Research Article  Open Access
Two Identification Methods for DualRate SampledData Nonlinear OutputError Systems
Abstract
This paper presents two methods for dualrate sampleddata nonlinear outputerror systems. One method is the missing output estimation based stochastic gradient identification algorithm and the other method is the auxiliary model based stochastic gradient identification algorithm. Different from the polynomial transformation based identification methods, the two methods in this paper can estimate the unknown parameters directly. A numerical example is provided to confirm the effectiveness of the proposed methods.
1. Introduction
System identification plays an important part in many engineering applications [1–6]. Many identification methods assume that the inputoutput data at every sampling instant are available for linear systems [7–11] and nonlinear systems [12–20], which is usually not the case in practice. When the input and output signals of the systems have different sampling rates, these systems are usually called irregularly sampleddata systems [21–27], for example, dualrate or multirate systems [28–30]. Dualrate/multirate systems in which the input and the output are sampled at different frequencies arise widely in robust filtering and control [31–33], adaptive control [34–37], and system identification [38–43]. In the literature of dualrate system identification, the socalled polynomial transformation technique is often used to transform the dualrate model [44, 45].
As far as we know, the identification methods based on the polynomial transformation technique cannot directly estimate the parameters of the dualrate system and the number of the unknown parameters to be estimated is more than the number of the unknown parameters of the original dualrate system.
The nonlinear system consisting of a static nonlinear block followed by a linear dynamic system is called a Hammerstein system [46–49]. The nonlinearity of the Hammerstein system is usually expressed by some known basis functions [50, 51] or by a piecewise polynomial function [52, 53]. When the Hammerstein system is a dualrate system and has a preload nonlinearity, to the best of our knowledge, there is no work on identification of such systems. The main contributions of this paper are presenting the two methods directly for estimating the parameters of the dualrate system. The proposed methods of this paper can combine the auxiliary model identification methods [54–57], the iterative identification methods [58–62], the multiinnovation identification methods [63–70], the hierarchical identification methods [71–83], and the twostage or multistage identification methods [84, 85] to study identification problems for other linear systems [86–90] or nonlinear systems [91–97].
The rest of this paper is organized as follows. Section 2 introduces the dualrate nonlinear outputerror systems. Section 3 gives a missing output identification model based stochastic gradient algorithm. Section 4 provides an auxiliary model based stochastic gradient algorithm. Section 5 introduces an illustrative example. Finally, concluding remarks are given in Section 6.
2. Problem Formulation
Let “” or “” stand for “ is defined as ,” let the norm of a column vector be , and let the superscript denote the matrix transpose.
Consider the following dualrate nonlinear outputerror system with colored noise: where is the system output, is the system input, is a stochastic white noise with zero mean, and are the polynomials in the unit backward shift operator , and is a preload nonlinearity shown in Figure 1 and can be expressed as [98, 99] where and are two preload points.
For the dualrate sampleddata system, all the input data , and only the scarce output data , are known. The intersample outputs or missing outputs , are unavailable.
Define a sign function Then the function can be expressed as Let Hence, we have Once and are estimated, the parameters and can be computed by , .
3. The Missing Outputs Identification Model Based Stochastic Gradient Algorithm
Substituting (7) into (1) gets Define the parameter vector and information vector as From (9) and (10), we get or Let be the estimate of . Defining and minimizing the cost function give the following stochastic gradient (SG) algorithm for estimating : Since the information on the righthand sides of (16) contains the unknown variables , , the SG algorithm in (14)–(18) is impossible to implement. In this section, we use the missing outputs identification model (MOI) to overcome this difficulty; these unknown are replaced with the output estimates of an MOI model, where represents the estimate of at time , represents the estimate of at time , and represents the estimate of .
Thus, we have the following missing output estimates based SG (MOESG) algorithm for estimating the parameter vector in (9): The steps of computing the parameter estimate by the MOESG algorithm are listed as follows. (1)Let , , , and give a small positive number .(2)Let , , and with being a column vector whose entries are all unity and .(3)Collect the input data , and collect the output data .(4)Let and compute by (22).(5)Form by (23).(6)Decrease by 1; if , go to step (4); otherwise, go to the next step.(7)Compute and by (24) and (25), respectively.(8)Update the parameter estimation vector by (20).(9)Compare and ; if , then terminate the procedure and obtain the ; otherwise, increase by and go to step (3).
The flowchart of computing the MOESG parameter estimate is shown in Figure 2.
4. The Auxiliary Model Based Stochastic Gradient Algorithm
Define From (8) and (26), we have Define the information vector as
Then we get
Assume is an integer multiple of and rewrite (30) as
Let be the estimate of . Defining and minimizing the cost function give the following SG algorithm of estimating : Because of the unknown variables in (33), the SG algorithm in (33)–(36) is impossible to implement. In this section, we use the auxiliary model; these unknown are replaced with the outputs of an auxiliary model, where is the estimate of and is the estimate of . We can obtain an auxiliary model based stochastic gradient (AMSG) algorithm: The steps of computing the parameter estimate by the AMSG algorithm are listed as follows.(1)Let , , , , and give a small positive number .(2)Let , , and with being a column vector whose entries are all unity and .(3)Collect the input data , and collect the output data .(4)Let and compute by (40).(5)Form by (41).(6)Decrease by 1; if , go to step (4); otherwise, go to next step.(7)Compute and by (42) and (43), respectively.(8)Update the parameter estimation vector by (38).(9)Compare and ; if , then terminate the procedure and obtain the ; otherwise, increase by and go to step (3).
The flowchart of computing the AMSG parameter estimate is shown in Figure 3.
Remark 1. Compared with the polynomial transformation technique, the MOESG method and the AMSG method can estimate the unknown parameters directly.
5. Example
Consider the following nonlinear outputerror system with the updating period : the input is taken as a persistent excitation signal sequence with zero mean and unit variance and is a white noise sequence with zero mean and variance . The unknown parameters are as follows: Applying the MOESG algorithm and the AMSG algorithm to estimate the parameters, the parameter estimates and their errors based on the MOESG algorithm and the AMSG algorithm are shown in Tables 1 and 2 and the parameter estimation errors versus are shown in Figures 4 and 5.


From Tables 1 and 2 and Figures 4 and 5, we can draw the following conclusions.(1)Both the MOESG algorithm and the AMSG algorithm can estimate the unknown parameters directly.(2)The parameter estimation errors become smaller and smaller and go to zero with increasing.
6. Conclusions
Two identification methods for dualrate nonlinear outputerror systems are presented to estimate the unknown parameters directly and can avoid estimating more parameters than the original systems. Furthermore, the two methods can also be extended to other systems such as
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China and supported by the Natural Science Foundation of Jiangsu Province (no. BK20131109).
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Copyright © 2014 Jing Chen and Ruifeng Ding. 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.