20th Iranian Conference on Electrical Engineering, (ICEE2012), May 15-17, Tehran, Iran

Gradient Based Iterative Identification of Multivariable

Hammerstein-Wiener Models with Application to a Steam

Generator Boiler

Masoumeh J afari *, Maryam Salimifard * *, and Maryam Dehghani * * *

Department of Power and Control, School of Electrical and Computer Engineering,

Shiraz University, Shiraz, Iran.

*jafarim@shirazu.ac.ir, * *msalimifard@shirazu.ac.ir, * * *mdehghani@shirazu.ac.ir

Abstract: Most of the real industrial systems are nonlinear and multivariable which might be correlated with some noises. Therefore, considering a model which can effectively characterize these types of systems are very appealing. In this regard, this paper presents a multivariable Hammerstein­Wiener model for identification of nonlinear systems with moving average noises. For this purpose, this model is first re­expressed as a multivariable pseudo-linear regression problem. Then, a gradient based iterative learning algorithm is invoked which can successfully estimate the matrix of unknown parameters as well as the noises. The efficiency of the proposed identification scheme is investigated through data for a real multivariable nonlinear process as a case study. This process is a Steam Generator Boiler at Abbott Power Plant in Champaign IL which has characteristics of instabilities, nonlinearity, non­minimum phase behaviour, time delays, noise spectrum in the same frequency range of the plant dynamics, and load disturbances. As the results verify, this approach is quite efficient for identification of multivariable nonlinear systems.

Keywords: Nonlinear system identification, multivariable systems, Hammerstein-Wiener model, moving average noises, gradient based iterative algorithm.

1. Introduction System identification is one of the most appealing

areas in different fields of science and technology. In fact, accurate identification of a system is the basic step for control, optimization, fault detection, etc. [1,2]. While efficiency of linear identification approaches is clear for most of researches, e.g. [1,3,4], they might not be sufficient enough to describe a complex nonlinear system in the whole operating range. Furthermore, many real systems which are inherently nonlinear, have multi inputs and multi outputs. Therefore, multivariable nonlinear identification techniques seem quite necessary.

In the literature, there are several approaches on identification of linear or nonlinear multivariable systems such as [1,5-9]. Most of the approaches address the independently distributed noises or the so-called "white" noises. However, the outputs of real systems are usually correlated with noises which might not satisty the assumption of white noises. As such, some iterative

methods have been developed for identifying pseudo­linear regressive models, e.g., output error systems, and output error moving average systems. Since, the iterative algorithms use all the collected data in each iteration, they can obtain highly accurate parameter estimates. In this aspect, Ding et al. [10,11] proposed a gradient based identification algorithm for nonlinear Hammerstein systems. Later, their iterative methods were extended to Wiener nonlinear systems by Wang and Ding [12], and to multivariable linear systems by Zhang et al. [l3] and Bao et al. [14]. Besides, Ding and Chen [15], and Han et al. [16] proposed a hierarchical least squares based identification algorithm for linear multi variable systems which were extended to multivariable Hammerstein systems by Salimifard et al. [17].

In this paper, the gradient based iterative identification methods [1O,II,l3] are extended for nonlinear multivariable Hammerstein-Wiener systems with moving average noises. The Hammerstein-Wiener model [18-20] is one of the powerful approaches which benefits from the advantages of the both popular Hammerstein and Wiener models. The proposed approach extends the current methods from different aspects such as single-input single-output models [10-12,20], and linear models [l3-16]. Furthermore, in the previous paper [17], for description of the L TI block in the Hammerstein model, only the transfer matrix models with least common denominators was considered. Therefore, all the model outputs are limited to have the same autoregressive dynamics. But the proposed Hammerstein-Wiener model is more flexible in the sense of description of the multivariable linear dynamics.

To investigate the efficiency of the proposed approach, data from a real four-input four output industry system, a Steam Generator Boiler at Abbott Power Plant in Champaign IL is studied. Boilers are invoked to generate steam and deliver hot water to a turbine at constant rate, pressure and temperature [21]. This system has

978-1-4673-1148-9112/$31.00 ©2012 IEEE 916

characteristics of nonlinearity, non-mlmmum phase behaviour, instabilities, noise spectrum in the same frequency range of the plant dynamics, time delays, and load disturbances. Therefore, it is a quite useful benchmark problem for evaluation of the systems identification algorithms with actual data.

The remainder of the paper is organized as follows. Section 2 presents the identification problem of a nonlinear multi variable Hammerstein-Wiener structure with moving average noises. Gradient based iterative parameter estimation algorithm is presented in Section 3. To demonstrate the efficiency of the proposed approach, Section 4 provides the simulation results of a nonlinear real multivariable system. Finally, the paper concludes in Section 5.

2. Multivariable Hammerstein-Wiener Model Identification

Hammerstein-Wiener model as illustrated in Fig. 1, is an extension of the Hammerstein and Wiener models where a linear block is embedded between two static nonlinear functions [18-20]. In this paper, the dynamic linear block is described by a matrix form of ARMAX/CARMA like model which can well represent the moving average noises as well as the autoregressive and exogenous dynamics. The static nonlinear multivariable function N1 and NZ-i are approximated based on some arbitrary vector-based basis functions, such as Volterra bases, polynomials, radial basis functions, or wavelets as:

rl

N1(-) = I aigJ·) (Ia)

i= i r2

NZ-i(.) = I Cihi(') (lb)

i=i where Ci E lRl.nxn , hJ·) : lRl.n � lRl.n, i = 1, ... , rz, and ai E lRl.mxm ,gi ( -) : lRl.m � lRl.m, i = 1, ... , ri·

! vet)

U(t) �) �(t) ----+ .N10 � + N20

a(q)

Fig. I: Multivariable Hammerstein-Wiener model

This paper considers the following multi variable

na rz

yet) = -I Ai I Cjhj(y(t - 0) i=i j=i

nb rl

+ I Hi I ajgj(u(t - 0) i= i j=i nd

+ I DiV(t - 0 + vet) i= i

(2)

Therefore, the following pseudo-linear-in-the-parameter model is obtained:

yet) = (JT lfJ(t) + vet). (3)

Definition of the parameter matrix (J E lRl.zxn, and the information matrix lfJ(t) E lRl.z are as follows:

lfJ(t) := [-hICy(t -1)), ... , -hi1 (y(t -na)),

gr (u(t -1)), ... , gi2 (u(t -nb))' (5)

vT(t-l), ... ,vT(t-nd)]

Where z = n x na x rz + m x nb x ri + n x nd, Rij = Aicj E lRl.nxn; i = 1, ... ,na, j = 1, ... ,rz, and

Pij = Hiaj E lRl.nxm; i = 1, ... ,nb, j = 1, ... ,ri. The next section extends an iterative identification

algorithm [10, II] to estimate the unknown parameter matrix as well as the noises in the proposed Hammerstein­Wiener model by using the available input-output data

{u(t),y(t)}.

3. Identification Algorithm Here, the gradient based iterative methods for

Hammerstein nonlinear ARMAX systems [10], and for linear systems [13] is developed to derive the gradient based identification method for the proposed nonlinear multivariable Hammerstein-Wiener model with moving average noises.

Consider L input-output samples from the system, and define the output matrix Y(L), the information matrix CP(L), and the white noise matrix VeL) as:

Y(L):= [y(1),y(2), ... ,y(L)] E lRl.nxL (6a)

CP(L) := [lfJ(l), lfJ(2), ... , lfJ(L)] E lRl.ZXL (6b)

VeL) := [v(l), v(2), ... , veL)] E lRl.nxL (6c)

Hammerstein-Wiener model with moving average noises. From (3) one can write:

Y(L) = (JTcp(L) + VeL). (7)

Define a quadratic cost function:

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Je = IIY(L) - OTC1>(L) 112

where IIXl12 := trace[XXT].

(8)

It is obvious that the information matrix cp(t), t =

1, ... , L) contains nd noise vectors v e t - 0, i = 1, ... , nd) which are not known a priori. Therefore, it is needed to replace the unmeasurable noise terms v e t) with their estimated residuals Vk(t). One can decompose the information matrix cp(t) = [CPs (t), CPn(t)] into two sub­matrices CPs(t) and CPn(t). The former, indexed by s, is related to all measurable data {u(t), y(t) : t = 1, ... ,L}. While the latter, indexed by n, is related to the unmeasurable noise terms v e t) as follows:

(9)

The estimate Vk(t) of vet) at iteration k is computed by

(10)

where

(11a)

(II b)

The gradient based iterative algorithm for the proposed nonlinear multivariable Hammerstein-Wiener model in the form of the pseudo-linear regression problem (3) can be summarized as follows.

Step 1. Collect L input-output samples from the system.

Step 2. To initialize, let k = 1, and 8 = Ozxn, where Ozxn is a z x n-dimensional matrix which entire elements are zero. Let VI be a n-dimensional random vector.

Step 3. Form fPk(t) by (11), and choose time-varying step-size or convergence factor 11k 2': 0 so that it satisfy [13]:

2 11k < T - Amax{C1>k(L)C1>k(L)}

Step 4. Update 8k by

L

(12)

8k = 8k-1 + 11k I fPk(O [y(O - 8I-IfPk(OY (13) i=1

Step 5. Estimate the noise Vk(t) by

Vk(t) = yet) - 8IfPk(t) (14)

Step 6. Compare 8k with 8k-1 with the following inequality:

118k - 8k_1112 ::; I:

If it holds, terminate the iterative algorithm, obtain 9k and Vk' Otherwise, let k = k + 1 and return to step 3.

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4. Simulation Results Here, to verify the performance of the proposed method, identification of a Steam Generator Boiler at Abbott Power Plant in Champaign IL is studied. This system as described in [21], has four inputs, namely the fuel flow rate, air flow rate, feedwater flow rate, and the disturbance defined by the load level. It has also four outputs including steam pressure (YI), excess oxygen in exhaust gases (Y2), level of water in the drum (Y3), and steam flow (Y4)' To make the open loop identification possible, the water level is stabilized by applying a feedforward action proportional to the steam flow with value 0.0403 and a PI action with values Kp = 0.258 and Ti = 1.1026e - 4 to the water flow input, where the reference of this controller is the third input. A set of 9600 samples of this system at sampling rate 3 seconds are available at DaISy [22].

For the identification purpose, the first 3000 input­output samples are used to estimate the multivariable Hammerstein-Wiener model, while the next 2000 samples are employed as a validation set to ascertain the predictive capability of the models on unseen data. The outputs of the steam generator boiler for estimation and validation data sets are depicted in Fig. 2. The parameters of the Hammerstein-Wiener model are chosen as na = 2, nb = 2, nd = 1, and r = 3. The basis functions that are considered for this system are vector polynomials up to order 3. The convergence factor I: in the iterative gradient based algorithm is set to 10-7.

Outputs of the Steam Generator Boiler

,;: :::� AI\, A�. "f"'/\.f\ J1 .A.ArViA .. A � 200 � � , VV "II,V � V- \IV \J - V -\1\.1

�� __ � __ L-� __ � __ �� __ -L __ �� o 500 1 000 1500 2000 2500 3000 3500 4000 4500 5000

500 1 000 1500 2000 2500 3000 3500 4000 4500 5000

� .: : o 500 1 000 1500 2000 2500 3000 3500 4000 4500 5000

�::� o 500 1 000 1500 2000 2500 3000 3500 4000 4500 5000 Samples

Fig. 2: The outputs of the Steam Generator Boiler

In the simulation results, the efficiency of the proposed method is presented through the mean square error (MSE) performance index which is defined as [1]:

L

MSE = � I (y(t) - y(t))2 (15)

t=1 where yet) and yet) denote the real and the model outputs, respectively. L is the number of data used for estimation or validation. Table I presents the performance index MSE for estimation and validation data sets. The

estimation quality of the proposed Hammerstein-Wiener model for the steam generator outputs are depicted in Figs. 3 to 6. While the prediction quality for validation data set is illustrated in Figs. 7 to 10. It should be noted that the output Yl is spanned in the range of 200 to 500 PSI. Therefore, its estimation and prediction errors are in the reasonable range. As the results verify, the proposed approach has quite satisfactory performance in the identification of nonlinear real multivariable systems.

Data set

TABLE I: Performance Index

MSE Yl MSE Yz MSE Y3

Estimation 94.2846 1.7189 0.4471

Validation 7l.2985 l.8914 0.3995

(a) Original \B. Estimation

MSEY4

0.6709

0.5393

1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 (b) Estimation Error

20

·40� ____ -L ____ �� ____ � ____ -L ______ L-____ � o 500 1000 1500 Samples

2000 2500 3000

Fig. 3: Estimation quality of output Yl of the Steam Generator; (a) real

output vs. estimated output, (b) estimation error

N "

(a) Original 'lIS. Estimation

(b) Estimation Error

Samples Fig. 4: Estimation quality of output Y2 of the Steam Generator; (a) real

output vs. estimated output, (b) estimation error

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(a) Original 'lIS. Estimation

1000 11 00 1200 1300 1400 1500 1600 1700 1800 1900 2000

500 1000

(b) Estimation Error

1500 Samples

2000 2500 3000

Fig. 5: Estimation quality of output Y3 of the Steam Generator; (a) real

output vs. estimated output, (b) estimation error

(a) Original \B. Estimation 40 ,---,---,---,---,---,---,---,---,---,---�

600 700 800 900 1000 1100 1200 1300 1400 1500

500 1000

(b) Estimation Error

1500 Samples

2000 2500 3000

Fig. 6: Estimation quality of output Y. of the Steam Generator; (a) real

output vs. estimated output, (b) estimation error

(a) Original 1,0$. Prediction

(b) Prediction Error

.::� ·40 � __ L-__ L-__ L-__ L-__ � __ � __ � __ � __ � __ � o 200 400 600 800 1000 1200 1400 1600 1800 2000

Samples Fig. 7: Prediction quality of output Yl of the Steam Generator; (a) real

output vs. predicted output, (b) prediction error

N "

(a) Original 1,0$. Prediction 20 ,--,---,---,--,---,---,---,--,---,---,--, 15

400 500 600 700 800 900 1000 1100 1200 1300 1400 1500

-10

(b) Prediction Error

200 400 600 800 1000 1200 1400 1600 1800 2000 Samples

Fig. 8: Prediction quality of output Y2 of the Steam Generator; (a) real

output vs. predicted output, (b) prediction error

M "

10

-2

(a) Original 1,0$. Prediction

200 400 600 800 1000 1200 1400 1600 1800 2000 (b) Prediction Error

200 400 600 800 1000 1200 1400 1600 1800 2000 Samples

Fig. 9: Prediction quality of output Y3 of the Steam Generator; (a) real

output vs. predicted output, (b) prediction error

,," a

-2

(a) Original 1,0$. Prediction -- Original

200 400 600 800 1000 1200 1400 1600 1800 2000 (b) Prediction Error

200 400 600 800 1 000 1200 1400 1600 1800 2000 Samples

Fig. I 0: Prediction quality of output Y. of the Steam Generator; (a) real

output vs. predicted output, (b) prediction error

5. Conclusions This paper presents a simple and efficient method for

identification of nonlinear multivariable systems in the presence of colored noises. In the proposed approach, Hammerstein-Wiener model was studied. Appropriate and flexible representations of this model led to a pseudo­linear-in-the-parameter problem which contains some terms related to the colored noises. Therefore, a gradient based iterative algorithm was successfully invoked to estimate the matrix of unknown parameters as well as the colored noises. The efficiency of the proposed identification scheme was investigated through a real industry steam generator process. As the results verify, the proposed method is quite efficient for the identification of nonlinear multivariable systems with actual data.

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