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Control of Switched Electrical Networks

Using Averaging on Lie Groups

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N.E. Leonard and P.S. Krishnaprasad

In * Proceedings of the 33rd IEEE Conference on Decision and
Control,* 1994, p.1919-1924.
#### Abstract

In this paper we apply the theory of averaging and motion control on
Lie groups \cite{NEEL_thesis} 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 \cite{Wood},
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 \cite{SiraRamirez87} 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.

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