalman earned his B.Sc degree in electrical engineering from the Massachusetts Institute of Technology (MIT) and subsequently attended postgraduate studies at Columbia University, where he received his doctoral degree and wrote his thesis “Analysis and Synthesis of Linear Systems Operating on Randomly Sampled Data” in 1958. 

 

Rudolf E. Kalman. Image courtesy of ETH

 

After a short research experience at IBM's research laboratory in Poughkeepsie, New York, Kalman took a position at the Research Institute for Advanced Studies in Baltimore, Maryland, where he produced some of his most significant work. 

In 1964, he left Baltimore to work as a professor of engineering mechanics, electrical engineering, and mathematical system theory at Stanford University. He spent his later academic years at the University of Florida, Gainesville, where he held positions as a graduate research professor in the departments of mathematics, electrical engineering, and industrial and systems engineering. Simultaneously, he was appointed the director of the Center for Mathematical System Theory and took yet another professorship at the Eidgenössische Technische Hochschule in Zürich. 

 

A Critic of Statistical Modeling

Kalman was a meticulous mathematical theorist, quoted thousands of times by his fellow colleagues. He was known to relentlessly question model accuracy, believing researchers should double and triple check whether they have dealt with real noisy data successfully or just projected their prejudices to the model.

 

Estimation of vehicle position using the Kalman filter. Image courtesy of Hindawi

 

His contributions to the space state concepts, including the notions of controllability, observability, the duality between control and estimation, minimality, linear-quadratic control, matrix Ricatti equations, and input/output realizability are now commonly used in control engineering.  

Because he was so deeply immersed in finding flaws in measurements and the representations of reality in models, Kalman criticized the concept of a statistical model as unscientific. On two occasions, he presented his thinking along these lines in two publications, called “Randomness and Probability” and “What is a Statistical Model.” 

Kalman argued that statistical models are just vague representations of reality because they are hypothetical guesswork lacking feedback from the reality they claim to present. He challenged the IID (independent and identically distributed) process because, as he said, “...nature does not seem to be like that.”

 

The Origin of Kalman Filters

Kalman filters detect signals in noise based on state space modeling and a recursive least-squares algorithm. These filters were a seminal improvement to the Wiener filtering model, which has proven impractical and difficult to apply. 

The Kalman filter algorithm uses data observed over time and polluted with noise and other inaccuracies to estimate unknown variables with greater accuracy. The algorithm sorts out navigation problems caused by raw data fed into computer control systems gathering multiple sensor measurements from gyroscopes, accelerometers, laser scanners, stereo cameras, and radars. Kalman filtering precisely calculates location, direction, and speed in the presence of noise.  

In the last 60 years, many improvements have been made to Kalman filters to adapt them to robotics and to correct filter consistency, convergence, and accuracy, expanding applications to autonomous navigation, economics, and biomedicine. 

 

A spacecraft updates its Kalman filter estimate at points in time and makes adjustments to stay on the right path. Image courtesy of Jack Trainer

 

The Kalman filtering algorithm provides the best estimate of the location and speed of moving objects. This is especially important for identifying the position of a person or an object within a few inches (approximately 10 cm) in space, such as with GPS trackers and smart objects in the IoT. Robotics has made the most use of Kalman filters, especially for parameter identification, robot control, and the autonomous navigation of mobile robots.

 

Awards and Recognition

Rudolf Kalman has received many of the highest prizes and awards obtainable for academic work in electrical engineering, including the IEEE Medal of Honor in 1974, the IEEE Centennial Medal in 1984, the Inamori foundation’s Kyoto Prize in Advanced Technology in 1985, the Richard E. Bellman Control Heritage Award in 1997, and the National Academy of Engineering’s Charles Stark Draper Prize in 2008. In 2009 he received the National Medal of Science from President Obama. 

 

President Obama awards the National Science Medal to Kalman. Image courtesy of the NSF

 

He has been elected to the American National Academy of Engineering, the National Academy of Sciences of the United States, and the American Academy of Arts and Sciences. Internationally, he has also been a member of the science academies in Hungary, France, and the former USSR. 

The American Mathematical Society awarded Kalman the Steele Prize for his contributions to three papers on the modern theory and practice of control systems: 

• “A new approach to linear filtering and prediction problems” (1960)
• “New results in linear filtering and prediction theory” (1961)
• “Mathematical description of linear dynamical systems” (1963)

He co-authored the last one with Richard S. Bucy. Hence, the Kalman recursive algorithm filters were once called Kalman-Bucy filters. The paper was initially met with skepticism. However, the NASA Ames Research Center in Mountain View later used the Kalman filter to estimate navigation for the Apollo spacecraft and make the moon landing mission a success. 

NASA later extended its work with Kalman's theory on non-linear systems and developed EKF (extended Kalman filters) based on Taylor’s series approximation. The algorithm was reformulated to process measurements at an arbitrary time interval and provide more accurate and reliable results.