Robust Hammerstein Spline Adaptive Filtering for Nonlinear System Identification Under Impulsive Noise

Nonlinear dynamics are inherent in a wide range of industrial processes and mechatronic systems, where impulsive noise—arising from sensor faults, switching transients, or electromagnetic interference—poses serious challenges to accurate measurement, system identification, and subsequent control performance. To address these challenges, this work presents a robust Hammerstein spline adaptive filtering method, termed HSAF-lncosh-MVC, aimed at achieving reliable nonlinear system identification in impulsive environments. The nonlinear module employs an lncosh-based adaptation mechanism that constrains the update direction within a bounded gradient region, thereby ensuring stable parameter estimation. Concurrently, the linear module integrates a maximum versoria criterion (MVC) that suspends weight updates in the presence of pronounced outliers, thus improving resilience against impulsive disturbances. This hybrid strategy of nonlinear and linear components is implemented through stochastic gradient operations without introducing significant additional computational cost. Theoretical analysis establishes the algorithm’s convergence and steady state characteristics. Extensive simulations, covering Gaussian, symmetric α-stable, and Cauchy noise scenarios as well as complex nonlinear dynamical systems, demonstrate that the proposed method achieves accelerated convergence and reduced steady state error relative to conventional Hammerstein spline filters, while maintaining computational efficiency. These results highlight the potential of HSAF-lncosh-MVC as an effective tool for robust signal processing and nonlinear system identification in measurement and instrumentation systems operating under impulsive noise conditions.