IEEE SIGNAL PROCESSING LETTERS, VOL. 19, NO. 12, DECEMBER 2012
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Hierarchical Least Squares Estimation Algorithm for Hammerstein–Wiener Systems Dong-Qing Wang and Feng Ding
Abstract—This letter focuses on identification problems of a Hammerstein–Wiener system with an output error linear element embedded between two static nonlinear elements. A hierarchical least squares algorithm is presented for the Hammerstein–Wiener system by using the auxiliary model identification idea and the hierarchical identification principle. The major contributions of the present study are that the identification model is formulated by using the auxiliary model identification idea (the estimate of the unknown internal variable is replaced with the output of an auxiliary model) and that the bilinear parameter vectors in the identification model are estimated by using the hierarchical identification principle. The proposed hierarchical identification approach is computationally more efficient than the existing over-parametrization method. Index Terms—Auxiliary model identification idea, Hammerstein–Wiener systems, hierarchical identification principle, least squares, parameter estimation.
I. INTRODUCTION
N
ONLINEARITIES exist widely in various physical systems, and nonlinear systems are of great importance in modeling [1]–[4]. Block structure models, including Hammerstein, Wiener, Hammerstein–Wiener, and Wiener–Hammerstein models, are convenient for modeling nonlinear systems. The Hammerstein-Wiener model is a nonlinear system with three blocks where a dynamic linear part is sandwiched between two static nonlinear parts. Many approaches have been proposed to identify block-oriented nonlinear models. The most commonly used approaches are the over-parametrization method [5]–[8], the iterative method [9], [10], the key term separation identification method [11], [12] and the blind identification method [13]. In the previous work, based on the over-parametrization method, Wang and Ding reported a stochastic gradient algorithm for Hammerstein–Wiener systems [6]; Ding and Chen Manuscript received June 26, 2012; revised August 12, 2012; accepted September 05, 2012. Date of publication October 02, 2012; date of current version October 10, 2012. This work was supported by the Shandong Provincial Natural Science Foundation (ZR2010FM024), the Qingdao Municipal Science and Technology Development Program (12-1-4-2-(3)-jch), the National Natural Science Foundation of China (61273194) and the 111 Project (B12018). The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Ricardo Merched. D.-Q. Wang is with the College of Automation Engineering, Qingdao University, Qingdao 266071, China (e-mail:
[email protected]). F. Ding is with the Control Science and Engineering Research Center, Jiangnan University, Wuxi 214122, China (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LSP.2012.2221704
presented recursive and iterative least squares estimation algorithms for Hammerstein nonlinear ARMAX systems [7]; Ding et al. reported an auxiliary model based recursive least squares algorithm for Hammerstein output-error systems [8]. By the iterative technique, Zhu proposed a relaxation iteration scheme and studied its convergence performance for Hammerstein–Wiener models [9]. The iterative identification method is usually used in the situation in which there exists unknown internal variables in the information vectors. Through the key term separation principle, Vörös studied the identification problems for the Hammerstein–Wiener models [11]. Bai presented a blind identification approach for Hammerstein–Wiener models using the over-parametrization method [13]. This letter investigates a hierarchical least squares method for a Hammerstein–Wiener system which consists of an output error linear subsystem and two static nonlinearities, and the nonlinear output element is assumed to be invertible. The sequential decomposition technique utilized in deriving the auxiliary functions was proposed by Vörös [11]. The computation of the separated key term is similar to that of the auxiliary model in the letter. By the auxiliary model identification idea [14], [15], we transform the Hammerstein–Wiener system into a bilinear parameter identification model and present a hierarchical least squares algorithm to estimate the bilinear parameter vectors of the Hammerstein–Wiener system by using the hierarchical identification principle [16], [17]. The hierarchical identification principle is to decompose a system into several subsystems, to identify the parameter vector of each subsystem, and to coordinate the associate items (coupled variables) between subsystems [18]. In this letter, we interactively estimate the parameter vectors in the bilinear parameter identification model of the Hammerstein–Wiener system by using the hierarchical identification principle. The proposed approach is computationally more efficient than the existing overparametrization methods [6]. The iterative method takes use of all data in each computational cycle, its convergence rate is fast [19]. The identification model based on the key term separation method contains only linear parameter vectors (without bilinear parameter vectors) [11]. The specialty of the blind method is that the input data is unobserved [13]. The rest of this letter is organized as follows. The problem formulation is described in Section II. The identification model and the identification algorithm are presented in Section III. Some simulation results are given in Section IV, and a brief summary of the main contents is given in Section V.
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III. THE IDENTIFICATION ALGORITHM Notice that parameters in (6) and (7) are not unique. Without loss of generality, assume that the first coefficient of the nonlinear part is unity, i.e., [6]. Combining (6) and (7) gives
Fig. 1. Hammerstein–Wiener output error system.
II. PROBLEM FORMULATION The Hammerstein-Wiener output error system in Fig. 1 can be expressed as (8) (1) (2)
Define
(3) (4) where and are the system input and output, , and are the internal variables, is stochastic white noise with zero mean, the linear block is an output error model, and are polynomials in the unit backward shift operator ( ), and defined by
Assume that the orders and are known and and for . The input nonlinearity modeled as a linear combination of basis functions :
.. .
.. .
.. .
.. .
.. .
, is
.. .
(5) .. .
where is the number of the basis functions. The output nonlinearity is considered to be invertible, and can be written as a linear combination of the basis functions [5]: (6)
.. .
.. .
Then we have (9)
From (1), (2) and (5), we have (10) The information vector in on the right-hand side of (10) contains the unknown internal variables , here we use the auxiliary model identification idea, to replace the unknown with the outputs of an auxiliary model [14], [15] which can be taken as the following form
Then we have (7)
(11) The objectives of this letter are two-fold: first, by means of the auxiliary model identification idea [14], [15], transform the Hammerstein–Wiener system into a bilinear parameter identification model; second, by means of the hierarchical identification principle [16], [17], present new algorithms to estimate the system parameters from available input-output data .
where in is the estimate of in and are the estimates of , and . be defined by
, and , in can
WANG AND DING: HIERARCHICAL LEAST SQUARES ESTIMATION ALGORITHM FOR HAMMERSTEIN–WIENER SYSTEMS
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Suppose the data length . Define a quadratic cost function containing three parameter vectors,
Minimizing the cost functions with respect to , and , and replacing with . By the hierarchical identification principle [16], when computing the estimate of one parameter vector, others are replaced with their estimates, then we can obtain the following hierarchical least squares algorithm for estimating , and of the Hammerstein–Wiener output error system (the HW-HLS algorithm for short):
(12) (13) (14) (15)
Fig. 2. The flowchart of computing the estimates
,
and
.
smaller, the HW-HLS algorithm is computationally more efficient than the over-parametrization method [7], [8] and other methods. IV. EXAMPLE
(16)
Consider the following Hammerstein–Wiener system,
(17) (18) (19) (20) .. .
.. .
.. . (21) (22) (23)
and the inversion of the output nonlinear block,
The input is taken as an uncorrelated persistent excitation signal sequence with zero mean and unit variance, and as a white noise sequence with zero mean and variance and , respectively. Applying the HW-HLS algorithm to estimate the parameters of this system, the parameter estimates and their errors
(24) (25) The initial values can be taken to be , , , and , , . The flowchart of computing the parameter estimates , and by the HW-HLS algorithm in (12)–(25) is shown in Fig. 2. By using the decomposition based hierarchical identification principle, the dimensions of the covariance matrices become
are shown in Table I and in Fig. 3. From Table I and Fig. 3, we can draw the following conclusions: 1) The HW-HLS algorithm can give highly accurate parameter estimates under the low noise levels; 2) The parameter estimation errors given by the HW-HLS algorithm become generally smaller and go to zero with the data length increasing. This shows that the proposed algorithm is effective.
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PARAMETER ESTIMATES
Fig. 3. The estimation errors
versus with different
TABLE I ERRORS WITH DIFFERENT
AND
.
V. CONCLUSIONS The letter discusses identification problems for a Hammerstein–Wiener output error system. The identification model is formulated by using the auxiliary model identification idea, and a hierarchical least squares algorithm is derived to estimate the bilinear parameter vectors of the system by using the hierarchical identification principle. The computer simulation demonstrates the effectiveness of the proposed method. REFERENCES [1] U. Remes, K. J. Palomaki, T. Raiko, A. Honkela, and M. Kurimo, “Missing-feature reconstruction with a bounded nonlinear state-space model,” IEEE Signal Process. Lett., vol. 18, no. 10, pp. 563–566, Oct. 2011. [2] L. Du, P. Stoica, J. Li, and L. N. Cattafesta, “Computationally efficient approaches to aeroacoustic source power estimation,” IEEE Signal Process. Lett., vol. 18, no. 1, pp. 11–14, Jan. 2011. [3] P. V. Vu and D. M. Chandler, “A fast wavelet-based algorithm for global and local image sharpness estimation,” IEEE Signal Process. Lett., vol. 19, no. 7, pp. 423–426, 2012.
[4] G. L. Santosuosso, K. Benzemrane, and G. Damm, “Nonlinear speed estimation of a GPS-free UAV,” Int. J. Contr., vol. 84, no. 11, pp. 1873–1885, 2011. [5] E. W. Bai, “An optimal two-stage identification algorithm for Hammerstein-Wiener nonlinear systems,” Automatica, vol. 34, no. 3, pp. 333–338, 1998. [6] D. Q. Wang and F. Ding, “Extended stochastic gradient identification algorithms for Hammerstein-Wiener ARMAX systems,” Comput. Math. Applicat., vol. 56, no. 12, pp. 3157–3164, 2008. [7] F. Ding and T. Chen, “Identification of Hammerstein nonlinear ARMAX systems,” Automatica, vol. 41, no. 9, pp. 1479–1489, 2005. [8] F. Ding, Y. Shi, and T. Chen, “Auxiliary model based least-squares identification methods for Hammerstein output-error systems,” Syst. Contr. Lett., vol. 56, no. 5, pp. 373–380, 2007. [9] Y. C. Zhu, “Estimation of an N-L-N Hammerstein-Wiener model,” Automatica, vol. 38, no. 9, pp. 1607–1614, 2002. [10] Y. Liu and E. W. Bai, “Iterative identification of Hammerstein systems,” Automatica, vol. 43, no. 2, pp. 346–354, 2007. [11] J. Vörös, “An iterative method for Hammerstein-Wiener systems parameter identification,” J. Elect. Eng., vol. 55, no. 11-12, pp. 328–331, 2004. [12] J. Vörös, “Identification of nonlinear cascade systems with time-varying backlash,” J. Elect. Eng., vol. 62, no. 2, pp. 87–92, 2011. [13] E. W. Bai, “A blind approach to the Hammerstein–Wiener model identification,” Automatica, vol. 38, no. 6, pp. 967–979, 2002. [14] F. Ding, G. Liu, and X. P. Liu, “Parameter estimation with scarce measurements,” Automatica, vol. 47, no. 8, pp. 1646–1655, 2011. [15] D. Q. Wang, “Least squares-based recursive and iterative estimation for output error moving average systems using data filtering,” IET Contr. Theory Applicat., vol. 5, no. 14, pp. 1648–1657, 2011. [16] F. Ding and T. Chen, “Hierarchical least squares identification methods for multivariable systems,” IEEE Trans. Automat. Contr., vol. 50, no. 3, pp. 397–402, 2005. [17] Z. N. Zhang, F. Ding, and X. G. Liu, “Hierarchical gradient based iterative parameter estimation algorithm for multivariable output error moving average systems,” Comput. Math. Applicat., vol. 61, no. 3, pp. 672–682, 2011. [18] J. Ding, F. Ding, X. P. Liu, and G. Liu, “Hierarchical least squares identification for linear SISO systems with dual-rate sampled-data,” IEEE Trans. Automat. Contr., vol. 56, no. 11, pp. 2677–2683, 2011. [19] J. H. Li, R. F. Ding, and Y. Yang, “Iterative parameter identification methods for nonlinear functions,” Appl. Math. Model., vol. 26, no. 6, pp. 2739–2750, 2012.
