Sensitivity-Based Reference-Time Scaling for Local Orthogonalization of Fractional-Order Model Parameters

Fractional-order models are useful for describing systems with memory, anomalous relaxation, and non-classical dynamic behavior. However, parameter estimation in these models may be affected by strong covariance between the kinetic coefficient and the fractional order, reducing the independent interpretability of the estimated parameters. This work proposes a reference-time scaling strategy for an unforced fractional-order decay model and a forced fractional-order step response model, aiming to improve the local conditioning of the estimation problem by making the sensitivity vectors of the dimensionless coefficient and the fractional order locally orthogonal. The covariance structure is analyzed through the local sensitivity matrix, and a sensitivity-based expression for the reference time is derived to set the off-diagonal term of the approximate Gauss–Newton covariance matrix to zero, thereby reducing first-order linear dependencies. The methodology is evaluated using previously reported experimental data from Amiodarone plasma concentration-time profiles for fractional pharmacokinetic modeling and from the temperature response of a didactic thermal system to a step change in the manipulated variable for fractional-order system identification. The results show that the fitted trajectories, residual sums of squares, dimensional kinetic coefficients, and fractional orders remain invariant under reference-time scaling. Nevertheless, the choice of reference time strongly affects the covariance and correlation between the dimensionless parameter μ or κ and β, while the recovered dimensional coefficients m and k remain invariant. Conventional choices, such as the maximum, arithmetic mean, geometric mean, and harmonic mean of the experimental times, did not systematically reduce parameter correlation. In contrast, the proposed reference time, selected from the local sensitivity structure, reduced the first-order local correlation in both applications. The results indicate that reference-time scaling is a simple reparameterization tool that improves local statistical interpretability under the Gauss–Newton approximation of fractional-order parameter estimates without modifying the physical model or the quality of the fit.