Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 516374, 8 pages
http://dx.doi.org/10.1155/2015/516374

Research Article

Least Squares Based and Two-Stage Least Squares Based Iterative Estimation Algorithms for H-FIR-MA Systems

Zhenwei Shi1,2 and Zhicheng Ji¹

¹School of Internet of Things Engineering, Jiangnan University, Wuxi 214122, China
²Wuxi Electrical and Higher Vocational School, Wuxi 214028, China

Received 25 May 2014; Accepted 17 September 2014

Academic Editor: Shifei Ding

Copyright © 2015 Zhenwei Shi and Zhicheng Ji. 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.

This paper studies the identification of Hammerstein finite impulse response moving average (H-FIR-MA for short) systems. A new two-stage least squares iterative algorithm is developed to identify the parameters of the H-FIR-MA systems. The simulation cases indicate the efficiency of the proposed algorithms.

1. Introduction

System modeling [1–5] and parameter estimation [6–10] are basic for controller design [11, 12]. Nonlinear Hammerstein model identification has received much attention due to its ability to describe a wide class of nonlinear systems and has extensive applications in many engineering problems [13, 14]. The Hammerstein models are special class of nonlinear systems; the nonlinear block is usually static nonlinearity and is followed by a linear system [15]. For example, Wang et al. discussed the identification problem for a Hammerstein nonlinear system with a dynamic subspace state space [16]; Greblicki investigated a class of continuous time Hammerstein system identification [17].

There are a lot of research topics about linear or nonlinear system identification [18, 19] and control [20, 21]. For example, Ding et al. derived the gradient search based and the Newton based identification methods for Hammerstein systems [22]; Wang and Ding proposed a hierarchical least squares identification method for Hammerstein-Wiener systems by using the hierarchical identification principle and the auxiliary model identification idea [23]; Based on the data filtering technique and the key-term separation principle, Wang et al. investigated a filtering based recursive least squares identification algorithm for Hammerstein output error moving average systems [24]. The proposed algorithm can identify not only the system model parameters but also the noise model parameters and the internal variables.

The iterative algorithm is one of the basic methods for system analysis and synthesis, and nonlinear optimization [25–28]. In [29], Wang and Ding presented a gradient based and least squares based iterative identification algorithms for Wiener systems through the use of the hierarchical identification principle. In [30], Ding et al. discussed the Newton iterative identification algorithm of a class of Wiener nonlinear systems with moving average noises from input-output measurement data. Li et al. derived iterative parameter identification methods for nonlinear functions [31]. Pan et al. proposed a digital image correlation using iterative least squares and pointwise least squares for displacement field and strain field measurements [32]. In the field of control, Zhang et al. applied the iterative algorithm to the predictive control field [33, 34].

Recently, the multistage identification strategy is widely applied to the system identification field [35, 36]. For example, Ding and Duan studied a new-type two-stage least squares based iterative algorithm for identifying the system model parameters and the noise model parameters [37].

The main concern of this paper is to investigate the parameter identification problem of Hammerstein finite impulse response moving average (H-FIR-MA) systems. The memoryless polynomial input nonlinearity is followed by a linear dynamical system, as is explained in Figure 1. Both the least squares iterative and the two-stage least squares iterative algorithms are proposed to estimate the parameters of the H-FIR-MA systems.

Figure 1: The LSI estimation errors versus (, , and ).

The layout of this paper is organized as follows. Section 2 describes the identification model of H-FIR-MA systems. Section 3 provides the least squares iterative algorithm for the H-FIR-MA systems. Section 4 introduces the two-stage least squares iterative algorithm for the H-FIR-MA systems. In Section 5, we apply the proposed algorithms to an example to illustrate their implementation. Finally, concluding remarks are offered in Section 6.

2. System Description and Identification Model

Some notation is given. stands for the norm of a matrix ; “” or “” expresses that “ is defined as ”; represents the identity matrix of appropriate sizes and is defined as an -dimensional identity column vector.

Consider an H-FIR-MA system, which is described by

where and are the input and output sequences of the systems, is an uncorrelated stochastic noise sequence with zero mean and variance , and , , and are unmeasurable. The output of the nonlinear block is a linear combination, with unknown coefficients , of a known basis in the system input , and can be expressed as

Assume that the orders , , and are known in (1) and (4) and , , , and for . In order to get unique parameter estimates, here we let [24]. The item in (1) is chosen as the key term; substituting (4) into (1) gets

Define the parameter vector and the information vector as follows:

From (5), we obtain the following identification model:

Define the cost function:

In what follows, we derive the algorithms for identifying the H-FIR-MA system using the least squares and two-stage least squares iterative estimation algorithms.

3. The Least Squares Iterative Estimation Algorithm

In this section, referring to the method in [27], we give simply the least squares iterative (LSI) estimation algorithm for the H-FIR-MA system for comparison.

Consider the data from to and define the stacked information matrices , the stacked output vector , and the stacked white noise vector as

Hence, (7) can be rewritten as

According to the estimation model in (12), the cost function in (8) can be written as

To minimize , letting its partial derivative of with respect to be zero, we have

It is impossible to obtain the estimate , because the information matrix (i.e., in (6)) contains the unmeasurable inner variables and the noise terms . Here we adopt the auxiliary model idea and the hierarchical identification principle: let … be iteration variable, let be the iterative estimate of at iterative and , and let be the iterative estimates of   and . We replace and in (6) with their estimates and obtain the estimates and as follows:

Replacing in (14) with and combining (10) and (15), we can obtain the LSI estimation algorithm of identifying for the H-FIR-MA system as follows [27]:

The computation procedures of the LSI algorithm in (16)–(22) are summarized as follows.

Step 1. Let and set the initial values , , ; is a large number (i.e., ).

Step 2. Collect the input and output data and (), compute by (20), structure and by (17) and (19), respectively, and form by (18).

Step 3. Update the parameter identification by (16).

Step 4. Compute and by (21) and (22), respectively.

Step 5. Increase by 1 and jump to Step 2.

4. The Two-Stage Least Squares Iterative Estimation Algorithm

Here, we derive a two-stage least squares iterative (TS-LSI) estimation algorithm for the H-FIR-MA system. From (5) and (6), we can obtain the following identification model:

Define the information vector and as

Define two intermediate variables and ; then the system in (23) can be decomposed into two “suppositional” subsystems:

The estimates of two “suppositional” subsystems in (25) can be obtained by minimizing the cost function:

Consider the data from to and in (25) define the stacked information matrices and and the stacked vector and as

Two intermediate variables can be rewritten as

From (25), we have

According to the estimation model in (31), the cost function in (26) can be written as

To minimize , let its partial derivative of with respect to be zero:

From (34), the least squares estimate of the parameter vector can be expressed as

Here, put (29) into (35) and (35) gives

In accordance with the same derivation process of , we can easily get the estimation formula of

However, (36) and (37) contain the unknown parameter and , respectively, it is impossible to and . According to the method in Section 3, we can summarize the two-stage least squares iterative estimation algorithm for estimating and of the H-FIR-MA systems as follows:

The computation procedures of the TS-LSI algorithm in (38)–(46) are summarized as follows.

Step 1. Let , and set the initial values , , , , (i.e., ).

Step 2. Collect the input and output data and  , compute by (44), and form by (40) and by (42).

Step 3. Structure and , respectively, by (41) and (43).

Step 4. Update the parameter identification by (38) and (39), respectively.

Step 5. Compute and by (45) and (46), respectively.

Step 6. Increase by 1 and jump to Step 2.

5. Simulations and Case Study

In this section, we consider an H-FIR-MA system, where the static nonlinearity is chosen as polynomials. More precisely

In simulation, the input is taken as an uncorrelated measured stochastic signal sequence with zero mean, the noise is a white noise sequence with zero mean and variances and , respectively, and the corresponding noise-to-signal ratios are and . The noise-to-signal ratios can be calculated by the following formula:

where and are, respectively, expressed as the variances of and in (1).

Take two different data lengths and . The parameter values are estimated using the two different methods described in the paper, namely, the LSI and the TS-LSI methods in Sections 3 and 4. We apply the LSI method to estimate the parameters of this case; the parameter estimation with different data length and noise variances are shown in Tables 1 and 2, and the estimation errors versus iteration are shown in Figures 1 and 2, where .

Table 1: The LSI estimation and errors with and ().

0.5
1
−1.4978
0.5020
−0.0174
0.8217
0.5027
0.3252
31.5090

5
−1.4950
0.5016
−0.6198
0.8202
0.4979
0.3228
1.5288

10
−1.4953
0.5014
−0.6199
0.8202
0.4972
0.3228
1.5276

1.0
1
−1.4956
0.5040
−0.0348
0.8435
0.5055
0.3195
30.694

5
−1.4890
0.5027
−0.6198
0.8405
0.4954
0.3147
2.5092

10
−1.4896
0.5022
−0.6199
0.8406
0.4935
0.3146
2.5156

True values

−1.5000
0.5000
−0.6400
0.8000
0.5000
0.3310

Table 2: The LSI estimation and errors with and ().

0.5
1
−1.4976
0.4977
−0.0028
0.8038
0.5034
0.3325
32.2250

5
−1.4951
0.4973
−0.6296
0.7956
0.5050
0.3345
0.7077

10
−1.4946
0.4969
−0.6296
0.7957
0.5052
0.3345
0.7215

1.0
1
−1.4953
0.4955
−0.0058
0.8077
0.5069
0.3340
32.084

5
−1.4891
0.4943
−0.6296
0.7914
0.5094
0.3380
1.1290

10
−1.4883
0.4935
−0.6291
0.7912
0.5091
0.3380
1.0966

True values

−1.5000
0.5000
−0.6400
0.8000
0.5000
0.3310

Figure 2: The LSI estimation errors versus (, , and ).

Similarly, the parameter estimation and estimation errors of the TS-LSI method with different data length and noise variances are shown in Tables 3 and 4 and Figures 3 and 4.

Table 3: The TS-LSI estimation and errors with and ().

0.5
1
−1.5067
0.5064
−0.0874
0.8803
0.4947
0.2939
28.307

5
−1.4961
0.5008
−0.6194
0.8185
0.4955
0.3235
1.4828

10
−1.4955
0.5033
−0.6192
0.8182
0.5015
0.3237
1.4736

1.0
1
−1.5048
0.5083
−0.1033
0.9010
0.4971
0.2882
27.708

5
−1.4904
0.5016
−0.6196
0.8370
0.4927
0.3161
2.3477

10
−1.4901
0.5048
−0.6197
0.8366
0.5000
0.3163
2.3106

True values

−1.5000
0.5000
−0.6400
0.8000
0.5000
0.3310

Table 4: The TS-LSI estimation and errors with and ().

0.5
1
−1.4949
0.4998
0.0034
0.7899
0.4984
0.3378
32.552

5
−1.4940
0.4977
−0.6350
0.7964
0.5023
0.3344
0.5547

10
−1.4934
0.4977
−0.6339
0.7962
0.5015
0.3341
0.4968

1.0
1
−1.4913
0.4982
0.0077
0.7919
0.5004
0.3408
32.772

5
−1.4867
0.4951
−0.6350
0.7929
0.5045
0.3379
1.0305

10
−1.4858
0.4950
−0.6349
0.7927
0.5038
0.3360
0.9382

True values

−1.5000
0.5000
−0.6400
0.8000
0.5000
0.3310

Figure 3: The TS-LSI estimation errors versus (, , and ).

Figure 4: The TS-LSI estimation errors versus (, , and ).

From the simulation results in Tables 1–4 and Figures 1–4, we can draw the following conclusions.

(1)The parameter estimation errors given by the LSI and TS-LSI algorithms become small as iterations increase.(2)The parameter estimation errors given by the LSI and TS-LSI algorithms become closer to their true values with the data length increasing.(3)It is easy to see that a high noise level results in a low consistence rate of the parameter estimates to the true parameters for both of the proposed algorithms.(4)When the data length goes to infinity, the estimation errors converge to zero. The simulations of results in Tables 1–4 and Figures 1–4 indicate that the proposed algorithms based iterative algorithm should stop for about . The fluctuation of the estimation errors is caused for large due to the stationary of noise.

All in all, this shows that the proposed algorithms are effective.

6. Conclusions

The LSI and the TS-LSI identification algorithms are developed for H-FIR-MA systems. The simulation results indicate that the proposed algorithms can obtain highly accurate parameter estimates and fast convergence rate and illustrate the proposed algorithms’ performance. Compared with other methods, the LSI and TS-LSI methods must compute the matrix inversion. The proposed methods are simple in principle and the basic idea can be applied to other fields [38–41].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported in part by the National High Technology Research and Development Program of China (China 863 Program) (no. 2013AA040405), the Guangdong-Hong Kong Breakthrough Bidding Project in Key Areas (no. 2012A080107015), the fundamental research funds for the central universities (JUSRP51310A), and the Chinese State Grain Administration commonweal research project (no. 201313012).

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