Systems with associative dynamics

This paper introduces a class of nonlinear discrete-time dynamic models that generalize familiar linear model structures; our motivation is to explore the extent to which known results for the linear case do or do not extend to this nonlinear class. The results presented here are based on a complete characterization of the solution of the associative functional equation F[F(x, y),z] — F[x, F(y, z)] due to J. Aczel, leading to a class of invertible binary operators that includes addition, multiplication, and infinitely many others. We present some illustrative examples of these dynamic models, give a simple explicit representation for their inverses, and present sufficient conditions for bounded-input, bounded-output stability. Finally, we propose a generalization of this model class and we demonstrate that these models have classical state-space realizations, unlike arbitrarily structured NARMA models.