Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 240929, 8 pages

http://dx.doi.org/10.1155/2013/240929

Research Article

Recursive Least-Squares Estimation for Hammerstein Nonlinear Systems with Nonuniform Sampling

Xiangli Li,1,2 Lincheng Zhou,¹ Ruifeng Ding,³ and Jie Sheng⁴

¹Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Jiangnan University, Wuxi 214122, China

²Jiangsu College of Information Technology, Wuxi 214153, China

³School of Internet of Things Engineering, Jiangnan University, Wuxi 214122, China

⁴Institute of Technology, University of Washington, Tacoma, WA 98402-3100, USA

Received 17 June 2013; Accepted 2 September 2013

Academic Editor: Victoria Vampa

Copyright © 2013 Xiangli Li 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.

This paper focuses on the identification problem of Hammerstein nonlinear systems with nonuniform sampling. Using the key-term separation principle, we present a discrete identification model with nonuniform sampling input and output data based on the frame period. To estimate parameters of the presented model, an auxiliary model-based recursive least-squares algorithm is derived by replacing the unmeasurable variables in the information vector with their corresponding recursive estimates. The simulation results show the effectiveness of the proposed algorithm.

1. Introduction

In actual industrial processes, there exist widely nonlinear systems which are described by block-oriented nonlinear systems [1–3]. Block-oriented nonlinear models are in general divided into Hammerstein systems and Wiener systems [4]. A Hammerstein system, which consists of a static nonlinear subsystem followed by a linear dynamic subsystem, can represent some nonlinear systems [5]. Many publications have been reported for the identification of the Hammerstein systems [6, 7]. For example, Chen et al. studied identification problems for the Hammerstein systems with saturation and dead-zone nonlinearities by choosing an appropriate switching function [8]; Ding et al. presented the projection, the stochastic gradient, and the Newton recursive and the Newton iterative identification algorithms for the Hammerstein nonlinear systems, and then they analyzed and compared the performances of these approaches by numerical examples [9]. Li et al. derived a least-squares based iterative algorithm for the Hammerstein output error systems with nonuniform sampling by using the overparameterization model [10].

The different input-output updating period (or called multirate sampling) is inevitable in discrete-time systems [11–13]. The identification of multirate sampled systems have attracted much attention of many researchers. Recently, Liu et al. proposed a novel hierarchical least-squares algorithm for a class of nonuniformly sampled systems based on the hierarchical identification principle [14]. Shi et al. presented a crosstalk identification algorithm for multirate xDSL FIR systems [15]. Han et al. gave state-space models for multirate multi-input sampled-data systems and derived an auxiliary model-based recursive least-squares algorithm for identifying the parameters of multirate systems [16].

The recursive least-squares algorithm is a class of basic parameter estimation approaches which are suitable for online applications. In this literature, Wang adopted a filtering auxiliary model-based recursive least-squares identification algorithm for output error moving average systems [17]. Differing from the work in [14, 16], this paper discusses the parameter estimation problem for nonuniformly sampled Hammerstein nonlinear systems. The basic idea is, to combine the auxiliary model identification idea [18–24] and the key-term separation principle to derive the auxiliary model-based recursive least-squares algorithm for the Hammerstein nonlinear systems with nonuniform sampling.

The rest of this paper is organized as follows. Section 2 establishes the identification model of the Hammerstein nonlinear systems with nonuniform sampling. Section 3 derives a recursive least-squares parameter estimation algorithm based on the auxiliary model identification idea. Section 4 provides an example to illustrate the effectiveness of the proposed algorithm. The conclusions of the paper are summarized in Section 5.

2. The Identification Model

Let us introduce some notations. The superscript denotes the matrix transpose; stands for an identity matrix of appropriate sizes; represents an -dimensional column vector whose elements are 1; “” stands for “ is defined as ”; and is a unit backward shift operator; that is, .

Consider a Hammerstein nonlinear system with nonuniform sampling shown in Figure 1, where is a nonuniform zero-order hold with irregularly updating intervals , dealing with a discrete-time signal and producing the input of the nonlinear subsystem ; is the output of the nonlinear subsystem; is a continuous-time process; is the true output of but is unmeasurable; is a sampler that produces a discrete-time signal with period ; and is the system output but is corrupted by the additive noise .

Figure 1: Hammerstein systems with nonuniform sampling.

Assuming that the input has the updating intervals , we have [11, 25]

where is the frame period. The nonlinear subsystem in the Hammerstein nonlinear system is a polynomial of a known order:

where is the polynomial order.

Suppose that has the following state-space representation:

where is the state vector, and are the input and output of the continuous-time process, respectively, and , , , and are matrices of appropriate sizes. Referring to [25] and discretizing (3) with the frame period , we have

where

The output at the sampling instant can be expressed as

Hence, the system output is written as

Referring to [26] and from (4) and (6), we have

where

Equation (8) can be transformed into

Substituting (10) into (7), the system output can be expressed as

Equation (11) can be rewritten equivalently as

Here, substituting (2) into (12) results in a complex expression containing the products of parameters. To solve this problem, we use the key-term separation principle presented in [27], and let . Then, the identification model of the proposed system is as follows:

The objective of this paper is to develop a recursive least-squares algorithm for estimating the parameters of the nonuniformly sampled Hammerstein systems by using the auxiliary model identification idea in [11].

3. The Recursive Least-Squares Algorithm

In this section, we derive the recursive least-squares estimation algorithm for the Hammerstein nonlinear systems with nonuniform sampling, referring to the method in [1].

Define the information vectors and the parameter vectors as

Equation (13) can be written in a regressive form as

Define a quadratic criterion function as

Let be the estimate of at time . Minimizing gives the following recursive least-squares algorithm:

Note that the information vector in (17) contains unknown inner variables and ; the parameter vector cannot be estimated by the standard least-squares method. The solution is based on the auxiliary model identification idea [11]: to replace the unmeasurable term in with its estimate

Replacing in (2) with its estimate , we can obtain the estimate of as follows:

Define the estimate of as

Using in place of in (17) and (18), we have

Equations (19) to (22) form the AM-RLS algorithm for the not uniformly sampled Hammerstein nonlinear systems, which can be summarized as

To initialize the algorithm, we take to be a small real vector; for example, and with normally a large positive number (e.g., ).

4. Example

An example is given to demonstrate the feasibility of the proposed algorithm. Assume that the dynamical linear subsystem has the following state-space representation:

and the static nonlinear subsystem is denoted by

Let , s, and s; that is, s and s. Discretizing with the frame period , we obtain

Thus, the corresponding input-output expression is given by

In simulation, the inputs and are taken as persistent excitation signal sequences with zero mean and unit variance; is a white noise with zero mean and variance . Applying the proposed algorithm to estimate the parameters of this system, the estimates of and their errors with different noise variances are shown in Tables 1 and 2, and the parameter estimation errors versus are shown in Figure 2. When and , the corresponding signal-to-noise ratios (the square root of the ratio of output and noise variances) are and , respectively.

Table 1: The parameter estimates and their errors ().

δ (%)

100
−0.03122
−0.05592
0.02344
0.39192
−0.08169
0.14407
1.52981
0.68962
0.02254
73.36523

500
−0.53614
0.31821
−0.43136
0.63245
−0.30871
0.53689
1.30985
0.69341
0.10358
26.81256

1000
−0.55071
0.44816
−0.41426
0.78764
−0.24332
0.62087
1.12565
0.56690
0.22360
12.86475

2000
−0.57852
0.42159
−0.43335
0.71714
−0.22310
0.63026
1.04567
0.49863
0.23534
8.87717

3000
−0.60057
0.39569
−0.44560
0.68205
−0.23745
0.60417
1.09320
0.49337
0.20626
10.44562

4000
−0.61945
0.41573
−0.44904
0.68292
−0.24136
0.60799
1.03387
0.50662
0.24914
7.82199

5000
−0.66253
0.42857
−0.49385
0.68309
−0.26702
0.62299
1.00550
0.50873
0.25535
5.03493

6000
−0.67482
0.46019
−0.52100
0.71556
−0.26552
0.64442
0.99935
0.53599
0.26184
2.87761

True values
−0.68000
0.47241
−0.52674
0.73948
−0.25070
0.66221
1.00000
0.50000
0.25000

Table 2: The parameter estimates and their errors ().

δ (%)

100
−0.69605
0.48092
−0.59484
0.83269
−0.30638
0.59981
1.17677
0.54701
0.18587
13.23523

500
−0.68503
0.51969
−0.54542
0.81494
−0.30331
0.68632
1.05813
0.54706
0.22836
7.30117

1000
−0.65503
0.49678
−0.50768
0.79029
−0.26915
0.68531
1.01708
0.51876
0.25274
4.13831

2000
−0.64233
0.46233
−0.49229
0.74295
−0.25140
0.66911
1.00467
0.50079
0.25083
2.89926

3000
−0.64733
0.45069
−0.49445
0.72782
−0.25286
0.65764
1.01820
0.49893
0.24231
3.07166

4000
−0.65262
0.45294
−0.49564
0.72445
−0.25257
0.65608
1.00431
0.50179
0.25233
2.68214

5000
−0.66319
0.45510
−0.50654
0.72291
−0.25789
0.65842
0.99803
0.50215
0.25341
2.01651

6000
−0.66697
0.46155
−0.51409
0.72921
−0.25706
0.66243
0.99700
0.50881
0.25474
1.45453

True values
−0.68000
0.47241
−0.52674
0.73948
−0.25070
0.66221
1.00000
0.50000
0.25000

Figure 2: The estimation errors versus .

From Tables 1 and 2 and Figure 2, we can draw the following conclusions.

(i)The parameter estimation errors of the AM-RLS algorithm become (generally) smaller as increases; see the estimation errors of the last columns of Tables 1 and 2 and Figure 2. (ii)Under different noise levels, the parameter estimates can converge to the true value, and a lower noise level results in a faster convergence rate of the parameter estimates to the true parameters; see the error curves in Figure 2 and the estimation errors in Tables 1 and 2. (iii)The proposed recursive algorithm differs from the iterative identification approach in [10] and can be used as an online identification.

5. Conclusions

In this paper, we have established the identification model of the Hammerstein nonlinear systems with nonuniform sampling by using the key-term separation principle. To estimate the parameters of the proposed model, the recursive least-squares parameter estimation algorithm is derived based on the auxiliary model identification idea. The proposed algorithm can simultaneously estimate the parameters of the linear and nonlinear subsystems of the Hammerstein nonlinear systems with nonuniform sampling. The simulation results show that the parameters of the Hammerstein systems with nonuniform sampling can be estimated effectively by the proposed algorithm. Although the algorithm is presented for a class of nonuniformly sampled Hammerstein nonlinear systems, the basic idea can also be extended to identify other linear and nonlinear systems [28, 29] and can combine the hierarchical identification methods [30–34], the multi-innovation identification methods [35–44], and other identification methods [45–58] to present new identification algorithms for linear or nonlinear and scalar or multivariable systems [59].

Acknowledgments

This work was supported by the Fundamental Research Funds for the Central Universities (JUDCF11042 and JUDCF12031) and the PAPD of Jiangsu Higher Education Institutions and the 111 Project (B12018).

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