Adaptive Parameter Estimation under Finite Excitation

TL;DR

This work addresses adaptive parameter estimation for nonlinear systems with unknown constant parameters under finite excitation (FE) rather than persistent excitation (PE). It introduces a Newton-based estimator (NBE) that uses pre-filtering to obtain a linear-in-parameters model and employs time-varying forgetting factor $l(t)$ and weight $\\ell(t)$ along with online information matrices $\\mathcal{M},\\mathcal{N}$ and a dynamic gain $\\Gamma$ to ensure boundedness and exponential convergence under FE. The convergence analysis relies on a Lyapunov function $V(\\tilde{\\Theta})=\\tfrac{1}{2}\\tilde{\\Theta}^T\\tilde{\\Theta}$ and shows that $\\Gamma\\mathcal{M}$ remains positive definite under FE (and PE), yielding exponential decay of the estimation error for $t$ beyond the excitation horizon; robustness to disturbances yields convergence to a bounded set. Numerical examples on a mass-spring-damper system illustrate superior convergence speed and robustness of the NBE under FE compared with gradient-, least-squares-, and robust-adaptive estimators, highlighting potential for faster, more reliable online parameter adaptation in practical settings.

Abstract

Although persistent excitation is often acknowledged as a sufficient condition to exponentially converge in the field of adaptive parameter estimation, it must be noted that in practical applications this may be unguaranteed. Recently, more attention has turned to another relaxed condition, i.e., finite excitation. In this paper, for a class of nominal nonlinear systems with unknown constant parameters, a novel method that combines the Newton algorithm and the time-varying factor is proposed, which can achieve exponential convergence under finite excitation. First, by introducing pre-filtering, the nominal system is transformed to a linear parameterized form. Then the detailed mathematical derivation is outlined from an estimation error accumulated cost function. And it is given that the theoretical analysis of the proposed method in stability and robustness. Finally, comparative numerical simulations are given to illustrate the superiority of the proposed method.

Figures (9)

• Figure 1: Comparative results of $\log_{10}{\|\tilde{\varTheta}}\|$ under the PE condition
• Figure 2: Comparative results of $\tilde{\varTheta}$ under the PE condition
• Figure 3: Comparative results of $\log_{10}{\|\tilde{\varTheta}}\|$ under the FE condition
• Figure 4: Comparative results of $\tilde{\varTheta}$ under the FE condition
• Figure 5: Comparative results of $\log_{10}{\|\tilde{\varTheta}}\|$ under the PE condition in presence of disturbance