Complexity
Volume 2018 (2018), Article ID 9598307, 11 pages
https://doi.org/10.1155/2018/9598307

Research Article

Adaptive Gradient-Based Iterative Algorithm for Multivariable Controlled Autoregressive Moving Average Systems Using the Data Filtering Technique

Jian Pan,¹ Hao Ma,¹ Xiao Jiang,¹ Wenfang Ding,¹ and Feng Ding1,2,3

¹Hubei Collaborative Innovation Center for High-Efficiency Utilization of Solar Energy, Hubei University of Technology, Wuhan 430068, China
²College of Automation and Electronic Engineering, Qingdao University of Science and Technology, Qingdao 266042, China
³School of Internet of Things Engineering, Jiangnan University, Wuxi 214122, China

Correspondence should be addressed to Jian Pan; jpan@163.com

Received 31 March 2018; Accepted 3 June 2018; Published 24 July 2018

Academic Editor: Jing Na

Copyright © 2018 Jian Pan 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.

The identification problem of multivariable controlled autoregressive systems with measurement noise in the form of the moving average process is considered in this paper. The key is to filter the input–output data using the data filtering technique and to decompose the identification model into two subidentification models. By using the negative gradient search, an adaptive data filtering-based gradient iterative (F-GI) algorithm and an F-GI with finite measurement data are proposed for identifying the parameters of multivariable controlled autoregressive moving average systems. In the numerical example, we illustrate the effectiveness of the proposed identification methods.

1. Introduction

Parameter estimation plays an important role in system control [1–4], system analysis [5–8], and signal processing [9–13]. Parameter estimation is significant in system modeling [14, 15]. Multi-input multi-output systems widely exist in industrial control areas, which are also called multivariate systems or multivariable systems [16–18]. They are more complex in model structures than single-input single-output systems and always have high dimensions and numerous parameters, which make the parameter estimation more difficult. In this literature, Ding et al. proposed a filtering decomposition-based least squares iterative algorithm for multivariate pseudolinear ARMA systems [19]. Ma et al. studied the parameter estimation problem of multivariate Hammerstein systems and presented a modified Kalman filter-based recursive least squares algorithm to give the parameter estimates [20]. Pan et al. proposed a filtering-based multi-innovation extended stochastic gradient algorithm for multivariable systems [21].

The data filtering technique is an important approach in system identification [22] and state estimation. Chen and Ding applied the data filtering technique to identify the multi-input and single-output system based on the maximum likelihood recursive least squares algorithm [23]. Mao et al. derived an adaptive filtering-based multi-innovation stochastic gradient algorithm for the input nonlinear system with autoregressive noise [24]. They introduced a linear filter to filter the input and output signals and decomposed the identification model into two subidentification models (i.e., a noise model and a system filtered model), which can improve the convergence rate and computation efficiency [25]. The identification methods can be applied to many areas [26–29].

The gradient search is useful for identification as an optimization method [30, 31]. Many gradient-based algorithms, including the stochastic gradient algorithms [32–34] and the gradient-based iterative algorithms, have been developed using the multi-innovation identification theory, the maximum likelihood estimation methods [35, 36], the key-term separation principle [37, 38], and the data filtering theory. For example, Ma et al. presented an iterative variational Bayesian method to identify the Hammerstein varying systems with parameter uncertainties. Chen et al. studied the identification problem of bilinear-in-parameter systems and presented a gradient-based iterative algorithm by using the hierarchical identification principle and the gradient search [39]. Deng and Ding developed a Newton iterative identification method for an input nonlinear finite impulse response system with moving average noise [40]. Other methods can be referred as to the transfer function identification [41–45], linear system identification [46–51], and nonlinear system identification [52–59].

This paper uses the hierarchical identification principle to study the data filtering-based iterative identification methods for a multivariable controlled autoregressive moving average (M-CARMA) system. The basic idea is to introduce a linear filter to decompose the original identification model into two subidentification models and then obtain the parameter estimates using the negative gradient search. The main contributions are as follows:

(i)A filtering-based gradient iterative (F-GI) algorithm is proposed using the data filtering technique and the gradient search.(ii)A filtering-based gradient iterative algorithm with finite measurement data is developed to obtain the parameter estimates.

The layout of the remainder of this paper is as follows. Section 2 derives the identification model for the M-CARMA system. In Section 3, we derive a data filtering-based gradient iterative algorithm based on the data filtering technique. A filtering-based gradient iterative algorithm with finite measurement data is developed to estimate the unknown parameters in Section 4. A numerical example is shown in Section 5 to illustrate the benefits of the proposed methods in this paper. Finally, some concluding remarks are given in Section 6.

2. The Problem Formulation

Some notation is introduced for convenience: denotes the estimate of at time ; “” or “” stands for“ is defined as ”; the symbol () represents an identity matrix of appropriate size (); the symbol represents an -dimensional column vector whose elements are 1; denotes a unit forward shift operator like and ; the superscript T symbolizes the vector/matrix transpose; and the norm of a matrix is defined by .

The following multivariable controlled autoregressive moving average system in Figure 1 is considered,

where is the system input vector, is the system output vector, is a white noise vector with zero mean, and are the matrix polynomials in the unit backward shift operator , and is the polynomial in .

Figure 1: A multivariable controlled autoregressive moving average system.

Assume that the orders , , and are known, and , , and for . The intermediate variable is defined as

The system information vector , the noise information vector , the system parameter matrix , and the noise parameter vector are defined as

Equations (3) and (1) can be written as

Equation (5) is the noise identification model. For the M-CARMA system in (1), choose the polynomial as a filter. Define the filtered input vector , the filtered output vector , and the filtered information vector as

Multiplying both sides of (1) by obtains

or

Then we have

Equations (10) and (5) form the filtered identification models of the M-CARMA system.

3. The F-GI Algorithm

In this section, a linear filter is applied to deal with the moving average noise. A gradient-based iterative identification algorithm is proposed for M-CARMA systems by using the data filtering technique [60–63].

Considering the newest data from to , the stacked filtered output matrix , the stacked filtered information matrix the stacked noise vector , and the stacked noise information matrix are defined as

Define a quadratic criterion function:

Let be an iterative variable. Let and be the estimates of and at iteration . Minimizing and and using the negative gradient search will give the following iterative relations for obtaining the parameter estimates of and :

where and are the iterative step size or the convergence factor. However, the difficulty is that the noise information matrix (i.e., ) contains the unmeasured vector . So the gradient-based iterative algorithm in (13) and (14) cannot give the parameter estimate directly. The solution is to use the hierarchical identification principle and to replace the unknown variable with its corresponding estimates at iteration , and to define the estimate of as

Using to construct the estimate of obtains

Replacing in (6) with gives

Replacing in (17) with obtains the estimate of at iteration :

From (6), we have

Replacing , , and with , , and obtains the iterative estimate of at iteration :

Then, using to construct the iterative estimate of at iteration gives

Use to construct the estimate of as

Using to filter and gives the filtered estimates and of and :

Furthermore, we use to construct the estimate of , use and to construct the estimate of , and use to construct the estimate of at iteration :

From the above derivation, we can summarize a filtering-based multi-innovation gradient iterative identification algorithm:

The identification steps of the algorithm in (25), (26), (27), (28), (29), (30), (31), (32), (33), (34), (35), (36), (37), (38), (39), and (40) to compute and are listed as follows:

(1)Set the initial values: let , give the data length , and give a small positive number . Set the initial values , , .(2)Collect the input–output data and and construct using (33).(3)Let and set the initial values ,(4)Construct and using (31) and (32).(5)Compute using (36) and form the stacked noise vector using (29) and the stacked noise matrix using (30).(6)Choose using (40) and update the noise parameter estimates using (28).(7)Read from in (38). Compute and using (34) and (35).(8)Construct using (27) and using (26).(9)Choose using (39) and update the system parameter estimates using (25).(10)Compute using (37).(11)Compare with and compare with : if and , increase by 1 and turn to Step 4; otherwise, obtain and the parameter estimate vector and , let , , increase by 1 and turn to Step 2.

The flowchart of computing and from the F-GI algorithm is shown in Figure 2.

Figure 2: The flowchart of computing the F-GI parameter estimates.

4. The F-GI Algorithm with Finite Measurement Data

Consider the data from to and define the stacked filtered output matrix , the stacked filtered information matrix , the stacked noise vector , and the stacked noise information matrix as

Note that and contain all the measured data .

The two gradient criterion functions are defined as

Similarly, minimizing and , we can derive a filtering-based gradient iterative (F-GI) algorithm with the data length for the M-CARMA system:

The identification steps of the F-GI algorithm with finite measurement data in (43), (44), (45), (46), (47), (48), (49), (50), (51), (52), (53), (54), (55), (56), (57), and (58) to compute and are listed as follows.

(1)Collect the input–output data and give a small positive number . Construct using (51).(2)Let and set the initial values (3)Construct and using (49) and (50).(4)Compute using (54), construct using (47), and construct using (48).(5)Choose using (58) and update using (46).(6)Read from in (56) and compute and using (52) and (53).(7)Construct using (45) and construct using (44).(8)Choose using (57) and update using (43).(9)Compute using (55).(10)Compare with and compare with : if and , increase by 1 and turn to Step 3; otherwise, obtain and the parameter estimate vector and .

The flowchart of computing and from the F-GI algorithm with finite measurement data is shown in Figure 3.

Figure 3: The flowchart of computing the F-GI parameter estimates with finite measurement data.

The algorithm in (43), (44), (45), (46), (47), (48), (49), (50), (51), (52), (53), (54), (55), (56), (57), and (58) for multivariable CARMA systems is based on the filtering technique and the gradient search and can be extended to more complex multivariable systems with cooled noises.

5. Numerical Example

Consider a two-input two-output CARMA system:

where

In simulation, and are taken as two persistent excitation signal sequences with zero mean and unit variance, and and as two white noise sequences with . Take the data length and and apply the F-GI algorithm with finite measurement data in (43), (44), (45), (46), (47), (48), (49), (50), (51), (52), (53), (54), (55), (56), (57), and (58) to estimate the parameters of this M-CARMA system. The parameter estimates and errors are shown in Table 1 with , , and , and the parameter estimation errors versus are shown in Figure 4. For comparison with the different data length , the simulation results are shown in Table 2 and Figure 5.

Table 1: The F-GI parameter estimates and errors ().

1
0.14294
0.09843
0.00868
0.00915
0.00296
0.08637
0.00937
−0.00661
−0.27864
86.50364

10
0.42310
0.34081
0.15604
0.12072
−0.02498
0.29864
0.20824
−0.15361
−0.70856
56.91872

50
0.63701
0.59049
0.44414
0.30015
−0.21622
0.83432
0.36331
−0.32901
−0.77686
16.06327

200
0.65476
0.60674
0.46465
0.31821
−0.31707
1.02684
0.38855
−0.39407
−0.56444
3.58517

500
0.65477
0.60673
0.46466
0.31824
−0.31710
1.02689
0.38857
−0.39418
−0.56441
3.58655

1
0.14514
0.09935
0.00228
0.00199
0.00410
0.08912
0.00243
−0.00127
−0.28184
86.71589

10
0.52198
0.41436
0.13005
0.04549
−0.05221
0.40777
0.16182
−0.15682
−0.91038
54.85474

50
0.65005
0.60460
0.37591
0.17468
−0.29039
0.97368
0.33485
−0.29105
−0.60764
13.34916

200
0.65652
0.60385
0.46751
0.25681
−0.31942
1.02762
0.38992
−0.36323
−0.56348
6.73257

500
0.65706
0.60351
0.48224
0.27468
−0.31970
1.02832
0.39775
−0.37791
−0.56279
5.95620

True values
0.65000
0.60000
0.45000
0.35000
−0.30000
1.00000
0.38000
−0.40000
−0.60000

Figure 4: The CARMA-FGI estimation errors with different noise variances

Table 2: The F-GI parameter estimates and errors ().

1
0.14346
0.09808
0.01067
0.00762
0.00511
0.08406
0.01113
−0.00704
−0.27896
86.56006

10
0.39890
0.32288
0.22390
0.15247
−0.01745
0.27349
0.21267
−0.15857
−0.68297
57.02995

50
0.63521
0.59550
0.44755
0.31132
−0.21207
0.82865
0.34932
−0.32757
−0.77919
16.41676

200
0.65416
0.60726
0.45998
0.33687
−0.31265
1.01818
0.37903
−0.40383
−0.58336
1.98279

500
0.65417
0.60725
0.45998
0.33691
−0.31269
1.01824
0.37904
−0.40396
−0.58332
1.98657

1
0.14560
0.09899
0.00315
0.00157
0.00466
0.08794
0.00329
−0.00156
−0.28174
86.73735

10
0.51092
0.41373
0.17192
0.06708
−0.05673
0.42188
0.14417
−0.12991
−0.91458
53.92889

50
0.64851
0.60875
0.38838
0.20796
−0.28513
0.96729
0.30507
−0.28172
−0.62099
12.59343

200
0.65613
0.60516
0.46036
0.30206
−0.31371
1.01919
0.36807
−0.38499
−0.58216
3.60147

500
0.65684
0.60452
0.47158
0.31970
−0.31423
1.02001
0.37658
−0.40343
−0.58126
2.91324

True values
0.65000
0.60000
0.45000
0.35000
−0.30000
1.00000
0.38000
−0.40000
−0.60000

Figure 5: The CARMA-FGI estimation errors with different noise variances .

From Tables 1–3 and Figures 4 and 5, we can draw the following conclusions.

(1)The parameter estimation errors obtained by the presented algorithms gradually become smaller with the iterative variable increasing. Thus, the proposed algorithms for M-CARMA systems are effective.(2)The system parameter estimates converge to their true values with the increasing of the data length.(3)Under the same data length, a smaller noise variance leads to higher parameter estimation accuracy and a faster convergence rate.(4)A longer data length leads to a smaller estimation error under the same noise level.

Table 3: The F-GI parameter estimates and errors ().

1000
0.65477
0.60673
0.46466
0.31824
−0.31710
1.02689
0.38857
−0.39418
−0.56441
3.58655

2000
0.65417
0.60725
0.45998
0.33691
−0.31269
1.01824
0.37904
−0.40396
−0.58332
1.98657

1000
0.65706
0.60351
0.48224
0.27468
−0.31970
1.02832
0.39775
−0.37791
−0.56279
5.95620

2000
0.65684
0.60452
0.47158
0.31970
−0.31423
1.02001
0.37658
−0.40343
−0.58126
2.91324

True values
0.65000
0.60000
0.45000
0.35000
−0.30000
1.00000
0.38000
−0.40000
−0.60000

6. Conclusions

An F-GI algorithm and an F-GI algorithm with finite measurement data are proposed for identifying the multivariable controlled autoregressive system with measurement noise in this paper. The linear filter is introduced to filter the input–output data, and the hierarchical identification principle is applied to decompose the identification model into two subidentification models. The simulation results show that the proposed algorithms can generate accurate estimates. The proposed approaches in the paper can combine other mathematical tools [64–69] and statistical strategies [70–75] to study the performances of some parameter estimation algorithms and can be applied to other multivariable systems with different structures and disturbance noises and other literature [76–86] such as system identification [87–92].

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 61471162) and the National First-Class Discipline Program of Light Industry Technology and Engineering (LITE2018-26).

References

W. Zhang, X. Lin, and B. S. Chen, “LaSalle-type theorem and Its applications to infinite horizon optimal control of discrete-time nonlinear stochastic systems,” IEEE Transactions on Automatic Control, vol. 62, no. 1, pp. 250–261, 2017.

Y. Lin and W. Zhang, “Necessary/sufficient conditions for pareto optimum in cooperative difference game,” Optimal Control, Applications and Methods, vol. 39, no. 2, pp. 1043–1060, 2018.

D. Q. Wang, Z. Zhang, and J. Y. Yuan, “Maximum likelihood estimation method for dual-rate Hammerstein systems,” International Journal of Control, Automation and Systems, vol. 15, no. 2, pp. 698–705, 2017.

D. Wang, L. Mao, and F. Ding, “Recasted models based hierarchical extended stochastic gradient method for MIMO nonlinear systems,” IET Control Theory and Applications, vol. 11, no. 4, pp. 476–485, 2017.

Y. Wang, D. Zhao, Y. Li, and S. X. Ding, “Unbiased minimum variance fault and state estimation for linear discrete time-varying two-dimensional systems,” IEEE Transactions on Automatic Control, vol. 62, no. 10, pp. 5463–5469, 2017.

J. Na, M. N. Mahyuddin, G. Herrmann, X. Ren, and P. Barber, “Robust adaptive finite-time parameter estimation and control for robotic systems,” International Journal of Robust and Nonlinear Control, vol. 25, no. 16, pp. 3045–3071, 2015.

J. Na, G. Herrmann, and K. Zhang, “Improving transient performance of adaptive control via a modified reference model and novel adaptation,” International Journal of Robust and Nonlinear Control, vol. 27, no. 8, pp. 1351–1372, 2017.

W. Zhang, Y. Zhao, and L. Sheng, “Some remarks on stability of stochastic singular systems with state-dependent noise,” Automatica, vol. 51, pp. 273–277, 2015.

J. Na, J. Yang, X. Wu, and Y. Guo, “Robust adaptive parameter estimation of sinusoidal signals,” Automatica, vol. 53, pp. 376–384, 2015.

L. Xu and F. Ding, “Recursive least squares and multi-innovation stochastic gradient parameter estimation methods for signal modeling,” Circuits, Systems, and Signal Processing, vol. 36, no. 4, pp. 1735–1753, 2017.

L. Xu and F. Ding, “Iterative parameter estimation for signal models based on measured data,” Circuits, Systems, and Signal Processing, vol. 37, no. 7, pp. 3046–3069, 2018.

L. Xu, “The parameter estimation algorithms based on the dynamical response measurement data,” Advances in Mechanical Engineering, vol. 9, no. 11, pp. 1–12, 2017.

J. Na, J. Yang, X. Ren, and Y. Guo, “Robust adaptive estimation of nonlinear system with time-varying parameters,” International Journal of Adaptive Control and Signal Processing, vol. 29, no. 8, pp. 1055–1072, 2015.

L. Xu and F. Ding, “Parameter estimation algorithms for dynamical response signals based on the multi-innovation theory and the hierarchical principle,” IET Signal Processing, vol. 11, no. 2, pp. 228–237, 2017.

L. Xu and F. Ding, “Parameter estimation for control systems based on impulse responses,” International Journal of Control, Automation and Systems, vol. 15, no. 6, pp. 2471–2479, 2017.

P. Ma, F. Ding, and Q. Zhu, “Decomposition-based recursive least squares identification methods for multivariate pseudo-linear systems using the multi-innovation,” International Journal of Systems Science, vol. 49, no. 5, pp. 920–928, 2018.

J. Ding, “Recursive and iterative least squares parameter estimation algorithms for multiple-input-output-error systems with autoregressive noise,” Circuits, Systems, and Signal Processing, vol. 37, no. 5, pp. 1884–1906, 2018.

J. Ding, “The hierarchical iterative identification algorithm for multi-input-output-error systems with autoregressive noise,” Complexity, vol. 2017, Article ID 5292894, 11 pages, 2017.

F. Ding, F. Wang, L. Xu, and M. Wu, “Decomposition based least squares iterative identification algorithm for multivariate pseudo-linear ARMA systems using the data filtering,” Journal of the Franklin Institute, vol. 354, no. 3, pp. 1321–1339, 2017.

J. Ma, W. Xiong, J. Chen, and D. Feng, “Hierarchical identification for multivariate Hammerstein systems by using the modified Kalman filter,” IET Control Theory and Applications, vol. 11, no. 6, pp. 857–869, 2017.

J. Pan, X. Jiang, X. Wan, and W. Ding, “A filtering based multi-innovation extended stochastic gradient algorithm for multivariable control systems,” International Journal of Control, Automation and Systems, vol. 15, no. 3, pp. 1189–1197, 2017.

F. Ding, X. Liu, and Y. Gu, “An auxiliary model based least squares algorithm for a dual-rate state space system with time-delay using the data filtering,” Journal of the Franklin Institute, vol. 353, no. 2, pp. 398–408, 2016.

F. Chen and F. Ding, “The filtering based maximum likelihood recursive least squares estimation for multiple-input single-output systems,” Applied Mathematical Modelling, vol. 40, no. 3, pp. 2106–2118, 2016.

Y. Mao, F. Ding, and E. Yang, “Adaptive filtering-based multi-innovation gradient algorithm for input nonlinear systems with autoregressive noise,” International Journal of Adaptive Control and Signal Processing, vol. 31, no. 10, pp. 1388–1400, 2017.

C. Wang and L. Zhu, “Parameter identification of a class of nonlinear systems based on the multi-innovation identification theory,” Journal of the Franklin Institute, vol. 352, no. 10, pp. 4624–4637, 2015.

Y. Zhang, Y. Cao, Y. Wen, L. Liang, and F. Zou, “Optimization of information interaction protocols in cooperative vehicle-infrastructure systems,” Chinese Journal of Electronics, vol. 27, no. 2, pp. 439–444, 2018.

F. Liu, “A note on Marcinkiewicz integrals associated to surfaces of revolution,” Journal of the Australian Mathematical Society, vol. 104, no. 3, pp. 380–402, 2018.

P. Li, R. Dargaville, Y. Cao, D. Y. Li, and J. Xia, “Storage aided system property enhancing and hybrid robust smoothing for large-scale PV systems,” IEEE Transactions on Smart Grid, vol. 8, no. 6, pp. 2871–2879, 2017.

Y. Cao, L. Ma, S. Xiao, X. Zhang, and W. Xu, “Standard analysis for transfer delay in CTCS-3,” Chinese Journal of Electronics, vol. 26, no. 5, pp. 1057–1063, 2017.

M. Li, X. Liu, and F. Ding, “Least-squares-based iterative and gradient-based iterative estimation algorithms for bilinear systems,” Nonlinear Dynamics, vol. 89, no. 1, pp. 197–211, 2017.

M. Li, X. Liu, and F. Ding, “The gradient based iterative estimation algorithms for bilinear systems with autoregressive noise,” Circuits, Systems, and Signal Processing, vol. 36, no. 11, pp. 4541–4568, 2017.

F. Ding, X. Wang, L. Mao, and L. Xu, “Joint state and multi-innovation parameter estimation for time-delay linear systems and its convergence based on the Kalman filtering,” Digital Signal Processing, vol. 62, pp. 211–223, 2017.

X. Zhang, F. Ding, F. E. Alsaadi, and T. Hayat, “Recursive parameter identification of the dynamical models for bilinear state space systems,” Nonlinear Dynamics, vol. 89, no. 4, pp. 2415–2429, 2017.

X. Zhang, L. Xu, F. Ding, and T. Hayat, “Combined state and parameter estimation for a bilinear state space system with moving average noise,” Journal of the Franklin Institute, vol. 355, no. 6, pp. 3079–3103, 2018.

M. Li, X. Liu, and F. Ding, “The maximum likelihood least squares based iterative estimation algorithm for bilinear systems with autoregressive moving average noise,” Journal of the Franklin Institute, vol. 354, no. 12, pp. 4861–4881, 2017.

L. Ma and X. Liu, “Recursive maximum likelihood method for the identification of Hammerstein ARMAX system,” Applied Mathematical Modelling, vol. 40, no. 13-14, pp. 6523–6535, 2016.

J. Vörös, “Identification of nonlinear cascade systems with output hysteresis based on the key term separation principle,” Applied Mathematical Modelling, vol. 39, no. 18, pp. 5531–5539, 2015.

F. Ding, H. Chen, L. Xu, J. Dai, Q. Li, and T. Hayat, “A hierarchical least squares identification algorithm for Hammerstein nonlinear systems using the key term separation,” Journal of the Franklin Institute, vol. 355, no. 8, pp. 3737–3752, 2018.

M. Chen, F. Ding, L. Xu, T. Hayat, and A. Alsaedi, “Iterative identification algorithms for bilinear-in-parameter systems with autoregressive moving average noise,” Journal of the Franklin Institute, vol. 354, no. 17, pp. 7885–7898, 2017.

K. Deng and F. Ding, “Newton iterative identification method for an input nonlinear finite impulse response system with moving average noise using the key variables separation technique,” Nonlinear Dynamics, vol. 76, no. 2, pp. 1195–1202, 2014.

L. Xu, “A proportional differential control method for a time-delay system using the Taylor expansion approximation,” Applied Mathematics and Computation, vol. 236, pp. 391–399, 2014.

L. Xu, “Application of the Newton iteration algorithm to the parameter estimation for dynamical systems,” Journal of Computational and Applied Mathematics, vol. 288, pp. 33–43, 2015.

L. Xu, L. Chen, and W. Xiong, “Parameter estimation and controller design for dynamic systems from the step responses based on the Newton iteration,” Nonlinear Dynamics, vol. 79, no. 3, pp. 2155–2163, 2015.

L. Xu, “The damping iterative parameter identification method for dynamical systems based on the sine signal measurement,” Signal Processing, vol. 120, pp. 660–667, 2016.

F. Ding, D. Meng, J. Dai, Q. Li, A. Alsaedi, and T. Hayat, “Least squares based iterative parameter estimation algorithm for stochastic dynamical systems with ARMA noise using the model equivalence,” International Journal of Control, Automation and Systems, vol. 16, no. 2, pp. 630–639, 2018.

L. Xu, F. Ding, Y. Gu, A. Alsaedi, and T. Hayat, “A multi-innovation state and parameter estimation algorithm for a state space system with d-step state-delay,” Signal Processing, vol. 140, pp. 97–103, 2017.

F. Ding, F. Wang, L. Xu, T. Hayat, and A. Alsaedi, “Parameter estimation for pseudo-linear systems using the auxiliary model and the decomposition technique,” IET Control Theory and Applications, vol. 11, no. 3, pp. 390–400, 2017.

F. Ding, L. Xu, and Q. Zhu, “Performance analysis of the generalised projection identification for time-varying systems,” IET Control Theory and Applications, vol. 10, no. 18, pp. 2506–2514, 2016.

Y. Wang and F. Ding, “Novel data filtering based parameter identification for multiple-input multiple-output systems using the auxiliary model,” Automatica, vol. 71, pp. 308–313, 2016.

Y. Wang and F. Ding, “Filtering-based iterative identification for multivariable systems,” IET Control Theory and Applications, vol. 10, no. 8, pp. 894–902, 2016.

Y. Wang and F. Ding, “The auxiliary model based hierarchical gradient algorithms and convergence analysis using the filtering technique,” Signal Processing, vol. 128, pp. 212–221, 2016.

H. Chen, Y. Xiao, and F. Ding, “Hierarchical gradient parameter estimation algorithm for Hammerstein nonlinear systems using the key term separation principle,” Applied Mathematics and Computation, vol. 247, pp. 1202–1210, 2014.

Y. Mao and F. Ding, “A novel parameter separation based identification algorithm for Hammerstein systems,” Applied Mathematics Letters, vol. 60, pp. 21–27, 2016.

M. Gan, H.-X. Li, and H. Peng, “A variable projection approach for efficient estimation of RBF-ARX model,” IEEE Transactions on Cybernetics, vol. 45, no. 3, pp. 462–471, 2015.

M. Gan, C. L. P. Chen, G. Y. Chen, and L. Chen, “On some separated algorithms for separable nonlinear squares problems,” IEEE Transactions on Cybernetics, 2018.

F. Ding, Y. Wang, J. Dai, Q. Li, and Q. Chen, “A recursive least squares parameter estimation algorithm for output nonlinear autoregressive systems using the input-output data filtering,” Journal of the Franklin Institute, vol. 354, no. 15, pp. 6938–6955, 2017.

Y. Wang and F. Ding, “A filtering based multi-innovation gradient estimation algorithm and performance analysis for nonlinear dynamical systems,” IMA Journal of Applied Mathematics, vol. 82, no. 6, pp. 1171–1191, 2017.

Y. Wang and F. Ding, “Recursive least squares algorithm and gradient algorithm for Hammerstein-Wiener systems using the data filtering,” Nonlinear Dynamics, vol. 84, no. 2, pp. 1045–1053, 2016.

Y. Wang and F. Ding, “Recursive parameter estimation algorithms and convergence for a class of nonlinear systems with colored noise,” Circuits, Systems, and Signal Processing, vol. 35, no. 10, pp. 3461–3481, 2016.

M. Li and X. Liu, “The least squares based iterative algorithms for parameter estimation of a bilinear system with autoregressive noise using the data filtering technique,” Signal Processing, vol. 147, pp. 23–34, 2018.

M. Li and X. Liu, “Auxiliary model based least squares iterative algorithms for parameter estimation of bilinear systems using interval-varying measurements,” IEEE Access, vol. 6, pp. 21518–21529, 2018.

Y. Wang, F. Ding, and L. Xu, “Some new results of designing an IIR filter with colored noise for signal processing,” Digital Signal Processing, vol. 72, pp. 44–58, 2018.

F. Ding, L. Xu, F. E. Alsaadi, and T. Hayat, “Iterative parameter identification for pseudo-linear systems with ARMA noise using the filtering technique,” IET Control Theory and Applications, vol. 12, no. 7, pp. 892–899, 2018.

F. Liu, Q. Xue, and K. Yabuta, “Rough maximal singular integral and maximal operators supported by subvarieties on Triebel-Lizorkin spaces,” Nonlinear Analysis, vol. 171, pp. 41–72, 2018.

F. Liu, “Continuity and approximate differentiability of multisublinear fractional maximal functions,” Mathematical Inequalities & Applications, vol. 21, no. 1, pp. 25–40, 2018.

F. Liu and H. Wu, “Singular integrals related to homogeneous mappings in triebel-lizorkin spaces,” Journal of Mathematical Inequalities, vol. 11, no. 4, pp. 1075–1097, 2017.

F. Liu and H. X. Wu, “Regularity of discrete multisublinear fractional maximal functions,” Science China Mathematics, vol. 60, no. 8, pp. 1461–1476, 2017.

F. Liu and H. Wu, “On the regularity of maximal operators supported by submanifolds,” Journal of Mathematical Analysis and Applications, vol. 453, no. 1, pp. 144–158, 2017.

F. Liu, “On the Triebel-Lizorkin space boundedness of Marcinkiewicz integrals along compound surfaces,” Mathematical Inequalities & Applications, vol. 20, no. 2, pp. 515–535, 2017.

C. Yin and J. Zhao, “Nonexponential asymptotics for the solutions of renewal equations, with applications,” Journal of Applied Probability, vol. 43, no. 3, pp. 815–824, 2006.

H. Gao and C. Yin, “The perturbed sparre Andersen model with a threshold dividend strategy,” Journal of Computational and Applied Mathematics, vol. 220, no. 1-2, pp. 394–408, 2008.

C. Yin and C. Wang, “The perturbed compound Poisson risk process with investment and debit interest,” Methodology and Computing in Applied Probability, vol. 12, no. 3, pp. 391–413, 2010.

C. Yin and K. C. Yuen, “Optimality of the threshold dividend strategy for the compound Poisson model,” Statistics & Probability Letters, vol. 81, no. 12, pp. 1841–1846, 2011.

C. Yin, Y. Shen, and Y. Wen, “Exit problems for jump processes with applications to dividend problems,” Journal of Computational and Applied Mathematics, vol. 245, pp. 30–52, 2013.

C. Yin and Y. Wen, “Optimal dividend problem with a terminal value for spectrally positive Lévy processes,” Insurance: Mathematics and Economics, vol. 53, no. 3, pp. 769–773, 2013.

Y. Ji and F. Ding, “Multiperiodicity and exponential attractivity of neural networks with mixed delays,” Circuits, Systems, and Signal Processing, vol. 36, no. 6, pp. 2558–2573, 2017.

J. Pan, X. Yang, H. Cai, and B. Mu, “Image noise smoothing using a modified Kalman filter,” Neurocomputing, vol. 173, pp. 1625–1629, 2016.

X. Wan, Y. Li, C. Xia, M. Wu, J. Liang, and N. Wang, “A T-wave alternans assessment method based on least squares curve fitting technique,” Measurement, vol. 86, pp. 93–100, 2016.

N. Zhao, M. Wu, and J. Chen, “Android-based mobile educational platform for speech signal processing,” International Journal of Electrical Engineering Education, vol. 54, no. 1, pp. 3–16, 2016.

N. Zhao, Y. Chen, R. Liu, M. Wu, and W. Xiong, “Monitoring strategy for relay incentive mechanism in cooperative communication networks,” Computers & Electrical Engineering, vol. 60, pp. 14–29, 2017.

P. Gong, W.-Q. Wang, F. Li, and H. C. So, “Sparsity-aware transmit beamspace design for FDA-MIMO radar,” Signal Processing, vol. 144, pp. 99–103, 2018.

Z. H. Rao, C. Y. Zeng, M. H. Wu et al., “Research on a handwritten character recognition algorithm based on an extended nonlinear kernel residual network,” KSII Transactions on Internet and Information Systems, vol. 12, no. 1, pp. 413–435, 2018.

N. Zhao, R. Liu, Y. Chen et al., “Contract design for relay incentive mechanism under dual asymmetric information in cooperative networks,” Wireless Networks, 2018.

G. Xu, Y. Shekofteh, A. Akgül, C. Li, and S. Panahi, “A new chaotic system with a self-excited attractor: entropy measurement, signal encryption, and parameter estimation,” Entropy, vol. 20, no. 2, pp. 1–23, 2018.

D. Q. Zhu, X. Cao, B. Sun, and C. M. Luo, “Biologically inspired self-organizing map applied to task assignment and path planning of an AUV system,” IEEE Transactions on Cognitive and Developmental Systems, vol. 10, no. 2, pp. 304–313, 2018.

X.-F. Li, Y.-D. Chu, A. Y. T. Leung, and H. Zhang, “Synchronization of uncertain chaotic systems via complete-adaptive-impulsive controls,” Chaos, Solitons & Fractals, vol. 100, pp. 24–30, 2017.

J. Li, W. X. Zheng, J. Gu, and L. Hua, “A recursive identification algorithm for Wiener nonlinear systems with linear state-space subsystem,” Circuits, Systems, and Signal Processing, vol. 37, no. 6, pp. 2374–2393, 2018.

J. Li, W. X. Zheng, J. Gu, and L. Hua, “Parameter estimation algorithms for Hammerstein output error systems using Levenberg–Marquardt optimization method with varying interval measurements,” Journal of the Franklin Institute, vol. 354, no. 1, pp. 316–331, 2017.

X. Wang and F. Ding, “Recursive parameter and state estimation for an input nonlinear state space system using the hierarchical identification principle,” Signal Processing, vol. 117, pp. 208–218, 2015.

X. Wang and F. Ding, “Convergence of the recursive identification algorithms for multivariate pseudo-linear regressive systems,” International Journal of Adaptive Control and Signal Processing, vol. 30, no. 6, pp. 824–842, 2016.

D. Wang, L. Li, Y. Ji, and Y. Yan, “Model recovery for Hammerstein systems using the auxiliary model based orthogonal matching pursuit method,” Applied Mathematical Modelling, vol. 54, pp. 537–550, 2018.

D. Wang and Y. Gao, “Recursive maximum likelihood identification method for a multivariable controlled autoregressive moving average system,” IMA Journal of Mathematical Control and Information, vol. 33, no. 4, pp. 1015–1031, 2016.