Special Issue

## Stochastic Systems 2014

View this Special Issue

Research Article | Open Access

Volume 2014 |Article ID 329437 | 10 pages | https://doi.org/10.1155/2014/329437

# Two Identification Methods for Dual-Rate Sampled-Data Nonlinear Output-Error Systems

Accepted20 Mar 2014
Published10 Apr 2014

#### 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 . Many identification methods assume that the input-output data at every sampling instant are available for linear systems  and nonlinear systems , 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 , for example, dual-rate or multirate systems . Dual-rate/multirate systems in which the input and the output are sampled at different frequencies arise widely in robust filtering and control , adaptive control , and system identification . 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 . 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 , the iterative identification methods , the multi-innovation identification methods , the hierarchical identification methods , and the two-stage or multistage identification methods [84, 85] to study identification problems for other linear systems  or nonlinear systems .

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.

 1000 2000 3000 4000 5000 True values 0.30790 0.43409 0.48162 0.49513 0.49505 0.49000 −0.16601 −0.20319 −0.20626 −0.20656 −0.20341 −0.20000 0.19508 0.19548 0.19462 0.19665 0.19816 0.20000 0.36487 0.39043 0.39879 0.40105 0.39987 0.40000 0.09729 0.09384 0.08995 0.08769 0.08705 0.08000 0.13565 0.14818 0.15401 0.15931 0.15867 0.16000 0.02161 0.02602 0.02558 0.02764 0.02770 0.02000 0.02641 0.03181 0.03127 0.03378 0.03385 0.04000 (%) 26.70140 8.46344 2.72656 2.15284 1.91759
 1000 2000 3000 4000 5000 True values 0.39201 0.46141 0.50310 0.49802 0.48917 0.49000 −0.18980 −0.19696 −0.19784 −0.20113 −0.20307 −0.20000 0.18974 0.19349 0.19872 0.20192 0.20281 0.20000 0.40122 0.41674 0.39648 0.40109 0.40350 0.40000 0.09799 0.08924 0.08427 0.08475 0.08276 0.08000 0.14716 0.15484 0.15489 0.16514 0.16040 0.16000 0.02005 0.02781 0.02034 0.02761 0.02600 0.02000 0.02674 0.03708 0.02712 0.03682 0.03467 0.04000 (%) 14.27547 5.08770 2.79209 1.91002 1.41209

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).

1. F. Ding, System Identification—New Theory and Methods, Science Press, Beijing, China, 2013.
2. F. Ding, System Identification—Performances Analysis for Identification Methods, Science Press, Beijing, China, 2014.
3. Y. Liu, Y. Xiao, and X. Zhao, “Multi-innovation stochastic gradient algorithm for multiple-input single-output systems using the auxiliary model,” Applied Mathematics and Computation, vol. 215, no. 4, pp. 1477–1483, 2009. View at: Publisher Site | Google Scholar
4. Y. Liu, L. Xie, and F. Ding, “An auxiliary model based on a recursive least-squares parameter estimation algorithm for non-uniformly sampled multirate systems,” Proceedings of the Institution of Mechanical Engineers: Journal of Systems and Control Engineering, vol. 223, no. 4, pp. 445–454, 2009. View at: Publisher Site | Google Scholar
5. Y. Liu, J. Sheng, and R. F. Ding, “Convergence of stochastic gradient estimation algorithm for multivariable ARX-like systems,” Computers and Mathematics with Applications, vol. 59, no. 8, pp. 2615–2627, 2010. View at: Publisher Site | Google Scholar
6. Y. J. Liu and R. Ding, “Consistency of the extended gradient identification algorithm for multi-input multi-output systems with moving average noises,” International Journal of Computer Mathematics, vol. 90, no. 9, pp. 1840–1852, 2013. View at: Google Scholar
7. F. Ding and T. Chen, “Performance bounds of forgetting factor least-squares algorithms for time-varying systems with finite measurement data,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 52, no. 3, pp. 555–566, 2005. View at: Publisher Site | Google Scholar
8. F. Ding, Y. Shi, and T. Chen, “Performance analysis of estimation algorithms of nonstationary ARMA processes,” IEEE Transactions on Signal Processing, vol. 54, no. 3, pp. 1041–1053, 2006. View at: Publisher Site | Google Scholar
9. F. Ding, T. Chen, and L. Qiu, “Bias compensation based recursive least-squares identification algorithm for MISO systems,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 53, no. 5, pp. 349–353, 2006. View at: Publisher Site | Google Scholar
10. F. Ding, H. Yang, and F. Liu, “Performance analysis of stochastic gradient algorithms under weak conditions,” Science in China F: Information Sciences, vol. 51, no. 9, pp. 1269–1280, 2008. View at: Publisher Site | Google Scholar
11. F. Ding, “Coupled-least-squares identification for multivariable systems,” IET Control Theory and Applications, vol. 7, no. 1, pp. 68–79, 2013. View at: Google Scholar
12. F. Ding, X. G. Liu, and J. Chu, “Gradient-based and least-squares-based iterative algorithms for Hammerstein systems using the hierarchical identification principle,” IET Control Theory and Applications, vol. 7, pp. 176–184, 2013. View at: Google Scholar
13. D. Wang and F. Ding, “Least squares based and gradient based iterative identification for Wiener nonlinear systems,” Signal Processing, vol. 91, no. 5, pp. 1182–1189, 2011. View at: Publisher Site | Google Scholar
14. D. Q. Wang and F. Ding, “Hierarchical least squares estimation algorithm for Hammerstein-Wiener systems,” IEEE Signal Processing Letters, vol. 19, no. 12, pp. 825–828, 2012. View at: Google Scholar
15. D. Q. Wang, F. Ding, and Y. Y. Chu, “Data filtering based recursive least squares algorithm for Hammerstein systems using the key-term separation principle,” Information Sciences, vol. 222, pp. 203–212, 2013. View at: Google Scholar
16. D. Q. Wang, F. Ding, and X. M. Liu, “Least squares algorithm for an input nonlinear system with a dynamic subspace state space model,” Nonlinear Dynamics, vol. 75, no. 1-2, pp. 49–61, 2014. View at: Google Scholar
17. J. Chen, Y. Zhang, and R. F. Ding, “Auxiliary model based multi-innovation algorithms for multivariable nonlinear systems,” Mathematical and Computer Modelling, vol. 52, no. 9-10, pp. 1428–1434, 2010. View at: Publisher Site | Google Scholar
18. J. Chen, X. Wang, and R. F. Ding, “Gradient based estimation algorithm for Hammerstein systems with saturation and dead-zone nonlinearities,” Applied Mathematical Modelling, vol. 36, no. 1, pp. 238–243, 2012. View at: Publisher Site | Google Scholar
19. J. Chen and F. Ding, “Least squares and stochastic gradient parameter estimation for multivariable nonlinear Box-Jenkins models based on the auxiliary model and the multi-innovation identification theory,” Engineering Computations, vol. 29, no. 8, pp. 907–921, 2012. View at: Google Scholar
20. J. Chen, Y. Zhang, and R. F. Ding, “Gradient-based parameter estimation for input nonlinear systems with ARMA noises based on the auxiliary model,” Nonlinear Dynamics, vol. 72, no. 4, pp. 865–871, 2013. View at: Google Scholar
21. F. Ding, L. Qiu, and T. Chen, “Reconstruction of continuous-time systems from their non-uniformly sampled discrete-time systems,” Automatica, vol. 45, no. 2, pp. 324–332, 2009. View at: Publisher Site | Google Scholar
22. F. Ding and J. Ding, “Least-squares parameter estimation for systems with irregularly missing data,” International Journal of Adaptive Control and Signal Processing, vol. 24, no. 7, pp. 540–553, 2010. View at: Publisher Site | Google Scholar
23. F. Ding, P. X. Liu, and G. Liu, “Multiinnovation least-squares identification for system modeling,” IEEE Transactions on Systems, Man, and Cybernetics B: Cybernetics, vol. 40, no. 3, pp. 767–778, 2010. View at: Publisher Site | Google Scholar
24. Y. J. Liu, F. Ding, and Y. Shi, “Least squares estimation for a class of non-uniformly sampled systems based on the hierarchical identification principle,” Circuits, Systems and Signal Processing, vol. 31, no. 6, pp. 1985–2000, 2012. View at: Google Scholar
25. F. Ding, G. Liu, and X. P. Liu, “Partially coupled stochastic gradient identification methods for non-uniformly sampled systems,” IEEE Transactions on Automatic Control, vol. 55, no. 8, pp. 1976–1981, 2010. View at: Publisher Site | Google Scholar
26. J. Ding, F. Ding, X. P. Liu, and G. Liu, “Hierarchical least squares identification for linear SISO systems with dual-rate sampled-data,” IEEE Transactions on Automatic Control, vol. 56, no. 11, pp. 2677–2683, 2011. View at: Publisher Site | Google Scholar
27. F. Ding, G. Liu, and X. P. Liu, “Parameter estimation with scarce measurements,” Automatica, vol. 47, no. 8, pp. 1646–1655, 2011. View at: Publisher Site | Google Scholar
28. J. Chen, “Several gradient parameter estimation algorithms for dual-rate sampled systems,” Journal of the Franklin Institute. Engineering and Applied Mathematics, vol. 351, no. 1, pp. 543–554, 2014. View at: Google Scholar
29. J. Chen and R. Ding, “An auxiliary-model-based stochastic gradient algorithm for dual-rate sampled-data Box-Jenkins systems,” Circuits Systems and Signal Processing, vol. 32, no. 5, pp. 2475–2485, 2013. View at: Google Scholar
30. J. Ding, Y. Shi, H. Wang, and F. Ding, “A modified stochastic gradient based parameter estimation algorithm for dual-rate sampled-data systems,” Digital Signal Processing: A Review Journal, vol. 20, no. 4, pp. 1238–1247, 2010. View at: Publisher Site | Google Scholar
31. Y. Shi and B. Yu, “Output feedback stabilization of networked control systems with random delays modeled by Markov chains,” IEEE Transactions on Automatic Control, vol. 54, no. 7, pp. 1668–1674, 2009. View at: Publisher Site | Google Scholar
32. Y. Shi and H. Fang, “Kalman filter-based identification for systems with randomly missing measurements in a network environment,” International Journal of Control, vol. 83, no. 3, pp. 538–551, 2010. View at: Publisher Site | Google Scholar
33. Y. Shi and B. Yu, “Robust mixed H2/H control of networked control systems with random time delays in both forward and backward communication links,” Automatica, vol. 47, no. 4, pp. 754–760, 2011. View at: Publisher Site | Google Scholar
34. F. Ding and T. Chen, “Least squares based self-tuning control of dual-rate systems,” International Journal of Adaptive Control and Signal Processing, vol. 18, no. 8, pp. 697–714, 2004. View at: Publisher Site | Google Scholar
35. F. Ding and T. Chen, “A gradient based adaptive control algorithm for dual-rate systems,” Asian Journal of Control, vol. 8, no. 4, pp. 314–323, 2006. View at: Google Scholar
36. F. Ding, T. Chen, and Z. Iwai, “Adaptive digital control of Hammerstein nonlinear systems with limited output sampling,” SIAM Journal on Control and Optimization, vol. 45, no. 6, pp. 2257–2276, 2007. View at: Publisher Site | Google Scholar
37. J. Zhang, F. Ding, and Y. Shi, “Self-tuning control based on multi-innovation stochastic gradient parameter estimation,” Systems and Control Letters, vol. 58, no. 1, pp. 69–75, 2009. View at: Publisher Site | Google Scholar
38. Y. Shi, F. Ding, and T. Chen, “Multirate crosstalk identification in xDSL systems,” IEEE Transactions on Communications, vol. 54, no. 10, pp. 1878–1886, 2006. View at: Publisher Site | Google Scholar
39. Y. J. Liu, F. Ding, and Y. Shi, “An efficient hierarchical identification method for general dual-rate sampled-data systems,” Automatica, vol. 50, no. 3, pp. 962–973, 2014. View at: Google Scholar
40. F. Ding and T. Chen, “Identification of dual-rate systems based on finite impulse response models,” International Journal of Adaptive Control and Signal Processing, vol. 18, no. 7, pp. 589–598, 2004. View at: Publisher Site | Google Scholar
41. F. Ding and T. Chen, “Combined parameter and output estimation of dual-rate systems using an auxiliary model,” Automatica, vol. 40, no. 10, pp. 1739–1748, 2004. View at: Publisher Site | Google Scholar
42. F. Ding and T. Chen, “Parameter estimation of dual-rate stochastic systems by using an output error method,” IEEE Transactions on Automatic Control, vol. 50, no. 9, pp. 1436–1441, 2005. View at: Publisher Site | Google Scholar
43. F. Ding and T. Chen, “Hierarchical identification of lifted state-space models for general dual-rate systems,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 52, no. 6, pp. 1179–1187, 2005. View at: Publisher Site | Google Scholar
44. F. Ding, P. X. Liu, and Y. Shi, “Convergence analysis of estimation algorithms for dual-rate stochastic systems,” Applied Mathematics and Computation, vol. 176, no. 1, pp. 245–261, 2006. View at: Publisher Site | Google Scholar
45. F. Ding, P. X. Liu, and H. Yang, “Parameter identification and intersample output estimation for dual-rate systems,” IEEE Transactions on Systems, Man, and Cybernetics A: Systems and Humans, vol. 38, no. 4, pp. 966–975, 2008. View at: Publisher Site | Google Scholar
46. J. Li and F. Ding, “Maximum likelihood stochastic gradient estimation for Hammerstein systems with colored noise based on the key term separation technique,” Computers and Mathematics with Applications, vol. 62, no. 11, pp. 4170–4177, 2011. View at: Publisher Site | Google Scholar
47. X. L. Li, L. C. Zhou, R. Ding, and J. Shing, “Recursive least-squares estimation for Hammerstein nonlinear systems with nonuniform sampling,” Mathematical Problems in Engineering, vol. 2013, Article ID 240929, 8 pages, 2013. View at: Publisher Site | Google Scholar
48. F. Ding, X. P. Liu, and G. Liu, “Identification methods for Hammerstein nonlinear systems,” Digital Signal Processing: A Review Journal, vol. 21, no. 2, pp. 215–238, 2011. View at: Publisher Site | Google Scholar
49. F. Ding, “Hierarchical multi-innovation stochastic gradient algorithm for Hammerstein nonlinear system modeling,” Applied Mathematical Modelling, vol. 37, no. 4, pp. 1694–1704, 2013. View at: Google Scholar
50. J. H. Li, “Parameter estimation for Hammerstein CARARMA systems based on the Newton iteration,” Applied Mathematics Letters, vol. 26, no. 1, pp. 91–96, 2013. View at: Google Scholar
51. D. Wang, Y. Chu, and F. Ding, “Auxiliary model-based RELS and MI-ELS algorithm for Hammerstein OEMA systems,” Computers and Mathematics with Applications, vol. 59, no. 9, pp. 3092–3098, 2010. View at: Publisher Site | Google Scholar
52. J. Vörös, “Modeling and parameter identification of systems with multisegment piecewise-linear characteristics,” IEEE Transactions on Automatic Control, vol. 47, no. 1, pp. 184–188, 2002. View at: Publisher Site | Google Scholar
53. J. Vörös, “Modeling and identification of systems with backlash,” Automatica, vol. 46, no. 2, pp. 369–374, 2010. View at: Google Scholar
54. F. Ding, Y. Shi, and T. Chen, “Auxiliary model-based least-squares identification methods for Hammerstein output-error systems,” Systems and Control Letters, vol. 56, no. 5, pp. 373–380, 2007. View at: Publisher Site | Google Scholar
55. F. Ding, P. X. Liu, and G. Liu, “Auxiliary model based multi-innovation extended stochastic gradient parameter estimation with colored measurement noises,” Signal Processing, vol. 89, no. 10, pp. 1883–1890, 2009. View at: Publisher Site | Google Scholar
56. F. Ding and Y. Gu, “Performance analysis of the auxiliary model based least squares identification algorithm for one-step state delay systems,” International Journal of Computer Mathematics, vol. 89, no. 15, pp. 2019–2028, 2012. View at: Google Scholar
57. F. Ding and Y. Gu, “Performance analysis of the auxiliary model-based stochastic gradient parameter estimation algorithm for state space systems with one-step state delay,” Circuits, Systems and Signal Processing, vol. 32, no. 2, pp. 585–599, 2013. View at: Google Scholar
58. F. Ding, Y. Liu, and B. Bao, “Gradient-based and least-squares-based iterative estimation algorithms for multi-input multi-output systems,” Proceedings of the Institution of Mechanical Engineers I: Journal of Systems and Control Engineering, vol. 226, no. 1, pp. 43–55, 2012. View at: Publisher Site | Google Scholar
59. F. Ding, P. X. Liu, and G. Liu, “Gradient based and least-squares based iterative identification methods for OE and OEMA systems,” Digital Signal Processing, vol. 20, no. 3, pp. 664–677, 2010. View at: Publisher Site | Google Scholar
60. F. Ding, “Decomposition based fast least squares algorithm for output error systems,” Signal Processing, vol. 93, no. 5, pp. 1235–1242, 2013. View at: Google Scholar
61. Y. Liu, D. Wang, and F. Ding, “Least squares based iterative algorithms for identifying Box-Jenkins models with finite measurement data,” Digital Signal Processing, vol. 20, no. 5, pp. 1458–1467, 2010. View at: Publisher Site | Google Scholar
62. D. Q. Wang, “Least squares-based recursive and iterative estimation for output error moving average systems using data filtering,” IET Control Theory and Applications, vol. 5, no. 14, pp. 1648–1657, 2011. View at: Publisher Site | Google Scholar
63. F. Ding and T. Chen, “Performance analysis of multi-innovation gradient type identification methods,” Automatica, vol. 43, no. 1, pp. 1–14, 2007. View at: Publisher Site | Google Scholar
64. F. Ding, “Several multi-innovation identification methods,” Digital Signal Processing, vol. 20, no. 4, pp. 1027–1039, 2010. View at: Publisher Site | Google Scholar
65. F. Ding, H. Chen, and M. Li, “Multi-innovation least squares identification methods based on the auxiliary model for MISO systems,” Applied Mathematics and Computation, vol. 187, no. 2, pp. 658–668, 2007. View at: Publisher Site | Google Scholar
66. L. Han and F. Ding, “Multi-innovation stochastic gradient algorithms for multi-input multi-output systems,” Digital Signal Processing, vol. 19, no. 4, pp. 545–554, 2009. View at: Publisher Site | Google Scholar
67. D. Wang and F. Ding, “Performance analysis of the auxiliary models based multi-innovation stochastic gradient estimation algorithm for output error systems,” Digital Signal Processing, vol. 20, no. 3, pp. 750–762, 2010. View at: Publisher Site | Google Scholar
68. L. Xie, Y. J. Liu, H. Z. Yang, and F. Ding, “Modelling and identification for non-uniformly periodically sampled-data systems,” IET Control Theory and Applications, vol. 4, no. 5, pp. 784–794, 2010. View at: Publisher Site | Google Scholar
69. Y. Liu, L. Yu, and F. Ding, “Multi-innovation extended stochastic gradient algorithm and its performance analysis,” Circuits, Systems, and Signal Processing, vol. 29, no. 4, pp. 649–667, 2010. View at: Publisher Site | Google Scholar
70. L. Han and F. Ding, “Identification for multirate multi-input systems using the multi-innovation identification theory,” Computers and Mathematics with Applications, vol. 57, no. 9, pp. 1438–1449, 2009. View at: Publisher Site | Google Scholar
71. F. Ding and T. Chen, “Hierarchical gradient-based identification of multivariable discrete-time systems,” Automatica, vol. 41, no. 2, pp. 315–325, 2005. View at: Publisher Site | Google Scholar
72. F. Ding and T. Chen, “Hierarchical least squares identification methods for multivariable systems,” IEEE Transactions on Automatic Control, vol. 50, no. 3, pp. 397–402, 2005. View at: Publisher Site | Google Scholar
73. H. Han, L. Xie, F. Ding, and X. Liu, “Hierarchical least-squares based iterative identification for multivariable systems with moving average noises,” Mathematical and Computer Modelling, vol. 51, no. 9-10, pp. 1213–1220, 2010. View at: Publisher Site | Google Scholar
74. Z. Zhang, F. Ding, and X. Liu, “Hierarchical gradient based iterative parameter estimation algorithm for multivariable output error moving average systems,” Computers and Mathematics with Applications, vol. 61, no. 3, pp. 672–682, 2011. View at: Publisher Site | Google Scholar
75. D. Q. Wang, R. Ding, and X. Z. Dong, “Iterative parameter estimation for a class of multivariable systems based on the hierarchical identification principle and the gradient search,” Circuits, Systems and Signal Processing, vol. 31, no. 6, pp. 2167–2177, 2012. View at: Google Scholar
76. F. Ding and T. Chen, “On iterative solutions of general coupled matrix equations,” SIAM Journal on Control and Optimization, vol. 44, no. 6, pp. 2269–2284, 2006. View at: Publisher Site | Google Scholar
77. F. Ding, P. X. Liu, and J. Ding, “Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle,” Applied Mathematics and Computation, vol. 197, no. 1, pp. 41–50, 2008. View at: Publisher Site | Google Scholar
78. F. Ding and T. Chen, “Gradient based iterative algorithms for solving a class of matrix equations,” IEEE Transactions on Automatic Control, vol. 50, no. 8, pp. 1216–1221, 2005. View at: Publisher Site | Google Scholar
79. F. Ding and T. Chen, “Iterative least-squares solutions of coupled Sylvester matrix equations,” Systems and Control Letters, vol. 54, no. 2, pp. 95–107, 2005. View at: Publisher Site | Google Scholar
80. F. Ding, “Transformations between some special matrices,” Computers and Mathematics with Applications, vol. 59, no. 8, pp. 2676–2695, 2010. View at: Publisher Site | Google Scholar
81. L. Xie, J. Ding, and F. Ding, “Gradient based iterative solutions for general linear matrix equations,” Computers and Mathematics with Applications, vol. 58, no. 7, pp. 1441–1448, 2009. View at: Publisher Site | Google Scholar
82. J. Ding, Y. Liu, and F. Ding, “Iterative solutions to matrix equations of the form ${A}_{i}X{B}_{i}={F}_{i}$,” Computers and Mathematics with Applications, vol. 59, no. 11, pp. 3500–3507, 2010. View at: Publisher Site | Google Scholar
83. L. Xie, Y. Liu, and H. Yang, “Gradient based and least squares based iterative algorithms for matrix equations $AXB+C{X}^{T}D=F$,” Applied Mathematics and Computation, vol. 217, no. 5, pp. 2191–2199, 2010. View at: Publisher Site | Google Scholar
84. F. Ding, “Two-stage least squares based iterative estimation algorithm for CARARMA system modeling,” Applied Mathematical Modelling, vol. 37, no. 7, pp. 4798–4808, 2013. View at: Google Scholar
85. F. Ding and H. H. Duan, “Two-stage parameter estimation algorithms for Box-Jenkins systems,” IET Signal Processing, vol. 7, no. 8, pp. 646–654, 2013. View at: Google Scholar
86. J. Ding and F. Ding, “Bias compensation-based parameter estimation for output error moving average systems,” International Journal of Adaptive Control and Signal Processing, vol. 25, no. 12, pp. 1100–1111, 2011. View at: Publisher Site | Google Scholar
87. Y. Zhang, “Unbiased identification of a class of multi-input single-output systems with correlated disturbances using bias compensation methods,” Mathematical and Computer Modelling, vol. 53, no. 9-10, pp. 1810–1819, 2011. View at: Publisher Site | Google Scholar
88. Y. Zhang and G. Cui, “Bias compensation methods for stochastic systems with colored noise,” Applied Mathematical Modelling, vol. 35, no. 4, pp. 1709–1716, 2011. View at: Publisher Site | Google Scholar
89. F. Ding, “Combined state and least squares parameter estimation algorithms for dynamic systems,” Applied Mathematical Modelling, vol. 38, no. 1, pp. 403–412, 2014. View at: Google Scholar
90. F. Ding, X. M. Liu, H. B. Chen, and G. Y. Yao, “Hierarchical gradient based and hierarchical least squares based iterative parameter identification for CARARMA systems,” Signal Processing, vol. 97, pp. 31–39, 2014. View at: Google Scholar
91. P. P. Hu and F. Ding, “Multistage least squares based iterative estimation for feedback nonlinear systems with moving average noises using the hierarchical identification principle,” Nonlinear Dynamics, vol. 73, no. 1-2, pp. 583–592, 2013. View at: Google Scholar
92. J. Li, F. Ding, and G. Yang, “Maximum likelihood least squares identification method for input nonlinear finite impulse response moving average systems,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 442–450, 2012. View at: Publisher Site | Google Scholar
93. F. Ding and T. Chen, “Identification of Hammerstein nonlinear ARMAX systems,” Automatica, vol. 41, no. 9, pp. 1479–1489, 2005. View at: Publisher Site | Google Scholar
94. F. Ding, Y. Shi, and T. Chen, “Gradient-based identification methods for hammerstein nonlinear ARMAX models,” Nonlinear Dynamics, vol. 45, no. 1-2, pp. 31–43, 2006. View at: Publisher Site | Google Scholar
95. J. H. Li, F. Ding, and L. Hua, “Maximum likelihood Newton recursive and the Newton iterative estimation algorithms for Hammerstein CARAR systems,” Nonlinear Dynamics, vol. 75, no. 1-2, pp. 234–245, 2014. View at: Google Scholar
96. X. Luan, P. Shi, and F. Liu, “Stabilization of networked control systems with random delays,” IEEE Transactions on Industrial Electronics, vol. 58, no. 9, pp. 4323–4330, 2011. View at: Publisher Site | Google Scholar
97. X. L. Luan, S. Y. Zhao, and F. Liu, “H-infinity control for discrete-time markov jump systems with uncertain transition probabilities,” IEEE Transactions on Automatic Control, vol. 58, no. 6, pp. 1566–1572, 2013. View at: Google Scholar
98. J. Chen, L. X. Lu, and R. Ding, “Parameter identification of systems with preload nonlinearities based on the finite impulse response model and negative gradient search,” Applied Mathematics and Computation, vol. 219, no. 5, pp. 2498–2505, 2012. View at: Google Scholar
99. J. Chen, L. Lv, and R. F. Ding, “Multi-innovation stochastic gradient algorithms for dual-rate sampled systems with preload nonlinearity,” Applied Mathematics Letters, vol. 26, no. 1, pp. 124–129, 2013. View at: Publisher Site | Google Scholar

#### More related articles

We are committed to sharing findings related to COVID-19 as quickly and safely as possible. Any author submitting a COVID-19 paper should notify us at help@hindawi.com to ensure their research is fast-tracked and made available on a preprint server as soon as possible. We will be providing unlimited waivers of publication charges for accepted articles related to COVID-19. Sign up here as a reviewer to help fast-track new submissions.