On DREM regularization and unexcited linear regression estimation

The problem of estimation of unknown constant parameters in the linear regression with measurement noise is considered. Analysing different levels of excitation of the regressor, two notions of partial and feeble excitation are introduced. The former implies the absence of the persistent or interval excitation, while the latter property says that the excitation is just insufficient for an efficient estimation in a noisy setting. The dynamic extension and mixing method (DREM) is used for the problem solution, and in order to improve its estimation performance, regularization is proposed and the resulting improvement is investigated analytically. The theoretical findings are illustrated in the simulations.