  This work presents a sum-of-squares (SOS) based framework to perform
data-driven stabilization and robust control tasks on discrete-time linear
systems where the full-state observations are corrupted by L-infinity bounded
input, measurement, and process noise (error in variable setting). Certificates
of state-feedback superstability, quadratic stability or positive stability of
all plants in a consistency set are provided by solving a feasibility program
formed by polynomial nonnegativity constraints. Under mild compactness and
data-collection assumptions, SOS tightenings in rising degree will converge to
recover the true superstabilizing or positive stabilizing controller, with some
conservatism introduced for quadratic stabilizability. The performance of this
SOS method is improved through the application of a theorem of alternatives
while retaining tightness, in which the unknown noise variables are eliminated
from the consistency set description. This SOS feasibility method is extended
to provide worst-case-optimal robust controllers under H2 control costs. The
consistency set description may be broadened to include cases where the data
and process are affected by a combination of L-infinity bounded measurement,
process, and input noise.
