Non-linear Gradient Algorithm for Parameter Estimation: Extended version

TL;DR

This paper extends gradient algorithms to non-quadratic cost functions in parameter estimation, demonstrating global convergence and finite-time stability, thus improving convergence speed over traditional linear methods.

Contribution

It introduces a novel non-linear gradient algorithm for non-quadratic cost functions, proving global convergence and finite-time stability under persistence of excitation.

Findings

Accelerates convergence compared to linear algorithms

Achieves finite-time and fixed-time stability

Ensures global uniform asymptotic convergence

Abstract

Gradient algorithms are classical in adaptive control and parameter estimation. For instantaneous quadratic cost functions they lead to a linear time-varying dynamic system that converges exponentially under persistence of excitation conditions. In this paper we consider (instantaneous) non-quadratic cost functions, for which the gradient algorithm leads to non-linear (and non Lipschitz) time-varying dynamics, which are homogeneous in the state. We show that under persistence of excitation conditions they also converge globally, uniformly and asymptotically. Compared to the linear counterpart, they accelerate the convergence and can provide for finite-time or fixed-time stability.