# November 7, 2014, Roger Brockett, CISE Resident Scholar, Harvard University

Friday, November 7, 2014 at 3:00 PM to 4:00 PM
8 St. Mary’s Street, Room 210

Refreshments served at 2:45.

Roger Brockett
CISE Resident Scholar, Harvard University

## Nonlinear Integral Control:  Obtaining Infinite Gain at Zero Frequency When No Linear Relationship Suffices

We determine the controllability properties of systems of the form $\dot{x}=Ax+Bu\;;\; \dot{w}=q(x)$ with $q$ being a vector of quadratic functions of $x$. This has a direct relationship with many aspects of linear quadratic theory (passivity, optimality, etc.) but is also interesting because it is both remarkably tractable and because it is the second order approximation to a large class of nonlinear systems.   We not only give a complete description of the distribution generated by the vector fields associated with this system but, in important cases,  we are able to give a precise description of which points are reachable from a given initial state, distinguishing between those points that are reachable immediately and those that are only reachable after a sufficiently long period of time. Finally, the theory plays a crucial role in problems of the type to be discussed in the second lecture of this series.

Roger Brockett is An Wang research professor of Electrical Engineering and Computer Science at Harvard University. He was a student at Case Institute of Technology and did his Ph.D. work under the supervision of Mihajlo D. Mesarovic, in the Systems Research Center then led by Donald P. Eckman. Prior to joining the Harvard faculty in 1969, he taught for six years in the Electrical Engineering department at MIT, where he developed the textbook Finite Dimensional Linear Systems and involved graduate students in a range of topics centering on stability theory and applications. At Harvard, working along side of Y.C. Ho and an outstanding group of younger colleagues, he initially focused on the theory and applications of nonlinear systems emphasizing the use of differential geometric ideas. In the mid 1980s, fostered in part by the new NSF Engineering Research Center imitative and the ARO MURI program administered by Jagdish Chandra, the focus of his research turned to the application of control theoretic ideas to problems in robotics, computer vision and other aspects of intelligent machines. An important part of this transition was the development of a broadly inclusive robotics laboratory, engaging a number of Harvard faculty members as well as involving, long-term collaborations with colleagues and former students at Brown University, the University of Maryland, and MIT. His teaching has involved the development of courses for engineering students, ranging from a freshman design course to graduate level teaching across the field of control. In many cases his former Ph.D. students and post doctoral researchers have gone on to become leaders in their fields with their accomplishments being recognized, not only because of their work in their “day jobs” as teachers, researchers and managers, but also through their contributions to the management and editorial work of various IEEE societies, SIAM journals, etc.

Faculty Host: John Baillieul
Student Host: Shuai Wang