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

This article 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell ^\infty$</tex-math></inline-formula> 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell ^\infty$</tex-math></inline-formula> (extended) superstabilizing controllers. With some conservatism, quadratically stabilizing controllers with certified <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_{2}$</tex-math></inline-formula> 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_{2}$</tex-math></inline-formula> control costs.