  In an open-loop experiment, an input sequence is applied to an unknown linear
time-invariant system (in continuous or discrete time) affected also by an
unknown-but-bounded disturbance sequence (with an energy or instantaneous
bound); the corresponding state sequence is measured. The goal is to design
directly from the input and state sequences a controller that enforces a
certain performance specification on the transient behaviour of the unknown
system. The performance specification is expressed through a subset of the
complex plane where closed-loop eigenvalues need to belong, a so called LMI
region. For this control design problem, we provide here convex programs to
enforce the performance specification from data in the form of linear matrix
inequalities (LMI). For generic LMI regions, these are sufficient conditions to
assign the eigenvalues within the LMI region for all possible dynamics
consistent with data, and become necessary and sufficient conditions for
special LMI regions. In this way, we extend classical model-based conditions
from a seminal work in the literature to the setting of data-driven control
from noisy data. Through two numerical examples, we investigate how these
data-based conditions compare with each other.
