Robust Data-Driven Control of Discrete-Time Linear Systems with Errors in Variables

This work presents a (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
$\ell ^\infty$
bounded input, measurement, and process noise (error in variable setting). Certificates of full-state-feedback robust performance, superstabilization or quadratic stabilization 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 worst-case optimal
$\ell ^\infty$
(extended) superstabilizing controllers. With some conservatism, quadratically stabilizing controllers with certified
$\mathcal {H}_{2}$
performance bounds can also be found. 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
$\mathcal {H}_{2}$
control costs.