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PROCESS DESIGN AND CONTROL A Nonlinear Gain Scheduling Control Strategy Based on Neuro-Fuzzy Networks Jie Zhang* Centre for Process Analytics and Control Technology, Department of Chemical & Process Engineering, University of Newcastle, Newcastle upon Tyne NE1 7RU, U.K.
A nonlinear gain scheduling control strategy based on neuro-fuzzy network models is proposed. In neuro-fuzzy-network-based modeling, the process operation is partitioned into several fuzzy operating regions, and within each region, a local linear model is used to model the process. The global model output is obtained through center-of-gravity defuzzification. Process knowledge is used to initially set up the network structure, and process input-output data are used to train the network. Based on a neuro-fuzzy network model, a nonlinear controller can be developed by combining several local linear controllers that are tuned on the basis of the local model parameters. This strategy represents a nonlinear gain scheduled controller. The techniques have been successfully applied to the modeling and control of pH dynamics in a simulated continuous stirred tank reactor. 1. Introduction The most widely used controllers in industrial processes are proportional-integral-derivative (PID) controllers because of their simple structure and robust performance over a wide range of process operating conditions. Indeed, a well-tuned PID controller can give satisfactory control performance provided that the controlled process is not significantly nonlinear. PID controllers are best understood by process operators, and well-established tuning rules are available.1-3 When the controlled process is highly nonlinear, however, a fixed-gain PID controller usually cannot give satisfactory control performance as the controller gain has to be detuned to ensure stability. One way to improve the control performance of a PID controller on a highly nonlinear process is to vary the controller parameters according to the process operating conditions. This is known as gain scheduling control.4 In conventional gain scheduling control, the controller parameters are scheduled according to some monitored process conditions. Such control is simpler to implement than automatic tuning or adaptation. Its drawback is that the parameter changes can be abrupt across the operating region boundaries, which can result in unsatisfactory or even unstable control performance across the region boundaries. To overcome this difficulty, fuzzy gain scheduling, where the controller parameter variations can be achieved in a smooth fashion, was proposed.5,6 The experience of process operators can be represented in rules, and fuzzy inference can be used to interpolate the controller parameters across the operating region boundaries. It has been shown by Ling and Edgar5 that, if appropriate fuzzy gain scheduling * E-mail:
[email protected]. Tel.: +44-191-2227240. Fax: +44-191-2225292.
rules and membership functions can be obtained, fuzzy gain scheduling controllers can outperform conventional gain scheduling controllers. The determination of appropriate gain scheduling rules and membership functions is, however, a difficult and time-consuming task. To overcome this difficulty, Tan et al.7 proposed neuralnetwork- and neuro-fuzzy-network-based gain scheduling control, where appropriate controller design, interpolation rules, and membership functions can be learned from a set of controller parameters tuned under different operating conditions. Because neural networks usually lack extrapolation capability, care must be taken when generating training data that should cover a wide range of process operating conditions. In this paper, a nonlinear gain scheduled control strategy based on a neuro-fuzzy network model of the controlled process and the heterogeneous control framework suggested by Kuipers and Astrom8 is proposed. In neuro-fuzzy network modeling,9,10 the process operation is partitioned into several operating regions. Within each operating region, a reduced-order linear model is used to describe the local dynamics of the process. Fuzzy sets are used to represent the partition of operating regions, and the transition from one region to another region is smooth. The global model output is obtained through center-of-gravity (COG) defuzzification, which is essentially the weighted average of local model outputs. Process knowledge is used to initially partition the process operation into several local operating regions and to set up the initial membership functions, and process data are used to train the neuro-fuzzy network. During training, membership functions are refined, and local models are identified. On the basis of the identified local linear models, a local linear PI or PID controller is developed and tuned for each local operating region. The global controller output is then obtained by com-
10.1021/ie990866h CCC: $20.00 © 2001 American Chemical Society Published on Web 06/13/2001
Ind. Eng. Chem. Res., Vol. 40, No. 14, 2001 3165
bining the local controller outputs in the framework of COG defuzzification. An advantage of this nonlinear gain scheduling control strategy is that the complex task of tuning a nonlinear controller is decomposed into the tuning of several local linear PID controllers using the identified local model parameters. Furthermore, it eliminates the difficult task of finding appropriate gain scheduling rules and membership functions, which have been taken into account in the procedure of local controller tuning and local model identification. 2. Nonlinear Gain Scheduling Control Based on Neuro-Fuzzy Network Models 2.1. Neuro-Fuzzy Network Modeling. The fuzzy modeling approach described in this section was originally proposed by Takagi and Sugeno.11 The global operation of a nonlinear process is divided into several local operating regions, and within each local region, Ri, a reduced-order linear model is used to represent the process behavior. Fuzzy sets are utilized to define the process operating conditions such that the fuzzy dynamic model of a nonlinear process can be described in the following way
Ri: IF operating condition i no
THEN yˆ i(t) )
∑ j)1
ni
aijy(t - j) +
biju(t - j) ∑ j)1
i ) 1, 2, ..., nr (1)
The final model output is obtained through COG defuzzification as nr
∑ i)1
yˆ (t) ) [
nr
µi yˆ i(t)]/(
µi) ∑ i)1
(2)
In the above model, y is the process output; yˆ is the model prediction; u is the process input; yˆ i is the prediction of process output in the ith operating region; nr is the number of fuzzy operating regions; ni and no are the time lags in the input and output, respectively; µi is the membership function for the ith model; aij and bij are the local model parameters; and t represents discrete time. The operating regions of a process can usually be defined by one or several process variables. A number of fuzzy sets, such as “low”, “medium”, and “high”, can be defined for each of these process variables. An operating condition can then be defined through logical combinations of the fuzzy sets of those variables. Suppose that x and y are the process variables used to define the process operating regions and that they are assigned the fuzzy sets low, medium, and high. The ith operating region can be defined, for example, as
x is high AND y is medium The membership function for this operating region can be calculated in one approach as
µi ) min[µh(x), µm(y)]
(3)
or in another approach as
µi ) µh(x) µm(y)
(4)
In the above equations, µi is the membership function
Figure 1. Neuro-fuzzy network.
of the ith operating region, µh(x) is the membership function of x being high, and µm(y) is the membership function of y being medium. This fuzzy model can be represented by a neuro-fuzzy network. Fuzzy reasoning is capable of handling uncertain and imprecise information, whereas a neural network is capable of learning from examples. Neurofuzzy networks intend to combine the advantages of both fuzzy reasoning and neural networks. Neuro-fuzzy networks have been studied by several researchers, and several different types of neuro-fuzzy networks have been proposed.9,10,12 Figure 1 shows a neuro-fuzzy network that contains four layers: a fuzzification layer, a rule layer, a function layer, and a defuzzification layer. Inputs to the fuzzification layer are the process variables that are used to define fuzzy operating regions. Each of these variables is transformed into several fuzzy sets in the fuzzification layer with the membership function being given by the neuron output. Three types of neuron activation functions are used. They are the sigmoidal function, the Gaussian function, and the complement sigmoidal function. Weights in this layer determine the membership functions of the corresponding fuzzy sets. Through network training, appropriately shaped membership functions can be obtained. In some industrial processes, the partition of local operating regions could be affected by unmeasured disturbances. In these cases, the effect of unmeasured disturbances could be reflected by some measured process variables that can be used as the inputs to the fuzzification layer for defining local operating regions. It is also possible to use principal components, which are linear combinations of measured process variables, as fuzzification layer inputs. Each neuron in the rule layer corresponds to a fuzzy operating region of the process being modeled. Its inputs are the fuzzy sets that determine the corresponding operating region, and its output is the membership function of that region. Neurons in the rule layer implement the fuzzy intersection defined by eq 4, and the construction of the rule layer is guided by some process knowledge, such as how each operating region is characterized. Each neuron in the function layer corresponds to a particular local linear model. Its output is a summation of its inputs, which are weighted process variables. A bias is included to represent a constant term in a local model. Weights in the function layer are the local linear model parameters. The defuzzification layer performs the COG defuzzification defined by eq 2 and gives the final network output.
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Neuro-fuzzy networks can be trained using any of a number of training methods, such as the back-propagation method13 and Levenberg-Marquardt optimization.14 In this study, the neuro-fuzzy networks are trained using the Levenberg-Marquardt algorithm with regularization. The training objective function can be defined as
J)
N
1
∑[yˆ (t) - y(t)] N t)1
2
+ λ||W||
2
(5)
where N is the number of training data points, yˆ is the network prediction, y is the target value, W is a vector of network weights, and λ is a regularization parameter used to penalize excessively large network weights that do not contribute significantly to the reduction of model errors. The appropriate value of λ is obtained through a cross-validation procedure. In the Levenberg-Marquardt method, the network weight adjustments are calculated from
∆W(k+1) ) -η
[
1
N
∑
∂yˆ (t)
( ) ] ∂yˆ (t)
-1
T
+ δI
N i)1 ∂W(k) ∂W(k)
[
t
ui(t) ) Ki e(t) +
e(j) ∑ j)1
+
TIi
TDi ∆e(t) Ts
(7)
]
(8)
where ui is the control action from the ith local controller, e is the control error, Ki is the controller gain, TIi is the integral time constant, TDi is the derivative time constant, and Ts is the sampling time. The local controllers can be tuned using a number of methods based on the local model parameters, such as the Ziegler-Nichols rule,1 the Cohen and Coon rule,2 and the RZN (refined Ziegler-Nichols) tuning rule.3 Let µi and ui be the membership function and output, respectively, of the local controller for the ith operating region. Then, the global controller output, u(t), can be obtained by combining the local controller outputs through COG defuzzification as nr
u(t) ) [
nr
µiui(t)]/(∑µi) ∑ i)1 i)1
[
µˆ iKi e(t) + ∑ i)1 t
Ts
)K ˜ e(t) +
e(j) ∑ j)1 T ˆI
+
T ˜D Ts
]
TIi
∆e(t)
TDi ∆e(t) Ts
]
(10)
nr
K ˜ )
µˆ iKi ∑ i)1
(11)
( )
(12)
µˆ iKiTDi)/K ˜ ∑ i)1
(13)
nr
˜/ T ˆI ) K
µˆ iKi
∑ i)1 T
Ii
nr
It can be seen that the neuro-fuzzy-network-based PID controller is equivalent to a gain-scheduled PID controller where the controller parameters are scheduled according to eqs 11-13. Equation 11 can be implemented by simple fuzzy gain scheduling rules. However, the implementation of eqs 12 and 13 requires complicated gain scheduling rules. To schedule the integral time constant, the scheduled controller gain needs to be calculated first and then divided by the interpolated ratio Ki/TIi. To schedule the derivative time constant, the product of Ki and TDi is interpolated first and then divided by the scheduled controller gain. Thus, the neuro-fuzzy PID controller represents a nonlinear fuzzy gain scheduling PID controller. It can be seen that the normalized local operating region membership functions, µˆ i, are a crucial element in the above scheduling rules. Although the above scheduling rules are essentially the same as those derived by Ling and Edgar,5 those authors did not discuss how to obtain the appropriate local operating region membership functions, which can be obtained through neuro-fuzzy-network-based modeling, as demonstrated in this paper. 3. Applications to a pH Reactor 3.1. The pH Reactor. The proposed technique has been applied to pH control in a neutralization process taken from ref 15. The neutralization process takes place in a continuous stirred tank reactor (CSTR). There are two input streams to the CSTR. One is acetic acid of concentration C1 at flow rate F1, and the other is sodium hydroxide of concentration C2 at flow rate F2. The mathematical equations of the CSTR can be described by assuming that the tank level is perfectly controlled15 as
V
dξ ) F1C1 - (F1 + F2)ξ dt
(14)
V
dζ ) F2C2 - (F1 + F2)ζ dt
(15)
(9)
Let µˆ i be the normalized membership function for the nr µj. Then, it ith operating region such that µˆ i ) µi/∑j)1 follows from eqs 8 and 9 that
+
where
∂J
where W(k) and ∆W(k) are the vectors of weights and weight adaptations at training step k, respectively; η is the learning rate; and δ is a parameter for controlling the searching step size. Training is terminated by a cross-validation-based “early stopping” criterion.10 2.2. Neuro-Fuzzy-Network-Based Gain Scheduling Control. Because the neuro-fuzzy network model presented in this paper explicitly gives the linear local models, a heterogeneous controller can be developed on the basis of a neuro-fuzzy network model. For each local operating region, a PID controller of the following form can be developed
Ts
∑ i)1
nr
µˆ iui(t) )
e(j) ∑ j)1
T ˜D ) (
∂W(k) (6)
W(k+1) ) W(k) + ∆W(k+1)
nr
u(t) )
t
Ts
[H+]3 + (Ka + ζ)[H+]2 + {Ka(ζ - ξ) - Kw}[H+] KwKa ) 0 (16)
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pH ) -log10 [H+]
(17)
where
ξ ) [HAC] + [AC-]
(18)
ζ ) [Na+]
(19)
Table 1 gives the meaning and the initial setting of each variable. It is well-known that pH dynamics are highly nonlinear, and the steady-state relationship between the acid flow rate and the pH in the reactor for different values of C2 is plotted in Figure 2. It can be seen that the process gain is very high in the medium pH region while being quite low in both the low and high pH regions. 3.2. Neuro-Fuzzy Network Models. A neuro-fuzzy network was used to model the pH dynamics in the reactor from the simulated process input-output data. In the simulation, the sampling time was set to 0.1 min, and a 0.1-min measurement time lag was used. The simulated pH values were corrupted with measurement noise with the distribution N(0, 0.1). To generate training, testing, and validation data, multilevel random perturbations were added to the flow rate of acetic acid while other inputs to the reactor were kept constant. Three sets of data were generated. One set was used as training data, another set was used as testing data, and the remaining set was used as unseen validation data. Based on the process characteristics, the nonlinear operating region was divided into three local regions: “pH low”, “pH medium”, and “pH high”. The fuzzification layer weights were initialized on the basis of the titration curve. Weights for the function layer were initialized as random numbers in the range (-0.2, 0.2). A second-order linear model was initially used within each local region. If the identified model was not adequate, then the number of fuzzy operating regions or the local model order was increased. The LevenbergMarquardt training method, with regularization and the cross-validation-based stopping criterion, was used to train the neuro-fuzzy network. The training process took about 15 min on a Pentium III 550 PC. With the current dramatic increase in computer capacity, network training time would be less and less of an issue in industrial applications. After neuro-fuzzy network training and model validation, the identified model is
IF pH low THEN yˆ (t) ) 0.5651y(t-1) + 0.2712y(t-2) 4.92u(t-2) - 0.52u(t-3) + 1.424 IF pH medium THEN yˆ (t) ) 0.3011y(t-1) + 0.2289y(t-2) 100.39u(t-2) - 30.07u(t-3) + 14.695 IF pH high THEN yˆ (t) ) 0.667y(t-1) + 0.2463y(t-2) 0.78u(t-2) - 3.87u(t-3) + 1.323 In the above model, yˆ is the predicted pH in the reactor, y is the pH in the reactor, and u is the flow rate of acetic acid. The identified membership functions (normalized) for the three fuzzy sets pH low, pH medium, and pH high are plotted in Figure 3. It can be seen that the partition of the process operating regions generally
Table 1. Physical Parameters Used in the Simulations variable
meaning
initial setting
V F1 F2 C1 C2 Ka Kw
volume of tank flow rate of acid flow rate of base concentration of acid in F1 concentration of acid in F2 acid equilibrium constant water equilibrium constant
1L 0.081 L/min 0.512 L/min 0.32 mol/L 0.05005 mol/L 1.8 × 10-5 1.0 × 10-14
Figure 2. Titration curve.
Figure 3. Learned membership functions.
agrees with the operating regions indicated by the titration curve. Figure 3 shows that there is a small contribution of the pH high model in the pH low region. This could be due to the fact that the pH low and pH high regions have some similar characteristics, such as much lower gains than the pH medium region. Nevertheless, this contribution is small (less than 0.2) and does not affect the overall interpretation of the identified neuro-fuzzy network model. Comparing the coefficients of the local models, one can find that the local model associated with the operating region pH medium has a much higher gain than the local models for the other two operating regions. This is exactly what one would expect from the titration curve. Hence, neuro-fuzzy network models can provide certain insight into the modeled processes and can be easily interpreted. The neuro-fuzzy network model was tested on the unseen
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Ind. Eng. Chem. Res., Vol. 40, No. 14, 2001 Table 2. Parameters of Local Controllers Kc TI
pH low
pH medium
pH high
-0.0339 0.6765
-0.0005 0.1615
-0.0227 1.3375
Table 3. Parameters of the Five Local Controllers
Figure 4. Predictions on validation data.
validation data, and the results are presented in Figure 4, which shows that the neuro-fuzzy network model is very accurate. The identified model can be transformed into transfer function form so that the process dynamics can be interpreted in the frequency domain. The discrete-time transfer functions are readily obtained from the above neuro-fuzzy network model as
IF pH low THEN G ˆ (z) )
0.6765 0.4615 0.1615 0.6615 1.3375
{ {
Kc,i-1 +
-100.39z-2 - 30.07z-3 1 - 0.3011z-1 - 0.2289z-2
Kc,n
IF pH high
if y e y1
(y - yi-1)(Kc,i - Kc,i-1) if yi-1 < y e yi yi - yi-1 if y > yn (23)
if y e y1 (y - yi-1)(TI,i - TI,i-1) TI ) TI,i-1 + if yi-1 < y e yi yi - yi-1 TI,n if y > yn (24) TI,1
-2
THEN G ˆ (z) )
-3
-0.78z - 3.87z 1 - 0.667z-1 - 0.2463z-2
3.3. Nonlinear Gain Scheduling Control. A neurofuzzy-network-based heterogeneous controller was developed for the pH reactor. It consists of three PI controllers with different controller parameters that were determined on the basis of the corresponding local model parameters. In this study, the local controllers were tuned using the discrete-time-model-based tuning rule given by Smith and Corripio.16 For the second-order plus dead time discrete transfer function model
G(z) )
(b0 + b1z-1 + b2z-2)z-N-1
Smith and Corripio
Kc )
TI
-0.032 -0.0015 -0.0005 -0.0019 -0.0227
Kc,1
IF pH medium THEN G ˆ (z) )
Kc
5.7 5.9 6.5 10.5 11.8
adjustable parameter to maintain robustness. In this study, the values of q for the pH low, pH medium, and pH high regions were set to 0.8, 0.9, and 0.9, respectively. The local PI controller settings are given in Table 2. Because the process has high gain in the medium pH region, the local PI controller corresponding to this region has low gain. The local PI controllers for the low and high pH regions have high gains because the process gains in these regions are low. For the purpose of comparison, a conventional gain scheduling PI controller was also developed for this reactor. Several operating points, characterized by the measured pH values in the reactor, were chosen to cover the possible operating ranges. A PI controller was developed for each operating point. The controller gain and integral time constant were then scheduled through linear interpolation as
Kc )
-4.92z-2 - 0.52z-3 1 - 0.5651z-1 - 0.2712z-2
pH
1 - a1z-1 + a2z-2 16
(20)
suggest the tuning rules
(1 - q)(a1 + 2a2) (b0 + b1 + b2)[1 + N(1 - q)] TI ) Ts
a1 + 2a2 1 - a1 - a2
(21)
(22)
where N is the time delay and q (0 e q < 1) is an
where yi is the ith operating point; y is the scheduling variable (pH in the reactor); Kc,i and TI,i are controller gain and integral time constant, respectively, at the ith operating point; and n is the number of operating points. An attempt was made at scheduling the three controllers given in Table 2, but the resulting controller was not stable. This is because the operating regions for this process cannot be adequately represented by the simple interpolation of three operating points. To get acceptable control performance, five PI controllers had to be developed for operating points at the pH values of 5.7, 5.9, 6.5, 10.5, and 11.8. The selection of these operating points was based on the titration curve shown in Figure 2. These controllers were tuned using the above discretetime-based tuning rule and the controller parameters given in Table 3. These controller parameters were scheduled through linear interpolation to derive the gain scheduling PI controller. However, because the process has steep changes in process gain, this gain scheduling PI controller does not give satisfactory performance across the operating region boundaries, as can be seen in Figure 5, which shows the control performance under
Ind. Eng. Chem. Res., Vol. 40, No. 14, 2001 3169
Figure 5. Control performance for the conventional gain scheduling PI controller.
Figure 6. Control parameter variations.
Figure 8. Distribution of control errors.
process operations. In both Figures 5 and 7, the effect of a disturbance is simulated by changing the base concentration (C2) from 0.05 to 0.055 mol/L at 45 min and changing it back to 0.05 mol/L at 60 min. It can be seen that the responses under the neuro-fuzzy-networkbased controller are much better than those shown in Figure 5. The MSE (mean squared error) for the control from the neuro-fuzzy-network-based controller is 0.2803, whereas that from the gain scheduling PI controller is 0.6301. The observations in Figures 5 and 7 suggest that conventional gain scheduling through linear interpolation cannot handle steep changes in process characteristics unless a large number of controllers at various operating points are developed whereas the neuro-fuzzynetwork-based PI controller can. The effectiveness of the neuro-fuzzy-network-based approach is mainly due to the fact that appropriate membership functions for local operating regions are identified through network training. Figure 8 shows the distribution of control errors from the two controllers. It can be seen that the control errors from the neuro-fuzzy-network-based gain scheduling control have a narrower distribution (around zero) than those from the linear gain scheduling controller. 4. Conclusions
Figure 7. Control performance for the neuro-fuzzy-network-based controller.
setpoint changes and disturbances. Figure 6 shows the variations in the controller gain and the integral time constant, clearly demonstrating that abrupt variations exist. Figure 7 shows the performance of the neurofuzzy-network-based heterogeneous controller for setpoint changes and disturbances over a wide range of
Neuro-fuzzy-network-based process models provide a basis for building a novel type of nonlinear controller that is composed of several local linear controllers. The global controller output is obtained by combining local controller outputs on the basis of the membership functions of the local linear models. PID controllers can be used as local controllers and tuned on the basis of the identified local model parameters. In this way, the complex task of tuning a nonlinear controller can be achieved through the tuning of several local linear PID controllers. The proposed neuro-fuzzy-network-based PID controller is equivalent to a nonlinear fuzzy gain scheduled PID controller. However, it eliminates the difficult task of finding appropriate gain scheduling rules and membership functions, which have been taken into account in the procedure of local controller tuning and local model identification. Acknowledgment This work was partly supported by the U.K. EPSRC through the Grant GR/N13319.
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Literature Cited (1) Ziegler, J. G.; Nichols, N. B. Optimum Settings for Automatic Controllers. Trans. ASME 1942, 64, 759-768. (2) Cohen, G. H.; Coon, G. A. Theoretical Considerations of Retarded Control. Trans. ASME 1953, 75, 827. (3) Hang, C. C.; Astrom, K. J.; Ho, W. K. Refinements of the Ziegler-Nichols Tuning Formula. IEE Proc. D: Control Theory Appl. 1991, 138, 111-118. (4) Astrom, K. J.; Wittenmark, B. Adaptive Control, 2nd ed.; Addison-Wesley: Reading, MA, 1989. (5) Ling, C.; Edgar T. F. A New Fuzzy Gain Scheduling Algorithm for Process Control. Asia Pac. Eng. J., Part A 1993, 3, 129-141. (6) Zhao, Z. Y.; Tomizuka, M.; Isaka, S. Fuzzy Gain Scheduling of PID Controllers. IEEE Trans. Syst., Man, Cybernetics 1993, 23, 1392-1398. (7) Tan, S.; Hang, C. C.; Chai, J. S. Gain Scheduling: From Conventional to Neuro-Fuzzy. Automatica 1997, 33, 411-419. (8) Kuipers, B.; Astrom, K. The Composition and Validation of Heterogeneous Control Laws. Automatica 1994, 30, 233-249. (9) Zhang, J.; Morris, A. J. Fuzzy Neural Networks for Nonlinear Systems Modelling. IEE Proc. D: Control Theory Appl. 1995, 142, 551-561. (10) Zhang, J.; Morris, A. J. Reccurent Neuro-fuzzy Networks
for Nonlinear Process Modelling. IEEE Trans. Neural Networks 1999, 11, 313-326. (11) Takagi, T.; Sugeno, M. Fuzzy Identification of Systems and Its Application to Modelling and Control. IEEE Trans. 1985, SMC15, 116-132. (12) Jang, J. S. R. Self-Learning Fuzzy Controllers Based on Temporal Back Propagation. IEEE Trans. Neural Networks 1992, 3, 714-723. (13) Rumelhart, D. E.; Hinton, G. E.; Williams, R. J. Learning Internal Representations by Error Propagation. In Parallel Distributed Processing; Rumelhart, D. E., McClelland, J. L., Eds.; MIT Press: Cambridge, MA, 1986. (14) Marquardt, D. An Algorithm for Least Squares Estimation of Nonlinear Parameters. SIAM J. Appl. Math. 1963, 11, 431441. (15) McAvoy, T. J.; Hsu, E.; Lowenthal, S. Dynamics of pH in Controlled Stirred Tank Reactor. Ind. Eng. Chem. Process Des. Dev. 1972, 11, 68-70. (16) Smith, C. A.; Corripio, A. B. Principles and Practice of Automatic Process Control; John Wiley & Sons: New York, 1997.
Received for review December 1, 1999 Revised manuscript received March 13, 2001 Accepted May 8, 2001 IE990866H
