Combined Parameter and State Estimation Algorithms for Multivariable Nonlinear Systems Using MIMO Wiener Models

Salhi, Houda; Kamoun, Samira

doi:https://doi.org/10.1155/2016/9614167

Journal of Control Science and Engineering

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Research Article | Open Access

Volume 2016 | Article ID 9614167 | https://doi.org/10.1155/2016/9614167

Combined Parameter and State Estimation Algorithms for Multivariable Nonlinear Systems Using MIMO Wiener Models

Houda Salhi¹and Samira Kamoun¹

Academic Editor: James Lam

Received29 Nov 2015

Revised20 Apr 2016

Accepted12 May 2016

Published06 Jun 2016

Abstract

This paper deals with the parameter estimation problem for multivariable nonlinear systems described by MIMO state-space Wiener models. Recursive parameters and state estimation algorithms are presented using the least squares technique, the adjustable model, and the Kalman filter theory. The basic idea is to estimate jointly the parameters, the state vector, and the internal variables of MIMO Wiener models based on a specific decomposition technique to extract the internal vector and avoid problems related to invertibility assumption. The effectiveness of the proposed algorithms is shown by an illustrative simulation example.

1. Introduction

Over the last years, modeling, identification, and parameter estimation theories have received much attention by various research teams [1–4]. Blocks-oriented nonlinear models, in particular, which consist of interconnected linear dynamic subsystems and memory less nonlinear elements, have been widely used for modeling a large variety of nonlinear systems in such different fields as mechanical dynamics [5], chemical process [6], biotechnologies [7], signal filtering [8], and so on. In fact, this class of nonlinear models is able to describe the dynamic of complex systems, with a relatively simple structure. They can even simplify the identification, control, or diagnostic problems [9–12]. Added to that, several approaches developed in the linear case can be applied with an appropriate practical implementations [13–19]. In recent years, many identification methods have been studied for blocks-oriented systems and a large amount of works have been published in the literature. For example, Vörös [20] proposed a least squares based iterative algorithm for Hammerstein-Wiener systems with a backlash output; Hu et al. [21] developed an extended least squares parameter estimation algorithm for Wiener systems based on the overparameterization method; Ding et al. [22] used the hierarchical identification principal to identify Hammerstein systems; Mao and Ding [8] proposed a multi-innovation stochastic gradient algorithm for Hammerstein systems using the key term separation principal; Li [23] studied the maximum likelihood estimation algorithm for Hammerstein CARARMA systems; Guo and Bretthauer [24] proposed a recursive identification method for Wiener models, based on the prediction error method. Furthermore, Chaudhary et al. [25] explored an adaptive algorithm based on fractional signal processing for parameter estimation of Hammerstein autoregressive models; Wu et al. [26] presented a robust Hammerstein adaptive filtering algorithm based on the Maximum Correntropy Criterion, which aims at maximizing the similarity between the model output and the reel response; Falck et al. [27] proposed an identification method of NARX Wiener-Hammerstein models using kernel-based estimation technique; Kibangou and Favier [28] developed a new approach for estimating a Parallel-Cascade Wiener System using a joint diagonalization of the th-order Volterra kernel slices to identify linear subsystems and using the least square algorithm to identify nonlinear subsystems. This approach has been extended to other blocks-oriented models with polynomials nonlinearities [29].

Recently, much attention has been paid to blocks-oriented state-space systems which have been successfully used for control algorithms, identification schemes, and signal filtering [30, 31]. However, the parameter estimation has become more difficult because the blocks-oriented models not only include the unknown parameter of linear and nonlinear subsystems but also include the unmeasurable state variables [32–35]. In this framework, Wang and Ding proposed a recursive parameter and state estimation for Hammerstein state-space systems [36] and for Hammerstein-Wiener state-space systems [37], using the hierarchical principal; Wang et al. [38] discussed an iterative identification algorithm Hammetstein state-space system, by combining the iterative least square and the hierarchical identification method. However, for Wiener state-space models, there is a little contribution in the literature that addresses the parameter estimation problems or the state estimation problems. In fact, Westwick and Verhaegen [39] proposed a subspace identification method for MIMO Wiener subsystems with odd and even nonlinearities and a Gaussian input system; Bruls et al. [40] derived separable least squares algorithms for a state-space Wiener model with Chebyshev polynomials nonlinearity; Lovera et al. [41] developed a recursive subspace identification method for Wiener state-space models using the singular-value decomposition technique; Glaria Lopez and Sbarbaro [42] proposed an observer design for a Wiener model with known parameters.

The main difficulty in the identification of Wiener models is that the internal variables, acting between linear and nonlinear blocks, are almost unavailable and the input-output available data are not enough to provide all information on these unknown variables. To overcome this difficulty, most published works, addressing the identification of Wiener systems, assume one of these assumptions: the invertibility of the unknown nonlinear element [43], an a priori knowledge of the nonlinearity [44], an approximation of the nonlinearity as a piecewise linear function [44], and a specific input signal [39]. However, these assumptions, and especially the invertibility assumption, severely limit the applicability of Wiener models because the output nonlinearity, in several real cases, is noninvertible or is quite complicated to find the inverse nonlinearity especially for multivariable systems.

This paper introduces a recursive identification method for MIMO Wiener model. This model is characterised by a linear dynamic block as an observer state-space model and a nonlinear block as combined and arbitrary (reversible or irreversible) nonlinearities. A recursive algorithm which combines the least squares technique, the adjustable model, and the Kalman filter principle is developed to resolve the parameters and state estimation problem with less computational effort and a fast convergence rate. Indeed, in the proposed method, the parameters of the linear part and nonlinear part of the MIMO Wiener model are estimated separately in order to decrease the dimension of the unknown parameters matrices and reduce the parameter redundancy. Moreover, a modified Kalman filter and a specific decomposition technique are developed to extract and estimate the unknown internal vector without any research of the inverse nonlinear functions.

The remainder of this paper is organized as follows. Section 2 describes the problem formulation for MIMO Wiener state-space models. The least squares based and adjustable model based recursive parameter estimation algorithm and a new recursive state estimation algorithm based on Kalman filter theorem are presented in Section 3. Section 4 provides an illustrative example to show the efficiency of the proposed algorithms. Finally, some concluding remarks are given in Section 5.

2. Problem Formulation

Consider the MIMO discrete-time Wiener model Figure 1 where the linear dynamic part is given by the following state-space equation:where , , and are, respectively, the state vector, the input vector, and the internal vector at the discrete-time , and are two noise vectors, , , and and are defined, respectively, by

Assume that the degrees , , and are known and the internal vector and the state vector are unmeasurable. The static nonlinear function of the MIMO Wiener model is defined aswhere is the output system vector, is the nonlinear function vector that depends on the unknown internal vector , and is an error vector.

In the rest of this paper, we propose to rewrite the system output vector into two submodels.

The first submodel is given by the following equation:where and are defined by

However, the second submodel is based on a decomposition technique of nonlinear functions , given in (3):with and .

Using (6), the system output vector can be written aswhere (if , ; is an identity matrix) and is defined byIt is worth noting that the application of this decomposition technique to Wiener model has led to useful modelisation because it allows the extraction of the inaccessible internal vector and then the identification and control schemes become easier and more efficient.

From (1) and (4), it is clear that the number of unknown parameters is quite large, causing difficulties in the identification algorithm implementation. In order to reduce these difficulties and have a precise estimation quality, we propose to estimate the parameters of the linear dynamic part and those of the static nonlinear part, in two separate and recursive steps. The objective of this paper is to present a new recursive method to estimate jointly the parameters (, , , ), the state vector , and the intermediate vector using the measured input-output data .

3. The Recursive Parametric and State Estimation Algorithm

In order to simplify the formulation of the parametric and state estimation problem, this section is divided into two subsections. The basic idea is to estimate recursively the dynamic linear part parameters and the static nonlinear part parameters of the considered MIMO Wiener model and then to estimate the state vector and the internal vector .

3.1. The Parameter Estimation Algorithm

If the state vector and the internal vector are known, then we can apply the following adjustable model to generate the linear dynamic subsystem estimate (1):Using the least squares principle and minimizing the prediction errors functionswe can obtain the following recursive algorithm:where and are definite positive symmetrical matrices and and are, respectively, the maximum eigenvalues of and . In order to guarantee the convergence of the parameter estimate matrices , , , and , the gains parameter and must be chosen such that and . It should be noted that these gains can be chosen as time-varying parameters in order to improve the parameter estimation quality.

To avoid the parametric redundancy problem and construct the estimate parameters of the static nonlinear part (4), we suggest using the following recursive least squares (RLS) algorithm:where represent the estimate of in the regression equation (4) and is a definite positive symmetrical matrix where is an identity matrix and is generally taken as a large positive number, for example, .

In reality, the regression matrix contains the unmeasurable internal vector , so the parametric estimation algorithms (11) and (12) cannot be implemented directly. Added to that, the state vector is also supposed to be unknown. The proposed solution is to replace and in (11) and (12) with their estimated vectors and and then to define

3.2. Modified Kalman Filter

In the area of state estimation algorithms, the extended Kalman filter (EKF) is the most widely used nonlinear estimation method [45]. The EKF is based on the linearization of nonlinear models and calculation of the Jacobian matrices with each iteration, which may cause significant implementation difficulties and high estimation errors. Therefore, we propose to use a new filtering technique, developed in our previous work [15], based on the linear Kalman filter (KF). This filter yields the same performance as KF and can be applied to blocks-oriented models without any linearization steps. Let and represent, respectively, the a priori and a posteriori estimate of at a discrete-time and represents the a priori estimate of .

This part is based essentially on the Kalman filter principle, which is defined by the following.

Theorem 1. Define and as two random vectors, with a multinomial distribution vector with a mean vector and a variance-covariance matrix . Thus, the conditional probability density is a Gaussian function, where its mean and variance are given, respectively, by

In a first step, applying this theorem to the linear state-space equation (1) yieldswhere and are variance-covariance matrices of noise vectors and , respectively, and is the variance-covariance matrices.

However, the a posteriori estimate contains the unknown internal vector , so the algorithm (15) is impossible to implement. The solution is to replace the unknown internal vector with its estimate . For this reason, we propose to use the second form (7) of the system output vector and apply the KF to the following state-space model:Using Theorem 1 gives the following equations to generate the recursive estimate vector :where is the variance-covariance matrix of the noise vector and is the variance-covariance matrix.

Combining (15) and (17), we can summarize the recursive algorithm to generate the estimates and of the state vector and the internal vector :The details and the convergence analysis of (18) is treated in [15].

Thus, we can replace and in (11) and (12) with their estimates and .

Combining (11), (12), and (18), we can form a recursive state and parameter estimation algorithm for state-space MIMO Wiener systems (which is abbreviated as the RPSE algorithm). To initialize the RPSE algorithm, we take , , , , , , and as small real matrices and vectors; for example, with is a matrix whose all elements are equal to 1. We also take , , , with , , , and as large positive numbers and is an identity matrix with appropriate sizes. It should be noted that the choice of the initial conditions and the different gains must be chosen adequately in order to improve the estimation quality and the convergence rapidity of the various parameters.

The procedure for computing the parameter, the state, and the internal vector estimates using the RPSE algorithm is listed as follows:(1)To initialize, let , , , , , , , , , , , , , , , , , and .(2)Collect the input-output vectors and .(3)Form and , compute and , and update , , , using (11).(4)Compute the covariance matrix and construct using (12).(5)Compute the state gains and and the covariance matrix and update the state estimate and the internal vector estimate using (18).(6)Increase by and go to step .

The flowchart of the recursive learning algorithms which are used for the parameters estimation of nonlinear MIMO Wiener models is shown in Figure 2.

4. Example

Consider the following state-space model:and the system outputs are defined asThese two outputs can be grouped into the following matrix form:where and are given by Using the decomposition technique, (21) can be written in a second matrix form:where and are defined as In simulation, the inputs and are taken as two square sequences of levels and , respectively; the variance-covariance matrices are , , and . Applying the RPSE algorithm to estimate the parameters, the state, and internal variables of this system, the gain parameters and the initial conditions are chosen properly. The internal variables and and their estimates and and the outputs and with the predicted outputs and are shown in Figure 3. The estimation errors , , , and are shown in Figure 4. The evolution curve of the variance and of the system outputs and is given in Figure 5, where with the statistical mean of the output prediction error and and the initial and final discrete time. The parameter estimates and their estimation errors with different data length and different noise variances are shown in Tables 1 and 2, where the parameter estimation error is defined by

From the simulation results in Tables 1, 2, and 3 and Figures 3, 4, 5, and 6, we can draw the following conclusions:(i)The estimated parameters converge to real ones and the parameter estimation errors given by the RPSE algorithm become smaller when increases and the output variances decrease; see Tables 1 and 2.(ii)The estimated internal outputs and and the predicted system outputs and can track the actual outputs without the computation step of the inverse nonlinear function and with small estimation errors; see Figures 3 and 4.(iii)The output variances and rapidly drop to law values when the noise variances decrease; see Figure 5.(iv)The estimation quality is better when the parametric gains and are chosen as a time-varying parameters; see Figure 6.(v)The variances of the parameters estimates, using the Monte Carlo simulation, are small which improves the effectiveness of the RPSE algorithm; see Table 3.(vi)The proposed algorithm can achieve a satisfactory estimation quality through appropriately choosing the parametric gains and the innovation length.

5. Conclusions

This paper presents a recursive parameter and state estimation algorithm by combining the least square technique, the adjustable model, and the Kalman filter principal for estimating jointly the parameters, the state vector, and the internal variables of MIMO Wiener state-space models. By estimating the parameters of the linear and nonlinear parts separately and using a specific decomposition technique, we can remove the redundant parameters and avoid problems related to computing the inverse nonlinear functions. The proposed algorithm can be combined with adaptive control schemes and extended to other blocks-oriented models.

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

Acknowledgments

This work was supported by the ministry of higher education and scientific research of Tunisia.

References

H. Han, L. Xie, F. Ding, and X. Liu, “Hierarchical least-squares based iterative identification for multivariable systems with moving average noises,” Mathematical and Computer Modelling, vol. 51, no. 9-10, pp. 1213–1220, 2010.
View at: Publisher Site | Google Scholar
R. Q. Fuentes, I. Chairez, A. Poznyak, and T. Poznyak, “3D nonparametric neural identification,” Journal of Control Science and Engineering, vol. 2012, Article ID 618403, 10 pages, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
F. Ding, L. Qiu, and T. Chen, “Reconstruction of continuous-time systems from their non-uniformly sampled discrete-time systems,” Automatica, vol. 45, no. 2, pp. 324–332, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Liu, F. Ding, and Y. Shi, “An efficient hierarchical identification method for general dual-rate sampled-data systems,” Automatica, vol. 50, no. 3, pp. 962–970, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
A. Fathi and A. Mozaffari, “Identification of a dynamic model for shape memory alloy actuator using Hammerstein-Wiener gray box and mutable smart bee algorithm,” International Journal of Intelligent Computing and Cybernetics, vol. 6, no. 4, pp. 328–357, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
A. D. Kalafatis, L. Wang, and W. R. Cluett, “Linearizing feedforward-feedback control of pH processes based on the Wiener model,” Journal of Process Control, vol. 15, no. 1, pp. 103–112, 2005.
View at: Publisher Site | Google Scholar
A. Bhattacharjee and A. Sutradhar, “Online identification and internal model control for regulating hemodynamic variables in congestive heart failure patient,” International Journal of Pharma Medicine and Biological Sciences, vol. 4, no. 2, pp. 85–89, 2015.
View at: Google Scholar
Y. Mao and F. Ding, “Multi-innovation stochastic gradient identification for Hammerstein controlled autoregressive autoregressive systems based on the filtering technique,” Nonlinear Dynamics, vol. 79, no. 3, pp. 1745–1755, 2015.
View at: Publisher Site | Google Scholar
W. Yu, D. Wilson, and B. Young, “Control performance assessment for block-oriented nonlinear systems,” in Proceedings of the 8th IEEE International Conference on Control and Automation (ICCA '10), pp. 1151–1156, Xiamen, China, June 2010.
View at: Publisher Site | Google Scholar
S. I. Biagiola and J. L. Figueroa, “Wiener and Hammerstein uncertain models identification,” Mathematics and Computers in Simulation, vol. 79, no. 11, pp. 3296–3313, 2009.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
F. Guo, A new identification method for wiener and hammerstein systems [Ph.D. thesis], Karlsruhe University, 2004.
M. Salimifard, M. Jafari, and M. Dehghani, “Identification of nonlinear MIMO block-oriented systems with moving average noises using gradient based and least squares based iterative algorithms,” Neurocomputing, vol. 94, pp. 22–31, 2012.
View at: Publisher Site | Google Scholar
H. Salhi, S. Kamoun, N. Essounbouli, and A. Hamzaoui, “Adaptive discrete-time sliding-mode control of nonlinear systems described by Wiener models,” International Journal of Control, vol. 89, no. 3, pp. 611–622, 2016.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
H. Salhi and S. Kamoun, “A recursive parametric estimation algorithm of multivariable nonlinear systems described by Hammerstein mathematical models,” Applied Mathematical Modelling Journal, vol. 39, no. 16, pp. 4951–4962, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
H. Salhi and S. Kamoun, “State and parametric estimation of nonlinear systems described by wiener sate-space mathematical models,” in Handbook of Research on Advanced Intelligent Control Engineering and Automation, chapter 4, pp. 107–147, 2014.
View at: Publisher Site | Google Scholar
S. I. Biagiola and J. L. Figueroa, “Identification of uncertain MIMO Wiener and Hammerstein models,” Computers and Chemical Engineering, vol. 35, no. 12, pp. 2867–2875, 2011.
View at: Publisher Site | Google Scholar
S. Lakshminarayanan, S. L. Shah, and K. Nandakumar, “Identification of Hammerstein models using multivariate statistical tools,” Chemical Engineering Science, vol. 50, no. 22, pp. 3599–3613, 1995.
View at: Publisher Site | Google Scholar
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 Site | Google Scholar | MathSciNet
Z. Zhang, F. Ding, and X. Liu, “Hierarchical gradient based iterative parameter estimation algorithm for multivariable output error moving average systems,” Computers & Mathematics with Applications, vol. 61, no. 3, pp. 672–682, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
J. Vörös, “Iterative identification of nonlinear dynamic systems with output backlash using three-block cascade models,” Nonlinear Dynamics, vol. 79, no. 3, pp. 2187–2195, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Hu, B. Liu, Q. Zhou, and C. Yang, “Recursive extended least squares parameter estimation for Wiener nonlinear systems with moving average noises,” Circuits, Systems, and Signal Processing, vol. 33, no. 2, pp. 655–664, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
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 & Applications, vol. 7, no. 2, pp. 176–184, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
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 | MathSciNet
F. Guo and G. Bretthauer, “Identification of MISO Wiener and Hammerstein systems,” in Proceedings of the 7th European Control Conference (TEE '03), pp. 2144–2149, University of Cambridge, September 2003.
View at: Google Scholar
N. I. Chaudhary, M. A. Z. Raja, J. A. Khan, and M. S. Aslam, “Identification of input nonlinear control autoregressive systems using fractional signal processing approach,” The Scientific World Journal, vol. 2013, Article ID 467276, 13 pages, 2013.
View at: Publisher Site | Google Scholar
Z. Wu, S. Peng, B. Chen, and H. Zhao, “Robust Hammerstein adaptive filtering under maximum correntropy criterion,” Entropy, vol. 17, no. 10, pp. 7149–7166, 2015.
View at: Google Scholar | MathSciNet
T. Falck, P. Dreesen, K. De Brabanter, K. Pelckmans, B. De Moor, and J. A. K. Suykens, “Least-squares support vector machines for the identification of Wiener-Hammerstein systems,” Control Engineering Practice, vol. 20, no. 11, pp. 1165–1174, 2012.
View at: Publisher Site | Google Scholar
A. Y. Kibangou and G. Favier, “Identification of parallel-cascade Wiener systems using joint diagonalization of third-order Volterra kernel slices,” IEEE Signal Processing Letters, vol. 16, no. 3, pp. 188–191, 2009.
View at: Publisher Site | Google Scholar
A. Y. Kibangou and G. Favier, “Tensor analysis-based model structure determination and parameter estimation for block-oriented nonlinear systems,” IEEE Journal on Selected Topics in Signal Processing, vol. 4, no. 3, pp. 514–525, 2010.
View at: Publisher Site | Google Scholar
X. Wang and F. Ding, “Modelling and multi-innovation parameter identification for Hammerstein nonlinear state space systems using the filtering technique,” Mathematical and Computer Modelling of Dynamical Systems, vol. 22, no. 2, pp. 113–140, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
F. Ding, X. Liu, and X. Ma, “Kalman state filtering based least squares iterative parameter estimation for observer canonical state space systems using decomposition,” Journal of Computational and Applied Mathematics, vol. 301, pp. 135–143, 2016.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
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 | MathSciNet
X. Ma and F. Ding, “Gradient-based parameter identification algorithms for observer canonical state space systems using state estimates,” Circuits, Systems, and Signal Processing, vol. 34, no. 5, pp. 1697–1709, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
F. Ding and T. Chen, “Hierarchical identification of lifted state-space models for general dual-rate systems,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 52, no. 6, pp. 1179–1187, 2005.
View at: Publisher Site | Google Scholar | MathSciNet
X. Ma and F. Ding, “Recursive and iterative least squares parameter estimation algorithms for observability canonical state space systems,” Journal of the Franklin Institute. Engineering and Applied Mathematics, vol. 352, no. 1, pp. 248–258, 2015.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
X. Wang and F. Ding, “Recursive parameter and state estimation for an input nonlinear state space system using the hierarchical identification principle,” Signal Processing, vol. 117, pp. 208–218, 2015.
View at: Publisher Site | 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: Publisher Site | Google Scholar
D. Wang, F. Ding, and L. Ximei, “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 Site | Google Scholar | MathSciNet
D. Westwick and M. Verhaegen, “Identifying MIMO Wiener systems using subspace model identification methods,” Signal Processing, vol. 52, no. 2, pp. 235–258, 1996.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
J. Bruls, C. T. Chou, B. R. J. Haverkamp, and M. Verhaegen, “Linear and non-linear system identification using separable least-squares,” European Journal of Control, vol. 5, no. 1, pp. 116–128, 1999.
View at: Publisher Site | Google Scholar
M. Lovera, T. Gustafsson, and M. Verhaegen, “Recursive subspace identification of linear and non-linear Wiener state-space models,” Automatica, vol. 36, no. 11, pp. 1639–1650, 2000.
View at: Publisher Site | Google Scholar | MathSciNet
T. Á. Glaria Lopez and D. Sbarbaro, “Observer design for nonlinear processes with Wiener structure,” in Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC '11), pp. 2311–2316, IEEE, Orlando, Fla, USA, December 2011.
View at: Publisher Site | Google Scholar
E.-W. Bai, “Identification of linear systems with hard input nonlinearities of known structure,” Automatica, vol. 38, no. 5, pp. 853–860, 2002.
View at: Publisher Site | Google Scholar | MathSciNet
J. Vörös, “Parameter identification of Wiener systems with discontinuous nonlinearities,” Systems and Control Letters, vol. 44, no. 5, pp. 363–372, 2001.
View at: Publisher Site | Google Scholar | MathSciNet
K. Xiong, C. L. Wei, and L. D. Liu, “Robust Kalman filtering for discrete-time nonlinear systems with parameter uncertainties,” Aerospace Science and Technology, vol. 18, no. 1, pp. 15–24, 2012.
View at: Publisher Site | Google Scholar

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Copyright © 2016 Houda Salhi and Samira Kamoun. 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.

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