#
A Robust Control Framework for Linear, Time-Invariant,
Spatially Distributed Systems

## J. Reinschke, M.W. Cantoni and M.C. Smith

## Abstract

A robust control framework for linear, time-invariant,
spatially-distributed systems is outlined in this paper. We adopt an
input-output approach which takes account of the spatially-distributed
nature of the input and output signals for such systems. The approach
is a generalisation of H-infinity control in the sense that the 2-norm
(in both time and space) is used to quantify the size of signals. It
is shown that a frequency-domain representation, in the form of a
graph symbol, exists for every linear, time-invariant,
spatially-distributed system under very mild assumptions. The graph
symbol gives rise to left and right coprime representations if the
system is also stabilisable. We investigate fundamental issues of
feedback control such as feedback stability and robust stability to
plant and/or controller uncertainty quantified in the gap-metric. This
includes a generalisation of the Sefton-Ober gap formula to the
infinite-dimensional operator case. A design example in which an
electrostatically destabilised membrane is feedback-stabilised
concludes the paper.