Making Sense of Nonlinearity: Parrilo's techniques offer hope for real world solutions

February 26, 2010

"Explained: Linear and nonlinear systems. Much scientific research across a range of disciplines tries to find linear approximations of nonlinear behaviors. But what does that mean?" -- Larry Hardesty, MIT News Office, February 26, 2010. Read more.

In the second of a two part series, the MIT News Office has explored the work of EECS faculty member Pablo Parrilo, the Finmeccanica Career Development Professor of Engineering and principal investigator in the Laboratory for Information and Decision Systems Laboratory, LIDS, on the understanding of nonlinear phenomena and ways of expressing them for real world application.

As described in part one of this series, Parrilo has developed a new set of techniques that make it easier to get a handle on nonlinear systems. Moreover, in many cases, his techniques provide algorithms — step-by-step instructions — for analyzing those systems, taking away much of the guesswork. (See the January 29, 2010 MIT News Office article).

In the second MIT News Office article Parillo states: “I think that it’s a reasonable statement that we mostly understand linear phenomena,” and he continues, “There’s this famous quote — I’m not sure who said it first — that the theory of nonlinear systems is like a theory of non-elephants.” Because linear equations are so much easier to solve than nonlinear ones, much research across a range of disciplines is devoted to finding linear approximations of nonlinear phenomena.

In the real world--of robots being developed by EECS Professor Russ Tedrake in the Computer Science and Artificial Intelligence Lab, CSAIL at MIT--Parrilo’s theoretical tools enable Tedrake to create novel control systems for his robots. A walking robot’s gait could be the result of a number of mechanical systems working together in a nonlinear way. The collective forces exerted by all those systems might be impossible to calculate on the fly; but within a narrow range of starting conditions, a linear equation might describe them well enough for practical purposes.

Parrilo’s theoretical tools allow Tedrake to determine how well a given linear approximation will work within how wide a range of starting conditions. His control system thus consists of a whole battery of linear control equations, one of which is selected depending on the current state of the robot.