MODELLING AND CONTROL OF MULTIVARIABLE PROCESSES USING GENERALIZED HAMMERSTEIN MODEL Y.M. Hlaing, M.-S. Chiu and S. Lakshminarayanan Department of Chemical & Biomolecular Engineering, National University of Singapore, Singapore.
Abstract: In this paper, a generalized Hammerstein model consisting of a static polynomial function in series with time-varying linear model is proposed in order to model the Hammerstein-like multivariable processes whose linear dynamics vary over the operating space. An iteration procedure is proposed to identify the generalized Hammerstein model by using the just-in-time learning (JITL) technique. Unlike the conventional Hammerstein model, only the static nonlinear part of generalized Hammerstein model, thus identified is retained in its subsequent application in controller design. Consequently, the on-line use of proposed model requires the identification of linear model by using the JITL technique and current process data. An adaptive decentralized PID control strategy based on the proposed model is also developed. The controller parameters are adjusted on-line by using the gradient descent algorithm and information provided by the JITL. Simulation results show that the proposed methods have better prediction accuracy and control performance than those achieved by the conventional Hammerstein model and its associated adaptive decentralized PID controller design, respectively. Keywords: just-in-time learning; Hammerstein model; adaptive control; decentralized control.
INTRODUCTION
Correspondence to: Professor M.-S. Chiu, Department of Chemical & Biomolecular Engineering, National University of Singapore, 4 Engineering Drive 4, Singapore, 117576. E-mail:
[email protected]
DOI: 10.1205/cherd06210 0263–8762/07/ $30.00 þ 0.00 Chemical Engineering Research and Design Trans IChemE, Part A, April 2007 # 2007 Institution of Chemical Engineers
proposed a two-stage identification algorithm to extract the model parameters. Sung (2002) used a special test signal that enables the decoupling of the identification of the linear dynamics part from that of the static nonlinear part. Identification of Hammerstein models using multivariate statistical tools was proposed by Laksminarayanan et al. (1995). Al–Duwaish and Karim (1997) used a hybrid model consisting of a neural network to identify the static nonlinear part in series with autoregressive moving average (ARMA) model for identification of single-input singleoutput and multi-input multi-output (MIMO) Hammerstein model. Several other identification and controller design methods for Hammerstein model were developed by Su and McAvoy (1993), Abonyi et al. (2000) and Lee et al. (2004). However, Hammerstein model is restricted to the situation where the process gain changes with the operating conditions while the linear dynamics remain fairly constant over the region of process operation. As a result, the conventional Hammerstein model is not adequate for modelling the process when both process gain and linear dynamics change over the region of plant operation (Lakshminarayanan et al., 1997). To overcome
The Hammerstein model is a block-oriented nonlinear model as depicted in Figure 1, which consists of a static nonlinear part (NL) in series with a linear dynamics block G(z), where the former is modelled by different methods such as polynomials or neural networks. Many chemical processes have been modelled with Hammerstein model, for example pH neutralization processes (Lakshminarayanan et al., 1995; Fruzzetti et al., 1997), distillation columns (Eskinat et al., 1991; Pearson and Pottmann, 2000), polymerization reactor (Su and McAvoy, 1993; Ling and Rivera, 1998) and dryer process (Rollins et al., 2003). Various system identification methods have been proposed to identify the Hammerstein model. Narendra and Gallman (1966) were the first to develop an iterative identification procedure for Hammerstein model. Recently, Jia et al. (2005) developed an updating algorithm based on the Lyapunov approach to guarantee the global convergence of the model parameters in an iterative manner. In addition, several approaches have been proposed to identify Hammerstein model in a noniterative fashion. For examples, Pottmann et al. (1993) 445
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Figure 2. MIMO Hammerstein model with separate nonlinearities. Figure 1. MIMO Hammerstein model with combined nonlinearities.
this drawback, a generalized Hammerstein model consisting of varying linear dynamics preceded by a static nonlinear part is considered in this paper. An iterative procedure is developed to identify the generalized Hammerstein model. In addition, an adaptive decentralized PID control strategy based on the proposed new model is developed. The controller parameters are adjusted on-line by using the gradient descent method. Simulation results are presented to illustrate that the proposed methods have better prediction accuracy and control performance than those achieved by the Hammerstein model and its associated adaptive decentralized PID controller design.
IDENTIFICATION OF GENERALIZED HAMMERSTEIN MODEL Two possible structures as depicted in Figures 1 and 2 can be used to describe a MIMO Hammerstein model depending on whether the nonlinearities are separate or combined (Lakshminarayanan et al., 1995; Al-Duwaish and Karim, 1997). The combined nonlinearity case is more general, but it can cause a very challenging parameter estimation problem because of the large number of parameters to be estimated. Therefore, the generalized Hammerstein model with separate nonlinearities will be considered in this paper. The proposed generalized Hammerstein model is an extension of the conventional Hammerstein model by replacing the fixed linear model by the time-varying linear model. To identify the linear dynamics in the generalized Hammerstein model, just-in-time learning (JITL) technique (Atkeson et al., 1997; Bontempi et al., 1999, Cheng and Chiu, 2004) is employed in this paper. As a result, a low-order model is adequate to describe the linear dynamics in the proposed model owing to the features of JITL technique. Without loss of generality, a multivariable process with two inputs and two outputs will be utilized to detail the proposed identification procedure. For a 2 2 generalized Hammerstein model with separate nonlinearities, it can be described by the following equations: # " k y1 (k) y1 (k 1) a11 ak12 ¼ y2 (k) y2 (k 1) ak21 ak22 " # k v1 (k nd1 1) b1 0 þ (1) v2 (k nd2 1) 0 bk2 v1 (k) ¼g11 u1 (k) þ g12 u21 (k) þ þ g1m1 u1m1 (k)
(2)
v2 (k) ¼g21 u2 (k) þ g22 u22 (k) þ þ g2m2 u2m2 (k)
(3)
where yi (k) and ui (k) (i ¼ 1, 2) denote the process outputs
and inputs at the kth sampling instant respectively, v1 (k) and v2 (k) are unmeasurable internal variables, ak11 , ak12 , ak21 , ak22 , bk1 and bk2 are the parameters of linear dynamics of generalized Hammerstein model at the kth sampling instant, g1i (i ¼ 1 m1 ), g2j ( j ¼ 1 m2 ) are the parameters of static nonlinear part, and nd1 and nd2 are the process time-delays. Similar to the iterative identification procedure developed by Narendra and Gallman (1966), the proposed identification method for generalized Hammerstein model obtains the parameters by separating the estimation problem of the static nonlinear part from that of the linear dynamics. Let { y1 (k), y2 (k), u1 (k), u2 (k)}k¼1N denote the process data collected from the off-line identification test. When the parameters of linear dynamics for y1 (k) in equation (1) are available, parameters of the corresponding static nonlinear part in equation (2) are obtained by solving the following objective function: Min
gˆ 11 ,gˆ 12 ,..., gˆ 1m1
¼
E1 (gˆ 11 , gˆ 12 , . . . , gˆ 1m1 )
N 1X (y1 (k) y^ 1 (k; gˆ 11 , gˆ 12 , . . . ,gˆ 1m1 ))2 N k¼1
(4)
where y^ 1 (k) is the first predicted output of generalized Hammerstein model: k y^ 1 (k) ¼ aˆ k11 y1 (k 1) þ aˆ k12 y2 (k 1) þ bˆ 1 vˆ 1 (k nd1 1)
v^ 1 (k) ¼ gˆ 11 u1 (k) þ gˆ 12 u21 (k) þ þ gˆ 1m1 um1 1 (k) (5) k where aˆ k11 , aˆ k12 , bˆ 1 are the known linear model parameters and gˆ 1i (i ¼ 1 m1 ) are the nonlinear parameters to be determined. By differentiating the objective function E1 with respect to gˆ 1i obtains N @E1 2X ¼ y1 (k) y^ 1 (k; gˆ 11 , gˆ 12 ,..., gˆ 1m1 ) bk11 N k¼1 @gˆ 11
(6)
N @E1 2X ¼ y1 (k) y^ 1 (k; gˆ 11 , gˆ 12 ,..., gˆ 1m1 ) bk12 N k¼1 @gˆ 12
(7)
.. . N @E1 2X ¼ y1 (k) y^ 1 (k; gˆ 11 , gˆ 12 ,..., gˆ 1m1 ) bk1m1 (8) N k¼1 @gˆ 1m1 k where bk1i ¼ bˆ 1 ui1 (k nd1 1), i ¼ 1 m1 , k ¼ 1 N:
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MODELLING AND CONTROL OF MULTIVARIABLE PROCESSES By setting equations (6) –(8) to zero, the nonlinear parameters are solved by
gˆ 11 , gˆ 12 , . . . ,gˆ 1m1
T
T ¼ A1 1 c11 , c12 , . . . , c1m1
By setting equations (14)–(16) to zero, the nonlinear parameters are solved by
(9)
where
447
gˆ 21 , gˆ 22 , . . . ,gˆ 2m2
T
T ¼ A1 2 c21 , c22 , . . . , c2m2
(17)
where 2
N N P P bk11 bk11 bk12 bk11 6 6 k¼1 k¼1 6 N N 6 P k P 6 b11 bk12 bk12 bk12 6 A1 ¼ 6 k¼1 k¼1 6 .. .. 6 6 . . 6 N N P 4P k b11 bk1m1 bk12 bk1m1 k¼1
k¼1
3
N P
2
bk1m1 bk11 7 7 k¼1 7 N 7 P k k 7 b1m1 b12 7 k¼1 7 7 .. 7 7 .. . . 7 N P k 5 .. . b1m1 bk1m1
N P
bk21 bk21
N P
bk22 bk21
6 6 k¼1 k¼1 6 N N 6 P k P 6 b21 bk22 bk22 bk22 6 A2 ¼ 6 k¼1 k¼1 6 .. .. 6 6 . . 6 N N P 4P k k k b21 b2m2 b22 bk2m2
k¼1
k¼1
k¼1
N P
k¼1
(10) and
3
bk2m2 bk21 7 7 k¼1 7 N 7 P bk2m2 bk22 7 7 k¼1 7 7 .. 7 7 ... . 7 N P 5 k k ... b2m2 b2m2 (18)
and c1i ¼
N X
y1 (k) aˆ k11 y1 (k 1) aˆ k12 y2 (k 1) bk1i ,
c2j ¼
k¼1
N X y2 (k) aˆ k21 y1 (k 1) aˆ k22 y2 (k 1) bk2j , k¼1
i ¼ 1 m1
j ¼ 1 m2 (11)
Similarly, when the parameters of linear dynamics for y2(k) in equation (1) are available, parameters of the static nonlinear part in equation (3) are obtained by solving the following objective function: Min
gˆ 21 ,gˆ 22 ,..., gˆ 2m2
¼
E2 (gˆ 21 , gˆ 22 , .. . , gˆ 2m2 )
N 1X (y2 (k) y^ 2 (k; gˆ 21 , gˆ 22 , . .. ,gˆ 2m2 ))2 N k¼1
(12)
where y^ 2 (k) is the second predicted output of generalized Hammerstein model:
(19) On the other hand, when parameters of the static nonlinear parts, gˆ 1i (i ¼ 1 m1 ) and gˆ 2j ( j ¼ 1 m2 ), are known, both intermediate variables v1(k) and v2(k) can be calculated from equations (2) and (3). Subsequently, two reference databases, {(y1 (k), x1 (k))}k¼1N and {(y2 (k),x2 (k))}k¼1N where xi (k) ¼ { y1 (k), y2 (k),vi (k)} (i ¼ 1, 2), are constructed for the JITL algorithms in order to calculate the parameters of 2 N local models corresponding to N query data for predicting y1 and y2. The JITL algorithm adopted in this paper is briefly reviewed in the Appendix. The following summarizes the proposed off-line iterative identification procedure for a 2 2 generalized Hammerstein model:
k
y^ 2 (k) ¼ aˆ k21 y2 (k 1) þ aˆ k22 y2 (k 1) þ bˆ 2 vˆ 2 (k nd2 1) v^ 2 (k) ¼ gˆ 21 u2 (k) þ gˆ 22 u22 (k) þ þ gˆ 2m2 um2 2 (k) (13) k where aˆ k21 , aˆ k22 , bˆ 2 are the known linear model parameters and gˆ 2j ( j ¼ 1 m2 ) are the nonlinear parameters to be determined. By differentiating the objective function E2 with respect to gˆ 2j obtains N @E2 2X ¼ y2 (k) y^ 2 (k; gˆ 21 , gˆ 22 ,..., gˆ 2m2 ) bk21 (14) @gˆ 21 N k¼1 N @E2 2X ¼ y2 (k) y^ 2 (k; gˆ 21 , gˆ 22 ,..., gˆ 2m2 ) bk22 (15) N k¼1 @gˆ 22
.. . N @E2 2X ¼ y2 (k) y^ 2 (k; gˆ 21 , gˆ 22 ,..., gˆ 2m2 ) bk2m2 N k¼1 @gˆ 2m2
(16) k where bk2j ¼ bˆ 2 uj2 (k nd2 1), j ¼ 1 m2 , k ¼ 1 N:
(1) Given the data set {y1 (k), y2 (k), u1 (k), u2 (k)}k¼1n obtained from the off-line identification experiment, the parameters of static nonlinear parts are initialized as zero except that gˆ 11 ¼ gˆ 21 ¼ 1. (2) Calculate v1(k) and v2(k) from equations (2) and (3) and construct the reference databases f(y1 (k), x1 (k))gk¼1N and f(y2 (k), x2 (k))gk¼1N for the JITL algorithms, by which the parameters of a set of linear k k models, aˆ k11 , aˆ k12 , aˆ k21 , aˆ k22 , bˆ 1 , bˆ 2 are obtained. (3) The parameters of static nonlinear parts are calculated by using equations (9) and (17) and the linear models obtained in step 2. (4) When the convergence criterion is met, stop; otherwise, go to step 2 by using the updated parameters gˆ 1i (i ¼ 1 m1 ) and gˆ 2j (j ¼ 1 m2 ) obtained in step 3. It is worth noting that only the parameters of static nonlinear functions obtained by the aforementioned iterative identification procedure are fixed as part of the model parameters of generalized Hammerstein model, while those of the linear dynamics are all discarded. When the generalized Hammerstein model is used for on-line process modeling and model-based controller design purposes, the parameters of linear dynamics are calculated by using the JITL algorithm
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and current process data at each sampling instant. Again, parameters for linear dynamics thus identified are discarded and the same computation procedure repeats at the next sampling instant to obtain the most up-to-date model parameters for linear dynamics. The following procedure summarizes how to obtain the predicted outputs of generalized Hammerstein model during its on-line application: (1) Given the identical database { y1 (k), y2 (k), u1 (k), u2 (k)}k¼1N previously used in the off-line identification procedure and the parameters for static nonlinear function obtained in the aforementioned iterative identification procedure. (2) Compute v1 (k) and v2 (k) from equations (2) and (3) and construct the reference databases {(y1 (k), x1 (k))} k¼1N and {(y2 (k), x2 (k))}k¼1N for the JITL algorithms. (3) Given the on-line process data {yp,1 ( j), yp,2 ( j), up,1 ( j), up,2 ( j)} at the jth sampling instant, compute vp,1 (j) and vp,2 (j) from equations (2) and (3) and subsequently calculate the predicted outputs of generalized Hammerstein model by using { yp,1 ( j), yp,2 ( j), vp,1 ( j)} and {yp,1 ( j), yp,2 ( j), vp,2 ( j)} as the query data for the JITL algorithms, respectively.
ADAPTIVE DECENTRALIZED PID CONTROLLER DESIGN In the proposed controller design, the reciprocals of nonlinearities, NL121 and NL21 2 , are used to remove the effect of static nonlinear part of the process as depicted in Figure 3, where two JITL algorithms provide necessary information for on-line tuning the PID parameters. The control laws of two PID controllers are given by vi (k) ¼ vi (k 1) þ Dvi (k) Dvi (k) ¼
k wi,1 (k)ei (k)
þ
(20)
k wi,2 (k)Dei (k)
þ
k wi,3 (k)(ei (k)
2ei (k 1) þ ei (k 2))
(21)
for i ¼ 1, 2: ei (k) is the error between ith process output and its set-point at the kth sampling instant, and k k Dei (k) ¼ ei (k) ei (k 1). The PID parameters wi,1 , wi,2 and k wi,3 are tuned online by the updating formula to be derived shortly. To this end, the following objective function is used to update PID parameters: 2
Min J ¼ (r1 (k þ 1) y^ 1 (k þ 1))
(22)
þ (r2 (k þ 1) y^ 2 (k þ 1))2
where r1 (k þ 1) and r2 (k þ 1) are the set-points, y^ 1 (k þ 1) and y^ 2 (k þ 1) are the predicted outputs of generalized Hammerstein model. Since controller parameters are constrained to be positive or negative, the following mapping function is employed. ( wi,jk
¼
(k)
ezi,j , (k)
ezi,j ,
if
wi,jk 0
if
wi,jk 0
; i ¼ 1, 2, j ¼ 1 3
Figure 3. Decentralized adaptive PID control system for 2 2 generalized Hammerstein processes.
By using the steepest decent method, the updating equations for the mapping parameter vectors z1 (k) ¼ ½z1,1 (k) z1,2 (k) z1,3 (k)T and z2 (k) ¼ ½z2,1 (k) z2,2 (k) z2,3 (k)T are derived as follows (Chen, 2001):
zi (k þ 1) ¼ zi (k) þ 2hi (ri (k þ 1) zi (k) y^ i (k þ 1)) 1 þ ziT (k)zi (k)
(24)
for, i ¼ 1, 2 and the parameters h1 and h2 are the learning rates. The vectors z1(k) and z2(k) are defined as @y^ i (k þ 1) ¼ ½zi,1 (k) zi,2 (k) zi,3 (k)T , i ¼ 1, 2 (25) @zi (k) k @y^ i (k þ 1)@vi (k)@wi,1 k ei (k)wi,1 (26) ¼ bkþ1 zi,1 (k) ¼ i k @vi (k)@wi,1 @zi,1 (k) zi (k) ¼
k zi,2 (k) ¼ bikþ1 Dei (k)wi,2
(27)
k zi,3 (k) ¼ bikþ1 (ei (k) 2ei (k 1) þ ei (k 2))wi,3
(28)
for i ¼ 1, 2 the parameters bkþ1 and bkþ1 are obtained from 2 1 the JITL algorithms dedicated to predict y1 and y2, respectively. The implementation of the proposed adaptive decentralized PID controller is summarized as follows: (1) Initialize w k1,j and w k2,j ( j ¼ 1 3) and learning rate parameters h1 and h2. (2) Given the current errors e1(k) and e2(k), compute v1(k) and v2(k) from equation (20) and calculate manipulated variables u1 (k) and u2 (k). (3) Update two linear models by using the current process data and JITL algorithms and subsequently adjust z1 (k) and z2 (k) according to equation (24). (4) Obtain PID parameters for the next sampling instant using equation (23) and go to step 2.
(23)
EXAMPLE
where z1,j (k) and z2,j (k) are real number and adjusted on-line according to their respective updating equations and information provided by the JITL algorithms. Subsequently, PID parameters can be obtained by equation (23).
A bench-scale pH neutralization system studied by Henson and Seborg (1994) is shown in Figure 4, where NaOH with concentration 0.003 M is used as the base stream (q1), NaHNO3 with concentration 0.03 M as the buffer stream
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Figure 4. The pH neutralization process.
(q2), and HNO3 with concentration 0.003 M as the acid stream (q3). The model and model parameters used in the present simulation study are identical to those given in Henson and Seborg (1994) and thus are omitted here due to the space limitation. This process was previously modeled as a 2 2 Hammerstein model by Lakshminarayanan et al. (1995). Two outputs of this process are h1 and pH (denoted by y1 and y2 in the following development), while the process inputs are q1 and q3 (denoted by u1 and u2). The operating space considered is y1 [ ½10:5 16:5 and y2 [ ½4:5 9:5. To proceed with the proposed identification procedure, random input signals and corresponding process outputs as shown in Figure 5 are collected to construct the databases for the JITL algorithms. By using kmin ¼ 25, kmax ¼ 90 and V ¼ 0.98 for the JITL, the static nonlinear part of generalized Hammerstein model is identified as v1 (k) ¼ 2:8745u1 (k)
0:2689u21 (k)
þ
v2 (k) ¼ 1:5495u2 (k) þ 6:2594u22 (k) þ 13:08u32 (k) For comparison purpose, the following Hammerstein model is obtained by using the Narendra-Gallman method and the identical input and output data given in Figure 5. "
y1 (k)
#
"
0:9608
0:0395
#"
y1 (k 1)
#
¼ y2 (k)
y2 (k 1) 0:0607 0:9048 " #" # 0:0699 0 v1 (k 1) þ 0
0:1748
Figure 5. Input– output data for construction of JITL databases.
0:0567u31 (k)
v2 (k 1)
v1 (k) ¼ 0:9782u1 (k) þ 0:2735u21 (k) þ 2:3471u31 (k) v2 (k) ¼ 0:9459u2 (k) 2:0381u22 (k) þ 10:869u32 (k) The predictive performances of these two models are compared in Figure 6 for the step changes of 1.5 ml s21 and 22.5 ml s21 in base flow rate (u1), respectively, and the corresponding prediction errors calculated by meanabsolute-error (MAE) are summarized in Table 1. It is clear that generalized Hammerstein model has superior predictive performance than the conventional Hammerstein model. Likewise, generalized Hammerstein model gives a marked improvement in predicting the open-loop response
corresponding to +3 ml s21 step changes in acid flow rate (u2) as illustrated in Figure 7 and Table 1. Next, to illustrate the adaptive capability of the proposed model, a perturbed process with model parameters Cv4, Wa2 and Wb2 subject to 5%, 20% and 30% perturbations from their respective nominal values is considered. In this case, the database of JITL is made adaptive by adding those new process data that are not adequately represented by the present database. To this end, the criterion employed to update the database at each sampling instant is to check whether the difference between the predicted output by JITL and actual process output is within of 3% the process output. Figure 8 shows that generalized Hammerstein model predicts the open-loop responses reasonably accurate when the perturbed process is subject to the aforementioned step changes in acid flow rate (u2). Note that the symbol ‘ þ ’ in the figure denotes the sampling instants at which database is updated. To proceed with the proposed decentralized controller design, the pairing between the process outputs and inputs needs to be considered. After performing the RGA analysis (Bristol, 1966), the tank level (h1) is paired by base flow rate and the effluent pH is paired by acid flow rate. To design the proposed adaptive decentralized PID controller, the initial controller parameters for the first control loop are
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Figure 7. Open-loop responses of level (top) and pH (bottom) for +3 ml s21 changes in q3. Solid line: process; dotted line: generalized Hammerstein model; dash-dot line: Hammerstein model.
Figure 6. Open-loop responses of level (top) and pH (bottom) for 1.5 ml s21 and –2.5 ml s21 changes in q1. Solid line: process; dotted line: generalized Hammerstein model; dash-dot line: Hammerstein model.
Table 1. Prediction error for open-loop responses in Figures 6 and 7. Hammerstein model y1 þ1.5 ml s21 change in u1 22.5 ml s21 change in u1 þ3 ml s21 change in u2 23 ml s21 change in u2
y2
Generalized Hammerstein model y1
y2
1.38 1021 1.50 1021 1.12 1023 5.76 1023 3.89 1021 1.99 1022 1.74 1023 1.91 1023 9.19 1022 1.57 1022 3.51 1023 4.37 1024 6.52 1022 1.28 1021 3.21 1023 9.75 1023
w 01,1 ¼ 0.6, w 01,2 ¼ 4.5 and w 01,3 ¼ 2, and for the second control loop w 02,1 ¼ 20.09, w 02,2 ¼ 21 and w 02,3 ¼ 20.5, respectively, whereas the respective learning rates are fixed as h1 ¼ 1.6 and h2 ¼ 1.85. For comparison purpose, a decentralized PID controller is also design based on the 2 2 Hammerstein model previously identified. The controller
Figure 8. Open-loop responses of level (top) and pH (bottom) for + 3ml s21 step changes in q3 (with modelling errors). Solid line: process; dotted line: generalized Hammerstein model; dash-dot line: Hammerstein model ‘ þ ’: database update.
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Figure 9. Closed-loop responses for set-point changes in y1: (a) 14 to 15, (b) 14 to 13. Solid line: proposed model based PID design; dotted line: Hammerstein model based PID design.
parameters for the first control loop are w1,1 ¼ 0.15, w1,2 ¼ 1.1 and w1,3 ¼ 0.5, and for the second control loop w2,1 ¼ 20.08, w2,2 ¼ 20.8 and w2,3 ¼ 20.1. To evaluate the servo performance of two controllers, +1 set-point changes in y1 and +2 set-point changes in y2 are considered. As can be seen from Figures 9 and 10 and Table 2, the proposed adaptive decentralized PID controller has superior control performance than that achieved by its counterpart designed based on the Hammerstein model. In addition, their load performances for the step disturbance
of 20.35 in buffer flow rate are compared in Figure 11 and Table 2. It is evident that the proposed controller outperforms its counterpart.
CONCLUSION In this paper, a generalized Hammerstein model capable of modelling the Hammerstein-like multivariable processes whose linear dynamics vary over the operating space is proposed. An iterative identification procedure for generalized
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Figure 10. Closed-loop responses for set-point changes in y2: (a) 7 to 9, (b) 7 to 5. Solid line: proposed model based PID design; dotted line: Hammerstein model based PID design.
Table 2. Summary of MAEs for closed-loop responses in Figures 9– 11. Hammerstein model based PID design y2
y1 þ1 set-point change in y1 21 set-point change in y1 þ2 set-point change in y2 22 set-point change in y2 20.35 step change in buffer flow rate
Proposed model based PID design
22
8.94 10 9.55 1022 6.83 1022 1.81 1021 3.50 1022
y1 22
3.85 10 4.42 1022 1.14 1021 1.93 1021 1.73 1022
y2 22
5.56 10 5.72 1022 7.13 1022 1.19 1021 2.37 1022
4.39 1022 3.66 1022 9.15 1022 1.63 1021 1.54 1022
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Figure 11. Closed-loop responses for step disturbance in buffer stream. Solid line: proposed model based PID design; dotted line: Hammerstein model based PID design.
Hammerstein model and an adaptive decentralized PID control strategy based on the proposed model are developed. The controller parameters are adjusted on-line by using the gradient descent learning algorithm and information provided by the JITL. Simulation results show that the proposed methods have better prediction accuracy and control performance than those achieved by the Hammerstein model and its associated adaptive decentralized PID design, respectively.
REFERENCES Abonyi, J., Babuska, R., Botto, M.A., Szeifert, F. and Nagy, L., 2000, Identification and control of nonlinear systems using fuzzy hammerstein models, Ind Eng Chem Res, 39: 4302– 4314. Al-Duwaish, H. and Karim, M.N., 1997, A new method for the identification of hammerstein model, Automatica, 33: 1871–1875. Atkeson, C.G., Moore, A.W. and Schaal, S., 1997, Locally weighted learning, Artificial Intelligence Review, 11: 11– 73. Bontempi, G., Birattari, M. and Bersini H., 1999, Lazy learning for local modelling and control design, Int J Control, 72: 643– 658. Bristol, E.H., 1966, On a new measure of interaction for multivariable process control, IEEE Trans on Automatic Control, 11: 133 –134. Chen, C.T., 2001, Direct adaptive control of chemical process systems, Ind Eng Chem Res, 40: 4121–4140. Cheng, C. and Chiu, M.S., 2004, A new data-based methodology for nonlinear process modeling, Chem Eng Sci, 59: 2801–2810. Eskinat, E., Johnson, S.H. and Luyben, W.L., 1991, Use of hammerstein models in identification of nonlinear systems, AIChE J, 37: 255– 268. Fruzzetti, K.P., Palazoglu, A. and McDonald, K.A., 1997, Nonlinear model predictive control using hammerstein models, J Proc Control, 7: 31–41. Henson, M.A. and Seborg, D.E., 1994, Adaptive nonlinear control of a pH neutralization process, IEEE Trans on Control Systems Technology, 2: 169– 182. Jia, L., Chiu, M.S. and Ge, S.S., 2005, Iterative identification of neuro-fuzzy based hammerstein model with global convergence, Ind Eng Chem Res, 44: 1823–1831. Lakshminarayanan, S., Shah, S.L. and Nandakumar, K., 1995, Identification of hammerstein models using multivariate statistical tools, Chem Eng Sci, 50: 3599– 3613.
Lakshminarayanan, S., Shah, S.L. and Nandakumar, K., 1997, Modeling and control of multivariable process: the dynamic projection to latent structures approach, AIChE J, 43: 2307–2323. Lee, M.W., Huang, H.P. and Jeng, J.C., 2004, Identification and controller design for nonlinear processes using relay feedback, J Chem Eng Japan, 37: 1194– 1206. Ling, W.M. and Rivera, D.E, 1998, Control relevant model reduction of volterra series models, J Proc Control, 8: 79–88. Narendra, K.S. and Gallman, P.G., 1966, An iterative method for the identification of the nonlinear systems using the hammerstein model, IEEE Trans. on Automatic Control, 12: 546– 550. Pearson, R.K. and Pottmann, M., 2000, Gray-box identification of block-oriented nonlinear models, J Proc Control, 10: 301–315. Pottmann, M., Unbehauen, H. and Seborg, D.E., 1993, Application of general multimodel approach for identification of highly nonlinear processes, Int J Control, 57: 97– 120. Rollins, D.K, Bhanddari, N., Bassily, A.M., Colver, G.M. and Chin, S.T., 2003, A continuous-time nonlinear dynamic predictive modeling method for hammerstein processes, Ind Eng Chem Res, 42: 860– 872. Su, H.T. and McAvoy, T.J., 1993, Integration of multilayer percepton networks and linear dynamic models: a hammerstein modeling approach, Ind Eng Chem Res, 26: 1927– 1936. Sung, S.W., 2002, System identification method for hammerstein processes, Ind Eng Chem Res, 41: 4295–4302. The manuscript was received 30 October 2006 and accepted for publication 18 December 2006.
APPENDIX: JITL ALGORITHM Suppose that a database consisting of N process data (yi, x i)i¼1N, yi [ R, x i [ R n, is collected. Given a specific query data xq [ R n, the objective of JITL is to predict the model output y^ q ¼ f (xq ) according to the known database (yi, x i)i¼1N. To do so, the relevant data are selected from the database first by using the following similarity measure si, which was recently proposed to improve the predictive accuracy of JITL technique (Cheng and Chiu, 2004). pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 si ¼ V ejjxq xi jj þ(1V)cos(ui ), if cos(ui ) 0
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(A1)
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where V is a weight parameter and is constrained between 0 and 1, and ui is the angle between Dx q and Dx i, where Dx i ¼ x i 2 x i21 and Dx q ¼ x q 2 x q21. The value of si is bounded between 0 and 1. To apply JITL in the modeling of dynamic systems, a firstorder or second-order ARX model is employed. Firstly, all si are computed by equation (A1) and for each l [ ½kmin kmax , where kmin and kmax are prespecified parameters as the minimum and maximum number of relevant data, the relevant data set (y l, Fl), where yl [ Rll and Fl [ Rln , is constructed by selecting the l most relevant data (yi , xi )
corresponding to the largest si to the lth largest s i. Next, the leave-one-out cross validation test is conducted and the validation error is calculated. Upon the completion of the above iteration, the optimal l, l , is determined by that giving the smallest validation error. Subsequently, the predicted output for query data is calculated as xTq (PlT Pl )1 PlT Wl yl , where Pl ¼ Wl Fl and Wl is a diagonal matrix with entries being the first l largest si , provided that this optimal model satisfies the stability constraint. Otherwise, one needs to resort to an optimization procedure to recompute the optimal model.
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