The paper presents a procedure to estimate the unknown values of known nonlinear mappings, which arguments are unknown parameters of a linear regression equation (LRE). Archetypical examples of such mappings are analytical equations to calculate the parameters of full information-based ideal control laws. So, it is possible to apply developed estimation procedure to the general tasks of adaptive control of linear and nonlinear affine systems, the right-hand side of which is factorizable in the LRE form. Particularly, the proposed procedure allows obtaining a fundamentally new scheme of adaptive control, which does not require: (i) existence of an adaptive control Lyapunov function and/or to meet the restrictive matching conditions, (ii) any a priori information about the parameters of the nonlinear system. These conditions are replaced by the assumption of a) existence of an asymptotically stabilizing controller, which parameters depends from unknown plant parameters in a manner that allows one to apply a proposed estimation procedure, b) identifiability of plant parameters. The obtained theoretical results are validated via simulations.