**1. Introduction**

The accuracy of a system model affects the performance and safety of industrial control systems [1–5], and system identification is a theory and method for constructing mathematical model of systems and has been widely implemented in practice [6–9]. The behavior of most modern industrial control systems and synthetic systems are nonlinear by nature. Presently, an important research field in modern signal processing is the research of parameter identification for nonlinear systems, in which the block-structure systems, such as the Hammerstein model, are among the most current nonlinear systems due to their efficiency and accuracy to model complex nonlinear systems [10–12]. The representative feature of a Hammerstein model is that its architecture consists of two blocks: a static nonlinear model followed by a linear dynamic model. The simplicity in structure makes it provide a good compromise between the accuracy of nonlinear systems and the tractability of linear systems, and thus promoting its use in different nonlinear applications such as automatic control [13–15], fault detection and diagnosis [16–18], and so on.

Recently, several new system identification methods and theories have been developed for nonlinear models in the literature, including the least squares methods [19], the gradientbased methods [20], the iterative methods [21],the subspace identification methods [22], the hierarchical identification theory [23], the auxiliary model and the multi-innovation (MI) identification theories [24]. One well-known algorithm is the stochastic gradient (SG) algorithm, which has lower computational cost and complexity than the recursive least squares algorithm, whereas slow-convergence phenomena are often observed. Therefore, different modifications of the SG algorithm were developed to enhance its performance [25–30]. In particular, by extending scalar innovation into innovation vectors, the MI identification theory was proposed to improve the convergence speed and estimation accuracy in [31], and the fractional-order calculus method was introduced to show that it can achieve more satisfactory performance in [32,33].

**Citation:** Xu, C.; Mao, Y. Auxiliary Model-Based Multi-Innovation Fractional Stochastic Gradient Algorithm for Hammerstein Output-Error Systems. *Machines* **2021**, *9*, 247. https://doi.org/10.3390/ machines9110247

Academic Editors: Hongtian Chen, Kai Zhong, Guangtao Ran and Chao Cheng

Received: 29 August 2021 Accepted: 21 October 2021 Published: 23 October 2021

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To the best of our knowledge, different fractional-order gradient methods have been produced [34–36]. For example, in [37], a fractional-order SG algorithm was designed to identify the Hammerstein nonlinear ARMAX systems by an improved fractional-order gradient method. Based on the MI theory and the fractional-order calculus, an MI fractional least mean squares identification algorithm was presented for the Hammerstein controlled autoregressive systems, where the update mechanism was composed of the first-order gradient and the fractional gradient [38]. However, the above-discussed papers only consider the Hammerstein equation-error systems, and the cross-products between the parameters in the linear block and nonlinear block can lead to many redundant parameters. When the dimensions of parameter vectors are large, it will cause high computational complexity and deteriorate the identification accuracy.

In this work, we study the identification problem of the Hammerstein output-error moving average (OEMA) systems, which have been less studied due to the difficulty in identification [39,40]. To avoid estimating the redundant parameters, the Hammerstein model is parameterized using the key-term separation principle [41]. Furthermore, based on the identification model, the fractional-order SG algorithm is extended to the identification of Hammerstein OEMA systems and an auxiliary model-based multi-innovation fractional stochastic gradient (AM-MIFSG) algorithm is presented by the auxiliary model identification idea. The proposed algorithm can generate higher estimation accuracy than the common multi-innovation stochastic gradient (MISG) algorithm, with fewer parameters required to be estimated.

The paper is structured as follows. Section 2 gives a description for Hammerstein OEMA systems. Section 3 introduces the multi-innovation identification theory and drives an auxiliary model-based multi-innovation stochastic gradient (AM-MISG) identification algorithm for a comparison purpose. Section 4 presents the AM-MIFSG identification algorithm for the Hammerstein OEMA systems. Section 5 gives the convergence analysis of the proposed AM-MIFSG algorithm. Section 6 verifies the results in this paper using a simulation example. Finally, concluding remarks are given in Section 7.
