Research Article  Open Access
Jing Chen, Ruifeng Ding, "Two Identification Methods for DualRate SampledData Nonlinear OutputError Systems", Mathematical Problems in Engineering, vol. 2014, Article ID 329437, 10 pages, 2014. https://doi.org/10.1155/2014/329437
Two Identification Methods for DualRate SampledData Nonlinear OutputError Systems
Abstract
This paper presents two methods for dualrate sampleddata nonlinear outputerror systems. One method is the missing output estimation based stochastic gradient identification algorithm and the other method is the auxiliary model based stochastic gradient identification algorithm. Different from the polynomial transformation based identification methods, the two methods in this paper can estimate the unknown parameters directly. A numerical example is provided to confirm the effectiveness of the proposed methods.
1. Introduction
System identification plays an important part in many engineering applications [1–6]. Many identification methods assume that the inputoutput data at every sampling instant are available for linear systems [7–11] and nonlinear systems [12–20], which is usually not the case in practice. When the input and output signals of the systems have different sampling rates, these systems are usually called irregularly sampleddata systems [21–27], for example, dualrate or multirate systems [28–30]. Dualrate/multirate systems in which the input and the output are sampled at different frequencies arise widely in robust filtering and control [31–33], adaptive control [34–37], and system identification [38–43]. In the literature of dualrate system identification, the socalled polynomial transformation technique is often used to transform the dualrate model [44, 45].
As far as we know, the identification methods based on the polynomial transformation technique cannot directly estimate the parameters of the dualrate system and the number of the unknown parameters to be estimated is more than the number of the unknown parameters of the original dualrate system.
The nonlinear system consisting of a static nonlinear block followed by a linear dynamic system is called a Hammerstein system [46–49]. The nonlinearity of the Hammerstein system is usually expressed by some known basis functions [50, 51] or by a piecewise polynomial function [52, 53]. When the Hammerstein system is a dualrate system and has a preload nonlinearity, to the best of our knowledge, there is no work on identification of such systems. The main contributions of this paper are presenting the two methods directly for estimating the parameters of the dualrate system. The proposed methods of this paper can combine the auxiliary model identification methods [54–57], the iterative identification methods [58–62], the multiinnovation identification methods [63–70], the hierarchical identification methods [71–83], and the twostage or multistage identification methods [84, 85] to study identification problems for other linear systems [86–90] or nonlinear systems [91–97].
The rest of this paper is organized as follows. Section 2 introduces the dualrate nonlinear outputerror systems. Section 3 gives a missing output identification model based stochastic gradient algorithm. Section 4 provides an auxiliary model based stochastic gradient algorithm. Section 5 introduces an illustrative example. Finally, concluding remarks are given in Section 6.
2. Problem Formulation
Let “” or “” stand for “ is defined as ,” let the norm of a column vector be , and let the superscript denote the matrix transpose.
Consider the following dualrate nonlinear outputerror system with colored noise: where is the system output, is the system input, is a stochastic white noise with zero mean, and are the polynomials in the unit backward shift operator , and is a preload nonlinearity shown in Figure 1 and can be expressed as [98, 99] where and are two preload points.
For the dualrate sampleddata system, all the input data , and only the scarce output data , are known. The intersample outputs or missing outputs , are unavailable.
Define a sign function Then the function can be expressed as Let Hence, we have Once and are estimated, the parameters and can be computed by , .
3. The Missing Outputs Identification Model Based Stochastic Gradient Algorithm
Substituting (7) into (1) gets Define the parameter vector and information vector as From (9) and (10), we get or Let be the estimate of . Defining and minimizing the cost function give the following stochastic gradient (SG) algorithm for estimating : Since the information on the righthand sides of (16) contains the unknown variables , , the SG algorithm in (14)–(18) is impossible to implement. In this section, we use the missing outputs identification model (MOI) to overcome this difficulty; these unknown are replaced with the output estimates of an MOI model, where represents the estimate of at time , represents the estimate of at time , and represents the estimate of .
Thus, we have the following missing output estimates based SG (MOESG) algorithm for estimating the parameter vector in (9): The steps of computing the parameter estimate by the MOESG algorithm are listed as follows. (1)Let , , , and give a small positive number .(2)Let , , and with being a column vector whose entries are all unity and .(3)Collect the input data , and collect the output data .(4)Let and compute by (22).(5)Form by (23).(6)Decrease by 1; if , go to step (4); otherwise, go to the next step.(7)Compute and by (24) and (25), respectively.(8)Update the parameter estimation vector by (20).(9)Compare and ; if , then terminate the procedure and obtain the ; otherwise, increase by and go to step (3).
The flowchart of computing the MOESG parameter estimate is shown in Figure 2.
4. The Auxiliary Model Based Stochastic Gradient Algorithm
Define From (8) and (26), we have Define the information vector as
Then we get
Assume is an integer multiple of and rewrite (30) as
Let be the estimate of . Defining and minimizing the cost function give the following SG algorithm of estimating : Because of the unknown variables in (33), the SG algorithm in (33)–(36) is impossible to implement. In this section, we use the auxiliary model; these unknown are replaced with the outputs of an auxiliary model, where is the estimate of and is the estimate of . We can obtain an auxiliary model based stochastic gradient (AMSG) algorithm: The steps of computing the parameter estimate by the AMSG algorithm are listed as follows.(1)Let , , , , and give a small positive number .(2)Let , , and with being a column vector whose entries are all unity and .(3)Collect the input data , and collect the output data .(4)Let and compute by (40).(5)Form by (41).(6)Decrease by 1; if , go to step (4); otherwise, go to next step.(7)Compute and by (42) and (43), respectively.(8)Update the parameter estimation vector by (38).(9)Compare and ; if , then terminate the procedure and obtain the ; otherwise, increase by and go to step (3).
The flowchart of computing the AMSG parameter estimate is shown in Figure 3.
Remark 1. Compared with the polynomial transformation technique, the MOESG method and the AMSG method can estimate the unknown parameters directly.
5. Example
Consider the following nonlinear outputerror system with the updating period : the input is taken as a persistent excitation signal sequence with zero mean and unit variance and is a white noise sequence with zero mean and variance . The unknown parameters are as follows: Applying the MOESG algorithm and the AMSG algorithm to estimate the parameters, the parameter estimates and their errors based on the MOESG algorithm and the AMSG algorithm are shown in Tables 1 and 2 and the parameter estimation errors versus are shown in Figures 4 and 5.


From Tables 1 and 2 and Figures 4 and 5, we can draw the following conclusions.(1)Both the MOESG algorithm and the AMSG algorithm can estimate the unknown parameters directly.(2)The parameter estimation errors become smaller and smaller and go to zero with increasing.
6. Conclusions
Two identification methods for dualrate nonlinear outputerror systems are presented to estimate the unknown parameters directly and can avoid estimating more parameters than the original systems. Furthermore, the two methods can also be extended to other systems such as
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China and supported by the Natural Science Foundation of Jiangsu Province (no. BK20131109).
References
 F. Ding, System Identification—New Theory and Methods, Science Press, Beijing, China, 2013.
 F. Ding, System Identification—Performances Analysis for Identification Methods, Science Press, Beijing, China, 2014.
 Y. Liu, Y. Xiao, and X. Zhao, “Multiinnovation stochastic gradient algorithm for multipleinput singleoutput systems using the auxiliary model,” Applied Mathematics and Computation, vol. 215, no. 4, pp. 1477–1483, 2009. View at: Publisher Site  Google Scholar
 Y. Liu, L. Xie, and F. Ding, “An auxiliary model based on a recursive leastsquares parameter estimation algorithm for nonuniformly 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
 Y. Liu, J. Sheng, and R. F. Ding, “Convergence of stochastic gradient estimation algorithm for multivariable ARXlike systems,” Computers and Mathematics with Applications, vol. 59, no. 8, pp. 2615–2627, 2010. View at: Publisher Site  Google Scholar
 Y. J. Liu and R. Ding, “Consistency of the extended gradient identification algorithm for multiinput multioutput systems with moving average noises,” International Journal of Computer Mathematics, vol. 90, no. 9, pp. 1840–1852, 2013. View at: Google Scholar
 F. Ding and T. Chen, “Performance bounds of forgetting factor leastsquares algorithms for timevarying 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
 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
 F. Ding, T. Chen, and L. Qiu, “Bias compensation based recursive leastsquares 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
 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
 F. Ding, “Coupledleastsquares identification for multivariable systems,” IET Control Theory and Applications, vol. 7, no. 1, pp. 68–79, 2013. View at: Google Scholar
 F. Ding, X. G. Liu, and J. Chu, “Gradientbased and leastsquaresbased 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
 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
 D. Q. Wang and F. Ding, “Hierarchical least squares estimation algorithm for HammersteinWiener systems,” IEEE Signal Processing Letters, vol. 19, no. 12, pp. 825–828, 2012. View at: Google Scholar
 D. Q. Wang, F. Ding, and Y. Y. Chu, “Data filtering based recursive least squares algorithm for Hammerstein systems using the keyterm separation principle,” Information Sciences, vol. 222, pp. 203–212, 2013. View at: Google Scholar
 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. 12, pp. 49–61, 2014. View at: Google Scholar
 J. Chen, Y. Zhang, and R. F. Ding, “Auxiliary model based multiinnovation algorithms for multivariable nonlinear systems,” Mathematical and Computer Modelling, vol. 52, no. 910, pp. 1428–1434, 2010. View at: Publisher Site  Google Scholar
 J. Chen, X. Wang, and R. F. Ding, “Gradient based estimation algorithm for Hammerstein systems with saturation and deadzone nonlinearities,” Applied Mathematical Modelling, vol. 36, no. 1, pp. 238–243, 2012. View at: Publisher Site  Google Scholar
 J. Chen and F. Ding, “Least squares and stochastic gradient parameter estimation for multivariable nonlinear BoxJenkins models based on the auxiliary model and the multiinnovation identification theory,” Engineering Computations, vol. 29, no. 8, pp. 907–921, 2012. View at: Google Scholar
 J. Chen, Y. Zhang, and R. F. Ding, “Gradientbased 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
 F. Ding, L. Qiu, and T. Chen, “Reconstruction of continuoustime systems from their nonuniformly sampled discretetime systems,” Automatica, vol. 45, no. 2, pp. 324–332, 2009. View at: Publisher Site  Google Scholar
 F. Ding and J. Ding, “Leastsquares 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
 F. Ding, P. X. Liu, and G. Liu, “Multiinnovation leastsquares 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
 Y. J. Liu, F. Ding, and Y. Shi, “Least squares estimation for a class of nonuniformly 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
 F. Ding, G. Liu, and X. P. Liu, “Partially coupled stochastic gradient identification methods for nonuniformly sampled systems,” IEEE Transactions on Automatic Control, vol. 55, no. 8, pp. 1976–1981, 2010. View at: Publisher Site  Google Scholar
 J. Ding, F. Ding, X. P. Liu, and G. Liu, “Hierarchical least squares identification for linear SISO systems with dualrate sampleddata,” IEEE Transactions on Automatic Control, vol. 56, no. 11, pp. 2677–2683, 2011. View at: Publisher Site  Google Scholar
 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
 J. Chen, “Several gradient parameter estimation algorithms for dualrate sampled systems,” Journal of the Franklin Institute. Engineering and Applied Mathematics, vol. 351, no. 1, pp. 543–554, 2014. View at: Google Scholar
 J. Chen and R. Ding, “An auxiliarymodelbased stochastic gradient algorithm for dualrate sampleddata BoxJenkins systems,” Circuits Systems and Signal Processing, vol. 32, no. 5, pp. 2475–2485, 2013. View at: Google Scholar
 J. Ding, Y. Shi, H. Wang, and F. Ding, “A modified stochastic gradient based parameter estimation algorithm for dualrate sampleddata systems,” Digital Signal Processing: A Review Journal, vol. 20, no. 4, pp. 1238–1247, 2010. View at: Publisher Site  Google Scholar
 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
 Y. Shi and H. Fang, “Kalman filterbased 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
 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
 F. Ding and T. Chen, “Least squares based selftuning control of dualrate systems,” International Journal of Adaptive Control and Signal Processing, vol. 18, no. 8, pp. 697–714, 2004. View at: Publisher Site  Google Scholar
 F. Ding and T. Chen, “A gradient based adaptive control algorithm for dualrate systems,” Asian Journal of Control, vol. 8, no. 4, pp. 314–323, 2006. View at: Google Scholar
 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
 J. Zhang, F. Ding, and Y. Shi, “Selftuning control based on multiinnovation stochastic gradient parameter estimation,” Systems and Control Letters, vol. 58, no. 1, pp. 69–75, 2009. View at: Publisher Site  Google Scholar
 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
 Y. J. Liu, F. Ding, and Y. Shi, “An efficient hierarchical identification method for general dualrate sampleddata systems,” Automatica, vol. 50, no. 3, pp. 962–973, 2014. View at: Google Scholar
 F. Ding and T. Chen, “Identification of dualrate 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
 F. Ding and T. Chen, “Combined parameter and output estimation of dualrate systems using an auxiliary model,” Automatica, vol. 40, no. 10, pp. 1739–1748, 2004. View at: Publisher Site  Google Scholar
 F. Ding and T. Chen, “Parameter estimation of dualrate 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
 F. Ding and T. Chen, “Hierarchical identification of lifted statespace models for general dualrate systems,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 52, no. 6, pp. 1179–1187, 2005. View at: Publisher Site  Google Scholar
 F. Ding, P. X. Liu, and Y. Shi, “Convergence analysis of estimation algorithms for dualrate stochastic systems,” Applied Mathematics and Computation, vol. 176, no. 1, pp. 245–261, 2006. View at: Publisher Site  Google Scholar
 F. Ding, P. X. Liu, and H. Yang, “Parameter identification and intersample output estimation for dualrate 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
 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
 X. L. Li, L. C. Zhou, R. Ding, and J. Shing, “Recursive leastsquares 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
 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
 F. Ding, “Hierarchical multiinnovation stochastic gradient algorithm for Hammerstein nonlinear system modeling,” Applied Mathematical Modelling, vol. 37, no. 4, pp. 1694–1704, 2013. View at: Google Scholar
 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
 D. Wang, Y. Chu, and F. Ding, “Auxiliary modelbased RELS and MIELS algorithm for Hammerstein OEMA systems,” Computers and Mathematics with Applications, vol. 59, no. 9, pp. 3092–3098, 2010. View at: Publisher Site  Google Scholar
 J. Vörös, “Modeling and parameter identification of systems with multisegment piecewiselinear characteristics,” IEEE Transactions on Automatic Control, vol. 47, no. 1, pp. 184–188, 2002. View at: Publisher Site  Google Scholar
 J. Vörös, “Modeling and identification of systems with backlash,” Automatica, vol. 46, no. 2, pp. 369–374, 2010. View at: Google Scholar
 F. Ding, Y. Shi, and T. Chen, “Auxiliary modelbased leastsquares identification methods for Hammerstein outputerror systems,” Systems and Control Letters, vol. 56, no. 5, pp. 373–380, 2007. View at: Publisher Site  Google Scholar
 F. Ding, P. X. Liu, and G. Liu, “Auxiliary model based multiinnovation 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
 F. Ding and Y. Gu, “Performance analysis of the auxiliary model based least squares identification algorithm for onestep state delay systems,” International Journal of Computer Mathematics, vol. 89, no. 15, pp. 2019–2028, 2012. View at: Google Scholar
 F. Ding and Y. Gu, “Performance analysis of the auxiliary modelbased stochastic gradient parameter estimation algorithm for state space systems with onestep state delay,” Circuits, Systems and Signal Processing, vol. 32, no. 2, pp. 585–599, 2013. View at: Google Scholar
 F. Ding, Y. Liu, and B. Bao, “Gradientbased and leastsquaresbased iterative estimation algorithms for multiinput multioutput 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
 F. Ding, P. X. Liu, and G. Liu, “Gradient based and leastsquares 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
 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
 Y. Liu, D. Wang, and F. Ding, “Least squares based iterative algorithms for identifying BoxJenkins models with finite measurement data,” Digital Signal Processing, vol. 20, no. 5, pp. 1458–1467, 2010. View at: Publisher Site  Google Scholar
 D. Q. Wang, “Least squaresbased 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
 F. Ding and T. Chen, “Performance analysis of multiinnovation gradient type identification methods,” Automatica, vol. 43, no. 1, pp. 1–14, 2007. View at: Publisher Site  Google Scholar
 F. Ding, “Several multiinnovation identification methods,” Digital Signal Processing, vol. 20, no. 4, pp. 1027–1039, 2010. View at: Publisher Site  Google Scholar
 F. Ding, H. Chen, and M. Li, “Multiinnovation 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
 L. Han and F. Ding, “Multiinnovation stochastic gradient algorithms for multiinput multioutput systems,” Digital Signal Processing, vol. 19, no. 4, pp. 545–554, 2009. View at: Publisher Site  Google Scholar
 D. Wang and F. Ding, “Performance analysis of the auxiliary models based multiinnovation 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
 L. Xie, Y. J. Liu, H. Z. Yang, and F. Ding, “Modelling and identification for nonuniformly periodically sampleddata systems,” IET Control Theory and Applications, vol. 4, no. 5, pp. 784–794, 2010. View at: Publisher Site  Google Scholar
 Y. Liu, L. Yu, and F. Ding, “Multiinnovation 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
 L. Han and F. Ding, “Identification for multirate multiinput systems using the multiinnovation identification theory,” Computers and Mathematics with Applications, vol. 57, no. 9, pp. 1438–1449, 2009. View at: Publisher Site  Google Scholar
 F. Ding and T. Chen, “Hierarchical gradientbased identification of multivariable discretetime systems,” Automatica, vol. 41, no. 2, pp. 315–325, 2005. View at: Publisher Site  Google Scholar
 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
 H. Han, L. Xie, F. Ding, and X. Liu, “Hierarchical leastsquares based iterative identification for multivariable systems with moving average noises,” Mathematical and Computer Modelling, vol. 51, no. 910, pp. 1213–1220, 2010. View at: Publisher Site  Google Scholar
 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
 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
 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
 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
 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
 F. Ding and T. Chen, “Iterative leastsquares solutions of coupled Sylvester matrix equations,” Systems and Control Letters, vol. 54, no. 2, pp. 95–107, 2005. View at: Publisher Site  Google Scholar
 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
 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
 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
 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
 F. Ding, “Twostage 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
 F. Ding and H. H. Duan, “Twostage parameter estimation algorithms for BoxJenkins systems,” IET Signal Processing, vol. 7, no. 8, pp. 646–654, 2013. View at: Google Scholar
 J. Ding and F. Ding, “Bias compensationbased 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
 Y. Zhang, “Unbiased identification of a class of multiinput singleoutput systems with correlated disturbances using bias compensation methods,” Mathematical and Computer Modelling, vol. 53, no. 910, pp. 1810–1819, 2011. View at: Publisher Site  Google Scholar
 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
 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
 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
 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. 12, pp. 583–592, 2013. View at: Google Scholar
 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. 34, pp. 442–450, 2012. View at: Publisher Site  Google Scholar
 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
 F. Ding, Y. Shi, and T. Chen, “Gradientbased identification methods for hammerstein nonlinear ARMAX models,” Nonlinear Dynamics, vol. 45, no. 12, pp. 31–43, 2006. View at: Publisher Site  Google Scholar
 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. 12, pp. 234–245, 2014. View at: Google Scholar
 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
 X. L. Luan, S. Y. Zhao, and F. Liu, “Hinfinity control for discretetime markov jump systems with uncertain transition probabilities,” IEEE Transactions on Automatic Control, vol. 58, no. 6, pp. 1566–1572, 2013. View at: Google Scholar
 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
 J. Chen, L. Lv, and R. F. Ding, “Multiinnovation stochastic gradient algorithms for dualrate sampled systems with preload nonlinearity,” Applied Mathematics Letters, vol. 26, no. 1, pp. 124–129, 2013. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2014 Jing Chen and Ruifeng Ding. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.