Guaranteed Error Bounds on Approximate Model Abstractions through Reachability Analysis
This work provides a practical framework for robust model abstraction in dynamical systems, addressing sensitivity issues in exact equivalence for engineers and scientists.
The authors introduce approximate differential equivalence for nonlinear ODEs with polynomial derivatives, enabling model reduction with guaranteed error bounds computed via reachability analysis. They demonstrate the method on symmetric electric circuits, providing formal certificates on approximation quality.
It is well known that exact notions of model abstraction and reduction for dynamical systems may not be robust enough in practice because they are highly sensitive to the specific choice of parameters. In this paper we consider this problem for nonlinear ordinary differential equations (ODEs) with polynomial derivatives. We introduce approximate differential equivalence as a more permissive variant of a recently developed exact counterpart, allowing ODE variables to be related even when they are governed by nearby derivatives. We develop algorithms to (i) compute the largest approximate differential equivalence; (ii) construct an approximate quotient model from the original one via an appropriate parameter perturbation; and (iii) provide a formal certificate on the quality of the approximation as an error bound, computed as an over-approximation of the reachable set of the perturbed model. Finally, we apply approximate differential equivalences to study the effect of parametric tolerances in models of symmetric electric circuits.