Hermite Interpolatory Operator Framework for Stability-Constrained Transitions Between Integer-Order Dynamical Systems

Abstract

This paper develops the theoretical framework of the Hermite interpolatory derivative operator for constructing local paths between classical integer-order dynamical models. The main idea of this work is to interpolate not only the values of the operators at integer-order nodes but also the direction of change with respect to the tuning parameter. This additional Hermite condition allows the transition between models to be controlled while keeping the operator local, finitedimensional, and based only on ordinary derivatives. We prove exact recovery at the prescribed nodes, derive the polynomial Laplace symbol of the operator, and show that the resulting linear models have rational transfer functions. We also obtain a pole-velocity formula that links the chosen transition operators with the first-order motion of simple poles. This gives the framework a useful stability interpretation. Stability windows are then described using classical Routh-Hurwitz conditions, with special attention to possible order reduction when a leading coefficient vanishes. The proposed construction is not a fractional derivative and does not aim to reproduce memory effects. Instead, it provides a structured way to build rational, stability-aware transitions between integer-order models, electrical circuits, mechanical damping models, or model reduction where one wants rational transfer functions and local integer-order operators. A concrete transition matrix is computed with an identified stability window and representative pole trajectories as an illustrative example to show how the theoretical construction can be used in practice.