#### Abstract

This paper presents two methods for dual-rate sampled-data nonlinear output-error 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 input-output 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 sampled-data systems [21–27], for example, dual-rate or multirate systems [28–30]. Dual-rate/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 dual-rate system identification, the so-called polynomial transformation technique is often used to transform the dual-rate model [44, 45].

As far as we know, the identification methods based on the polynomial transformation technique cannot directly estimate the parameters of the dual-rate system and the number of the unknown parameters to be estimated is more than the number of the unknown parameters of the original dual-rate 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 piece-wise polynomial function [52, 53]. When the Hammerstein system is a dual-rate 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 dual-rate system. The proposed methods of this paper can combine the auxiliary model identification methods [54–57], the iterative identification methods [58–62], the multi-innovation identification methods [63–70], the hierarchical identification methods [71–83], and the two-stage 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 dual-rate nonlinear output-error 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 dual-rate nonlinear output-error 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 dual-rate sampled-data 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 right-hand 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 (MOE-SG) algorithm for estimating the parameter vector in (9): The steps of computing the parameter estimate by the MOE-SG 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 MOE-SG 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 (AM-SG) algorithm: The steps of computing the parameter estimate by the AM-SG 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 AM-SG parameter estimate is shown in Figure 3.

*Remark 1. *Compared with the polynomial transformation technique, the MOE-SG method and the AM-SG method can estimate the unknown parameters directly.

#### 5. Example

Consider the following nonlinear output-error 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 MOE-SG algorithm and the AM-SG algorithm to estimate the parameters, the parameter estimates and their errors based on the MOE-SG algorithm and the AM-SG 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 MOE-SG algorithm and the AM-SG 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 dual-rate nonlinear output-error 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).