Robust multivariable PI-controller for infinite dimensional systems

A robust multivariable controller is introduced for a class of distributed parameter systems. The system to be controlled is given as
\dot{x} = Ax + Bu, y = Cx
in a Banach space. The purpose of the control, which is based on the measurement
y
, is to stabilize and regulate the system so that
y(t) \rightarrow y_{r},
as
t \rightarrow \infty
, where y
r
is a constant reference vector. Under the assumptions that operator
A
generates a holomorphic stable semigroup,
B
is linear and bounded,
C
is linear and
A
-bounded, and the input and output spaces are of the same dimension; a necessary and sufficient condition is found for the existence of a robust multivariable controller. This controller appears to be a multivariable PI-controller. Also, a simple necessary criterion for the existence of a decentralized controller is derived. The tuning of the controller is discussed and it is shown that the I-part of the controller can be tuned on the basis of step responses, without exact knowledge of the system's parameters. The presented theory is then used as an example to control the temperature profile of a bar, with the Dirichlet boundary conditions.