Equivalence between Nonlinear Differential and Difference Equation Models Using Kernel Invariance Methods

Abstract

Actual modeling studies are based on discrete-time (sampled) data and yield discrete-time models, although the actual physiological processes may occur in continuous time. In the case of nonlinear parametric models (i.e., differential or difference equations), the physiological interpretation of the estimated model parameters may change considerably between discrete-time and continuous-time representations. Thus, methods for defining the equivalence between nonlinear parametric models in continuous-time (differential equations) and discrete-time (difference equations) are critically needed to assist in model interpretation. This paper presents the “kernel invariance method”, which is a conceptual extension of the “impulse invariance method” in linear system modeling. This method uses as a canonical representation the general Volterra model form of nonlinear systems and requires that the sampled continuous-time kernels (corresponding to the differential equation) be identical to the discrete-time kernels (corresponding to the equivalent difference equation). The actual implementation of this method may become unwieldy in the general case, but it appears to be tractable in certain cases of low-order nonlinear systems. Two illustrative examples of a quadratic system are presented that make use of 1st-order and 2nd-order kernel invariance.