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Complexity
Volume 2018, Article ID 9598307, 11 pages
https://doi.org/10.1155/2018/9598307
Research Article

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

1Hubei Collaborative Innovation Center for High-Efficiency Utilization of Solar Energy, Hubei University of Technology, Wuhan 430068, China
2College of Automation and Electronic Engineering, Qingdao University of Science and Technology, Qingdao 266042, China
3School 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 [14], system analysis [58], and signal processing [913]. 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 [1618]. 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 [2629].

The gradient search is useful for identification as an optimization method [30, 31]. Many gradient-based algorithms, including the stochastic gradient algorithms [3234] 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 [4145], linear system identification [4651], and nonlinear system identification [5259].

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 .

Figure 1: A multivariable controlled autoregressive moving average system.

Assume that the orders , , and are known, and , , and for . The intermediate variable is defined as

The system information vector , the noise information vector , the system parameter matrix , and the noise parameter vector are defined as

Equations (3) and (1) can be written as

Equation (5) is the noise identification model. For the M-CARMA system in (1), choose the polynomial as a filter. Define the filtered input vector , the filtered output vector , and the filtered information vector as

Multiplying both sides of (1) by obtains or

Then we have

Equations (10) and (5) form the filtered identification models of the M-CARMA system.

3. The F-GI Algorithm

In this section, a linear filter is applied to deal with the moving average noise. A gradient-based iterative identification algorithm is proposed for M-CARMA systems by using the data filtering technique [6063].

Considering the newest data from to , the stacked filtered output matrix , the stacked filtered information matrix the stacked noise vector , and the stacked noise information matrix are defined as

Define a quadratic criterion function:

Let be an iterative variable. Let and be the estimates of and at iteration . Minimizing and and using the negative gradient search will give the following iterative relations for obtaining the parameter estimates of and : where and are the iterative step size or the convergence factor. However, the difficulty is that the noise information matrix (i.e., ) contains the unmeasured vector . So the gradient-based iterative algorithm in (13) and (14) cannot give the parameter estimate directly. The solution is to use the hierarchical identification principle and to replace the unknown variable with its corresponding estimates at iteration , and to define the estimate of as

Using to construct the estimate of obtains

Replacing in (6) with gives

Replacing in (17) with obtains the estimate of at iteration :

From (6), we have

Replacing , , and with , , and obtains the iterative estimate of at iteration :

Then, using to construct the iterative estimate of at iteration gives

Use to construct the estimate of as

Using to filter and gives the filtered estimates and of and :

Furthermore, we use to construct the estimate of , use and to construct the estimate of , and use to construct the estimate of at iteration :

From the above derivation, we can summarize a filtering-based multi-innovation gradient iterative identification algorithm:

The identification steps of the algorithm in (25), (26), (27), (28), (29), (30), (31), (32), (33), (34), (35), (36), (37), (38), (39), and (40) to compute and are listed as follows: (1)Set the initial values: let , give the data length , and give a small positive number . Set the initial values , , .(2)Collect the input–output data and and construct using (33).(3)Let and set the initial values ,(4)Construct and using (31) and (32).(5)Compute using (36) and form the stacked noise vector using (29) and the stacked noise matrix using (30).(6)Choose using (40) and update the noise parameter estimates using (28).(7)Read from in (38). Compute and using (34) and (35).(8)Construct using (27) and using (26).(9)Choose using (39) and update the system parameter estimates using (25).(10)Compute using (37).(11)Compare with and compare with : if and , increase by 1 and turn to Step 4; otherwise, obtain and the parameter estimate vector and , let , , increase by 1 and turn to Step 2.

The flowchart of computing and from the F-GI algorithm is shown in Figure 2.

Figure 2: The flowchart of computing the F-GI parameter estimates.

4. The F-GI Algorithm with Finite Measurement Data

Consider the data from to and define the stacked filtered output matrix , the stacked filtered information matrix , the stacked noise vector , and the stacked noise information matrix as

Note that and contain all the measured data .

The two gradient criterion functions are defined as

Similarly, minimizing and , we can derive a filtering-based gradient iterative (F-GI) algorithm with the data length for the M-CARMA system:

The identification steps of the F-GI algorithm with finite measurement data in (43), (44), (45), (46), (47), (48), (49), (50), (51), (52), (53), (54), (55), (56), (57), and (58) to compute and are listed as follows. (1)Collect the input–output data and give a small positive number . Construct using (51).(2)Let and set the initial values (3)Construct and using (49) and (50).(4)Compute using (54), construct using (47), and construct using (48).(5)Choose using (58) and update using (46).(6)Read from in (56) and compute and using (52) and (53).(7)Construct using (45) and construct using (44).(8)Choose using (57) and update using (43).(9)Compute using (55).(10)Compare with and compare with : if and , increase by 1 and turn to Step 3; otherwise, obtain and the parameter estimate vector and .

The flowchart of computing and from the F-GI algorithm with finite measurement data is shown in Figure 3.

Figure 3: The flowchart of computing the F-GI parameter estimates with finite measurement data.

The algorithm in (43), (44), (45), (46), (47), (48), (49), (50), (51), (52), (53), (54), (55), (56), (57), and (58) for multivariable CARMA systems is based on the filtering technique and the gradient search and can be extended to more complex multivariable systems with cooled noises.

5. Numerical Example

Consider a two-input two-output CARMA system: where

In simulation, and are taken as two persistent excitation signal sequences with zero mean and unit variance, and and as two white noise sequences with . Take the data length and and apply the F-GI algorithm with finite measurement data in (43), (44), (45), (46), (47), (48), (49), (50), (51), (52), (53), (54), (55), (56), (57), and (58) to estimate the parameters of this M-CARMA system. The parameter estimates and errors are shown in Table 1 with , , and , and the parameter estimation errors versus are shown in Figure 4. For comparison with the different data length , the simulation results are shown in Table 2 and Figure 5.

Table 1: The F-GI parameter estimates and errors ().
Figure 4: The CARMA-FGI estimation errors with different noise variances
Table 2: The F-GI parameter estimates and errors ().
Figure 5: The CARMA-FGI estimation errors with different noise variances .

From Tables 13 and Figures 4 and 5, we can draw the following conclusions. (1)The parameter estimation errors obtained by the presented algorithms gradually become smaller with the iterative variable increasing. Thus, the proposed algorithms for M-CARMA systems are effective.(2)The system parameter estimates converge to their true values with the increasing of the data length.(3)Under the same data length, a smaller noise variance leads to higher parameter estimation accuracy and a faster convergence rate.(4)A longer data length leads to a smaller estimation error under the same noise level.

Table 3: The F-GI parameter estimates and errors ().

6. Conclusions

An F-GI algorithm and an F-GI algorithm with finite measurement data are proposed for identifying the multivariable controlled autoregressive system with measurement noise in this paper. The linear filter is introduced to filter the input–output data, and the hierarchical identification principle is applied to decompose the identification model into two subidentification models. The simulation results show that the proposed algorithms can generate accurate estimates. The proposed approaches in the paper can combine other mathematical tools [6469] and statistical strategies [7075] to study the performances of some parameter estimation algorithms and can be applied to other multivariable systems with different structures and disturbance noises and other literature [7686] such as system identification [8792].

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 61471162) and the National First-Class Discipline Program of Light Industry Technology and Engineering (LITE2018-26).

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