Research Article  Open Access
Ruifeng Ding, Linfan Zhuang, "Parameter and State Estimator for State Space Models", The Scientific World Journal, vol. 2014, Article ID 106505, 10 pages, 2014. https://doi.org/10.1155/2014/106505
Parameter and State Estimator for State Space Models
Abstract
This paper proposes a parameter and state estimator for canonical state space systems from measured inputoutput data. The key is to solve the system state from the state equation and to substitute it into the output equation, eliminating the state variables, and the resulting equation contains only the system inputs and outputs, and to derive a least squares parameter identification algorithm. Furthermore, the system states are computed from the estimated parameters and the inputoutput data. Convergence analysis using the martingale convergence theorem indicates that the parameter estimates converge to their true values. Finally, an illustrative example is provided to show that the proposed algorithm is effective.
1. Introduction
Parameter estimation and identification have had important applications in system modelling, system control, and system analysis [1–5] and thus have received much research attention in recent decades [6–11]. Several identification methods have been developed for state space models, for example, the subspace identification methods [12]. Gibson and Ninness presented a robust maximumlikelihood estimation for fully parameterized linear timeinvariant (LTI) state space models; the idea is to use the expectation maximization (EM) algorithm to estimate maximumlikelihood degrees [13]. Raghavan et al. studied the EMbased state space model identification problems with irregular output sampling [14].
The state space model includes not only the unknown parameter matrices/vectors, but also the unknown noise terms in the formation vector and unmeasurable state vector. Many algorithms can estimate the system states assuming that the system parameter matrices/vectors are available but such state estimation algorithm cannot work if the system parameters are unknown [15]. Recently, Ding presented a combined state and least squares parameter estimation algorithm for dynamic systems [16].
In the area of state space model identification, Ding and Chen proposed a hierarchical identification estimation algorithm for estimating the system parameters and states [17]. Li et al. assumed that the system states were available and used the measurable states and inputoutput data to estimate the parameters of lifted state space models for general dualrate systems [18]. Recently, some identification methods have been developed, for example, the least squares methods [19, 20], the gradientbased methods [21, 22], the bias compensation methods [23, 24], and the maximum likelihood methods [25–30]. The objective of this paper is to present a new parameter and state estimationbased residual algorithm from the given inputoutput data and further to analyze the convergence of the proposed algorithm.
The convergence analysis of identification algorithms has always been one of the important projects in the field of control. By using the stochastic martingale theory, Ding et al. studied the properties of stochastic gradient identification algorithms under weak conditions [31]. Ding and Liu discussed the gradientbased identification approach and convergence for multivariable systems with output measurement noise [32]. Other identification methods for linear or nonlinear systems [33–42] include the auxiliary model identification methods [43–57], the hierarchical identification methods [58–73], and the twostage or multistage identification methods [74–78].
This paper is organized as follows. Section 2 introduces the system description and its identification model paper. Section 3 derives a basic parameter identification algorithm for canonical state space systems and analyzes the performance of the proposed algorithm. Section 4 gives a state estimation algorithm. Section 5 provides an example for the proposed algorithm. Finally, concluding remarks are given in Section 6.
2. System Description and Identification Model
Let us introduce some notation [15]. “” or “” stands for “ is defined as ”; the symbol stands for an identity matrix of appropriate size ; the superscript denotes the matrix transpose; represents the determinant of a square matrix ; the norm of a matrix is defined by ; represents an vector whose elements are all ; represents the minimum eigenvalues of ; for , we write if there exists a positive constant such that .
In order to study the convergence of the algorithm proposed in [15], here we simply give that algorithm in [15]. Consider a linear system described by the following observability canonical state space model [15]: where is the state vector, is the system input, is the system output, and is a random noise with zero mean. Assume that the order is known, and , and for .
The system in (1) is an observability canonical form, and its observability matrix is an identity matrix; that is,
For the system in (1), the objective of this paper is to develop a new algorithm to estimate the parameter matrix/vector and (i.e., the parameters and ) and the system state vector from the available measurement inputoutput data .
Since the available measurement inputoutput data are known but the state vector is unknown, it is required to eliminate the state vector from (1) and obtain a new expression which only involves the input and output, in order to obtain the estimates of the parameters in (1). The following derives the identification model based on the method in [15].
Define some vectors/matrix, From (1), we have Combining (5) with (8) gives or Define the parameter vector and the information vector as Substituting (11) into (9) gives Replacing in (13) with yields which is called the identification model or identification expression of the statespace model.
3. The Parameter Estimation Algorithm and Its Convergence
The recursive least squares algorithm for estimating is expressed as This algorithm is commonly used for convergence analysis. To avoid computing the matrix inversion, this algorithm is equivalently expressed as where is the gain vector.
Define the parameter estimation error vector and the nonnegative function .
Theorem 1. For the system in (1) and algorithm in (15)–(18), assume that is a martingale difference sequence defined on a probability space , where is the algebra sequence generated by the observations up to and including time . The noise sequence satisfies the following assumptions: (A1), a.s.,(A2), a.s.,(A3) is strictly positive real.Then the following inequality holds: where
Proof. Define the innovation vector . Using (17), it follows that Subtracting from both sides of (15) and using (14), we have According to the definition of and using (16) and (29), we have Using (26), (27), and (29), and , we have Since , , are uncorrelated with and are measurable, taking the conditional expectation with respect to and using (A1)(A2) give The state space model in (1) can be transformed into an inputoutput representation, where is the adjoint matrix of , and are polynomials in a unit backward shift operator , and Referring to the proof of Lemma 3 in [43], using (33), we have Using (17), (26), and (35), from (27), we get Since is a strictly positive real function, referring to Appendix C in [79], we can obtain the conclusion . Adding both sides of (32) by gives the conclusion of Theorem 1.
Theorem 2. For the system in (1) and the algorithm in (15)–(18), assume that (A1)–(A3) hold and that is stable; that is, all zeros of are inside the unit circle; then the parameter estimation error satisfies
Proof. Using the formula , and from the definition of , we have Let Since is nondecreasing, using Theorem 1 yields Referring to the proof of Theorem 2 in [43], we have
Assume that there exist positive constants , , , and such that the following generalized persistent excitation condition (unbounded condition number) holds: Then for any , we have
4. The State Estimation Algorithm
Referring to the method in [15], the state estimate of the state vector can be expressed as To summarize, we list the steps involved in the algorithm in (19)–(23) and (44)–(51) to compute the parameter estimate and the state estimate .
(1)Let ; set the initial values , , , , , or for , . Give a small positive number .(2)Collect the inputoutput data and ; form using (23), using (45), and using (46).(3)Compute the gain vector using (20) and the covariance matrix using (21).(4)Update the parameter estimation vector using (19).(5)Compute using (22), and form using (47).(6)Determine using (51) and compute using (49); then form using (48).(7)Compute the state estimate using (44).(8)If they are sufficiently close, if , then terminate the procedure and obtain the estimate ; otherwise, increase by 1 and go to step 2.5. Example
Consider the following singleinput singleoutput secondorder system in canonical form: The simulation conditions are the same as in [15]. That is, the input is taken as an independent persistent excitation signal sequence with zero mean and unit variances and as a white noise sequence with zero mean and variances and , respectively. Apply the proposed parameter and state estimation algorithm in (19)–(23) and (44)–(51) to estimate the parameters and states of this example system; the parameter estimates and their estimation errors are shown in Tables 1 and 2; the parameter estimation errors versus are shown in Figure 1; the states and their estimates versus are shown in Figures 2 and 3, where () is the parameter estimation error.


From the simulation results of Tables 1 and 2 and Figures 1–3, we can draw the following conclusions.(1)A lower noise level leads to a faster rate of convergence of the parameter estimates to the true parameters.(2)The parameter estimation errors become smaller (in general) as the data length increases; see Tables 1 and 2 and Figure 1. In other words, increasing data length generally results in smaller parameter estimation errors.(3)The state estimates are close to their true values with increasing; see Figures 2 and 3. These indicate that the proposed parameter and state estimation algorithm are effective.
6. Conclusions
In this paper, the identification problems for linear systems based on the canonical state space models with unknown parameters and states are studied. A new parameter and state estimation algorithm has been presented directly from inputoutput data. The analysis using the martingale convergence theorem indicates that the proposed algorithms can give consistent parameter estimation. The simulation results show that the proposed algorithms are effective. The method in this paper can combine the multiinnovation identification methods [80–92], the iterative identification methods [93–100], and other identification methods [101–111] to present new identification algorithms or to study adaptive control problems for linear or nonlinear, singlerate or dualrate, scalar or multivariable systems [112–117].
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 PAPD of Jiangsu Higher Education Institutions and the 111 Project (B12018).
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. B. Hu, “Iterative and recursive least squares estimation algorithms for moving average systems,” Simulation Modelling Practice and Theory, vol. 34, pp. 12–19, 2013. View at: Google Scholar
 Z. Y. Wang, Y. X. Shen, Z. C. Ji, and F. Ding, “Filtering based recursive least squares algorithm for Hammerstein FIRMA systems,” Nonlinear Dynamics, vol. 73, no. 12, pp. 1045–1054, 2013. View at: Publisher Site  Google Scholar
 V. Singh, “Stability of discretetime systems joined with a saturation operator on the statespace: generalized form of LiuMichel's criterion,” Automatica, vol. 47, no. 3, pp. 634–637, 2011. View at: Publisher Site  Google Scholar
 X. L. Xiong, W. Fan, and R. Ding, “Leastsquares parameter estimation algorithm for a class of input nonlinear systems,” Journal of Applied Mathematics, vol. 2012, Article ID 684074, 14 pages, 2012. 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, “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 B. Yu, “Robust mixed ${H}_{2}/{H}_{\infty}$ 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
 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: Publisher Site  Google Scholar
 P. P. Hu, F. Ding, and J. Sheng, “Auxiliary model based least squares parameter estimation algorithm for feedback nonlinear systems using the hierarchical identification principle,” Journal of the Franklin Institute—Engineering and Applied Mathematics, vol. 350, no. 10, pp. 3248–3259, 2013. View at: Publisher Site  Google Scholar
 M. Viberg, “Subspacebased methods for the identification of linear timeinvariant systems,” Automatica, vol. 31, no. 12, pp. 1835–1851, 1995. View at: Publisher Site  Google Scholar
 S. Gibson and B. Ninness, “Robust maximumlikelihood estimation of multivariable dynamic systems,” Automatica, vol. 41, no. 10, pp. 1667–1682, 2005. View at: Publisher Site  Google Scholar
 H. Raghavan, A. K. Tangirala, R. B. Gopaluni, and S. L. Shah, “Identification of chemical processes with irregular output sampling,” Control Engineering Practice, vol. 14, no. 5, pp. 467–480, 2006. View at: Publisher Site  Google Scholar
 L. Zhuang, F. Pan, and F. Ding, “Parameter and state estimation algorithm for singleinput singleoutput linear systems using the canonical state space models,” Applied Mathematical Modelling, vol. 36, no. 8, pp. 3454–3463, 2012. 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: 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
 D. Li, S. L. Shah, and T. Chen, “Identification of fastrate models from multirate data,” International Journal of Control, vol. 74, no. 7, pp. 680–689, 2001. View at: Publisher Site  Google Scholar
 L. Ljung, System Identification: Theory for the User, PrenticeHall, Englewood Cliffs, NJ, USA, 2nd edition, 1999.
 Y. Xiao, F. Ding, Y. Zhou, M. Li, and J. Dai, “On consistency of recursive least squares identification algorithms for controlled autoregression models,” Applied Mathematical Modelling, vol. 32, no. 11, pp. 2207–2215, 2008. View at: Publisher Site  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
 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
 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
 W. Wang, J. Li, and R. F. Ding, “Maximum likelihood parameter estimation algorithm for controlled autoregressive autoregressive models,” International Journal of Computer Mathematics, vol. 88, no. 16, pp. 3458–3467, 2011. View at: Publisher Site  Google Scholar
 W. Wang, F. Ding, and J. Dai, “Maximum likelihood least squares identification for systems with autoregressive moving average noise,” Applied Mathematical Modelling, vol. 36, no. 5, pp. 1842–1853, 2012. 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
 J. Li, F. Ding, and G. W. 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
 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: 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: Publisher Site  Google Scholar
 F. Ding, H. Yang, and F. Liu, “Performance analysis of stochastic gradient algorithms under weak conditions,” Science in China F, vol. 51, no. 9, pp. 1269–1280, 2008. View at: Publisher Site  Google Scholar
 F. Ding and X.P. Liu, “Auxiliary modelbased stochastic gradient algorithm for multivariable output error systems,” Acta Automatica Sinica, vol. 36, no. 7, pp. 993–998, 2010. View at: Publisher Site  Google Scholar
 D. Q. 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, 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: 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
 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
 D. Q. Wang and F. Ding, “Extended stochastic gradient identification algorithms for HammersteinWiener ARMAX systems,” Computers and Mathematics with Applications, vol. 56, no. 12, pp. 3157–3164, 2008. View at: Publisher Site  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: Publisher Site  Google Scholar
 D. Q. Wang, T. Shan, and R. Ding, “Data filtering based stochastic gradient algorithms for multivariable CARARlike systems,” Mathematical Modelling and Analysis, vol. 18, no. 3, pp. 374–385, 2013. View at: Google Scholar
 D. Q. Wang, F. Ding, and D. Q. Zhu, “Data filtering based least squares algorithms for multivariable CARARlike systems,” International Journal of Control, Automation, and Systems, vol. 11, no. 4, pp. 711–717, 2013. View at: Google Scholar
 F. Ding, X. P. Liu, and G. Liu, “Identification methods for Hammerstein nonlinear systems,” Digital Signal Processing, vol. 21, no. 2, pp. 215–238, 2011. 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, 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 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 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 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: Publisher Site  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: Publisher Site  Google Scholar
 D. Q. Wang, Y. Chu, G. W. Yang, and F. Ding, “Auxiliary model based recursive generalized least squares parameter estimation for Hammerstein OEAR systems,” Mathematical and Computer Modelling, vol. 52, no. 12, pp. 309–317, 2010. View at: Publisher Site  Google Scholar
 D. Q. 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
 L. L. Han, J. Sheng, F. Ding, and Y. Shi, “Auxiliary model identification method for multirate multiinput systems based on least squares,” Mathematical and Computer Modelling, vol. 50, no. 78, pp. 1100–1106, 2009. View at: Publisher Site  Google Scholar
 L. L. Han, F. Wu, J. Sheng, and F. Ding, “Two recursive least squares parameter estimation algorithms for multirate multipleinput systems by using the auxiliary model,” Mathematics and Computers in Simulation, vol. 82, no. 5, pp. 777–789, 2012. View at: Publisher Site  Google Scholar
 Y. Gu and F. Ding, “Auxiliary model based least squares identification method for a state space model with a unit timedelay,” Applied Mathematical Modelling, vol. 36, no. 12, pp. 5773–5779, 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: Publisher Site  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
 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
 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
 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
 L. Wang, F. Ding, and P. X. Liu, “Convergence of HLS estimation algorithms for multivariable ARXlike systems,” Applied Mathematics and Computation, vol. 190, no. 2, pp. 1081–1093, 2007. 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
 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: 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: Publisher Site  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. J. 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. J. 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: Publisher Site  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: Publisher Site  Google Scholar
 H. Duan, J. Jia, and R. F. Ding, “Twostage recursive least squares parameter estimation algorithm for output error models,” Mathematical and Computer Modelling, vol. 55, no. 34, pp. 1151–1159, 2012. View at: Publisher Site  Google Scholar
 G. Yao and R. F. Ding, “Twostage least squares based iterative identification algorithm for controlled autoregressive moving average (CARMA) systems,” Computers and Mathematics with Applications, vol. 63, no. 5, pp. 975–984, 2012. View at: Publisher Site  Google Scholar
 S. J. Wang and R. Ding, “Threestage recursive least squares parameter estimation for controlled autoregressive autoregressive systems,” Applied Mathematical Modelling, vol. 37, no. 1213, pp. 7489–7497, 2013. View at: Google Scholar
 G. C. Goodwin and K. S. Sin, Adaptive Filtering, Prediction and Control, PrenticeHall, Englewood Cliffs, NJ, USA, 1984.
 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, 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
 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, “Several multiinnovation identification methods,” Digital Signal Processing, vol. 20, no. 4, pp. 1027–1039, 2010. 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: Publisher Site  Google Scholar
 F. Ding, H. B. 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. 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. Q. 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. J. 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
 Y. J. 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
 L. L. Han and F. Ding, “Parameter estimation for multirate multiinput systems using auxiliary model and multiinnovation,” Journal of Systems Engineering and Electronics, vol. 21, no. 6, pp. 1079–1083, 2010. View at: Publisher Site  Google Scholar
 L. 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, 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: Publisher Site  Google Scholar
 F. Ding, Y. J. Liu, and B. Bao, “Gradientbased and leastsquaresbased iterative estimation algorithms for multiinput multioutput systems,” Proceedings of the Institution of Mechanical Engineers, Part 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: Publisher Site  Google Scholar
 Y. J. Liu, D. Q. 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
 H. Y. Hu and F. Ding, “An iterative least squares estimation algorithm for controlled moving average systems based on matrix decomposition,” Applied Mathematics Letters, vol. 25, no. 12, pp. 2332–2338, 2012. View at: Publisher Site  Google Scholar
 D. Q. Wang, G. W. Yang, and R. F. Ding, “Gradientbased iterative parameter estimation for BoxJenkins systems,” Computers and Mathematics with Applications, vol. 60, no. 5, pp. 1200–1208, 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, 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
 F. Ding, “Coupledleastsquares identification for multivariable systems,” IET Control Theory and Applications, vol. 7, no. 1, pp. 68–79, 2013. View at: Publisher Site  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
 Y. J. 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
 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
 Y. J. 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, Part I: Journal of Systems and Control Engineering, vol. 223, no. 4, pp. 445–454, 2009. View at: Publisher Site  Google Scholar
 J. Ding, L. L. Han, and X. Chen, “Time series AR modeling with missing observations based on the polynomial transformation,” Mathematical and Computer Modelling, vol. 51, no. 56, pp. 527–536, 2010. 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, 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. 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
 J. Ding, Y. Shi, H. Wang, and F. Ding, “A modified stochastic gradient based parameter estimation algorithm for dualrate sampleddata systems,” Digital Signal Processing, vol. 20, no. 4, pp. 1238–1247, 2010. 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. J. Liu, F. Ding, and Y. Shi, “An efficient hierarchical identification method for general dualrate sampleddata systems,” Automatica, 2014. View at: Publisher Site  Google Scholar
 F. Ding, “Hierarchical parameter estimation algorithms for multivariable systems using measurement information,” Information Sciences, 2014. View at: Publisher Site  Google Scholar
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Copyright © 2014 Ruifeng Ding and Linfan Zhuang. 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.