A System Parametrization for Direct Data-Driven Analysis and Control with Error-in-Variables
Felix Brändle, Frank Allgöwer
TL;DR
This work addresses direct data-driven analysis and control for systems with error-in-variables by deriving an explicit linear fractional transformation (LFT) parametrization of all data-consistent systems using the Sherman–Morrison–Woodbury formula. It then formulate a single, data-length–independent semidefinite program (SDP) to compute a guaranteed upper bound on the unknown true system’s $\,\mathcal{H}_2$-norm, leveraging robust control techniques and a data-dependent feasibility condition via a right inverse $G$. The main contributions are the exact parametrization $\\Sigma_{\\Theta}^G$, the SDP-based $\,\mathcal{H}_2$ analysis, and the modular framework that accommodates controller synthesis and various noise models. The results demonstrate practical viability and show how the approach scales with data length and choice of $G$, providing reliable guarantees for direct data-driven analysis and control under measurement disturbances.
Abstract
In this paper, we present a new parametrization to perform direct data-driven analysis and controller synthesis for the error-in-variables case. To achieve this, we employ the Sherman-Morrison-Woodbury formula to transform the problem into a linear fractional transformation (LFT) with unknown measurement errors and disturbances as uncertainties. For bounded uncertainties, we apply robust control techniques to derive a guaranteed upper bound on the H2-norm of the unknown true system. To this end, a single semidefinite program (SDP) needs to be solved, with complexity that is independent of the amount of data. Furthermore, we exploit the signal-to-noise ratio to provide a data-dependent condition, that characterizes whether the proposed parametrization can be employed. The modular formulation allows to extend this framework to controller synthesis with different performance criteria, input-output settings, and various system properties. Finally, we validate the proposed approach through a numerical example.