A consequent and consistent continuous-time approach to system parameter estimation is introduced. Estimation algorithms, the underlying quality criteria and models of identified systems are described in the continuous-time domain, while suitable discretising operations are performed solely for the purpose of ultimate numerical realisation of estimation procedures. The considered indices of estimation quality take the form of integrals of absolute prediction errors rather than a common form of integrals or sums of square errors. In order to overcome the problem of analytical minimisation of such non-differentiable criteria, an approximate method is derived and applied in practical implementation of the resultant estimation schemes. Specific weighting mechanisms utilised in the algorithms allow for tracking the time-variant parameters of non-stationary systems, while with the employed instrumental variable the accuracy of estimates gets improved by means of suppression of the asymptotic bias. Following the so-called direct approach, an auxiliary discrete-time model that retains ‘physical’ parameterisation is obtained based on ‘finite-horizon’ spline-based integration of both sides of the presumed differential equation. In this aspect, application of splines makes the respective discrete-time processing resistant to cumulation of numerical errors. The attached numerical examples demonstrate the performance of the discussed estimation routines.