Recursive Identification of Nonlinear Hammerstein Systems With FIR Linear Parts Under Impulsive Noise With Model Selection and Order Determination

This paper proposes a new smoothly clipped absolute deviation (SCAD) regularized recursive identification algorithm for nonlinear Hammerstein systems having a finite duration impulse response (FIR) linear part in impulsive noise environment. It extends the conventional batch-mode over-determined two-stage algorithm (TSA) and alternating least squares (ALS) algorithms to recursive forms and integrates their advantages via a simplified model selection scheme, which significantly improves the convergence speed and steady-state mean squared error (MSE) performance. To facilitate tracking of time-varying systems, a variable forgetting factor (VFF) scheme is also incorporated to further improve its convergence and steady-state MSE performance. To improve the robustness against the possible impulsive noise, a robust statistics-based M-estimate objective function is employed to suppress the adverse effect of outliers. Moreover, the SCAD regularization is proposed for automatic model order determination, as it is asymptotically unbiased. An efficient implementation of the proposed algorithm is developed using QR decomposition, which simplifies the incorporation of the regularization and improves numerical stability. The convergence of the proposed robust regularized algorithm with VFF is also studied using the ordinary differential equation (ODE) approach. Computer simulations on stationary and non-stationary synthetic data as well as a real-world data for concentration and temperature prediction of a continuous stirred tank reactor demonstrate the effectiveness of the proposed algorithm over conventional algorithms. For the stirred tank reactor data, the proposed SCAD-VFF-TSA-ALS algorithm achieves the lowest mean absolute percentage errors (MAPEs) in both outputs, around 0.2237% in the concentration, and around 0.0140% in the temperature. Moreover, the proposed algorithm only requires $9pq+8p<^>{2}q<^>{2}+K(3p<^>{2}+3q<^>{2})+\mathcal {O}(p)+\mathcal {O}(q)$ arithmetic complexity per sample, where p is the order of the linear FIR part of the Hammerstein model and q is the order of the basis expansion of the nonlinearity. The conventional batch algorithm will require $\mathcal {O}(p<^>{3}q<^>{3})$ per sample for solving the least square problem plus a Singular Value Decomposition (SVD) with $\mathcal {O}(p<^>{3}q<^>{3})$ arithmetic complexity.