Estimation of Persistently Exciting Subspaces for Robust Parameter Adaptation

Recently we proposed the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mu$</tex-math></inline-formula>-modification for robust parameter adaptation, premised on the observation that only the parameter dynamics along the subspace with no persistent excitation must be rendered robust. Robustness thereby reduces to a problem of subspace estimation. This article proposes a new subspace estimator that recovers the non-persistently exciting (non-PE) subspace for a large class of regressors. This is achieved through a characterization of PE subspaces and the use of principal component analysis. Correctness of the design is proved using matrix perturbation theory, while an averaging analysis demonstrates that the design is best employed as a slow process. We develop a general error model capturing those commonly appearing in adaptive control, and we prove that the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mu$</tex-math></inline-formula>-modification provides a modular robust design, without compromising error regulation.