Robust and fast backbone tracking via phase-locked loops

TL;DR

The paper presents a systematic, model-free approach to fast and robust backbone tracking in nonlinear structures using phase-locked loops with an LMS-based adaptive phase detector. By applying a slow-time-scale averaging framework, it reduces the plant to a single nonlinear modal oscillator and derives closed-form PI gains that optimally balance speed and robustness of phase locking. The adaptive filter eliminates hold times and yields real-time estimates of phase and amplitude-dependent modal properties, validated by numerical Duffing simulations and a beam experiment showing substantial speedups over heuristic PLL designs. The method significantly improves robustness and speed of backbone tracking, enabling direct progression along the backbone with minimal pre-test modeling and without extensive post-processing. Practically, the approach reduces test time to roughly a few hundred cycles per backbone point and is readily implementable on standard shaker-based modal test setups.

Abstract

Phase-locked loops are commonly used for shaker-based backbone tracking of nonlinear structures. The state of the art is to tune the control parameters by trial and error. In the present work, an approach is proposed to make backbone tracking much more robust and faster. A simple PI controller is proposed, and closed-form expressions for the gains are provided that lead to an optimal settling of the phase transient. The required input parameters are obtained from a conventional shaker-based linear modal test, and an open-loop sine test at a single frequency and level. For phase detection, an adaptive filter based on the LMS algorithm is used, which is shown to be superior to the synchronous demodulation commonly used. Once the phase has locked, one can directly take the next step along the backbone, eliminating the hold times. The latter are currently used for recording the steady state, and to estimate Fourier coefficients in the post-process, which becomes unnecessary since the adaptive filter yields a highly accurate estimation at runtime.The excellent performance of the proposed approach is demonstrated for a doubly clamped beam undergoing bending-stretching coupling leading to a 20 percent shift of the lowest modal frequency. Even for fixed control parameters, designed for the linear regime, only about 100 periods are needed per backbone point, also in the nonlinear regime. This is much faster than what has been reported in the literature so far.

Figures (12)

• Figure 1: Schematic of considered problem setting.
• Figure 2: Model of the plant: (left) electrical and (middle) mechanical part of the exciter; (right) structure under test.
• Figure 3: Phase error (a) and frequency shift (b) over time for a unit initial frequency offset. The point labeled relevant extremum is used to define the optimal ${\lambda}_\mathrm{I}$. $\bar{\delta_{\mathrm{p}}} = 10^{-2}$, $\mu_{\mathrm{ex}}=0$ (implying $\delta_{\mathrm{p}}/\delta_{\mathrm{s}}=1$).
• Figure 4: Optimum ${\lambda}_\mathrm{I}$ leading to the proposed trade-off between control error and frequency overshoot during a step along the backbone. $\bar{\delta_{\mathrm{p}}}$ is the normalized decay rate.
• Figure 5: Simulated excitation level step leading to a frequency shift from $1.01\omega_{\mathrm{lin}}$ to $1.02\omega_{\mathrm{lin}}$: time evolution of (a) displacement, (b) excitation frequency, and (c) phase error. (d) is a zoom to the blue box in (c). The legend in (c) applies to all sub-figures. The black dots in (b) depict the natural frequency $\omega(a(t))/\omega_{\mathrm{lin}}$ computed based on harmonic balance using the instantaneous amplitude estimated by the adaptive filter. Control gains $k_{\mathrm{i}}$, $k_{\mathrm{p}}$ were chosen according to the proposed design procedure for each damping value. As time variable, the number of linear periods is used, $\omega_{\mathrm{lin}}t/(2\pi)$. $\omega_{\mathrm{LP}}/\omega_{\mathrm{lin}}=0.1$.