# A New Orthogonal Least Squares Identification Method for a Class of Fractional Hammerstein Models
###### Abstract  
It is known that fractional-order models can effectively represent complex high-order systems with fewer parameters. This paper focuses on the identification of a class of multiple-input single-output fractional Hammerstein models. When the commensurate order is assumed to be known, a greedy orthogonal least squares method is proposed to simultaneously identify the parameters and system orders, combined with a stopping rule based on the Bayesian information criterion. Subsequently, the commensurate order is determined by minimizing the normalized output error. The proposed method is validated by applying it to identify a CD-player arm system  
fractional Hammerstein systems, particularly those in which the linear component exhibits fractional characteristics.  
A wide range of methodologies have been proposed in the literature for the identification of classical integer-order Hammerstein systems [11; 12; 13; 14; 15]. For the fractional case, an adapted version of the simplified refined instrumental variable method has been introduced to estimate the parameters of fractional Hammerstein models under the assumption that all differentiation orders are known, with Monte Carlo simulations employed to evaluate the performance of the proposed approaches in [16]. [PERSON] et al. derived an adaptive immune algorithm based on a global search strategy, which was employed to determine initial values for the fractional Hammerstein model's coefficients and order, which were then refined using an auxiliary model recursive least squares method [17]. [PERSON] et al. developed an iterative linear optimization algorithm integrated with Lyapunov stability theory, which was applied to determine the fractional order and parameters of a neuro-fractional Hammerstein model [18]. Meanwhile, ref. [19] utilizes principal component analysis within a subspace identification framework to identify coefficient matrices of fractional systems and employs singular value decomposition to directly estimate the parameters of the nonlinear part. A Levenberg-Marquardt algorithm is applied to determine system parameters, where the linear part is represented by a fractional transfer function [20]. However, these studies either involve matrix inversion operations, which typically result in high computational complexity, or focus solely on parameter identification while neglecting the estimation of the fractional order.  
In this paper, a Householder transformation-based greedy orthogonal least squares (H-GOLS) algorithm is proposed for the identification of multi-input single-output (MISO) fractional Hammerstein models. The proposed methodology initiates by reformulating the system into a pseudo-linear regression model via over-parameterization, introducing upper bounds on the nonlinear order and regression length. The parameters are subsequently estimated using orthogonal least squares. In each iteration, the Householder transformation is applied to the information matrix, converting it into an upper triangular matrix. This approach eliminates the need for matrix inversion, thereby significantly reducing computational complexity and mitigating the risk of ill-conditioned solutions in high-dimensional matrices. The selection of columns in each iteration is simultaneously guided by a greedy criterion. Additionally, we adopt a modified stopping criterion based on the Bayesian information criterion (BIC) to determinate the sparsity level and model order. Finally, an output error criterion function is introduced to estimate the approximate fractional order.  
The remainder of this paper is organized as follows. Section 2 discusses the mathematical background on fractional differentiation, the structure of the fractional Hammerstein system with multiple inputs, and its identification problem formulation. Section 3 derives an algorithm for identifying the MISO fractional Hammerstein systems. Section 4 applies two examples to support the proposed algorithm. Finally, some conclusions are given in Section 5.