This article introduces the Tensor Network B-spline model for the regularized identification of nonlinear systems using a nonlinear autoregressive exogenous (NARX) approach. Tensor network theory is used to alleviate the curse of dimensionality of multivariate B-splines by representing the high-dimensional weight tensor as a low-rank approximation. An iterative algorithm based on the alternating linear scheme is developed to directly estimate the low-rank tensor network approximation, removing the need to ever explicitly construct the exponentially large weight tensor. This reduces the computational and storage complexity significantly, allowing the identification of NARX systems with a large number of inputs and lags. The proposed algorithm is numerically stable, robust to noise, guaranteed to monotonically converge, and allows the straightforward incorporation of regularization. The TNBS-NARX model is validated through the identification of the cascaded watertank benchmark nonlinear system, on which it achieves state-of-the-art performance while identifying a 16-dimensional B-spline surface in 4 seconds on a standard desktop computer. An open-source MATLAB implementation is available on GitHub.

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使用正则化张量网络 B 样条进行非线性系统识别

本文介绍了张量网络 B 样条模型，用于使用非线性自回归外生 (NARX) 方法对非线性系统进行正则化识别。张量网络理论用于通过将高维权重张量表示为低秩近似来缓解多元 B 样条的维数灾难。开发了一种基于交替线性方案的迭代算法来直接估计低秩张量网络近似值，从而无需明确构建指数级大权重张量。这显着降低了计算和存储复杂性，允许识别具有大量输入和滞后的 NARX 系统。所提出的算法数值稳定，对噪声具有鲁棒性，保证单调收敛，并允许直接合并正则化。TNBS-NARX 模型通过对级联水箱基准非线性系统的识别进行验证，在标准台式计算机上在 4 秒内识别 16 维 B 样条曲面的同时，它实现了最先进的性能。GitHub 上提供了开源 MATLAB 实现。