Mathematical Problems in Engineering

Volume 2014, Article ID 232848, 10 pages

http://dx.doi.org/10.1155/2014/232848

## Filtering Based Recursive Least Squares Algorithm for Multi-Input Multioutput Hammerstein Models

Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Jiangnan University, Wuxi 214122, China

Received 28 June 2014; Revised 7 September 2014; Accepted 25 September 2014; Published 16 October 2014

Academic Editor: Haranath Kar

Copyright © 2014 Ziyun Wang et al. 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 considers the parameter estimation problem for Hammerstein multi-input multioutput finite impulse response (FIR-MA) systems. Filtered by the noise transfer function, the FIR-MA model is transformed into a controlled autoregressive model. The key-term variable separation principle is used to derive a data filtering based recursive least squares algorithm. The numerical examples confirm that the proposed algorithm can estimate parameters more accurately and has a higher computational efficiency compared with the recursive least squares algorithm.

#### 1. Introduction

Parameter estimation is an important approach to model dynamical systems and has been widely used in estimating the parameters for nonlinear systems [1–3], deriving system identification methods [4–7], identifying state-space models [8, 9], and developing solutions for matrix equations [10–13]. For example, Dehghan and Hajarian discussed several solution methods for different matrix equations [14–16]. In the area of system control and modeling, Shi and Fang developed a Kalman filter based identification for systems with randomly missing measurements [17], gave output feedback stabilization [18], and presented a robust mixed control of networked control systems [19].

The least squares algorithm is a fundamental method [20–22] and many methods such as the iterative algorithm [23, 24] and the gradient algorithm [25] are widely used in the parameter estimation. In the field of Hammerstein system identification, several methods have been developed [26]. For example, a least squares based iterative algorithm and an auxiliary model based recursive least squares algorithm have been presented, respectively, for Hammerstein nonlinear ARMAX systems and Hammerstein output error systems [27, 28]; a Newton recursive algorithm and a Newton iterative algorithm for Hammerstein controlled autoregressive systems are presented in [29].

Consider a multi-input multioutput (MIMO) Hammerstein finite impulse response (FIR) system depicted by where is the nonlinear system input vector with zero mean and unit variances, is the measurement of but is corrupted by , is the white noise vector with zero mean, and and are polynomials in the unit backward shift operator : It is obvious that the relation between sizes and would influence the model identification of this multi-input multioutput Hammerstein system. For example, the dimension of the output vector is not less than that of the input vector if ; otherwise, when , the output size is smaller compared with that of the input vector. In this paper, we discuss the identification problem with .

The nonlinear block in the Hammerstein model is a linear combination of the known basis : where the superscript denotes the matrix transpose. The function in (3) is a nonlinear function of a known basis : where the coefficients are unknown. Substituting (4) into (3) yields Assume that , , and for , and the orders , , and are known but can be obtained by trial and error. In general, the orders of the Hammerstein model should be large when the nonlinear system is used in prediction; otherwise, the orders should be small if the system is applied for control.

The objective of this paper is to estimate the unknown parameter matrices: , , from the available input-output data of the multivariable Hammerstein finite impulse response moving average (FIR-MA) models [30].

Recently, the filtering idea has received much attention [31–33]. Xiao and Yue studied input nonlinear dynamical adjustment models and presented a recursive generalized least squares algorithm and a filtering based least squares algorithm by replacing the unknown terms in the information vectors with their estimates [34]. The overparameterization method in [34] leads to a redundant estimated product of the nonlinear systems and requires extra computation. Differing from the work in [30, 34, 35], this paper discusses the estimation problem of the MIMO Hammerstein systems using the data filtering idea and transfers the FIR-MA system to controlled autoregressive model by means of the key-term variable separate principle in [36–38]. The proposed algorithm used in this paper can extend to study parameter estimation problems of dual-rate/multirate sampled systems [39–42] and other linear or nonlinear systems [43–46].

Briefly, the rest of this paper is recognized as follows. Section 2 discusses a recursive least squares algorithm for the Hammerstein systems. Section 3 presents a filtering based recursive least squares algorithm by transferring an FIR-MA model to a controlled autoregressive model. Section 4 provides an illustrative example. Finally, some concluding remarks are offered in Section 5.

#### 2. The MRLS Algorithm

For comparison, the MRLS algorithm is listed in Section 2. Here we introduce some notations. represents the current time in this paper and “” or “” means that “ is defined as ”. The symbol represents an identity matrix of size followed by a null matrix of the last rows when and vice versa. The norm of a matrix (or a column vector) is defined by ; denotes the Kronecker product or direct product: if , , then ; col is supposed to be the vector formed by the column of the matrix : if , then col.

From (1)–(5), the intermediate variables and and output of the system can be expressed as Note that the subscripts (Roman) and denote the first letters of “system” and “noise” for distinguishing the types of the unknown parameter vectors or matrices, respectively. Define the parameter matrix , the parameter vectors , , and the information vectors , , and as Then, we have Distinguished from the hierarchical identification methods, we reparameterize the model in (7) by using the Kronecker product to get a parameter matrix and by gathering the input information vectors and and output information matrix into one information matrix to obtain an information matrix as follows: Thus, we obtain Equation (11) is the identification model of the multivariable Hammerstein FIR-MA system. Defining and minimizing the cost function and using the least squares search principle, we obtain the following recursive least squares algorithm [35] to obtain parameter estimates : Since the information matrix in (13) contains the unknown intermediate variables and the unmeasurable terms , the recursive algorithm in (13)–(15) cannot compute the parameter estimate . The solution here is replacing the unknown intermediate variables and the unmeasurable terms in with the variable estimates (or the outputs of the auxiliary model) and the estimates based on the auxiliary model identification idea. The replaced information matrices are defined as Define the parameter estimation matrices By replacing the parameters in (4) with , the output of the proposed auxiliary model is given by From (11), we obtain . Replacing and with and , the residual can be written as .

To summarize, we conclude the following recursive least squares algorithm for multivariable Hammerstein FIR-MA models (the MRLS algorithm for short):

#### 3. The F-MRLS Algorithm

The convergence rate of the MRLS algorithm in Section 2 is slow because the noise information intermediate variables contain unmeasurable time-delay noise . The solution here is to present a filtering based recursive least squares algorithm (the F-MRLS algorithm) for the multivariable Hammerstein models by filtering the rational function and transferring the FIR-MA model in (1) into a controlled autoregressive (CAR) model. Multiplying both sides of (1) by yields or where Thus, (21) can be rewritten as Define the filtered information matrices: Since the polynomial is unknown and to be estimated, it is impossible to use to construct in (25). Here, we adopt the principle of the MRLS algorithm in Section 2 and replace the unmeasurable variables and vectors with their estimates to derive the following algorithm.

By using the parameter estimates and , the estimates of polynomials and at time can be constructed as in model (1) can be rewritten as Let be the estimate of . Replacing , , , , , and with their estimates , , , , , and leads to Defining and minimizing the cost function and using the least squares search principle, we list the recursive least squares algorithm to compute : Let be the estimate of at time , filtering with to obtain the estimate : The estimate of can be computed by Define the estimate of by and construct the estimate of with and as follows: The filtered model in (21) can be rewritten in a matrix form: or Based on the MRLS search principle, we can obtain the estimate of by the following algorithm: The estimate can be computed by Filter by to obtain the estimate : Replacing , , , and in (37) with their estimates , , , and at time , the noise vector can be computed by In conclusion, we can summarize the following filtering based recursive least squares algorithm for multivariable Hammerstein models (the F-MRLS algorithm for short):

The steps involved in the F-MRLS algorithm for multivariable Hammerstein systems are listed in the following.(1)To initialize, let , set the initial values of the parameter estimation variables and covariance matrices as follows: , , , , , for , for and , , , , and give the basis functions .(2)Collect the input-output data and , and construct by (60), and , by (58), (59). Form the information vectors by (57), by (61), and by (56), respectively.(3)Compute by (55), the gain vector by (53), and the covariance matrix by (54), respectively. Update the parameter estimate by (52).(4)Compute by (51), by (49). Construct , , and by (48), (47), and (46), respectively. Compute by (45).(5)Compute the gain vector by (43) and the covariance matrix by (44). Update the parameter estimate by (42).(6)Construct the parameter vectors , , and by (66), (67), and (68). Form and by (65) and (64). Compute by (50), by (62), and by (63).(7)Increase by 1 and go to step 2.

#### 4. Examples

*Example 1. *Consider the following 2-input 2-output Hammerstein FIR-MA system:
where
In simulation, the inputs and are taken as two uncorrelated persistent excitation signal sequences with zero mean and unit variances and and as two white noise sequences with zero mean and variances . Applying the MRLS algorithm in (19) and the F-MRLS algorithm in (42)–(68) to estimate the parameters of this multivariable Hammerstein system, the F-MRLS parameter estimates and their estimation errors are shown in Table 1, the comparison between the F-MRLS algorithm and the MRLS algorithm in the estimation error versus is shown in Table 2, and the estimation errors versus are shown in Figures 1 and 2.

*Example 2. *Consider the following 2-input 2-output Hammerstein FIR-MA system:
where
In simulation, the inputs and and noise data and are settled in the same way as in Example 1, where variances ). Applying the MRLS algorithm and the F-MRLS algorithm in (42)–(68), the F-MRLS parameter estimates and their estimation errors are shown in Table 3, the comparison between the F-MRLS algorithm and the MRLS algorithm in the estimation error versus is shown in Table 4, and the estimation errors versus are shown in Figures 3 and 4.

To illustrate the advantages of the proposed algorithm, the numbers of multiplications and additions for each step of the F-MRLS algorithm and the MRLS algorithm are listed in Table 5.

From Tables 1–5, Figures 1–4, we can draw the following conclusions.(i)The parameter estimation errors are getting smaller with increasing, which proves that the proposed algorithms are effective.(ii)The F-MRLS algorithm is more accurate than the MRLS algorithm, which means the proposed F-MRLS algorithm has a better performance compared with the MRLS algorithm.(iii)The parameter estimates given by the F-MRLS algorithm have faster convergence than those given by the MRLS algorithm.

#### 5. Conclusions

This paper presents a data filtering based recursive least squares algorithm for MIMO nonlinear FIR-MA systems. The simulation results show that the proposed data filtering based recursive least squares algorithm is more accurate and reduces computational burden compared with the recursive least squares algorithm.

#### Conflict of Interests

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

#### Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (nos. 61174032 and 61202473), the Doctoral Foundation of Higher Education Priority Fields of Scientific Research (no. 20110093130001), and the Scientific Research Foundation of Jiangnan University (no. 1252050205135110) and by the PAPD of Jiangsu Higher Education Institutions.

#### References

- W.-H. Ho, J.-H. Chou, and C.-Y. Guo, “Parameter identification of chaotic systems using improved differential evolution algorithm,”
*Nonlinear Dynamics*, vol. 61, no. 1-2, pp. 29–41, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - M. D. Narayanan, S. Narayanan, and C. Padmanabhan, “Parametric identification of nonlinear systems using multiple trials,”
*Nonlinear Dynamics*, vol. 48, no. 4, pp. 341–360, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - W. Silva, “Identification of nonlinear aeroelastic systems based on the volterra theory: progress and opportunities,”
*Nonlinear Dynamics*, vol. 39, no. 1-2, pp. 25–62, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Y. Zhang and G. Cui, “Bias compensation methods for stochastic systems with colored noise,”
*Applied Mathematical Modelling: Simulation and Computation for Engineering and Environmental Systems*, vol. 35, no. 4, pp. 1709–1716, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · 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 Zentralblatt MATH · View at MathSciNet · View at Scopus - Y. S. Xiao, Y. Zhang, J. Ding, and J. Y. Dai, “The residual based interactive least squares algorithms and simulation studies,”
*Computers & Mathematics with Applications*, vol. 58, no. 6, pp. 1190–1197, 2009. View at Publisher · View at Google Scholar · View at Scopus - Q.-G. Wang, X. Guo, and Y. Zhang, “Direct identification of continuous time delay systems from step responses,”
*Journal of Process Control*, vol. 11, no. 5, pp. 531–542, 2001. View at Publisher · View at Google Scholar · View at Scopus - F. Ding, “State filtering and parameter estimation for state space systems with scarce measurements,”
*Signal Processing*, vol. 104, pp. 369–380, 2014. View at Publisher · View at Google Scholar · View at Scopus - Y. Gu, F. Ding, and J. H. Li, “State filtering and parameter estimation for linear systems with d-step state-delay,”
*IET Signal Processing*, vol. 8, no. 6, pp. 639–646, 2014. View at Publisher · View at Google Scholar - M. Dehghan and M. Hajarian, “Two algorithms for finding the HERmitian reflexive and skew-HERmitian solutions of Sylvester matrix equations,”
*Applied Mathematics Letters*, vol. 24, no. 4, pp. 444–449, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - M. Dehghan and M. Hajarian, “A lower bound for the product of eigenvalues of solutions to matrix equations,”
*Applied Mathematics Letters*, vol. 22, no. 12, pp. 1786–1788, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - M. Dehghan and M. Hajarian, “SSHI methods for solving general linear matrix equations,”
*Engineering Computations (Swansea, Wales)*, vol. 28, no. 8, pp. 1028–1043, 2011. View at Publisher · View at Google Scholar · View at Scopus - M. Dehghan and M. Hajarian, “An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation,”
*Applied Mathematics and Computation*, vol. 202, no. 2, pp. 571–588, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - M. Dehghan and M. Hajarian, “Finite iterative algorithms for the reflexive and anti-reflexive solutions of the matrix equation ${A}_{1}{X}_{1}{B}_{1}+{A}_{2}{X}_{2}{B}_{2}=C$,”
*Mathematical and Computer Modelling*, vol. 49, no. 9-10, pp. 1937–1959, 2009. View at Publisher · View at Google Scholar - M. Dehghan and M. Hajarian, “Fourth-order variants of Newton's method without second derivatives for solving non-linear equations,”
*Engineering Computations (Swansea, Wales)*, vol. 29, no. 4, pp. 356–365, 2012. View at Publisher · View at Google Scholar · View at Scopus - M. Dehghan and M. Hajarian, “Iterative algorithms for the generalized centro-symmetric and central anti-symmetric solutions of general coupled matrix equations,”
*Engineering Computations*, vol. 29, no. 5, pp. 528–560, 2012. View at Publisher · View at Google Scholar · View at Scopus - 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 Zentralblatt MATH · View at MathSciNet · 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 MathSciNet · 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 MathSciNet · View at Scopus - Y. 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 Publisher · View at Google Scholar · View at Scopus - Y. B. Hu, B. L. 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 · View at Google Scholar · View at MathSciNet · View at Scopus - C. Wang and T. Tang, “Recursive least squares estimation algorithm applied to a class of linear-in-parameters output error moving average systems,”
*Applied Mathematics Letters*, vol. 29, pp. 36–41, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - F. Ding, X. 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 · View at Google Scholar · View at MathSciNet · View at Scopus - C. Wang and T. Tang, “Several gradient-based iterative estimation algorithms for a class of nonlinear systems using the filtering technique,”
*Nonlinear Dynamics*, vol. 77, no. 3, pp. 769–780, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Y. B. Hu, B. L. Liu, and Q. Zhou, “A multi-innovation generalized extended stochastic gradient algorithm for output nonlinear autoregressive moving average systems,”
*Applied Mathematics and Computation*, vol. 247, pp. 218–224, 2014. View at Google Scholar - J. H. Li, “Parameter estimation for Hammerstein CARARMA systems based on the Newton iteration,”
*Applied Mathematics Letters*, vol. 26, no. 1, pp. 91–96, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - 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 MathSciNet · 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 Zentralblatt MATH · View at Scopus - F. Ding, X. P. Liu, and G. Liu, “Identification methods for Hammerstein nonlinear systems,”
*Digital Signal Processing: A Review Journal*, vol. 21, no. 2, pp. 215–238, 2011. View at Publisher · View at Google Scholar · View at Scopus - Z. Y. Wang, Y. X. Shen, Z. Ji, and R. 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 · View at MathSciNet · View at Scopus - M. S. Ahmad, O. Kukrer, and A. Hocanin, “Recursive inverse adaptive filtering algorithm,”
*Digital Signal Processing: A Review Journal*, vol. 21, no. 4, pp. 491–496, 2011. 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 & Applications*, vol. 5, no. 14, pp. 1648–1657, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - B. Yu, Y. Shi, and H. Huang, “${l}_{2}$-${l}_{\infty}$ filtering for multirate systems based on lifted models,”
*Circuits, Systems, and Signal Processing*, vol. 27, no. 5, pp. 699–711, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Y. Xiao and N. Yue, “Parameter estimation for nonlinear dynamical adjustment models,”
*Mathematical and Computer Modelling*, vol. 54, no. 5-6, pp. 1561–1568, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - F. Ding and 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 · View at Scopus - J. Vörös, “Iterative algorithm for parameter identification of Hammerstein systems with two-segment nonlinearities,”
*IEEE Transactions on Automatic Control*, vol. 44, no. 11, pp. 2145–2149, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - J. Vörös, “Identification of Hammerstein systems with time-varying piecewise-linear characteristics,”
*IEEE Transactions on Circuits and Systems II: Express Briefs*, vol. 52, no. 12, pp. 865–869, 2005. View at Publisher · View at Google Scholar · View at Scopus - J. Vörös, “Recursive identification of Hammerstein systems with discontinuous nonlinearities containing dead-zones,”
*IEEE Transactions on Automatic Control*, vol. 48, no. 12, pp. 2203–2206, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - 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 · View at Google Scholar · View at MathSciNet - J. Ding, C. X. Fan, and J. X. Lin, “Auxiliary model based parameter estimation for dual-rate output error systems with colored noise,”
*Applied Mathematical Modelling*, vol. 37, no. 6, pp. 4051–4058, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - L. Xie, H. Yang, and B. Huang, “FIR model identification of multirate processes with random delays using EM algorithm,”
*AIChE Journal*, vol. 59, no. 11, pp. 4124–4132, 2013. View at Publisher · View at Google Scholar · View at Scopus - J. Ding and J. Lin, “Modified subspace identification for periodically non-uniformly sampled systems by using the lifting technique,”
*Circuits, Systems, and Signal Processing*, vol. 33, no. 5, pp. 1439–1449, 2014. View at Publisher · View at Google Scholar · View at Scopus - F. Ding, “Hierarchical estimation algorithms for multivariable systems using measurement information,”
*Information Sciences*, vol. 277, pp. 396–405, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · 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 · View at MathSciNet · View at Scopus - X. Luan, P. Shi, and F. Liu, “Stabilization of networked control systems with random delays,”
*IEEE Transactions on Industrial Electronics*, vol. 58, no. 9, pp. 4323–4330, 2011. View at Publisher · View at Google Scholar · View at Scopus - X. Luan, S. Zhao, and F. Liu, “${H}_{\infty}$ control for discrete-time Markov jump systems with uncertain transition probabilities,”
*Institute of Electrical and Electronics Engineers. Transactions on Automatic Control*, vol. 58, no. 6, pp. 1566–1572, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus