Implicit Function Theorem: Estimates on the size of the domain

TL;DR

This paper provides explicit, easy-to-compute estimates for the domain size where the Implicit and Inverse Function Theorems hold, with applications in power systems, control theory, and feedback linearization.

Contribution

It introduces minimal-computation bounds for the theorem's domain size, applicable to various systems without optimization procedures.

Findings

Bounds depend on first and second derivatives at a point

Applications include power flow stability, Riccati equation robustness, and control system linearization

Estimates are computationally efficient and easy to apply

Abstract

In this article, we present explicit estimates of the size of the domain on which the Implicit Function Theorem and the Inverse Function Theorem are valid. For maps that are twice continuously differentiable, these estimates depend upon the magnitude of the first-order derivatives evaluated at the point of interest, and a bound on the second-order derivatives over a region of interest. One of the key contributions of this article is that the estimates presented require minimal numerical computation. In particular, these estimates are arrived at without any intermediate optimization procedures. We then present three applications in optimization and systems and control theory where the computation of such bounds turns out to be important. First, in electrical networks, the power flow operations can be written as Quadratically Constrained Quadratic Programs (QCQPs), and we utilize our bounds to compute the size of permissible power variations to ensure stable operations of the power system network. Second, the robustness margin of positive definite solutions to the Algebraic Riccati Equation (frequently encountered in control problems) subject to perturbations in the system matrices are computed with the aid of our bounds. Finally, we employ these bounds to provide quantitative estimates of the size of the domains for feedback linearization of discrete-time control systems.