In the process of actual system modeling, many systems exhibit nonlinear characteristics with memory. Thus, the parameter identification problem of the nonlinear system with memory usually appears in the system modeling. This report focuses on the nonlinear system identification of Wiener–Hammerstein-like model with memory hysteresis, in which a new recursive estimation way is introduced. In this algorithm, the estimation bias problem can be improved by introducing a data filtering technique. On the basis of the filtered data, some auxiliary matrices and vectors are proposed. Following this, the identification error variable is introduced by using auxiliary matrices and vectors with an adaptive forgetting factor. Afterward, the identification error variable is integrated into the design of parameter estimation adaptive law with recursive gain structure. By comparison with the classic estimation methods, the proposed algorithm shows an alternative identification algorithm design angle. In addition, it is strictly proved that the parameter estimation error converges to zero under a general excitation condition. Based on the results of indices , compared with the existing methods, the performance improvements of the proposed method are 33.9 %, 41.26%, and 53.5%, respectively. In terms of indices , the augmented performances of the developed scheme are 50%, 56.2%, and 68.4%, respectively, in comparison to the available schemes.

1. Introduction

High-accuracy and novel identification algorithm is increasingly popular due to the basis of the adaptive control. In a physical control system, the nonlinear system model based on sensor and data acquisition technology can better represent the characteristics of the actual system compared with a linear model [13]. Thus, nonlinear system identification has become a hot research subject in the automatic control community. Over the last couple of decades, much attention has been paid to the study of parameter identification for nonlinear Wiener, Hammerstein, and their combinations owing to its understandable, high efficiency, and separability [46]. Although the traditional Wiener, Hammerstein, and their combinations can describe the nonlinear characteristics of the actual system by replacing different memoryless nonlinear submodels, the modeling of the nonlinear actual system with memory characteristics is inaccurate or even failed. To increase the description capability of the conventional Wiener, Hammerstein, and their combinations, byusing the memory nonlinearity model to replace the memoryless nonlinearity model, the Wiener-like, Hammerstein-like, and their combinations-like models are proposed [7, 8]. Consequently, compared with the traditional systems, the study of system-like models has become a hot topic. On the other hand, the discussion of nonlinear systems’ identification with memory characteristics is helpful to understand the modeling process of actual systems.

Many available estimation schemes for the system-like models can be divided into two categories according to the optimization form: deterministic methods and nondeterministic methods. The deterministic method includes recursive structure [9], iterative structure [10], multi-innovation identification [11], subspace identification [12], two-stage identification [13], and experimental tests [14], etc. While nondeterministic method includes the improved PSO [15], hybrid metaheuristic algorithm [16], supervised learning method [17], gravitational search algorithm [18], weighted differential evolution [19], etc. Liu et al. [20] investigated a least-squares-like scheme for a Hammerstein–Wiener-like system using periodic input signals. Ding et al. [11] used a particle-filtering technique to optimize the weight of each innovation and applied a multi-innovation stochastic gradient algorithm to estimate the system parameters. Rayouf et al. [12] proposed a subspace identification algorithm to recover the parameter information of the Hammerstein model, in which the computational complexity of the identification method is reduced. In reference [21], the authors applied the multi-innovation theory to improve the accuracy of the gradient identification method for the Wiener–Hammerstein-like system. Vörös [22] used an iterative identification approach for the nonlinear cascade, in which the input is a memory backlash nonlinearity. Cao et al. [23] assumed that the electromagnetic micromirror consists of a dynamic system and a hysteresis model,so that a recursive identification method for the Hammerstein-like system is achieved. A nonsmooth Kalman filtering method is introduced by Li et al. [24] for the sandwich-like system, and the proposed method is demonstrated in a motor-driving mechanical transmission system. Fang et al. [8] report an overparameterized method based on the regularization technique for Hammerstein-like system, in which the persistently exciting condition is satisfied by using different input signals. Jin et al. [13] used an auxiliary model recursice least square to optimize the parameter values based on the obtained initial values. In reference [25], Krikelis used a recurrent artificial neural network to approximate the Hammerstein-like hysteresis model and introduced a mean squared error method to obtain the system parameter data. Zong et al. [15] used an auxiliary model to improve the estimation accuracy of the particle swarm optimization and applied the multi-innovation theory to add the convergence rate of the particle swarm optimization. Based on the above reports, most of their works focused on finding an identification method based on the prediction output error, observation error, or output error method, which leads to the bias estimation and extremum problem. According to the error correction principle, if the self-error is used to correct the error information, the identification performance will be improved. In other words, the identification error corrects the parameter identification update, and the identification performance will be improved. Hence, the purpose of this work is to extract the parameter estimation error data from the system data and use it to design a parameter estimation adaptive law, so as to provide an optional identification method.

Different memory models have been proposed to characterize the feature of the nonlinear system with memory nonlinearity, such as backlash model [26], backlash-like model [27], Prandtl–Ishlinskii model [28], Bouc–Wen model [29], and Duhem model [30]. Hysteresis nonlinearity widely exists in practical systems such as manufacturing machine, piezoelectric actuator, servo mechanical system, diesel engine injection, and image scanning system. Nevertheless, hysteresis nonlinearity exhibits highly undesirable behavior such as oscillations or inaccuracies if not considered appropriately. Therefore, when we use the nonlinear model to model the real system with hysteresis, we should consider the problem of hysteresis parameter identification, which can effectively compensate the adverse effect of hysteresis based on the parameter identification results. Hu and Hu [9] introduced an extended recursive estimation method to recover parameter information of hysteresis Wiener–Hammerstein-like model and chose the micropositioning stage system as a real process to test the usefulness of the presented estimator. Amini Tehrani et al. [31] proposed a standard deviation minimum estimator for the nonlinear system with hysteresis, in which the hysteresis curves are used to obtain the final parameters optimization. Ai et al. [32] used least squares based on the special input signal to estimate the Hammerstein-like model with hysteresis. In reference [33], an adaptive identification method using gradient law is reported to nonlinear hysteresis system and used parameter projection scheme to increase the convergence performance. Krekelis et al. [25] introduced a hysteresis observer to obtain the system hysteresis state data by using frequency-shaped optimization and design the integral sliding mode control scheme based on the obtained hysteresis information. Son et al. [34] reported the hysteresis parameter estimation by developing a hybrid optimization algorithm, and the Jaya operator is used to increase the convergence efficiency. Among all the hysteresis models, the Duhem model is a popular hysteresis model because the other hysteresis models can be described by choosing diverse functions of the Duhem model. In this study, as shown in Figure 1, parameter estimation of the Wiener–Hammerstein-like system with Duhem hysteresis model is studied due to the effective description ability for the actual systems such as micropositioning stage [35], crystal detector [36], electromagnetic scanning system [37], and motor servo drive system [21]. Hence, the research of parameter identification for Wiener–Hammerstein-like system with Duhem hysteresis has practical significance.

Based on the above discussion, parameter estimation with a new recursive way for Wiener–Hammerstein-like system is introduced in this study. The developed filter is proposed to improve the bias estimation problem and increase the utilization of the system data, and an adaptive attenuation factor is introduced. Then, by force of the auxiliary filtered variables with batch data, the estimated parameter error data can be induced. Finally, a new optional adaptive estimator is developed. The simulation of example and the experiment of the motor servo system demonstrate that the introduced identification method is suitable for Wiener–Hammerstein-like system, and the identification performance in terms of identification accuracy and convergence rate is superior to the existing identification algorithms.

In order to clearly state the contribution of this study, some error definitions are given as follows: parameter estimation error denotes the difference between the estimated value and expected value. Prediction error represents the difference between the model-predicted output and the realsystem output. The mean squared error (MSE) is definedby and mean of prediction error (PEM) is described by , where the prediction data length is described by , real system output variance is represented by , is real system output, and represents the prediction model output. The contributions of this work are stated as follows:(1)A developed filter with a miserly structure for reducing the noise data effect is introduced, so that the bias estimation problem can be improved and the assumption conditions of filter are relaxed compared with the common filter [24, 35].(2)Parameter estimation error data based on auxiliary filtered variables are obtained, such that the extraction method for the parameter estimation error from the system data is provided.(3)A new parameter estimation estimator using parameter estimation error data is derived, so as to provide an alternative way to design the identification algorithm by using other errors rather than the traditional prediction error [9, 13, 20].

Rest contents of our report are provided as follows: in Section 2, the system representation of Wiener–Hammerstein-like system with the Duhem model is shown. The proposed identification method for the considered system is introduced in Section 3. The convergence of the developed method is given in Section 4. Section 5 provides examples using the presented algorithm. Conclusion and some future works are given in Section 6.

2. Problem Statement

The diagram of Wiener–Hammerstein-like system with the Duhem hysteresis model in this study is described in Figure 1, in which the input and output sequences are displayed by and , respectively. The signals and are internal variables. The system considered is represented as follows:where and are hysteresis curve functions and . The polynomials , , , and with shift operator are denoted as follows:where , , , and are polynomial degrees and , , , and are polynomial coefficients.

From the form of Duhem model (2), it can be seen that the continuous model cannot be used for parameter identification. For this reason, the discrete form should be provided. To obtain the above purpose, equation (2) can be rewritten as follows [38, 39]:where is the absolute value of .

According to equations (2) and (6), we have

Based on the solution of model (2) [38], and are defined as follows:where , , , and denote the coefficients. The sign function is described by .

By substituting equations (8) and (9) into equation (3), the discrete-time form of the Duhem hysteresis model is described as follows [9]:where .

By multiplying on both side of equation (1) and combining the polynomials and , it obtains

Similarly, equation (3) can be written as follows:

Considering the addition noise , by substituting equations (10) and (11) into equation (12), the estimation model for the considered system is written as follows:where estimated parameter data can be described as follows:

The data variable is defined as follows:

In order to facilitate the subsequent parameter identification, the common assumptions are listed [4042].

Assumption 1. When the system is excited using the input excitation signal, the states of the system can be fully displayed.

Assumption 2. The linear subsystems and are stable subsystems.

Assumption 3. All states of the system are zero before the system is collected.

Assumption 4. The degrees , , , and are known, and coefficients , , , and need to estimated.

Assumption 5. The additional noise and the input signal are independent of each other.

Assumption 6. The and are predetermined one.
Assumption 1 indicates the work condition of the input signal. In Assumption 2, the transfer functions of linear subsystem do not have zero pole cancellation, which is the identification condition of linear subsystem. The system considered is the causal system, as shown in Assumption 3. The estimated parameters are described in Assumption 4. The work condition of the noise signal is provided in Assumption 5. The model uniqueness condition is displayed in Assumption 6.
The reason that the authors study this study is to introduce a new estimation approach using parameter error data for Wiener–Hammerstein-like system, identify the estimated parameter data , , , and , analyse the convergence performance of the developed estimation method, demonstrate the efficiency and practicality of the proposed method using some examples.

3. Estimation Algorithm

As stated in the purpose of the study, a new estimation algorithm is provided in this section.

The variables and are defined as follows:where represents developed filter operator.

Remark 1. Data filtering is achieved by introducing a filter operator , and a simple structure simplifies the filter assumption conditions.
We know that prediction error is the most widely used error data in the design of identification algorithm due to the accessibility of the prediction error. Other error data are difficult to be used in the design of parameter estimator because the other error data are difficult to obtain. In this study, we will introduce an estimation error extraction method based on the system data, so as to design the parameter estimation method. According to the principle of error correction, the parameter estimation error is the error data that are directly related to parameter estimation. If the parameter adaptive law can be corrected by parameter error data, the performance of parameter estimation will be enhanced. To achieve this objective, the auxiliary variables and are provided on the basis of the variables and :where is developed adaptive attenuation factor, . , , , denotes the batch data length.

Remark 2. The developed adaptive attenuation factor is a gradually decreasing factor to improve the addition of new system data and avoid the problem of data inundation.
Based on the variables and , the vector can be defined as follows:By substituting and into and defining parameter estimation error , , and it haswhere .

Remark 3. We regard the variable as an extended estimation error variable because it contains identification error data . By developing auxiliary vectors , , and with filter technique, the identification error data can be derived. Different from the traditional identification scheme design (e.g., predictive output error and output error methods), based on , a new method is introduced to construct a parameter estimation estimator using identification error data.
Based on equation (20), a new parameter estimation adaptive law can be defined as follows:where denotes the modified gain.
To increase the performance of modified gain and online, the following recursive form is provided:

Remark 4. Compared with the modified gain with constant values, the modified gain is proposed using a recursive structure in this study. This construction method improves the speed of parameter estimation and is convenient for online implementation.
When the proposed algorithm is implemented, one of the difficulties is that the internal variables and in system are unknown. As shown in Figure 2, one of the solutions is to convert the unmeasurable variables and into indirectly measurable variables by using the principle of reference model and replace the unknown variables and by using the output of reference model [43, 44].
According to equations (10) and (11), the reference models and can be defined as follows:

4. Convergence Performance

In this section, we will provide the convergence study of the proposed adaptive estimator using general persistent excitation.

Theorem 1. It is assumed that is used to construct an algebra sequence and satisfies the noise assumptions [45]:(H1) (H2) and the persistent excitation (PE) (H3) satisfies(H3) Based on the conditions (H1)–(H3), the parameter estimation error converges to zero, that is,

Proof. By subtracting on both sides of equation (21), the following expression can be obtained:where is denoted by .
By defining , the following equation can be yielded using equation (21):By using matrix inversion theory and equation (22), equation (27) can be rewritten as follows:where .
Because and and are not related, based on conditions (H1) and (H2), by applying a martingale convergence theorem to equation (28), it yieldswhere represents the conditional probability of under a given .
Making further efforts, is further defined. By using equation (24), it hasBy using the martingale convergence theorem to equation (30), it giveswhere describes a finite variable.
Equation (26) can be rewritten as follows:where describes a large finite variable.
Based on the definition of , obtainswhere the matrix minimum eigenvalue is denoted by , and the matrix trace is described by .
According to condition (H3), it yieldsandwhere is initial value.
Based on equations (29) and (30), we haveSo far, Theorem 1 has been completed.

5. Examples

In this section, two examples including a numerical example and a real-word process are provided that are chosen to check the effectiveness of the introduced estimation method. In addition, three existing identification methods are selected as comparison algorithms.

5.1. Simulation Example

In this subsection, the Wiener–Hammerstein-like system with the Duhem hysteresis model is described as follows:

In the simulation, the considered system is fed by using a random sequence with zero mean and unit variance . The output is polluted by choosing a white noise with and . The simulation data length is chosen as .

The initial values of the identification methods are summarized as follows:(1)Proposed algorithm: , , , , , , , .(2)Modified generalized recursive algorithm (MGRA) [23]: , , , , , .(3)Recursive least square method based multi-innovation theory (RLS-MI) [44]: , , , , .(4)Weighted multi-innovation stochastic gradient algorithm (WMISG) [11]: , , , , .

The simulation results for estimating parameters of the linear subsystems and Duhem hysteresis are provided in Figures 35 respectively. It can be observed that the developed scheme, WMISG, MGRA, and RLS-MI can realize the expected values with similar performance. Compared to RLS-MI, the other three estimators reach better estimation performance bringing higher estimation accuracy. Compared with MGRA, WMISG, and RLS-MI, the developed method requires less convergence time bringing a faster convergence speed. It implies that the developed scheme gives satisfactory identification precision and convergence performance.

The estimation error , , [44] for the four estimator methods is shown in Figure 6. It is noticed that all the estimation errors produced by four schemes can gradually decrease with the addition of sample data. The estimation error results by the developed approach decrease with less convergence time and higher estimation performance in comparison with MGRA, WMISG, and RLS-MI methods. It indicates that the advantage of the developed scheme is demonstrated based on estimation error curves.

The model tests for the four identification methods are shown in Figure 7. It is noticed that all the estimation models based on the parameter estimation results can derive the established models to produce the desired output results. However, the presented method yields better prediction performance giving a smaller prediction output error compared with the other three estimation methods. It demonstrates that the four identification methods are effective for estimating the Wiener–Hammerstein-like parameters, and the developed algorithm has excellent performance.

Some evaluation indicators are used to check the usefulness of the four estimators, as listed in Table 1. The mean squared error (MSE) and mean of prediction error (PEM) in reference [13], integral square error (ISE), and integral absolute error (IAE) [17] are chosen as evaluation indicators. The smaller the indicators, the better identification performance. In Table 1, the negative sign only indicates the degree of deviation from the zero point because can be a positive or negative sign. When the performance index values are compared, only the data part is used for comparison. It can be found from Table 1 that MGRA, WMISG, RLS-MI, and developed approach can obtain relatively small values, in which the effectiveness of the four estimators can be demonstrated. We take the indicators and as examples to evaluate the identification performance. Compared with the MGRA, WMISG, and RLS-MI, the improvements in the identification performance of the proposed method are 33.9%, 41.26%, and 53.5%, respectively, in terms of . According to the index , the increased performances of the developed scheme are 65.2, and 71.7%, respectively. Index results show that the designed scheme provided better estimation performance than the other two identification schemes.

5.2. Experiment

In this section, we consider a motor servo system, which includes a permanent magnet synchronous motor drives the skew-wheel, and which drives the manipulator thereafter to move according to a given trajectory, as shown in Figure 8. The platform consists of a permanent magnet synchronous motor (ZLAC60ASM200), digital signal processing (TMS320F2809), and encoder (HF154S-A48). The given signal is chosen as .

The dynamic model of the motor servo system is described as follows [4648]:where , , and .

It can be seen from equation (33) that the estimated parameter can be written as , , , and . The controller is chosen as PID, in which the parameters are . The parameter estimation results using the developed scheme, WMISG, MGRA, and RLS-MI are shown in Figure 9. It can be seen that the parameter estimation fluctuates greatly at the beginning, but with the addition of data, the parameter estimation curve tends to be stable or fluctuates slightly. Evidently, the developed method can obtain faster convergence performance because the less convergence time is taken compared with MGRA, WMISG, and RLS-MI. Compared to RLS-MI and WMISG, MGRA produces finer identification results owing to the smaller vibration. It shows the classy performance of the developed algorithm in terms of the convergence rate. The tracking output profiles are shown in Figure 10, where the tracking output by the four identification algorithms can rapidly achieve the real dynamic output. Nevertheless, the tracking errors obtained by RLS-MI, WMISG, and MGRA are larger and that provided by the developed scheme is smaller. It exhibits that the proposed identification method has outstanding dynamic tracking performance and proves the superiority of the proposed estimation algorithm.

As listed in Table 2, the indicators produced by the four estimators give small values, meaning the usefulness of the MGRA, WMISG, RLS-MI, and proposed method. The percentages of elevated identification performances obtained by the designed method are 29.7, 76.9%, respectively, based on the index as compared with the MGRA, WMISG, and RLS-MI. The improved performances of the presented approach are shown by using 50%, 56.2%, and 68.4% based on the index compared to the other three estimators, suggesting the outstanding identification performance.

6. Conclusion

In this study, we propose a parameter estimation with a new recursive way of overcoming the shortage of the prediction error method for the Wiener–Hammerstein-like system. It improves the bias estimation issue using the data filtering technique with petty form, in which the assumption condition on the filter can be slackened. It also solves the data inundating problem using the developed adaptive attenuation factor, in which the system’s new data can be further used by the estimator. The obtained identification error data are used to establish the parameter estimation adaptive law, in which a novel identification structure is proposed compared with the traditional estimation methods. The martingale convergence theorem-based analysis is applied to analyse the convergence of the estimator and it is strictly proven that the estimated parameters can converge to true values. The developed algorithm is used for two examples involving simulation example and real-world process, and the parameter estimation, model validation results and indicators , , indicate that the effectiveness and preponderance of the proposed estimation scheme.

7. Discussion

The parameter estimation results and estimation error curves show that the proposed scheme has a fast convergence speed because the developed adaptive attenuation factor can improve the data flooding problem. The model validation and indicators indicate that the developed method has high estimation accuracy due to the addition of the identification error data. In the future, we will use the time-varying system or MIMO system to verify the effectiveness of the proposed identification algorithm based on the modified version.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.


This paper was supported by the National Natural Science Foundation of China (nos. 61873246, 62102373, and 62006213) and Key Scientific Research Projects of Colleges and Universities in Henan Province (no. 21A120010).