Core Concepts
A framework for adapting continuous-time optimization algorithms to handle time-varying cost functions by incorporating a derivative estimation scheme, providing robust performance guarantees.
Abstract
The paper presents a framework for adapting continuous-time optimization algorithms to handle time-varying cost functions. The key insights are:
Lemma 1 shows how a continuous-time optimization algorithm that is input-to-state stable (ISS) for static cost functions can be adapted to handle time-varying cost functions, provided the time variations are known in the form of the derivative of the parameter vector.
Theorem 1 introduces a novel derivative estimation scheme based on the "dirty-derivative" concept, and provides explicit input-to-output stability (IOS) bounds on the estimation error.
Theorem 2 combines the results of Lemma 1 and Theorem 1 to show that the interconnection of the adapted optimization algorithm and the derivative estimator also results in an IOS system, providing robust performance guarantees for tracking the time-varying minimizer.
The framework allows incorporating time-variation information without requiring explicit knowledge of the derivative, and the performance can be improved by tuning the estimator's gain parameter. Simulation results demonstrate the effectiveness of the approach in a time-varying optimization task.