  Linear models with additive unknown-but-bounded input disturbances are
extensively used to model uncertainty in robust control systems design.
Typically, the disturbance set is either assumed to be known a priori or
estimated from data through set-membership identification. However, the problem
of computing a suitable input disturbance set in case the set of possible
output values is assigned a priori has received relatively little attention.
This problem arises in many contexts, such as in supervisory control, actuator
design, decentralized control, and others. In this paper, we propose a method
to compute input disturbance sets (and the corresponding set of states) such
that the resulting set of outputs matches as closely as possible a given set of
outputs, while additionally satisfying strict (inner or outer) inclusion
constraints. We formulate the problem as an optimization problem by relying on
the concept of robust invariance. The effectiveness of the approach is
demonstrated in numerical examples that illustrate how to solve safe reference
set and input-constraint set computation problems.
