  In this paper, a constructive and systematic strategy with more apparent
degrees of freedom to achieve the accurate estimation of unknown parameters via
a control perspective is proposed. By adding a virtual control in the final
equation of the gradient dynamics, the Gradient Estimator (GE) and Memory
Regressor and Extension (MRE) approaches are extended. The solution of the
virtual control law is identified by the P&I approach. The P&I approach is
based on the choice of an appropriate implicit manifold and the generation of a
suitable passive output and a related storage function. This facilitates the
virtual control law being obtained in a way that the parametric error converges
asymptotically to zero. Because the above ideas connect with the P&I approach
and GE, the developed methodology is labeled the passivity and immersion-based
modified gradient estimator (MGE). The proposed P&I-based modified gradient
estimator is extended via the MRE approach. This modification provides improved
transient response and fast convergence. Based on certain PE and non-PE
examples, a comparative analysis is carried out to show the efficacy of the
proposed approaches.
