Maximum Likelihood Iterative Algorithm for Hammerstein Systems with Hard Nonlinearities
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ISSN:1598-6446 eISSN:2005-4092 http://www.springer.com/12555
Maximum Likelihood Iterative Algorithm for Hammerstein Systems with Hard Nonlinearities Yan Pu, Yongqing Yang*, and Jing Chen Abstract: In this paper, we consider several iterative algorithms for Hammerstein systems with hard nonlinearities. The Hammerstein system is first simplified as a polynomial identification model through the key term separation technique, and then the parameters are estimated by using the maximum likelihood (ML) based gradient-based iterative algorithm. Furthermore, an ML least squares auxiliary variable algorithm and an ML bias compensation gradient-based iterative algorithm are developed to identify the saturation system with colored noise. Simulation results are included to illustrate the effectiveness of the proposed algorithms. Keywords: Gradient search, Hammerstein system, key term separation, maximum likelihood, saturation nonlinearity.
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INTRODUCTION
Nonlinear system modeling widely exists in system identification and has attracted a lot of attention in recent years [1, 2], inlcuding bilinear systems [3, 4]. The wellknown nonlinear systems include the Hammerstein model which is composed of a static nonlinear part followed by a linear time invariant (LTI) part, and the Wiener model is composed of a LTI part followed by a static nonlinear part [5, 6]. They are already widely used in the field of applications ranging from system modeling to chemical processes. Nonlinearities can be roughly divided into two types: the hard nonlinearity and the polynomial nonlinearity. Compared with the polynomial nonlinearity, the hard nonlinearity has various kinds of structures, which leads to the difficulties to choose a general model structure to represent data from the hard nonlinear system [7, 8]. The saturation nonlinearity, one kind of hard nonlinearity, is often encountered in engineer practice, and there exist lots of controller design methods for systems with saturation nonlinearity [9, 10]. Notice that a robust controller always has the assumption that the parameters of the nonlinear systems should be known in advance. However, there are a few literatures on parameter identification of hard nonlinear systems [11]. The focus of this paper is to develop some identification algorithms for such nonlinear systems with different kinds of noises. The existing estimation algorithms for nonlinear systems include the recursive algorithms [12], the iterative
algorithms [13] and the multi-innovation identification methods [14]. Among these algorithms, the ML algorithm has many optimal properties such as sufficiency, efficiency and consistency [15], which makes the ML algorithm be used widely in nonlinear system identification [16]. The idea of the ML algorithm is that a likelihood function can be constructed based on the input-output data and parameters, and then the estimators can be obtained by maximizing the likelihood functions. For instance, Schön and Wills provided a maximum likelihood method for nonlinear state-space systems [17],
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