The Scientific World Journal

Volume 2014, Article ID 106505, 10 pages

http://dx.doi.org/10.1155/2014/106505

## Parameter and State Estimator for State Space Models

^{1}Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Jiangnan University, Wuxi 214122, China^{2}School of Internet of Things Engineering, Jiangnan University, Wuxi 214122, China

Received 30 August 2013; Accepted 31 December 2013; Published 2 March 2014

Academic Editors: M. Hajarian and C. Saravanan

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.

#### Abstract

This paper proposes a parameter and state estimator for canonical state space systems from measured input-output 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 input-output 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 maximum-likelihood estimation for fully parameterized linear time-invariant (LTI) state space models; the idea is to use the expectation maximization (EM) algorithm to estimate maximum-likelihood degrees [13]. Raghavan et al. studied the EM-based 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 input-output data to estimate the parameters of lifted state space models for general dual-rate systems [18]. Recently, some identification methods have been developed, for example, the least squares methods [19, 20], the gradient-based 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 estimation-based residual algorithm from the given input-output 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 gradient-based 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 two-stage 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 input-output data .

Since the available measurement input-output 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 state-space 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 input-output 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*

*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 input-output 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*

*5. Example*

*Consider the following single-input single-output second-order 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*

*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 input-output 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, single-rate or dual-rate, scalar or multivariable systems [112–117].*

*Conflict of Interests*

*Conflict of Interests*

*The authors declare that there is no conflict of interests regarding the publication of this paper.*

*Acknowledgments*

*Acknowledgments*

*This work was supported by the PAPD of Jiangsu Higher Education Institutions and the 111 Project (B12018).*

*References*

*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 FIR-MA systems,”
*Nonlinear Dynamics*, vol. 73, no. 1-2, pp. 1045–1054, 2013. View at Publisher · View at Google Scholar - V. Singh, “Stability of discrete-time systems joined with a saturation operator on the statespace: generalized form of Liu-Michel's criterion,”
*Automatica*, vol. 47, no. 3, pp. 634–637, 2011. View at Publisher · View at Google Scholar · View at Scopus - X. L. Xiong, W. Fan, and R. Ding, “Least-squares 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 · View at Google Scholar - 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 · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - 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 Publisher · View at 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 · View at Google Scholar - M. Viberg, “Subspace-based methods for the identification of linear time-invariant systems,”
*Automatica*, vol. 31, no. 12, pp. 1835–1851, 1995. View at Publisher · View at Google Scholar · View at Scopus - S. Gibson and B. Ninness, “Robust maximum-likelihood estimation of multivariable dynamic systems,”
*Automatica*, vol. 41, no. 10, pp. 1667–1682, 2005. View at Publisher · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - L. Zhuang, F. Pan, and F. Ding, “Parameter and state estimation algorithm for single-input single-output linear systems using the canonical state space models,”
*Applied Mathematical Modelling*, vol. 36, no. 8, pp. 3454–3463, 2012. View at Publisher · View at Google Scholar · View at Scopus - 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 · View at Google Scholar - 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 · View at Google Scholar · View at Scopus - D. Li, S. L. Shah, and T. Chen, “Identification of fast-rate models from multirate data,”
*International Journal of Control*, vol. 74, no. 7, pp. 680–689, 2001. View at Publisher · View at Google Scholar · View at Scopus - L. Ljung,
*System Identification: Theory for the User*, Prentice-Hall, 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 auto-regression models,”
*Applied Mathematical Modelling*, vol. 32, no. 11, pp. 2207–2215, 2008. View at Publisher · View at Google Scholar · View at Scopus - 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 gradient-based identification of multivariable discrete-time systems,”
*Automatica*, vol. 41, no. 2, pp. 315–325, 2005. View at Publisher · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - 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. 3-4, pp. 442–450, 2012. View at Publisher · View at Google Scholar · View at Scopus - 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 · View at 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. 1-2, pp. 234–245, 2014. View at Publisher · View at 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 · View at Google Scholar · View at Scopus - F. Ding and X.-P. Liu, “Auxiliary model-based stochastic gradient algorithm for multivariable output error systems,”
*Acta Automatica Sinica*, vol. 36, no. 7, pp. 993–998, 2010. View at Publisher · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - 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 Publisher · View at Google Scholar - 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 - F. Ding and T. Chen, “Identification of Hammerstein nonlinear ARMAX systems,”
*Automatica*, vol. 41, no. 9, pp. 1479–1489, 2005. View at Publisher · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - D. Q. Wang and F. Ding, “Extended stochastic gradient identification algorithms for Hammerstein-Wiener ARMAX systems,”
*Computers and Mathematics with Applications*, vol. 56, no. 12, pp. 3157–3164, 2008. View at Publisher · View at Google Scholar · View at Scopus - 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 Publisher · View at Google Scholar - D. Q. Wang, T. Shan, and R. Ding, “Data filtering based stochastic gradient algorithms for multivariable CARAR-like 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 CARAR-like 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 · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - 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 Publisher · View at Google Scholar - 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 Publisher · View at 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. 1-2, pp. 309–317, 2010. View at Publisher · View at Google Scholar · View at Scopus - D. Q. 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 · View at Google Scholar · View at Scopus - L. L. Han, J. Sheng, F. Ding, and Y. Shi, “Auxiliary model identification method for multirate multi-input systems based on least squares,”
*Mathematical and Computer Modelling*, vol. 50, no. 7-8, pp. 1100–1106, 2009. View at Publisher · View at Google Scholar · View at Scopus - L. L. Han, F. Wu, J. Sheng, and F. Ding, “Two recursive least squares parameter estimation algorithms for multirate multiple-input systems by using the auxiliary model,”
*Mathematics and Computers in Simulation*, vol. 82, no. 5, pp. 777–789, 2012. View at Publisher · View at Google Scholar · View at Scopus - Y. Gu and F. Ding, “Auxiliary model based least squares identification method for a state space model with a unit time-delay,”
*Applied Mathematical Modelling*, vol. 36, no. 12, pp. 5773–5779, 2012. View at Publisher · View at Google Scholar · View at Scopus - 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 - 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 Publisher · View at Google Scholar - 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 · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - L. Wang, F. Ding, and P. X. Liu, “Convergence of HLS estimation algorithms for multivariable ARX-like systems,”
*Applied Mathematics and Computation*, vol. 190, no. 2, pp. 1081–1093, 2007. View at Publisher · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - 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 Publisher · View at 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 · View at Google Scholar · View at Scopus - 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 · 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 · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - F. Ding, “Transformations between some special matrices,”
*Computers and Mathematics with Applications*, vol. 59, no. 8, pp. 2676–2695, 2010. View at Publisher · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - 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 Publisher · View at Google Scholar - 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 Publisher · View at Google Scholar - H. Duan, J. Jia, and R. F. Ding, “Two-stage recursive least squares parameter estimation algorithm for output error models,”
*Mathematical and Computer Modelling*, vol. 55, no. 3-4, pp. 1151–1159, 2012. View at Publisher · View at Google Scholar · View at Scopus - G. Yao and R. F. Ding, “Two-stage 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 · View at Google Scholar · View at Scopus - S. J. Wang and R. Ding, “Three-stage recursive least squares parameter estimation for controlled autoregressive autoregressive systems,”
*Applied Mathematical Modelling*, vol. 37, no. 12-13, pp. 7489–7497, 2013. View at Google Scholar - G. C. Goodwin and K. S. Sin,
*Adaptive Filtering, Prediction and Control*, Prentice-Hall, Englewood Cliffs, NJ, USA, 1984. - 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 · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - F. Ding, “Several multi-innovation identification methods,”
*Digital Signal Processing*, vol. 20, no. 4, pp. 1027–1039, 2010. View at Publisher · View at Google Scholar · View at Scopus - 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 Publisher · View at Google Scholar - F. Ding, H. B. 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 · View at Google Scholar · View at Scopus - L. 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 · View at Google Scholar · View at Scopus - D. Q. 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 · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - Y. J. 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 · View at Google Scholar · View at Scopus - Y. J. 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 · View at Google Scholar · View at Scopus - L. L. Han and F. Ding, “Parameter estimation for multirate multi-input systems using auxiliary model and multi-innovation,”
*Journal of Systems Engineering and Electronics*, vol. 21, no. 6, pp. 1079–1083, 2010. View at Publisher · View at Google Scholar · View at Scopus - L. 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 · View at Google Scholar · View at Scopus - 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 Publisher · View at Google Scholar - F. Ding, Y. J. 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, Part I: Journal of Systems and Control Engineering*, vol. 226, no. 1, pp. 43–55, 2012. View at Publisher · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - 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 · View at Google Scholar - Y. J. Liu, D. Q. 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 · View at Google Scholar · View at Scopus - 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 · View at Google Scholar - D. Q. Wang, G. W. Yang, and R. F. Ding, “Gradient-based iterative parameter estimation for Box-Jenkins systems,”
*Computers and Mathematics with Applications*, vol. 60, no. 5, pp. 1200–1208, 2010. View at Publisher · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - F. Ding, “Coupled-least-squares identification for multivariable systems,”
*IET Control Theory and Applications*, vol. 7, no. 1, pp. 68–79, 2013. View at Publisher · View at Google Scholar - 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 · View at Google Scholar · View at Scopus - Y. J. 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 · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - Y. J. 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, Part I: Journal of Systems and Control Engineering*, vol. 223, no. 4, pp. 445–454, 2009. View at Publisher · View at Google Scholar · View at Scopus - 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. 5-6, pp. 527–536, 2010. View at Publisher · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - 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*, vol. 20, no. 4, pp. 1238–1247, 2010. View at Publisher · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - 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 · View at Scopus - 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 · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at Scopus - Y. J. Liu, F. Ding, and Y. Shi, “An efficient hierarchical identification method for general dual-rate sampled-data systems,”
*Automatica*, 2014. View at Publisher · View at Google Scholar - F. Ding, “Hierarchical parameter estimation algorithms for multivariable systems using measurement information,”
*Information Sciences*, 2014. View at Publisher · View at Google Scholar

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