Abstract
A cubic spline approximation-Bayesian composite quantile regression algorithm is proposed to estimate parameters and structure of the Wiener model with internal noise. Firstly, an ARX model with a high order is taken to represent the linear block; meanwhile, the nonlinear block (reversibility) is approximated by a cubic spline function. Then, parameters are estimated by using the Bayesian composite quantile regression algorithm. In order to reduce the computational burden, the Markov Chain Monte Carlo algorithm is introduced to calculate the expectation of parameters’ posterior distribution. To determine the structure order, the Final Output Error and the Akaike Information Criterion are used in the nonlinear block and the linear block, respectively. Finally, a numerical simulation and an industrial case verify the effectiveness of the proposed algorithm.
Highlights
Introduction eWiener model is a nonlinear system, which is composed of a dynamic linear block and a static nonlinear block
A blind identi cation method for the Wiener model was investigated [6]. e input signal of the system was given by a cyclostationary signal instead of a Gaussian random signal and the internal variables were recovered only based on the output. en, the order and parameters of the Wiener model were estimated by using the support vector machine regression
E Bayesian Composite Quantile Regression (CQR) (BCQR) algorithm used in this paper does not depend on the actual distribution of data, but on the likelihood function formed by the Asymmetric Laplace Distribution (ALD) [27]. e essence of BCQR is that the estimated parameter is regarded as a random variable, and the sampling distribution of parameter can be obtained by repeated sampling
Summary
The input signal u(k) firstly passes through the linear block of the Wiener model Bn(q− 1)/An(q− 1) and generates z1(k). Under the disturbance of the internal noise ε(k), z1(k) is further developed into z2(k) and becomes the input of the nonlinear block f(z2). (c) Unknown polynomial order can be defined by the Final Output Error (FOE) and the Akaike Information Criterion (AIC). An ARX model with a high order is employed to approximate the linear block of the Wiener model: Bn q− 1. Because the specific form of the expression is unknown, a stable linear transfer function is approximated by the finite impulse response (FIR) model. For the nonlinear block of the Wiener model, a cubic spline approximation (CSA) function is applied to fit the inverse function of the nonlinear clock: f−.
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