A sliding-window approach for latent restoring force modeling

Abstract

Restoring force surface (RFS) methods offer an attractive nonparametric framework for identifying nonlinear restoring forces directly from data, but their reliance on complete kinematic measurements at each degree of freedom limits scalability to multidimensional systems. The aim of this paper is to overcome these measurement limitations by proposing an identification framework with relaxed sensing requirements that exploits periodic multisine excitation. Starting from an initial linear model, a sliding-window feedback approach reconstructs latent states and nonlinear restoring forces nonparametrically, enabling identification of the nonlinear component through linear-in-parameters regression instead of highly non-convex optimization. Validation on synthetic and experimental datasets demonstrates high simulation accuracy and reliable recovery of physical parameters under partial sensing and noisy conditions.

Figures (11)

• Figure 1: The nllfr model structure as a feedback interconnection of a lti system and a static nonlinear mapping.
• Figure 2: Schematic overview of the bla framework. For random excitation signals with a Gaussian distribution, the response of a nonlinear system is replaced by the sum of an lti approximation $G_{\text{BLA}}(q)$, unmodeled nonlinear dynamics $y_{S}(n)$, and a disturbing noise source $v(n)$.
• Figure 3: Visualization of $\mathcal{D}_{wz}$ and corresponding polynomial fit for three hyperparameter combinations across all noise levels. The inferred values, obtained by solving \ref{['eq:mhe_time']}, generally suggest the correct cubic nature of the restoring force, but the influence of noise and hyperparameters is evident.
• Figure 4: Grid search over the horizon length $H$ and regularization strength $\lambda$. The left column shows nonparametric nrmse between measured and simulated output signals, while the right column shows parametric nrmse of the corresponding polynomial fits. Red dots mark hyperparameter combinations where the nonparametric nrmse are close to their lower bounds; these combinations also yield the most accurate polynomial fits.
• Figure 5: Evolution of the output simulation nrmse and the linear stiffness parameter $k$ over the iterations of the final optimization step. It can be seen that the optimization procedure first prioritizes reducing the simulation error, followed by correcting the bias in the stiffness parameter.

Theorems & Definitions (7)

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