Mathematical tools may help organize extremely large groups of systems with single controls

How do you control a large group of drones or robots so they do what you want, even if they each respond differently? 

That’s a problem Xudong Chen, associate professor of electrical & systems engineering in the McKelvey School of Engineering at Washington University in St. Louis, plans to solve with a three-year, $452,600 grant from the Air Force Office of Scientific Research.

Chen will investigate how to control numerous similar systems simultaneously using a finite set of controls. Researchers know how to control one system at a time yet are still addressing the challenging of controlling many systems together. Chen and his team want to solve the problem of feedback stabilization, or making the whole group reach a desired stable behavior and stay there.

To do so, he and his team plan to develop new mathematical tools to solve the problem, drawing from control theory, operator theory, perturbation theory, Lie theory and topology. The tools would be applicable to studying quantum control, robotics, machine learning and artificial intelligence.

“We would like to understand when systems can be controlled, when they cannot, and how to design better methods to do it,” Chen said.

They first plan to study linear ensemble systems to address stabilization. Using the tools in computational studies, they will extend their findings to networks and partial differential equations.

Secondly, they plan to take on nonlinear ensemble systems, investigating the relationship between stabilizing feedback and using control inputs to make a system’s state closer to zero over time. They will also determine whether there are mathematical barriers that stop the collection of systems from being stabilized together. 

Finally, Chen and his team expect to create new connections between controlling a large group of dynamical systems and estimating them using a finite number of measurements and thus closing the feedback loop. 

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