ph in cstr

Applying neural networks to on-line updated PID controllers fornonlinear process control

Junghui Chen*, Tien-Chih Huang

R&D Center for Membrane Technology, Department of Chemical Engineering, Chung-Yuan Christian University, Chung-Li, Taiwan 320,

Republic of China

Received 19 September 2002; received in revised form 16 April 2003; accepted 2 June 2003

Abstract

The inherent time-varying nonlinearity and complexity usually exist in chemical processes. The design of control structure shouldbe properly adjusted based on the current state. In this paper, an improved conventional PID control scheme using linearization

through a specified neural network is developed to control nonlinear processes. The linearization of the neural network model isused to extract the linear model for updating the controller parameters. In the scheme of the optimal tuning PID controller, theconcept of general minimum variance and constrained criterias are also considered. In order to meet most of the practical appli-cation problems, several variations of the proposed method, including the momentum filter, the updating criterion and the adjust-

ment of the step size of the control action, are presented to make the proposed algorithm more practical. To demonstrate thepotential applications of the proposed strategies, two simulation problems, including a pH neutralization and a batch reactor, areapplied.

# 2003 Elsevier Ltd. All rights reserved.

Keywords: Neural networks; Nonlinear modeling; PID controller

1. Introduction

Despite the advent of many complicated control theo-ries and techniques, more than 95% of the control loopsbased on proportional-Integral-Derivative (PID) con-trollers are still being used in the majority of industrialprocesses. It can be thus said to be the ‘‘bread andbutter’’ of control engineering [2] because of its simpli-city in structure, robustness in operation and easy com-prehension in principle. Nevertheless, the PIDalgorithm might be difficult to deal with in highly non-linear and time varying chemical processes. To improvethe control performance, several schemes of self-tuningPID controllers were proposed in the past. Wittenmark[25] proposed the control structure with the PID algo-rithm calculated via pole placement design. The methodwas limited in the order of the controlled processes. Theself-tuning PI or PID algorithms were automaticallyderived from the dynamic of the controlled processes[11]. An alterative self-tuning PID controller was based

on the generalized minimum variance control [6,8]. Thecontrol structure was orientated to have a PID struc-ture. The controller parameters were obtained using aparameter estimation scheme. Other forms of self-tun-ing PID can be found in literature [18,17,20]. However,the above self-tuning adaptive control approaches arelimited to linear system theory, i.e. these techniquesassume that the control model with the linear model isoperated in a linear region. If some changes in the pro-cess or environment occur, it must be manually checkedwhether the model is adequate to represent the realprocess or not since the control design is totally basedon a reliable model.Currently neural networks constitute a very large

research interest. They have great capability in solvingcomplex mathematical problems since they have beenproven to approximate any continuous function asaccurately as possible [12]. Hence, it has received con-siderable attention in the field of chemical process con-trol and has been applied to system identification andcontroller design [3,24]. All of these works show that theneural network can capture the characteristics of systempatterns and performance function approximation for

0959-1524/$ - see front matter # 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/S0959-1524(03)00039-8

Journal of Process Control 14 (2004) 211–230

www.elsevier.com/locate/jprocont

* Corresponding author. Fax:+886-3-265-4199.

E-mail address: [email protected] (J. Chen).

http://www.elsevier.com/locate/jprocont/a4.3d

mailto:[email protected]

nonlinear systems [15]. Thus, it is effectively used in thecontrol region for modeling nonlinear processes espe-cially in the model-based control, such as the direct andindirect neural network model based control [19], non-linear internal model control [14], and recurrent neuralnetwork model control [16]. Although the control per-formances of the above methods are satisfactory, thenonlinear iterative algorithm of the control design iscomputationally demanding because of the systembased on the nonlinear neural network model. This maymake the implementation strategy realistic only forcontrol of slow dynamic systems.In the linear control design theory, linearization of

nonlinear models is often used in the control field toalleviate the design of controllers for nonlinear systems.As we know, a model estimated through linearizationbased on the operating point can be considered validonly in a certain region around this point. Due to thecharacteristics of the nonlinearities and the size of theoperating region, it is necessary to consider whether touse a single linear model or to obtain more linearizationmodels around the different operating regions. The lat-ter called gain scheduling is often chosen to design andcontrol the nonlinear process. It selects a set of pre-defined linear controllers. Each controller is tuned for aspecific operating region. Gain scheduling can beimplemented in various ways according to the nature ofapplications under consideration.Based on the previous discussion of the linearization

of the nonlinear model, several combinational methodsbased on neural networks and the traditional linearcontroller design have been developed. Fuli et al. [10]proposed a compromised method with the neural net-work and pole placement design. This method assumedthat the plant could be linearized at each operatingpoint. It used the linear neural network to capture the

linear dynamic behavior of processes, and the multi-layered feedforward neural network to identify thenonlinear part as measured disturbance. In the con-troller design, it used pole placement as feedback con-trol, and the multilayered feedforward neural networkas feedforward control to eliminate the nonlinear dis-turbance. Ahmed and Tasadduq [1] mentioned a three-stage procedure for designing controllers by linearizationthrough a built neural network. The procedure imple-mented any linear controller on processes like pole-place-ment and optimal-control strategies. Finally, a neuralnetwork controller from the above control loop wasbuilt to replace the linear controller. Although the abovetwo methods are based on the linearization throughneural networks and pole placement design, they are notconcerned with the process with the PID controller.To improve the control performance, an on-line

updated PID algorithm is proposed. It combines thegeneral minimum variance (GMV) control law with theextracting the characteristic of the instantaneous linear-ized neural network model. With linearization of theneural network model, the PID algorithm can beimplemented directly without any modification. Thismethodology is good for controlling nonlinear processeswithout highly demanding computation, because thecontroller design is based on the linear model instead ofthe nonlinear model.

2. Design problem statement

The block diagram of the control system to be con-sidered is shown in Fig. 1. The controlled process is anynonlinear continuous or batch process. The PID con-troller from the process variable yðtÞ to the controlvariable uðtÞ is

Fig. 1. The scheme of the adaptive PID control based on the instantaneous linearization of the neural network model.

212 J. Chen, T.-C. Huang / Journal of Process Control 14 (2004) 211–230

uðtÞ ¼ us þ kc eðtÞ þ1

�i

ðeðtÞdtþ �d

deðtÞ

dt

� �ð1Þ

where us is bias value. eðtÞ ¼ yset tð Þ � y tð Þ is the outputerror deviated from the setpoint. kc, �i and �d are knownas the proportional gain, the integral time constant andderivative time constant respectively. A velocity form ofthe discrete PID control can be written as

DuðtÞ ¼ u tð Þ � u t� 1ð Þ

¼ kc

�e tð Þ � e t� 1ð Þð Þ þ

Dt2�i

e tð Þ þ eðt� 1Þð Þ

þ�dDt

e tð Þ � 2e t� 1ð Þ þ e t� 2ð Þð Þ

�ð2Þ

where the integral action of Eq. (2) is computed usingthe trapezoidal approximation. Rearrange the discreteform of the PID control to be in the following form

DuðtÞ ¼ k0eðtÞ þ k1eðt� 1Þ þ k2eðt� 2Þ

¼ eTðtÞkðtÞ ð3Þ

where kðtÞ ¼ k0 k1 k2� �T

, eðtÞ ¼ eðtÞ eðt� 1Þ eðt� 2Þ� �T

and

k0 ¼ kc 1þDt2�i

þ�dDt

� �; k1

¼ �kc 1�Dt2�i

þ2�dDt

� �; k2 ¼

kc�dDt

: ð4Þ

In Fig. 1, the control structure is similar to an adap-tive control structure. The parameters of the PID

controller are adjusted by an outer loop composed of aninstantaneously linearized neural network model esti-mator and a GMV control design calculation. In thetraditional adaptive control design, the time-varyingparameters of the linear model are estimated on-line bya recursive identification algorithm, including a forget-ting factor to place lighter emphasis on older data. Inthis study, an off-line neural network model is trained tomodel the nonlinear process and then an instantaneouslinearization of neural networks at each sampling pointis conducted to get a linearized model.In fact, the functional behavior of the proposed con-

trol structure looks similar to an adaptive controller ora gain schedule control whose model is chosen from aset of predefined linearized models, but in the instanta-neous linearization of the neural network model theprocess dynamic parameters can be changed quickly inresponse to process changes. Besides, the neural net-work model is trained off-line; extra computation loadis not needed to construct the current model in the on-line identification for the current control design. Themore important aspect is that the traditional neuralnetwork control design requires training the neuron-controller on-line as the performance error back-propagates through the network at every sample.Sometimes the emulator neural network is used due tothe requirement of the Jacobian of the process as shownin Fig. 2, since the process is unknown [22,26]. Thisnonlinear optimization problem may cause impropersolution for the control design.

2.1. Instantaneous linearization of the neural network

Assume a deterministic process where the generalform represents a discrete-time nonlinear system

Fig. 2. The configuration of neural network controller design based on an emulator neural network model.

J. Chen, T.-C. Huang / Journal of Process Control 14 (2004) 211–230 213

y tð Þ ¼ f yðt� 1Þ; . . . ; yðt� nyÞ; uðt� 1Þ; . . . ; uðt� nuÞ�

þ eðtÞ: ð5Þ

Here yðtÞ is the process output, uðtÞ the input, eðtÞ azero-mean disturbance term, and ny and nu indicate thenumber of output and input delay respectively. Theprocess model can be written as a deterministic neuralnetwork model

yNNARX tð Þ ¼ NNARXð�ðtÞÞ ð6Þ

where NNARX is a neural network ARX function. Thistype of model has been studied widely in non-linearsystem identification [7,27]. The regression vector isdefined as

� tð Þ ¼ y tð Þ; u tð Þ½ �T

¼ yðt� 1Þ; . . . ; yðt� nyÞ; uðt� 1Þ; . . . ; uðt� nuÞ� �T

:

ð7Þ

The goal of the neural network modeling is to find aparameterized structure that emulates the nonlinearprocess. The NNARX model used here is a three-layerfeedforward neural network with a linear activationfunction in the output neuron and with a hyperbolictangent activation function in the hidden neurons. Itcan be written as

yNNARX tð Þ ¼XNhiddenc¼1

woczc mc tð Þ½ � þ wob

mc tð Þ ¼Xnyi¼1iy¼i

whc;iyyðt� iÞ þXnui¼1

iu¼iþny

whc;iuuðt� iÞ þ whb;c ð8Þ

where zc is the transfer function for the hidden neuron c,woc and w

ob are the weights and the bias of the hidden-to-

output layer, whc;i and whb;c are the weights and the bias

of the input-to-hidden layer, and mc is the summation ofall products between inputs and input-to-hidden weightsin the input layer.The idea behind instantaneous linearization is to

extract a linear model from the nonlinear neural net-work model at each sample point. The approximatelinear model at time t ¼ � can be obtained by linearizingNNARX around the current state �ðt ¼ �Þ. Lineariza-tion means the first partial derivative term for each ele-ment of �ðt ¼ �Þ. The approximated model yINST-NN tð Þcan be written as:

yINST-NN tð Þ ¼ bias�Xnyi¼1

aiy t� ið Þ þXnui¼1

biu t� ið Þ ð9Þ

where ai and bi are the linear model coefficients of theoutput and the input,

ai ¼ �@NNARX

@y t� ið Þj� tð Þ¼�ð�Þi ¼ 1; 2; ; ny: ð10Þ

bi ¼@NNARX

@u t� ið Þj� tð Þ¼�ð�Þi ¼ 1; 2; ; nu: ð11Þ

The approximated linear model is affected by a con-stant bias, bias, depending on the current operatingpoint.

bias ¼ yð�Þ þXnyi¼1

aiy � � ið Þ þXnui¼1

biu � � ið Þ: ð12Þ

Based on the descriptions of Eqs. (10) and (11), thederivatives of the output with respect to the currentstate �ðt ¼ �Þ can be calculated as:

ai ¼@yNNARX tð Þ

@y t� ið Þ¼

XNhiddenc¼1

woc@zc tð Þ

@mc tð Þ

@mc tð Þ

@y t� ið Þ

¼XNhiddenc¼1

woc@zc tð Þ

@mc tð Þwhc;i ; i ¼ 1; 2; ; ny ð13Þ

bi ¼@yNNARX tð Þ

@u t� ið Þ¼

XNhiddenc¼1

woc@zc tð Þ

@mc tð Þ

@mc tð Þ

@u t� ið Þ

¼XNhiddenc¼1

woc@zc tð Þ

@mc tð Þwhc;ðiþnyÞ ; i ¼ 1; 2; ; nu: ð14Þ

The controller design based on the instantaneous lin-earization of the neural network model has two advan-tages:

2.1.1. Process modelingLinearization of the nonlinear model is a well-known

method often used among control designs. As we know,the model obtained through linearization around anoperating point can be considered valid only in a certainregion around this point. At a first glance, the linearizedmodel seems to be a crude description of the actualprocess, but the new model obtained by linearizationwill be immediately updated with the change of the sizeof the operating range and the character of non-linearities. In Fig. 3, the curve with a little fluctuationrepresents the actual process output with measurednoise. Due to the characteristic of the noise cancellationof the neural network, the trained neural network modelcan smooth the measured data and properly represent thedynamic process. Fig. 3 also indicates that the prediction


of the instantaneous linearization NNARXyINST-NN t ¼ � þ 1ð Þ at one-step ahead can closely followthe process behavior. In the traditional linear controltheory, the recursive identification is often used fornonlinear systems. The linear model is properly updatedwith the new data coming. However, the linear modelyREC-LIN tð Þ may not be quickly updated because themodel only accounts for the past information until nowwith little consideration of the current characteristic ofthe nonlinear process. The model error may be sig-nificantly large when the linear model can be consideredvalid only in a very narrow region around this point. Toget more impression of the difference of the predict-ability between yINST-NN tð Þ and yREC-LIN tð Þ, a simplenonlinear dynamic system is employed [15].

z tð Þ ¼z t� 1ð Þ

1þ z t� 1ð Þ2þ u t� 1ð Þ

3

yðtÞ ¼ zðtÞ þ vðkÞ ð15Þ

where yðtÞ and uðtÞ are the measurement of the processoutput and input variables at time t. The randommeasurement noise vðkÞ with Nð0; 0:1Þ is added to themeasured output.Our goal here is to predict one-step ahead y � þ 1ð Þ

based on the known values of the time series up to thecurrent time point. The sum of the square of the pre-diction errors (SSE) of these models are listed in Table 1.Here the recursive identifications with two different for-getting factors are also used. The SSE of the recursiveidentification would be reduced with a larger forgettingfactor that places heavier emphasis on more recent data.It is observed that SSE of the instantaneous lineariza-

tion of the trained neural network model is less thanthat of the other models.

2.1.2. Controller designIf the controller design is based on the nonlinear

model, there would be several problems, including thecomputation load of the iterative minimization, trap-ping in the local minimum of the criterion and repeatingdifferent initial points several times using the minimizedcriterion. It would significantly reduce interests tomatch the need of the realistic industrial problems. Oncea linearized neural network is obtained, it can be easilyapplied to the rich collection of well-understood lineardesign techniques in the final closed-loop system. Thedifficulties of nonlinear processes in applying the linearcontrol theory are eliminated, because there is an appro-priate time-invariant point for each local linearization.

2.2. General minimum variance based tuning

The goal of the controller design is to seek a controlsignal uðtÞ that will minimize the difference of the pro-cess output and the desired output at the next time step;i.e. the process output can reach the desired output atthe next time. Besides, from the operation point of view,the variance controller output should be minimized inorder to exert excessive control effort. The objectivefunction is expressed as

minkc;�i;�d

J ¼1

2minkc;�i;�d

e2 tþ 1ð Þ þ �Du2 tð Þ� �

ð16Þ

where � is the weighting penalty parameter. However,since e tþ 1ð Þ ¼ ysetðtþ 1Þ � yðtþ 1Þ, the objective

Fig. 3. Comparison of the predicted output from NNARX, INST-NN and recursive linear model at the next time step.


function involves a term in the future of the next timestep; namely y tþ 1ð Þ, which is not available at time t.To overcome this problem, a model based on thetrained NNARX can be used; that is,y tþ 1ð Þ ffi yNNARX tþ 1ð Þ. However, the objective func-tion based on the NNARX predicted model wouldinvolve the complicated computation for the nonlinearmodel. Therefore, the model can be further simplifiedfrom the linearization of the neural network model toprovide estimates of y tþ 1ð Þ ffi yINST-NN tþ 1ð Þ. Themodification objective is

minkc;�i;�d

J � minkc;�i;�d

L

¼1

2minkc;�i;�d

E eINST-NN tþ 1ð Þ� 2

þ�Du2 tð Þh i

ð17Þ

The prediction error eINST-NNðtþ 1Þ based on thelinearized model is eINST-NN tþ 1ð Þ ¼ yset tþ 1ð Þ�

yINST-NNðtþ 1Þ. After Eq. (9) is substituted into Eq.(17), it can be rearranged in the following form

eINST-NN tþ 1ð Þ ¼ yset tþ 1ð Þ �Xnyi¼1

aky t� iþ 1ð Þ

�Xnui¼1

bku t� iþ 1ð Þ � bias

¼�yset tþ 1ð Þ �

Xnyi¼1

aky t� kþ 1ð Þ

�Xnui¼2

bku t� iþ 1ð Þ � bias�

� b1u tð Þ

¼ CON� b1eT tð Þk tð Þ ð18Þ

where the last term of the above equation is found usingthe control action u tð Þ [Eq. (3)], and CONT ¼

yset tþ 1ð Þ �Pny

k¼1aky t� kþ 1ð Þ �Pnu

k¼2bku t� kþ 1ð Þ

� b1u t� 1ð Þ � bias. Let the updated control parametervector be

k tð Þ ¼ k t� 1ð Þ þ Dk tð Þ ð19Þ

where DkðtÞ is the space of change of the tuning para-meters at the sampling instant t and kðt� 1Þ is the oldcontrol parameter vector computed at the samplingt� 1.

Substituting Eqs. (3), (18) and (19) into the objectivefunction gives

L ¼1

2CONT � b1e

T tð Þ k t� 1ð Þ þ Dk tð Þ½ �� 2

þ�

2

� eT tð Þ k t� 1ð Þ þ Dk tð Þ½ �� 2

: ð20Þ

When minimizing L with respect to DkðtÞ, we areseeking a set of PID controller parameter in the quad-ratic function of this objective function. The gradient ofJ can be computed as

rL Dk tð Þð Þ ¼@L Dk tð Þð Þ

@Dk tð Þ

¼ A tð ÞDk tð Þ þ d tð Þ ð21Þ

where

A ¼

�b1ð Þ2þ�

� �e2 tð Þ �b1ð Þ

2þ�

� �e tð Þe t� 1ð Þ

�b1ð Þ2þ�

� �e t� 1ð Þe tð Þ �b1ð Þ

2þ�

� �e2 t� 1ð Þ

�b1ð Þ2þ�

� �e t� 2ð Þe tð Þ �b1ð Þ

2þ�

� �e t� 1ð Þe t� 2ð

2664

�b1ð Þ2þ�

� �e tð Þe t� 2ð Þ

�b1ð Þ2þ�

� �e t� 1ð Þe t� 2ð Þ

�b1ð Þ2þ�

� �e2 t� 2ð Þ

3775

ð22Þ

d tð Þ ¼ �

�b1CONTe tð Þ½ � þ ðb21 þ �Þe tð Þ eT tð Þ k t� 1ð Þ½ ��

�b1CONTe t� 1ð Þ½ � þ ðb21 þ �Þe t� 1ð Þ

eT tð Þ k t� 1ð Þ½ �� b1CONTe t� 2ð Þ½ � þ ðb21 þ �Þe t� 2ð Þ

eT tð Þ k t� 1ð Þ½ ��

26666664

37777775

ð23Þ

The optimal point will occur when the gradient isequal to zero. Thus, the required changes of the controlparameters are

Dk tð Þ ¼ �A�1 tð Þd tð Þ: ð24Þ

Using Eqs. (4) and (19), the corresponding PID con-trol parameters are

kc tð Þ ¼ � k1 tð Þ þ 2k2 tð Þ½ �

�i tð Þ ¼� k1 tð Þ þ 2k2 tð Þ½ �Dtk0 tð Þ þ k1 tð Þ þ k2 tð Þ

�d tð Þ ¼�k2 tð ÞDt

k1 tð Þ þ 2k2 tð Þ: ð25Þ

The optimum of the above objective function is not aproblem, but physically it is not suitable. The optimalPID parameters from Eq. (25) have negative values. Ifone wishes to obtain reasonable PID parameters, the

Table 1

Comparison among the different identification algorithms for one-

step-ahead prediction

Model
NNARX INST-NN REC-LIN
(f ¼ 0:8)

REC-LIN

(f ¼ 0:5)

SSE
0.0424 0.5989 2.1199 9.8620

Þ

:

allowable range of this controller parameter should bedefined. The constrained region is as follows:

ðiÞ kc > 0:

Dk1 tð Þ þ 2Dk2 tð Þð Þ < � k1 t� 1ð Þ þ 2k2 t� 1ð Þð Þ

ðiiÞ �i > 0:

Dk0 tð Þ þ Dk1 tð Þ þ Dk2 tð Þð Þ >

� k0 t� 1ð Þ þ k1 t� 1ð Þ þ k2 t� 1ð Þð Þ

ðiiiÞ �d > 0:

Dk2 tð Þ5 � k2 t� 1ð Þ:

Since the objective is a quadratic function[Eq. (20)]subject to the linear inequality constraints [Eq. (26)], thequadratic programming can applied here [9].

3. Heuristic rules to improve tuning

When the above-developed method is applied to apractical problem, non-smooth responses may happendue to the measurement noise and the model–processmismatch. These problems are a fact of life in complexand nonlinear chemical processes. In this section, sev-eral variations of the proposed methods will be devel-oped to overcome these practical problems.

3.1. Momentum filter

Application of the controller action will result inamplification of the noise when the controlled variableis obtained around the desired setpoint. In order toavoid oscillations of the control parameters in the con-trol action, the control parameters should be smoothedout. The momentum filter, like the first-order filter thatreduces the variations, has been added to the controlparameter changes. The updated control parameters forthe momentum modification are obtained

DkmðtÞ ¼ Dkmðt� 1Þ þ ð1� ÞDkðtÞ

kðtÞ ¼ kðt� 1Þ þ DkmðtÞ ð27Þ

where is the momentum coefficient between 0 and 1.

3.2. Updating criterion

Although the control design in the adaptive controlstructure should keep on-line calculating adaptive para-meters of the controller, the computation of the newcontrol action is redundant when the controlled outputis close to the desired setpoint. Besides, the new com-puted control action in the processes which are notusually free of noise is susceptible to process noise. To

overcome this problem, the concept of the statisticalprocess control algorithm can be applied. The updatedcriterion, by a cumulative sum of the past error with thefixed window size, is designed to detect the deterministicshift in the desired setpoint. The combination of themost current error data sets is defined as

SðtÞ ¼1

m

Xti¼t�mþ1

ðysetðtÞ � yðtÞÞ2 ð28Þ

where SðtÞ is the current control performance. m is thesize of moving window which contains m� 1 past out-puts until now. Fig. 4 shows the updated criterion basedon the performance of the current moving window. Anold error data would be removed and a new error datawould come into the picture with the moving of therectangular window at each sampling time. Conse-quently, whenever the current performance SðtÞ is belowits control limit SðtÞ4 2, the current controller para-meters are assumed to be fixed until SðtÞ > 2. 2 is thethreshold. It can be estimated from the steady-state processdata when there is no change in the control action or it isbased on the prior knowledge of the operating process.

3.3. Adjustment of the step size of control action

The control parameters obtained from the quadraticobjective function may not be particularly optimal,because the parameters are determined through anapproximation of the objective based on the instanta-neous linearization of the neural network model, LðkÞ.It is expected that the control parameters are valid onlyin a neighborhood around the current state. Here thepenalty term � in Eq. (16) is used to adjust the changeof the control action in order to let LðkÞ be close to the

Fig. 4. Moving window used to determine when the control para-

meters would be updated.


true criterion JðkÞ. The accurate measure of theapproximation can be defined

approx ¼ yNNARX tþ 1ð Þ � yINST-NN tþ 1ð Þ�� : ð29Þ

If the difference is close to a small value, LðkðtÞÞ ¼Lðkðt� 1Þ þ DkðtÞÞ is likely to be a reasonable approx-imation to JðkðtÞÞ ¼ Jðkðt� 1Þ þ DkðtÞÞ. This indicatesthat the approximated model is good enough in thecurrent design region; otherwise, the penalty should beadjusted by some factors to reduce or expand theregion. Thus, the algorithm provides a nice compromisebetween the approximated linear model and the neuralnetwork model to compute the feasible control actions.

4. Procedures for the PID controller design

The proposed strategy is consisted of two phases. Thefirst phase involves identifying the relationship of thedynamic process between the input process variable andthe output process variable. NNARX is trained toderive these relationships in order to accurately pre-dict output behavior of the possible operating condi-tion. Thus, adding the current regression vector �ðtÞto the trained network allows us to estimate the cor-responding output. In Phase Two, a controller tuneris directly computed based on the instantaneous line-arization of the NNARX model so as to solve thePID control parameter design problem. Some heur-istic rules are incorporated to polish the calculatedPID control parameters for matching the practical

Fig. 5. The scheme of the optimal PID controller parameter design based on the instantaneous linearization of NNARX.


applications. Fig. 5 schematically depicts the on-linedesign procedure. The detailed procedure is summar-ized as follows:Phase One: Identify the relationship between the

input and the output to predict the output at the nextstep.Step 1: Train a NNARX model based on the experi-

mental data. Once trained, the NNARX model repre-sents a nonlinear or complex function for the outputthat it learned. Set k=1.Phase Two: Determine the current optimal PID con-

troller parameters.Step 2: At the new sampling time k, compute the

performance of the current data window [Eq. (28)] anddetermine whether the control parameters should beupdated. If the current performance SðtÞ is below itscontrol limit, the PID controller parameters are kepton the same values; and go to Step 7. Otherwise, theupdate control parameters are carried out; and go toStep 3.

Step 3: Extract the linearized model through the line-arization of the NNARX model around the currentinput–output pair.Step 4: With the current linearized model and the

difference between the predicted output and the processoutput, determine the update control parameters usingEqs. (24) and (25).Step 5: Measure the accuracy of the approximated

linear model by Eq. (29). If the weighted penalty needsto be changed in order to resize the control action, go toStep 4 to recompute the new updated parameters basedon the adjusted penalty; otherwise, go to Step 6. InFig. 5, the gray region for the adjustment proceduresof the penalty is zoomed in to describe the adjsutedpenalty procedure in Fig. 6.Step 6: Add the momentum filter to the change of the

controller parameters using Eq. (27).Step 7: Implement the new control action on the pro-

cess based on the calculated PID control parameters.Collect the current process input–output. Then set

Fig. 6. The scheme of the adjustment of the size of the control step by the change of the penalty parameter. If the control action step does not yield a

value less than "1 for approx, the control action is recomputed by�ðtþ 1Þ ¼ 2�ðtÞ in order to get a small control action. If the control action does

produce a smaller value "2 for approx, � is divided by �ðtþ 1Þ ¼ 0:5�ðtÞ in order to increase the size of the control action.


k ¼ kþ 1 and repeat the procedure from Step 2 toStep 7 for the next time step.

5. Illustration examples

In this section, two simulation examples involvingprocess of different dynamics are discussed to demon-strate the wide applicability of the proposed on-lineupdated tuning algorithm. The proposed algorithmshown as follows is called INST-NNPID, which standsfor the on-line updated PID controller with the instan-taneous linearized NNARX model. INST-NNPID pro-vides improved performance over the traditional self-tuning PID algorithms when applied to nonlinear pro-cesses. Each of these examples will be discussedseparately in the sub-sections as follows.

5.1. Example 1: nonlinear pH neutralization system

pH neutralization is quite common in the chemicalprocess. This example of dynamic modeling is based onthe case of Nahas et al. [14]. The pH CSTR systemshown in Fig. 7 has three input streams, including acid(HNO3), buffer (NaHCO3) and base stream (NaOH)respectively. The extremely nonlinear relationships areconsisted of two reaction invariants, three nonlinearordinary equations and one nonlinear algebraic equation.

Charge balance

Wa � Hþ� �

� OH�½ � � HCO�3

� �� 2 CO¼

3

� �ð30Þ

Carbonate ionbalance

Wb � H2CO3½ � þ HCO�3

� �þ CO¼

3

� �ð31Þ

dh

dt¼

1

Aðq1 þ q2 þ q3 � Cvh

0:5Þ ð32Þ

dWa4

dt¼

1

AhWa1 �Wa4ð Þq1 þ Wa2 �Wa4ð Þq2 þ Wa3 �Wa4ð Þq3½ �

ð33Þ

dWb4¼

1Wb1 �Wb4ð Þq1 þ Wb2 �Wb4ð Þq2 þ Wb3 �Wb4ð Þq3½ �

dt Ahð34Þ

Wa4 þ 1014-pH

þWb41þ 2� 10pH�pK2

1þ 10pK1�pH þ 10pH�pK2�10�pH ¼ 0: ð35Þ

In Eqs. (30)–(35), h is the liquid level, Wa4 and Wb4 arethe reaction invariants of the effluent stream, and q1; q2and q3 are the acid, buffer and base flow rates, respec-tively. The controlled variable is the pH value (pH). Thenominal conditions and the operating parameters arelisted in Table 2.In the initial phase, a set of training data representing

the process is obtained. Then a NNARX model isdeveloped for the process under study. The dynamicmathematical model is for the response of pH to chan-ges in q3 flow rate. When a pseudo-random variation q3between 850 and 1050 ml/min is added to the process togenerate the variation of pH between 6.5 and 9. Twopast inputs and two past outputs are used to constructthe NNARX model. A test data is also undertaken toverify the prediction capability of the trained NNARXin order to see if the NNARX model is good for the useof the proposed method [4,13].

5.1.1. Setpoint changeThe setpoints of pH values have been changed from 7

to 9 and then from 9 to 8. First the response to the
Fig. 7. The scheme diagram of pH CSTR.
Table 2

Simulation conditions for Example 1

A ¼ 207 cm2
Cv ¼ 8:75 ml cm�1 s�1 pK1=6.35
pK2=10.25
Wa1 ¼ 3� 10�3 M Wa2 ¼ �3� 10�2 M
Wa3 ¼ �3:05� 10�3 M
Wb3 ¼ 5� 10�5 M Wb1 ¼ 0
Wb2 ¼ 3� 10�2 M
q1 ¼ 16:6 ml s�1 q2 ¼ 0:55 ml s�1
q3 ¼ 15:6 ml s�1
C1 ¼ 0:003 M HNO3 C2 ¼ 0:03 M NaHCO3
C3 ¼ 0:003 M NaOH0:00005 M NaHCO3

Steady state:
h ¼ 14:0 cm pH4 ¼ 7:0

controlled system with the fixed controller parameterswithout updating is tested. When the system is operatedat a pH around 8, the controller parameters are com-puted based on the minimum ITAE (integral of thetime-weighted absolute error) tuning formula [23]. Theplot in Fig. 8 shows that the fixed tuning parametersresult in significant overshoot and oscillatory behavior.As would be expected, the adaptive tuning parametersare needed for the nonlinear process. Fig. 9 demon-strates the performance using INST-NNPID. The pen-alty parameter is fixed at a larger value (� ¼ 20) due tothis high nonlinearity of the pH CSTR system. Moretime is required to adjust the process behavior after thesetpoint is changed. However, the controlled results willaggressively react to the change of the setpoint before-hand if a smaller value of � is selected. Using the pro-posed rule to automatically adjust �, when the adaptivepenalty is taken into consideration, the control responseand each updated control parameter are separatelyshown in Figs. 10 and 11. Unlike the previous studywith fixed penalty, the PID controller with the adaptivepenalty of the process output follows the desired set-point much more quickly. Fig. 12 also provides thereason why the variable penalty should be changed.When the setpoint is changed at time 25, there is asignificant difference between the predictions fromyINST-NN tþ 1ð Þ and yNNARX tþ 1ð Þ. The penalty tendsto increase for the reduced aggressive control action.When the difference between yINST-NN tþ 1ð Þ andyNNARX tþ 1ð Þ is smaller enough between time 45 and

75, the process output is closed to the setpoint. Thepenalty tends to decrease for the increased aggressivecontrol action.In the past, many self-tuning controllers (STC) were

developed. STC emphasizes the combination of arecursive identification procedure and a selected con-troller synthesis. Despite intensive research activities,different branches of STC focus only on the controldesign method while the process model still follows aregression standard model form by applying the recur-sive least square method. The STC methods include self-tuning PID controllers (STPID) [6], modified Ziegler–Nichols PID controllers (MZNPID) [5], etc. MZNPIDleads to more aggressive control action because stabilityis considered rather than the control performance.Thus, MZNPID may not be good for our purposes. Onthe other hand, the STPID scheme requires carefulselection of the prefilter coefficients and the integralaction constant to obtain more stable responses. Thus, atrial-and-error procedure is required to obtain thesetuning parameters, sometimes causing bigger oscillatoryresponses. The behavior of the closed loop response inthe process for these design methods is shown in Fig. 13.Besides, the proposed design method with a recursivelinear model (called REC-LINPID) is also tested andshown in Fig. 13. This algorithm, unlike STPID, alsoperforms well without any prior design parameter. Forfair comparison of methods, the numerical measures ofthe performance based on SSE are listed in Table 3. Thelack of mechanism cannot quickly update the model to

Fig. 8. PID controller with the fixed controller parameters in setpoint change results for Example 1: (a) pH (dashed line) and setpoint (solid line); (b) q3.


Fig. 9. INST-NNPID with the fixed penalty (� ¼ 20) in setpoint change results for Example 1: (a) pH (dashed line) and setpoint (solid line); (b) q3.

Fig. 10. INST-NNPID with the adaptive penalty in setpoint change results for Example 1: (a) pH (dashed line) and setpoint (solid line); (b) q3.


Fig. 11. The self-tuning PID controller parameters in setpoint change results for Example 1: (a) kc; (b) �i; (c) �d.

Fig. 12. The adjustment � with the approximated error yNNARX tþ 1ð Þ � yINST-NN tþ 1ð Þ�� in the setpoint change results for Example 1: (a) �; (b) the

approximated error.


dt

cope with an actual nonlinear chemical process, so theperformances of STPID and REC-LINPID are worsethan that of the proposed method.

5.1.2. Disturbance changeThe disturbance variable, q2 changes from 990 to 1010

ml/min at time 50 and from 1010 to 1020 ml/s at time100. Fig. 14 depicts the rejected disturbance results.REC-LINPID causes unstable responses because thepenalty cannot be properly adjusted. It is not includedin Fig. 14. In Table 3, the corresponding SSE valuesalso indicate that, even without measured disturbances,the proposed strategy for the nonlinear control design isappropriate.Based on the above results, the proposed method not

only has less computation load when INST-NNPID isused but also has good performance at the setpointchange and the disturbance change even though the pHneutralization is highly nonlinear.

5.2. Example 2: nonlinear batch reactor

The batch reactor is sketched in Fig. 15. The reactantis fed into the reactor. Assume the mixture is blendedwell in the reactor and the liquid density is kept con-stant during operation. The consecutive reaction takesplace in the reactor as:

A!k1B!

k2C ð36Þ

where A ! B has second-order kinetics and B ! Chas first-order kinetics. It is noted that the energy dis-sipation induced by the shaft work and heat loss to thesurroundings is ignored. According to the aboveassumptions, the mass balances for components A andB and the energy balance are formulated as follows:

dCA¼ �k1C

2A; CA 0ð Þ ¼ CA0 ð37Þ

Fig. 13. Comparison between INST-NNPID, REC-LINPID and STPID in setpoint change results for Example 1.

Table 3

Comparison SSE among the different adaptive PID controller algorithms

Example
Model INST-NNPID REC-LINPID STPID
1
SSE (in setpoint change) 17.5477 19.4597 30.4284
SSE (in disturbance change)
0.1659 – 0.2782
2
SSE (in tracking trajectory) 13 764 24 373 –

Fig. 14. Comparison of INST-NNPID, REC-LINPID and STPID in disturbance change results for Example 1.

Fig. 15. The scheme diagram of the batch reactor.


V

dCBdt

¼ k1C2A � k2CB; CB 0ð Þ ¼ CB0 ð38Þ

dT

dt¼ 1k1C

2A þ 2k2CB þ �1 þ �2Tð Þ

þ 1 þ 2Tð Þu

ð39Þ

where

1 ¼ �DH1=�Cp

2 ¼ �DH2=�Cp

�1 ¼ UjAjTs;min þUc;maxAcTc�

=�CpV

�2 ¼ � UjAj þUc;maxAc�

=�CpV

1 ¼ UjAj Ts;max � Ts;min�

� Uc;max �Uc;min

� AcTc

� �=�Cp

2 ¼ Uc;max �Uc;min

� Ac=�CpV ð40Þ

k1 and k2 are the reaction rate constants which aretemperature-dependent,

k1 ¼ A10exp �E1=RTð Þ

k2 ¼ A20exp �E2=RTð Þ:ð41Þ

The parametric variable u lies in the region of 0 and 1.When u=0, it represents the maximum cooling andminimum heating; on the other hand, when u=1, itrepresents the minimum cooling and maximum heating.Hence, the input constraint (04 u4 1) of this caseneeds to be considered during control. The processmeasured T with noise Nð0; 1:0Þ is presented. Theparameter definitions are listed in Table 4. In this case,the control objective is to maximize the yield ofcomponent B by keeping the operation under thefollowing optimal temperature trajectory [21]:

Td tð Þ ¼ 54þ 71exp �2:5� 10�3t�

: ð42Þ

To train the NNARX model, the open-loop dataof output (T) driven by pseudo random binary sequenceinputs (u) are first collected. The time interval foreach input change is 30 s, the sampling time is 1 s andthe overall data number is 3000. The data set is imposed

Table 4

Simulation conditions for Example 2

A10 ¼ 1:1 m3 kmol�1 s�1
1 ¼ 41:8 �C s�1 2 ¼ 0:0515 �C s�1
A20 ¼ 172:2 s�1
2 ¼ 83:6�C s�1 CA 0ð Þ ¼ 1 kmol m�1
E1 ¼ 20900 kJ kg�1 K�1
�1 ¼ 4:3145 �C s�1 CB 0ð Þ ¼ 0 kmol m�1
E2 ¼ 41800 kJ kg�1 K�1
�2 ¼ �0:1099 �C s�1 T 0ð Þ ¼ 25�C
R ¼ 8:3143 kJ kmol�1 K�1
1 ¼ 1:4962 �C s�1
Fig. 16. INST-NNPID without the heuristic rules (i) and (ii) in tracking trajectory results for Example 2: (a) T (dashed line) and setpoint (solid line); (b) u.


on the NNARX model with different input lag termsto search for the appropriate weights in the model.The smallest overall prediction error results fromny ¼ nu ¼ 3. A set of validation data is also generated inthe same manner for verification of the developed modelbefore implementing the model for the rest of thecontrol study.

After the predicted model is developed, the self-tuningcontrol can be applied to the trajectory tracking of thebatch case. Fig. 16 demonstrates the tracking ability ofthe algorithms. The control action of the manipulatedsignal u is depicted at the bottom of the figure. Thecontrol output can follow the setpoint. Fig. 17 indicatesthat the large, high-frequency variation in the control

Fig. 18. INST-NNPID in tracking trajectory results for Example 2: (a) T (dashed line) and setpoint (solid line); (b) R.

Fig. 17. INST-NNPID controller parameters without the heuristic rules (i) and (ii) in tracking trajectory results for Example 2: (a) kc; (b) �i; (c) �d.


parameters still exists because of the noise measurementto a large extent even if the output temperature isaround the desired setpoint. By applying heuristic rules(i) and (ii), the momentum filter and the updated con-dition (with m ¼ 5), a much more satisfactory dynamicresponse and much smoother performance is achieved(Figs. 18 and 19).

Because of the linear model error and difficulty intuning STPID parameters, only the results of the self-tuning control design based on REC-LINPID are onlyconsidered here. In Fig. 20, the control outputs basedon REC-LINPID can follow the setpoint; however,REC-LINPID has significant variation in trajectorytracking due to the improperly updated ability in the

Fig. 20. REC-LINPID in tracking trajectory results for Example 2: (a) T (dashed line) and setpoint (solid line); (b) R.

Fig. 19. INST-NNPID controller parameters in tracking trajectory results for Example 2: (a) kc; (b) �i; (c) �d.


current operational model. The proposed method hasbetter trajectory tracking ability than REC-LINPIDsince the former is based on the instantaneous linearizedmodel, and the latter, based on the linear model.From a computational perspective, if the proposed

method is based on the neural network model, theoptimal control action is iterative in order to convergeto an acceptable accuracy. This strategy may becomeunrealistic for controlling the nonlinear dynamic pro-cess. By applying INST-NNPID, the solution has aunique minimum that can be found directly.

6. Conclusion

A new control approach using a minimum variationcontroller and a linearized neural network modelaround its current operation region is developed. Likegain scheduling, the instantaneous linearization tech-nique could be used to design a gain-scheduling type ofthe control system at each time interval. However, asopposed to gain scheduling control, the control para-meters based on the neural network model have aninfinite scheduling resolution for updating parameters.The proposed method has several advantages. First, thelinear control scheme applied to the nonlinear controldesign has less computation compared with the non-linear control counterpart based on the nonlinear neuralnetwork model. Second, the nonlinear characteristics ofneural networks can be incorporated into the controldesign. Third, this method provides a useful physicalinterpretation of the dynamics process. However, thereare several serious drawbacks. First, the linearizedmodel may be valid only in a narrow region when theprocess around the current operating range is verynonlinear. Second, the measurement noise and themodel–process mismatch may generate the defected lin-earized model. These drawbacks could have a seriousimpacts on controller design. To overcome these pro-blems, several variations of the heuristic rules, includingthe momentum filter, the updated condition and theadjusted step size of the control action, are developed tomake the proposed algorithm more practical. The con-trol strategy is the balance of the nonlinear and theconventional linear control designs to improve the con-trol performance for the nonlinear process. Without theconvergence problem, a unique minimum of the controldesign can be found directly because the model is linear.Besides, the implementation is much simpler and com-putationally less demanding. This strategy is particu-larly suitable for the chemical process that ischaracterized by high nonlinearity and analytical diffi-culty. The proposed method is demonstrated using twoexamples of a continuous pH neutralization process anda batch reactor. The results show that the proposedmethod with the heuristic rules is effective in following

the setpoint and reducing the variance of the outputcaused by unknown disturbance. Although the pro-posed method yields encouraging outcomes, all theresults presented in this paper are based on the simu-lation models. Additional work using a real plantshould be conducted in the future.

Acknowledgements

This work was partly sponsored by the NationalScience Council, R.O.C. and by the Ministry of Eco-nomic, R.O.C.

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