Modeling nonlinear systems using the tensor network B-spline and the multi-innovation identification theory

The nonlinear autoregressive exogenous (NARX) model shows a strong expression capacity for nonlinear systems since these systems have limited information about their structures. However, it is difficult to model the NARX system with a relatively high dimension by using the popular polynomial NARX and the neural network efficiently. This article uses the tensor network B-spline (TNBS) to model the NARX system, whose representation of the multivariate B-spline weight tensor can alleviate the computation and store burden for processing high-dimensional systems. Furthermore, applying the multi-innovation identification theory and the hierarchical identification principle, the recursive algorithm by combining the l2$$ {l}_2 $$-norm is proposed to the NARX system with Gaussian noise. Because of the local adjustability of the B-spline curve, the TNBS can fit nonlinear systems with strong nonlinearity by the meaning of setting a proper degree and knots number. Finally, a numerical experiment is given to demonstrate the effectiveness of the proposed algorithm.

Automated summary

This article presents a method for modeling NARX systems using tensor network B-spline (TNBS), which reduces the computational and storage burden for high-dimensional systems through the representation of multivariate B-spline weight tensors. The recursive algorithm proposed for NARX systems with Gaussian noise combines the multi-innovation identification theory and the hierarchical identification principle, using the l2-norm. TNBS can fit nonlinear systems with strong nonlinearity by adjusting the degree and number of knots. A numerical experiment is conducted to demonstrate the effectiveness of the algorithm.