An Adaptive RELAXation Method for Spectrum Identification in Active Vibration Control

Active vibration control (AVC) in marine and rotating machinery applications requires reliable identification of both the number and frequencies of discrete spectral lines embedded in noise and time-varying operating conditions. This article proposes an adaptive RELAXation (ARELAX) framework for multisinusoidal line-spectrum identification. The method integrates: 1) an iterative RELAX estimator that decouples closely spaced components by alternating single-component nonlinear least-squares updates and 2) a penalizing adaptive likelihood (PAL) criterion for model-order selection. PAL is constructed from generalized likelihood-ratio measures that quantify, respectively, the gain of increasing the order and the remaining gap to the maximum-order model, thereby enabling an adaptive penalty that is mild before the true order and becomes stringent beyond it. Simulation studies under abrupt frequency jumps, linear and nonlinear frequency sweeps, and amplitude variations, together with experimental validation on a cylindrical-shell vibration isolation platform, demonstrate that ARELAX provides improved robustness to nonstationary and low signal-to-noise ratio (SNR). Quantitative comparisons against short-time fast Fourier transform (STFT), multiple signal classification (MUSIC), and estimation of signal parameters via rotational invariance techniques (ESPRITs) are reported using global mean squared error (GMSE), steady-state mean squared error (SMSE), and runtime, showing that ARELAX achieves a favorable accuracy-complexity tradeoff for practical AVC deployment.