Time and place: Friday, May 14, 10:50-11:50, 202 Chernoff HallTitle: Reach Control Problem (1 hour)Abstract: The field of control theory has focused historically on
problems of tracking and regulation. An example of regulation would be
setting a desired speed in a vehicle cruise control system. Optimizing a
solar cell performance by rotating its surface to continually face the sun
during daylight hours is an example of tracking. Tracking and regulation are
very simple control system specifications. As control systems become more
integrated within high-end engineering systems as well as consumer products,
they are expected to achieve richer specifications that consider safety
requirements, startup procedures, and so forth. Good examples of complex
specifications would be the startup of a chemical or nuclear reaction,
requirements for an automatic parking system, or transition between different
gaits in a bipedal robot.
At the core of the research thrust to enrich control specifications is the Reach Control Problem. This problem is formulated on polyhedra or simplices in the state space and is focused on the class of affine systems. Roughly speaking, the problem is for an affine system defined on a simplex to reach a prespecified facet of the simplex in finite time without first exiting the simplex. The significance of the problem stems from its capturing the essential features of reachability problems for control systems: the presence of state constraints and the notion of trajectories reaching a goal in a guided and finite-time manner. Formulating such problems on a simplex is motivated by the fact that the simplex requires the minimal data set to define bounded n-space. Therefore, a deep
study of this problem will inform on many related and more complex problems
which have not been treated in a systematic, theoretical way.
In this talk, I will give an overview of results on the reach control problem. I will begin with basic principles and then move on to discussing what is known about solvability by affine feedbacks, continuous state feedbacks, and open-loop controls. |