Section: New Results

Integral control on Lie groups

Participant : Alain Sarlette.

A big challenge for the long-term control of interacting networks is their robustness to systematic biases. Integral control is a standard way to counter them when a target output can be measured. This method has been originally proposed, and extensively studied, for linear systems. However when the system (output) evolves on a nonlinear state space, the standard "integration" technique cannot be straightforwardly applied. Especially for global motions on spaces like the circle, sphere or (real or complex) rotation groups, the output integration viewpoint becomes problematic. We have hence proposed a new viewpoint on integral control, based on integrating the intended input [19] . For linear state spaces, it is equivalent to the standard definition. For nonlinear state spaces, this viewpoint can be transposed verbatim modulo introduction of a transport map on the tangent bundle, which is almost always present for control design purposes. In particular for systems on Lie groups, which are ubiquitous in robotics and in quantum physics, a full analysis of fully actuated systems has been proposed. The more challenging extension to underactuated systems is underway.