Control of switched electrical networks using averaging on Lie groups

We apply the theory of averaging and motion control on Lie groups to the problem of controlling energy transfers between dynamic storage elements in switched electrical networks. The switched networks of interest have bilinear state-space models in which the control u, representing the position of the switch, takes value in the set {0, 1}. The corresponding state transition matrix can be described by a right-invariant system evolving on a matrix Lie group, and as such we can use our theory to derive high-order average approximations to the evolution of the state transition matrix. We show how to use these average solutions to control energy transfers for a simple network that models the conversion portion of a dc-dc converter. Our approach provides an alternative to the feedback approach of Sira-Ramirez (1987) which is based on variable structure systems with sliding regimes. Our methodology is based on open-loop control (combined with feedback if desired) and thus ensures that prescribed energy transfers can be accomplished with a finite number of switchings. This avoids chattering problems sometimes associated with sustaining sliding motions