Complexity

Volume 2018, Article ID 9598307, 11 pages

https://doi.org/10.1155/2018/9598307

## Adaptive Gradient-Based Iterative Algorithm for Multivariable Controlled Autoregressive Moving Average Systems Using the Data Filtering Technique

^{1}Hubei Collaborative Innovation Center for High-Efficiency Utilization of Solar Energy, Hubei University of Technology, Wuhan 430068, China^{2}College of Automation and Electronic Engineering, Qingdao University of Science and Technology, Qingdao 266042, China^{3}School of Internet of Things Engineering, Jiangnan University, Wuxi 214122, China

Correspondence should be addressed to Jian Pan; moc.361@napj

Received 31 March 2018; Accepted 3 June 2018; Published 24 July 2018

Academic Editor: Jing Na

Copyright © 2018 Jian Pan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The identification problem of multivariable controlled autoregressive systems with measurement noise in the form of the moving average process is considered in this paper. The key is to filter the input–output data using the data filtering technique and to decompose the identification model into two subidentification models. By using the negative gradient search, an adaptive data filtering-based gradient iterative (F-GI) algorithm and an F-GI with finite measurement data are proposed for identifying the parameters of multivariable controlled autoregressive moving average systems. In the numerical example, we illustrate the effectiveness of the proposed identification methods.

#### 1. Introduction

Parameter estimation plays an important role in system control [1–4], system analysis [5–8], and signal processing [9–13]. Parameter estimation is significant in system modeling [14, 15]. Multi-input multi-output systems widely exist in industrial control areas, which are also called multivariate systems or multivariable systems [16–18]. They are more complex in model structures than single-input single-output systems and always have high dimensions and numerous parameters, which make the parameter estimation more difficult. In this literature, Ding et al. proposed a filtering decomposition-based least squares iterative algorithm for multivariate pseudolinear ARMA systems [19]. Ma et al. studied the parameter estimation problem of multivariate Hammerstein systems and presented a modified Kalman filter-based recursive least squares algorithm to give the parameter estimates [20]. Pan et al. proposed a filtering-based multi-innovation extended stochastic gradient algorithm for multivariable systems [21].

The data filtering technique is an important approach in system identification [22] and state estimation. Chen and Ding applied the data filtering technique to identify the multi-input and single-output system based on the maximum likelihood recursive least squares algorithm [23]. Mao et al. derived an adaptive filtering-based multi-innovation stochastic gradient algorithm for the input nonlinear system with autoregressive noise [24]. They introduced a linear filter to filter the input and output signals and decomposed the identification model into two subidentification models (i.e., a noise model and a system filtered model), which can improve the convergence rate and computation efficiency [25]. The identification methods can be applied to many areas [26–29].

The gradient search is useful for identification as an optimization method [30, 31]. Many gradient-based algorithms, including the stochastic gradient algorithms [32–34] and the gradient-based iterative algorithms, have been developed using the multi-innovation identification theory, the maximum likelihood estimation methods [35, 36], the key-term separation principle [37, 38], and the data filtering theory. For example, Ma et al. presented an iterative variational Bayesian method to identify the Hammerstein varying systems with parameter uncertainties. Chen et al. studied the identification problem of bilinear-in-parameter systems and presented a gradient-based iterative algorithm by using the hierarchical identification principle and the gradient search [39]. Deng and Ding developed a Newton iterative identification method for an input nonlinear finite impulse response system with moving average noise [40]. Other methods can be referred as to the transfer function identification [41–45], linear system identification [46–51], and nonlinear system identification [52–59].

This paper uses the hierarchical identification principle to study the data filtering-based iterative identification methods for a multivariable controlled autoregressive moving average (M-CARMA) system. The basic idea is to introduce a linear filter to decompose the original identification model into two subidentification models and then obtain the parameter estimates using the negative gradient search. The main contributions are as follows: (i)A filtering-based gradient iterative (F-GI) algorithm is proposed using the data filtering technique and the gradient search.(ii)A filtering-based gradient iterative algorithm with finite measurement data is developed to obtain the parameter estimates.

The layout of the remainder of this paper is as follows. Section 2 derives the identification model for the M-CARMA system. In Section 3, we derive a data filtering-based gradient iterative algorithm based on the data filtering technique. A filtering-based gradient iterative algorithm with finite measurement data is developed to estimate the unknown parameters in Section 4. A numerical example is shown in Section 5 to illustrate the benefits of the proposed methods in this paper. Finally, some concluding remarks are given in Section 6.

#### 2. The Problem Formulation

Some notation is introduced for convenience: denotes the estimate of at time ; “” or “” stands for“ is defined as ”; the symbol () represents an identity matrix of appropriate size (); the symbol represents an -dimensional column vector whose elements are 1; denotes a unit forward shift operator like and ; the superscript T symbolizes the vector/matrix transpose; and the norm of a matrix is defined by .

The following multivariable controlled autoregressive moving average system in Figure 1 is considered, where is the system input vector, is the system output vector, is a white noise vector with zero mean, and are the matrix polynomials in the unit backward shift operator , and is the polynomial in .