Experimental evaluation of combined model reference ...mduartem/PDFS/IJACSP2002.pdf · Manuel A....

*Correspondence to: Manuel A. Duarte-Mermoud, Department of Electrical Engineering, University of Chile, Av.Tupper 2007, Casilla 412-3, Santiago, Chile.

�E-mail: [email protected]

Contract/grant sponsor: CONICYT; contract/grant numbers: FONDECYT 1950502 and FONDECYT 1970351

Received 7 December 1999Revised 8 March 2000

Copyright � 2002 John Wiley & Sons, Ltd. Accepted 1 February 2001

INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSINGInt. J. Adapt. Control Signal Process. 2002; 16:85}106 (DOI: 10.1002/acs.674)

TECHNICAL NOTE

Experimental evaluation of combined model reference adaptivecontroller in a pH regulation process

Manuel A. Duarte-Mermoud��*�-, Franklin A. Rojo� and Ricardo PeH rez�

�Department of Electrical Engineering, University of Chile, Av. Tupper 2007, Casilla 412-3, Santiago, Chile�Department of Chemical and Bioprocess Engineering, P. Catholic University of Chile, Av. V. Mackenna 4860,

Casilla 306-22, Santiago, Chile

SUMMARY

An experimental evaluation of the combined model reference adaptive control (CMRAC) is presented in thispaper. This adaptive control scheme was used to control a relatively complex process like the pH ofa solution in a tank reactor at laboratory level. For comparison purposes, some very well-known controlstrategies were also implemented, which include PID control and standard model reference adaptive control(MRAC). Tracking and regulation capabilities of the control strategies studied were analysed and compared.Experimental results indicate that CMRAC behaves as well as the standard MRAC and a very well-tunedPID for a speci"c and known operating point. Advantages of the adaptive controllers are shown when theoperating point changes. Copyright � 2002 John Wiley & Sons, Ltd.

KEY WORDS: pH control; process control; PID control; model reference adaptive control; combined modelreference adaptive control

1. INTRODUCTION

The combined model reference adaptive control (CMRAC) is a relatively new control strategythat has been completely studied from the theoretical point of view [1}3] but only few applica-tions have been reported. In particular, the application of the CMRAC to a pH control processhas not been reported. This study is important since it will show new directions in whichtheoretical analysis of the method should go in order to apply adaptive control at the industriallevel in a more accurate and con"dent fashion.

The regulation of pH is a di$cult problem that arises in di!erent areas, like biotechnology,food processing, hydrometallurgy and environmental engineering. It is not a trivial task todesign an e!ective control system to keep pH within speci"ed values. This is so becausethe process is highly non-linear, sometimes presents non-minimum phase behaviour [4}6]and is usually subjected to periodic disturbances in inlet #owrates and concentrations.In addition, pH measurements su!er from drifts and noise. Then, pH regulation is a relevantand challenging case study to test new control algorithms. Several control algorithms havebeen proposed in the literature to design a good pH control system. Many of them usednon-linear model-based algorithms combined with some kind of adaptive feature [7}9]. Inthis sense, the controller design method usually consider the process nonlinear characteristic[10,11]. This non-linearity can be seen as a time-varying gain that leads to linear adaptivecontrollers. This is the case seen in the work by Gupta and Coughanowr [9] where theparameters of a "rst-order model are adjusted to maintain constant the product of the controllergain and the process gain (closed-loop gain). Buchholt and KuK mmel [12] propose self-tuningregulator based on recursive least-squares estimations together with a quadratic criterion. Jacobset al. [13] present a comparative study of several pH controllers, both classical and adaptive, fora strong acid}base system and the main conclusions are the necessity of continuously tuning thePID controller parameters which leads to the study of adaptive laws based on feedforward,feedback and linearization of the titration curve. Gustafsson and Waller [14] obtain a matrixformulation using non-measurable state variables independent of the chemical reaction, whichare estimated using RLS method and then controlled by using a PID controller [15]. Wright andKravaris [16] de"ne a control objective equivalent to pH (in terms of the pH measurement) andformulate a linear non-adaptive PI controller in terms of this new control objective. Besides usingthe concept of states independent of the chemical reaction given in Reference [7], the new controlobjective becomes linear with respect to these independent states which allow the designing oflinear controllers in terms of the strong acid equivalent. This new magnitude can be measuredwith special equipment or estimated on-line. When using a non-linear controller in terms of thepH, a global stabilizing controller can be derived [17], tuning two parameters. Pola andGonzaH lez [18] designed a model reference adaptive controller in a simulated reactor, obtainingsatisfactory results.

The experimental evaluation of the CMRAC and the comparison of its performance withconventional PID control and standard MRAC is reported in this work. A pH tank regulationprocess, at laboratory scale, was used as the experimental set-up. Obviously, better performancescan be achieved with non-linear model-based controllers. However, the aim of this paper is todetect the potentialities and limitations of CMRAC in order to de"ne their range of applicabilityand orient future theoretical developments. The plant model was obtained from "rst principlesfollowing the methodology of Gustafsson and Waller [14]. This model is characterized by a lineardynamical system in series with a static non-linear characteristic [14]. In the model, a term takinginto account some non-modelled dynamics has been added, such as the carbon dioxide absorp-tion [19]. The neutralization of sodium hydroxide with acetic acid is studied, trying to maintain,the pH at a pre-speci"ed value. The model validation using experimental data is then performed,prior to the implementation of control strategies from simulation and experimental viewpoints.The model of the plant was used to perform extensive simulations under di!erent controlstrategies. These simulations allowed the determination of the best set of control parameters(tuning) which were later used in the experimental set-up. Reference changes as well as step andpulse perturbations are produced to observe the behaviour of the controlled system. Mainly

86 M. A. DUARTE, F. A. ROJO AND R. PED REZ

Copyright � 2002 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. 2002; 16:85}106

Figure 1. Schematic diagram of the pH process.

experimental results are reported in this paper due to space constraint, although numeroussimulations were "rst performed.

2. EXPERIMENTAL SET-UP

The pH process consists of a glass tank or reactor R, with three inputs [20]. One is the water #ow(F

�), the second is a solution of sodium hydroxide (NaOH) acting as a perturbation (F

�) and the

third is a solution of acetic acid (HC�H

�O

�) used as control variable (F

�). The acetic acid is

commanded through a solenoid valve, V, varying the time (;�) during which it remains open. The

relationship between F�(1 min��) and ;

�(s) is given by F

�"0.06 ;

�. Figure 1 shows a diagram

of the whole process. Since the water #owrate is larger than that of the acid and base, the processtime constant will mainly depend on the former and on the reactor volume. In this process, thereis also a time delay depending basically on the time needed to get the #ows mixed homogeneouslyand the time that the sensor takes to get the pH measure.

As mentioned previously, the control action is performed by changing the #owrate of aceticacid added to the reactor over a period of time. In order to get a #owrate proportional to the timein which the solenoid valve remains open, an over#ow system has been incorporated so thata constant pressure is maintained at the valve input V. The over#ow system consists of two tanks.The main tank, P, is located at the lower part and its output is connected to an electric pump, B,that pumps acid towards the second tank, S, located at the upper part. The output of tank S isconnected to the dosi"er, D, which is permanently over#owing to obtain, in this way, a constantvolume of acid in D. The acid over#ow is sent to the main tank P and so on. The glass reactorR has two drainage pipes. The one on top is used in case the solution volume goes beyond reactorcapacity. The lower pipe is the #ow output at the desired pH.

MODEL REFERENCE ADAPTIVE CONTROLLER 87


Figure 2. Variables involved in the pH process.

3. MODELLING OF pH PROCESS

Over the last 25 years, numerous works related to modelling of pH in chemical reactions havebeen published. In this sense, it is important to mention the works by McAvoy et al. [4, 21],Gustafsson and Waller [14], Orava and Niemi [7], and Pajunen [8].

Even though the "nal evaluations were done in the laboratory, a simulator was used topre-tune the control algorithms, and perform many more evaluations than could possibly be doneby experimental runs (which are time consuming). These simulations helped to assess better theadaptive controllers and also to de"ne precisely the most di$cult situations for experimentallytesting the algorithms. Figure 2 shows the variables involved in the pH process, where F standsfor #ow and C for concentration.

To develop the model, we follow McAvoy et al. [4], assuming constant volume <�, constant

density, perfectly mixed and isothermal tank. The main species present in the reactor are: [H�],[OH�], [HC

�H

�O

�], [C

�H

�O�

�], and [Na�]. In addition, it is assumed that the water absorbs

CO�

from the air and that its concentration remains constant. Then other species, due to thecarbonic acid dissociation, are: [H

�CO

�], [HCO�

�] and [CO��

�]. If the following reaction

invariants are de"ned,

X�"[HC

�H

�O

�]#[C

�H

�O�

�] (1)

X�"[Na�] (2)

X�"[CO

�]"[H

�CO

�]#[HCO�

�]#[CO��

�] (3)

simple material balances can be written for them. The case of X�

is trivial, since it is assumedconstant. For X

�and X

�, the dynamic mass balances are

dX�

dt"�

F�<�C

�!�

F�#F

�< �X

�(4)

dX�

dt"�

F�<�C

�!�

F�#F

�< �X

�(5)

Eliminating the species using the chemical equilibrium expressions, and replacing them in theelectrical neutrality condition, the following implicit equation can be obtained for the protons



Table I. Parameters and process variables

Parameter/variable De"nition/value Parameter/variable De"nition/value

K�

Water dissociationconstant 1�10��

X�

Carbondioxide concentration5�10��M

K�

Acetic acid dissociation con-stant*1.76�10� [22]

F�

Acid #ow 0}0.8 lmin��

K��

Carbonic acid dissociationconstant*4.3�10� [22]

F�

Base #ow 0}0.05 lmin��

K��

Carbonic acid dissociationconstant*5.61�10�� [22]

F�

Water #ow 6 l min��

< Capacity 14 lNominal 10 l

C�

Acid concentration1.5�10��M

X�

Acetate concentration C�

Base concentration0}2�10�M

X�

Sodium concentration

concentration:

[H�]#X�"

K�

[H�]#

X�K

�[H�]#K

�

#

(K��

[H�]#2K��

K��

)X�

[H�]�#K��

[H�]#K��

K��

(6)

Finally, the pH is computed from the expression:

pH"!log��

[H�] (7)

Equations (4)}(7) determine the pH evolution inside the reactor. Table I gives the modelparameters and the nominal values of the main variables. K

�and K

�are the dissociation

constants of water and acetic acid, respectively. K��

, K��

are the two carbonic acid dissociationconstants associated with this chemical reaction.

This model was evaluated in steady-state and dynamic experiments. In Figure 3, the compari-son of model predictions and experimental data for two titration curves are shown. The initialconditions for these curves are given in Table II. As can be seen, there is no signi"cant di!erencebetween experiments and predictions.

The experimental evaluation in steady-state regime of the model obtained is done usingEquation (6) relating the pH of the solution with the concentration of the elements involved in thechemical reaction.

Let us assume that we have a volume<�of sodium hydroxide (NaOH) of concentration C

�and

a volume <�

of acetic acid (HAc) of concentration C�is added. Acetate concentration (X

�) and

sodium concentration (X�) of resultant solution in steady state are

X�"

C�<�

<�#<

�

(8)

X�"

C�<�

<�#<

�

(9)



Figure 3. Steady-state model performance.

Table II. Parameters for model validation

Experiment <�

(cm�) C�

(M) C�(M) X

�(M)

1 745 0.05 0.32 5.56�10��2 1000 0.05 0.32 3.66�10��

In this case <�, C

�, C

�are constant parameters, and therefore a static relationship between pH

solution and acid volume<�, called titration curve, is obtained. Two experiments were performed

with parameters shown in Table II. Figure 3 shows the titration curves obtained by replacingthese values in Equation (6). The value of the CO

�concentration (X

�) was determined by least

squares for each experiment.As observed in Figure 3, the model "ts reasonably well to the experimental data, and also, the

calibrated values of X�shown in Table II correspond to the solubility data found in the literature,

which range from 5.2�10��M at 103C to 3.8�10��M at 203C [22]. In our dynamic simula-tions, we used a constant value of 5�10��M.

In order to assess the e!ect of carbonic acid on the dynamic behaviour of the system, Figure 4compares the experimental data with two models: one that includes carbonic acid and the otherthat does not. Here, a pulse disturbance on the base #owrate from 0 to 1.98 l min�� is applied andthe system is controlled with a PID algorithm.



Figure 4. Dynamic response after an impulse disturbance under PID control. Model without CO�(***),

model with [CO�]"5�10�� (++++ ) and experimental data (z z z z ).

Figure 4 clearly shows that CO�

dissolution should be considered in the model in order torepresent properly the dynamic behaviour of the real system. The discrepancies observed betweenthe experimental data and the model that includes the CO

�e!ect, can be explained by the mixing

delay a!ecting the tank reactor, which is not included in the model. However, the model is goodenough to pre-tune the control algorithms and to evaluate the consequences of di!erentdisturbances.

4. DESCRIPTION OF CONTROL STRATEGIES

A brief description of the CMRAC algorithm used in this paper is given in what follows, just forthe sake of completeness. For a more detailed explanation of the algorithm, the reader is referredto the original sources [1}3]. The classical MRAC is also mentioned for comparison purposesonly.

4.1. Combined model reference adaptive control

The main objective of the MRAC is to minimize the error between the plant output and the modelreference output. The controller parameter adjustment is based on the control error in the directMRAC [1] and on the control and identi"cation error in the case of a combined MRAC [2].

For the particular case of the pH process under investigation, we will use a "rst-order model forthe plant around an operating point. Let the plant be de"ned by the equation

pH� (t)"a�pH(t)#b

�F�(t) (10)

and the asymptotically stable reference model

pH��(t)"a

�pH

�(t)#b

�r(t) (11)



Figure 5. MRAC scheme.

where pH(t) and pH�(t) are the plant output and the reference output, respectively, and u (t) and

r(t) are the plant input and the reference input, respectively.Under a well-known hypothesis, the control law for the direct MRAC has the form [1]

F�(t)"� (t)pH(t)#k(t)r (t)

�Q (t)"!sign(b�)�e

�(t)pH(t)

kR (t)"!sign(b�)�e

�(t)r (t) (12)

where e�(t)"pH

�(t)!pH(t), and � is the adaptive gain. The general scheme is shown in

Figure 5.For the CMRAC, the control and adaptive laws are given by [2]

F�(t)"� (t)pH(t)#k (t)r(t),

�Q (t)"!sign(b�)[�

�e�(t)pH(t)#�

�� (t)], aLQ (t)"�

�e�(t)y(t)#�

��(t)

kQ (t)"!sign(b�)[�

�e�(t)r (t)#�

��(t)], bK Q (t)"!�

�e�(t)u(t)!�

�[�(t)�� (t)#k(t)�

�(t)]

��(t)"!aL (t)#bK (t)�(t), ��(t)"bK (t)k (t)!b

�, (13)

where e�(t)"pHY (t)!pH(t), �

�, �

�, �

�, �

�are the adaptive gains and �

�(t), �

�(t) are the closed-loop

estimation errors. The general scheme of the CMRAC is shown in Figure 6.Although the continuous-time versions of MRAC and CMRAC mentioned above were used in

the simulations prior to the implementation of the control schemes, the discrete-time versions ofMRAC and CMRAC were used in the on-line control, by discretizing Equations (12) and (13),and choosing a su$cient small sampling period ¹

�. Nevertheless, the exact implementation of the



Figure 6. CMRAC scheme.

discrete-time versions of MRAC and CMRAC can be found in References [23] and [3],respectively, where theoretical stability is proved.

Since PID control is certainly the most used in industrial plants, in the following section thePID controller is described brie#y because it was also used for comparison purposes.

4.2. PID control

The digital PID controller (discrete-time) used in computer simulations and experimental results,is de"ned in terms of the incremental control and has the form

F�(k)"F

�(k!1)#K

��(e(k)!e(k!1))#¹

�¹

�

e (k)#¹

¹

�

(e(k)!2e (k!1)#e (k!2))�where F

�(t) is the control variable and e(t) is the control error. ¹

�is the sampling period and the

variable k corresponds to the instant ¹"k¹

�. This PID is of the parallel type.

Several techniques can be used to tune PID controller in the pH control [24]. Controllerparameters K

�, ¹

�, ¹

are tuned using Ziegler}Nichols [25,26], Smith}Corripio [26}28], minim-

ization of the integral of the control error [25] and IMC [29}31] methods.The simulations were performed mainly to adjust the controller parameters and to get an idea

of the overall performance of each controller before the implementation at laboratory level. Forthe sake of space, simulation results are not shown, in general, except for those directly connectedwith the CMRAC.

5. EXPERIMENTAL RESULTS

The three control algorithms described in Section 4 were evaluated from simulation andexperimental points of view. Digital PID controller, direct model reference adaptive controller



Table III. Parameters used in experimental results

Reactor volume 10 l

Acetic acid #ow (C�H

�O

�) [0}0.8] l min��

Acid concentration 1.5�10��MPerturbation base concentration 2�10�MCO

�concentration 5�10��M

(DMRAC) as well as a CMRAC were programmed in Turbo Pascal language in a PC. Accordingto simulation results a sampling period of ¹

�"20 s was chosen. The performance of these

controllers was assessed under di!erent kinds of disturbances: set point changes, step changes ininlet base #owrate and pulse disturbance in inlet base #owrate [32]. The details of the applieddisturbances are given next. All the algorithms were tuned in order to obtain the best perfor-mance for tracking and regulation. Initial tuning was based on simulations, but in some cases, thecontrol parameters were later adjusted according to experimental results.

The parameters used in all simulations and experiments are shown in Table III.

5.1. Evaluation of combined model reference adaptive control

Tuning of adaptive controllers was done by trial and error. The DMRAC tuning requires just oneparameter, the adaptive gain �. The value of this parameter used in the "nal experiments, was 30per cent smaller than the one obtained by simulations. On the other hand, in order to tune theCMRAC algorithm, four parameters need to be provided; the controller adaptive gains, �

�and �

�,

and the identi"cation adaptive gains, ��and �

�. In practice, however, only two parameters are

enough, since it is usually assumed that ��"�

�and �

�"�

�. In this case, the values used in the

experiments were also 30 per cent smaller than the ones obtained by simulations. It was veri"ed,using simulations, that the parameters of the reference model do not a!ect much the adaptivecontrollers' performances. In addition, it is not advisable to initialize these algorithms with zeroesin the model reference parameters.

The following typical changes were used to evaluate the controllers.Reference changes: Initially, the plant is driven to the operation point given by pH

��5.8. Being

at steady state, at time t"15.7 min a reference change to pH��

"7.2 is made and it is maintaineduntil time t"37min.

Step disturbance: Initially, the plant is driven to operate at pH��

"7.2. At instant t"15.7 minan alkaline #ow of concentration 0.1M is added for 20min. Then the plant is left operating withwater as in#uent #ow and acid as control #ow, while the perturbation vanishes.

Pulse disturbance: Similar to the previous perturbation, the plant is operating at pH��

"7.2until 15.7min. At this time, 100ml of sodium hydroxide 0.1M is added for 2s, and later the plantcontinues operating until 33 min from the beginning of the test.

5.1.1. Simulation Results. Prior to the experimental tests, several simulations for the CMRACwere performed using the mathematical model derived in Section 3. In order to "nd the bestcontroller parameters, changes in adaptive gains, initial conditions and model reference para-meters were studied. The simulations were performed under di!erent situations, includingreference changes, as well as step and pulse perturbations. It s worth noting that the relationshipbetween acid #ow F

�(lmin��) and the time the valve remains open;

�(s) is given by F

�"0.06;

�.



Figure 8. Simulation results for CMRAC. In#uence of adaptive gains with step perturbation.

Figure 7. Simulation results for CMRAC. In#uence of adaptive gains with reference changes.

(a) <ariations of adaptive gains. One important aspect of the CMRAC is the choice of theadaptive gains, since the transient behaviour of the controlled system depends heavily on this.Three of the numerous simulations carried out with di!erent values of adaptive gains are shownin Figures 7}9 under di!erent conditions. The values of adaptive gains used in the simulations areshown in Table IV.

Figure 7 shows the results when a change in the reference signal from pH 5.8 to 6.8 is made attime t"15.7min. The best response is obtained for �

�"�

�"3�10�� and �

�"�

�"3�10��.

The resulting simulations using the adaptive gains of Table IV and a step perturbation as thatde"ned in Section 5.1 are shown in Figure 8. The best response is obtained for �

�"�

�"3�10��

and ��"�

�"3�10��.

When a pulse perturbation like the one de"ned in Section 5.1 and the set of adaptive gainsgiven in Table IV are used, simulations are shown in Figure 9. The best response is obtained for��"�

�"3�10�� and �

�"�

�"3�10��.



Figure 9. Simulation results for CMRAC. In#uence of adaptive gains with pulse perturbation.

Table IV. Variation of adaptive gains

Simulation ��"�

��"�

�

1 1�10�� 3�10��2 3�10�� 3�10��3 5�10�� 1�10��

Table V. Variations in the model referenceparameters

Simulation a�

b�

1 0.4 0.42 0.04 0.043 0.004 0.004

(b) <ariation of model reference parameters. Another important choice in the CMRAC is themodel reference (see Equation (11)). Numerous simulations were performed for di!erent "rst-order reference models. A representative set of model reference parameters is given in Table V.

Simulation results when reference signal changes from pH 8 to 7.2 are plotted in Figure 10. Nobig di!erences are observed in all three cases.

(c) Initial condition variations. Initial conditions are also important in the behaviour ofthe resulting adaptive system when using CMRAC. Cases when y

��(0)"0 and y

��(0)"pH

�,

where pH�is a non-zero value, were studied. The simulation results when using the CMRAC are

presented in Figure 11, for pH�"8 and a

�"b

�"0.4, when changes in the reference signal from

pH 8 to 7.2 are applied. For zero initial condition, a larger overshoot is observed, since the initialerror is large with a consequently bigger e!ort on control variable.

5.1.2. Experimental tests. In this section, the experimental results of applying the CMRAC tothe pH laboratory plant are presented.



Figure 10. Simulation results for CMRAC. Variations in model reference parameters.

Figure 11. Simulation results for CMRAC. Variation of initial conditions.

(a) <ariation of adaptive gains. Several tests were performed with di!erent values of adaptivegains and changes in the set point. The values of adaptive gains used in the experiments are shownin Table VI.

Sub-index c and i stand for controller and identi"er, respectively. Initial conditions were allzero but y

�(0)"pH(0)"8. Experimental results when the set point is changed from pH 5.8 to 6.8

are shown in Figure 12. The best response is obtained for the case when adaptive gains are��"�

�"2�10�� and �

�"�

�"2�10��.

(b) Step perturbation. For this test, adaptive gains ��"�

�"2�10�� and �

�"�

�"2�10��

were used. The results are given in Figure 13 when a step perturbation like the one described inSection 5.1 was used. Initial conditions were all zero but y

�(0)"pH

�(0)"y

�(0)"pH(0)"7.7.

Set point was "xed in pH��

"7.2.(c) Pulse perturbation. In this case, adaptive gains were set to �

�"�

�"2�10�� and

��"�

�"2�10��. Experimental results obtained when a pulse disturbance like the one



Table VI. Variations in the adaptive gains

Experiment ��"�

��"�

�

1 1�10�� 1�10��2 2�10�� 2�10��3 5�10�� 5�10��

Figure 12. Experimental results for CMRAC. In#uence of adaptive gains with reference changes.

Figure 13. Experimental results for CMRAC. Step perturbation.

described in Section 5.1 are shown in Figure 14. Initial conditions were all zero buty�(0)"pH

�(0)"y

�(0)"pH(0)"8.1.

(d) Comparison of simulated and experimental results. The best results obtained for theCMRAC from simulation and experimental tests were chosen to compare both performances.Adaptive gains used in the comparison of the CMRAC under the simulated and experimentalconditions are given in Table VII.



Figure 14. Experimental results for CMRAC. Pulse perturbation.

Table VII. Adaptive gains for the CMRAC

Simulation Experiment

��"�

��"�

��"�

��"�

�3�10�� 3�10�� 2�10�� 2�10��

Figure 15. Experimental and simulation comparison of CMRAC. Reference changes.

The comparison of the results obtained from simulations and experimental set-up for theCMRAC are shown separately in Figures 15}17. Reference changes, step disturbance and pulsedisturbances are plotted, respectively.

A good agreement between simulations and experiments is found, especially in the referencechanges and pulse perturbation cases. In the case of step perturbation, a di!erence is observed inthe pH value once the perturbation ceases, since a lower pH is obtained in the simulation ascompared with the experiments.



Figure 16. Experimental and simulation comparison of CRMAC. Step perturbation.

Figure 17. Experimental and simulation comparison of CMRAC. Pulse perturbation.

5.2. PID control

PID algorithm was tuned using an IMC technique as described by Chien and Fruenhauf [29}31],giving better performances than other tuning methods also used such as Ziegler and Nichols[25,26], Smith and Corripio [26}28], and minimization of the error criteria [25]. Havingperformed extensive simulations of the PID controller, it was determined that the best values ofcontroller parameters were K

�"0.39, ¹

�"15 s and ¹

"100 s. It is important to mention that

the behaviour of the controlled system for reference changes, step perturbation and pulseperturbation, for both simulation and experimental tests, showed a good agreement particularlyfor step and pulse perturbations. Only the experimental results for PID controller are shown inFigures 18}21 under di!erent situations, for comparison with the CMRAC purposes.



Figure 18. Comparison of experimental responses under reference changes.

5.3. Direct MRAC

Similar to CMRAC and PID control, the DMRAC was extensively simulated and tested in thereal plant. The best value found for the adaptive gains for simulations and the experimentalsetting was �"2�10��.

Only the experimental results for DMRAC are shown in Figures 18}21 under di!erentsituations, as a comparison with the CMRAC.

5.4. Comparison of control strategies

In this section, a comparison of the experimental result is presented under the same conditionsand for the parameter values stated in previous sections.

5.4.1. Reference changes. With the initial condition pH"7.8 and set point in 5.8, the plantevolves until it reaches the set point (about 15 min). Then a reference change from 5.8 to 7.2 wasapplied on the controlled system. The experimental results are shown in Figure 18. It can beobserved that DMRAC and CMRAC exhibit a similar good behaviour, since the pH reaches itsreference 5.8 rapidly and with no oscillations. This is not the case for the PID control whereoscillations around the set point 5.8 are observed. When the set point is changed back to 7.2, allcontrol algorithms oscillate around the reference evidencing the di$culty of maintaining the pHnear the set point.



Figure 19. Experimental comparison of control algorithms under step perturbation.

5.4.2. Step perturbation. With the set point in 7.2, the step perturbation described in Section 5.1was applied to the experimental set-up. The responses of control schemes are shown in Figure 19.It is observed that PID behaves quite well under this type of perturbation. Adaptive controllerspresent similar behaviour between them before and after the step is applied. A similar behaviourfor the CMRAC and DMRAC is observed.

5.4.3. Pulse perturbation. Presented next is a comparison of controller's responses under experi-mental conditions when the pulse perturbation described in Section 5.1 is applied. The set pointwas "xed in 7.2. The results are summarized in Figure 20. The best response is obtained when thePID controller is used, since a smaller overshoot and the shortest settling time are obtained.There is no signi"cant di!erence between adaptive controllers, both present same overshoot andsettling time before and after the perturbation is applied.

5.5. Error comparisons

To further compare control strategies under experimental conditions, an error analysis isperformed. The criteria functions ISE, IAE and ITAE were used, which are de"ned as

ISE: ��

�

[e (t)]�dt,



Figure 20. Comparison of experimental result under pulse perturbation.

IAE: ��

�

�e(t) � dt,

ITAE: ��

�

�te(t) �dt.

The results are shown in Table VIII for all studied cases.It is observed that the PID controller exhibits the lowest criteria function of all control

strategies. Nevertheless, it has to be noticed that PID controller exhibits a good behaviour since itwas tuned with great e!ort around the operating point de"ned by the set point 7.2 and, of course,it has to be known beforehand. Experimental results, when set point was changed to 6.8 with thePID controller parameter tuned for 7.2, show an important deterioration in its behaviour asshown in Figure 21. Adaptive controllers instead give a good performance in spite of this changein the operating point.

From Figure 21, we observe that the PID controller exhibits oscillations around pH 6.8whereas the other controllers reach the steady-state regime after an overshoot. The error criteriafunctions for the di!erent controllers, shown in Table IX, illustrate this deterioration of PIDperformance.



Figure 21. Comparison of control schemes under a step perturbation with pH��

6.8.

Table VIII. Summary of criteria error for pH��

7.2

Direct CombinedCriteria PID MRAC MRAC

Reference changes ISE 28.8 56.5 57.4IAE 27.8 49.5 46ITAE 328 597 596

Step perturbation ISE 22.31 34.2 30.6IAE 26 44.1 40.1ITAE 767.4 1315.6 1207.7

Pulse perturbation ISE 15.5 24.6 28.8IAE 20 30.7 30.9ITAE 249 477.6 452.5

Table IX. Summary of error for pH��

6.8

Direct CombinedCriteria PID MRAC MRAC

Step perturbation ISE 26.9 22.8 23.8IAE 45.4 34.9 36ITAE 997.8 743.2 775.5



Similar results were also found in the case of reference changes and pulse perturbation, but theyare not shown here due to space constraint.

6. CONCLUSIONS

An experimental evaluation of the CMRAC applied to control the pH in a chemical reactor inwhich water reacts with acetic acid (weak acid) and sodium hydroxide (strong base) has beenpresented in this paper.

Its performance was compared with a classical PID controller and with another adaptivecontrol strategy (DMRAC). The algorithms were analysed under set point changes and step andpulse disturbances. The study was based on simulations of a specially developed non-linear modeland experiments at laboratory scale, though mainly experimental results have been shown.

Comparing the di!erent control strategies under set point changes, all of them result in a stableoperation bringing the pH to the desired value even in the critical case of pH 7.2 (see Figure 18).However, the PID controller presents oscillations, which is not the case in the adaptive control-lers. Nevertheless, the latter are slower than the PID controller. In the case of step perturbations,the PID controller performs better than the adaptive ones, since the latter exhibit large overshoot(see Figure 19). However, when this is done in a di!erent operating point, for which the non-adaptive controller was not tuned, the behaviour deteriorates (see Figure 21). For pulse perturba-tions, the PID controller performs the best since lesser overshoot and settling time are observed(see Figure 20). As far as the ISE, IAE and ITAE criteria are concerned, the PID controllerpresents the lowest value for the operating point de"ned by 7.2 (See Table VIII). However, itsbehaviour is highly dependent on parameter turning (See Table IX).

The adaptive controllers' performance is highly dependent on initial conditions and on thechoice of the adaptive gains. If some knowledge of the parameter values is available, it willimprove the adaptive controllers' performance, besides the fact that these controllers are also ableto account for changes in the operating point as opposite to PID control.

This study has revealed the necessity of addressing several theoretical issues related to theCMRAC, so that it can be successfully applied to industrial control processes. One is the in#uenceof the initial conditions on the transient behaviour of the adaptive system and the other isthe proper choice of the adaptive gains to obtain a better performance from tracking and regulationviewpoints. E!orts are currently being made to "nd reasonable answers to these questions.

ACKNOWLEDGEMENTS

The results reported in this paper have been funded by CONICYT through Grants FONDECYT 1950502and FONDECYT 1970351.

REFERENCES

1. Narendra KS, Annaswamy AM. Stable Adaptive System, Prentice-Hall: Englewood Cli!s, NJ, 1988.2. Duarte MA, Narendra KS. Combined direct and indirect approach to adaptive control. IEEE ¹ransactions on

Automatic Control 1989; 34 (10):1071}1075.3. Duarte MA, Ponce RF. Discrete-time combined model reference adaptive. International Journal of Adaptive Control

and Signal Processing 1997; 11(6):501}517.



4. McAvoy T, Hsu E, Lowenthal S. Dynamic of pH in controlled stirred tank reactor. Industrial and EngineeringChemistry, Process Design and Development 1972; 11(1):68}70.

5. Dumont G, Zervos C, Pageau G. Laguerre-based adaptive control of pH in a industrial bleach plant extraction stage.Automatica 1990; 26(4):781}787.

6. Gustafsson T, Waller K. Mythos about pH and pH control, A.I.Ch.E. Journal 1986; 32(2):335}337.7. Orava PJ, Niemi AJ. State model and stability of a pH control process. International Journal of Control 1974;

20:557}567.8. Pajunen G. Comparison of linear and nonlinear adaptive control of a pH-process. IEEE Control System Magazine

1987; 39}44.9. Gupta S, Coughanowr DR. On line gain identi"cation of #ow processes with application to adaptive pH control.

A.I.Ch.E. Journal 1978; (4):654}664.10. Shinskey FG. pH and pION Control in Process and=aste Streams. Wiley: New York, 1973.11. Henson MA, Seborg DE. Adaptive nonlinear control of a pH neutralization process. IEEE ¹ransactions on Control

System ¹echnology 1989; 2(3):169}182.12. Buchholt F, Kummel M. Self-tuning control of pH-neutralization process. Automatica 1979; 15:665}671.13. Jacobs O, Hewkinb PF, While C. Online computer control of pH in an industrial process. IEE Proceedings Pt.D.

1980; 127(4):161}168.14. Gustafsson T, Waller K. Dynamic modeling and reaction invariant control of pH. Chemical Engineering Science 1983;

38(3):389}398.15. Gustafsson TK. An experimental study of a class of algorithms for adaptive pH control. Chemical Engineering Science

1985; 40(5): 827}837.16. Wright R, Kravaris C. Nonlinear control of pH processes using strong acid equivalent. Industrial Engineering and

Chemical Research 1991; 30:1561}1572.17. Kravaris C, Chang-Bock Chung. Nonlinear state feedback synthesis by global input/output linearization. A.I.Ch.E

Journal 1987; 33 (4): 592}603.18. Pola M. Control adaptivo por modelo de referencia de un sistema no-lineal. Electrical Engineering ¹hesis. Santiago:

Electrical Engineering Department, University of Chile, 1981.19. Paszkiewics D, Prevot P. Modeling of a neutralization reaction and self-adaptive control of pH, Eighth IFAC/IFORS

Symposium on Identi,cation and System Parameter Estimation vol. 3, 1625}1630.20. E.E. Dept. Manual de operacioH n de la planta experimental para control de pH, ¹echnical Report, Electrical

Engineering Dept., University of Chile, 1988.21. McAvoy T. Time optimal and Ziegler}Nichols control, Industrial and Engineering Chemistry, Process Design and

Development, 1972; 11(1):72}78.22. Perry RH, Green D. Perry1s Chemical Engineers Handbook, (6th edn), Mc Graw-Hill International: New York, 1984.23. Narendra KS, Lin YH. Stable discrete adaptive control. IEEE ¹ransactions on Automatic Control 1980;

25(3):456}461.24. McMillan GK. pH Measurement and Control (2nd edn), Instrument Society of America (ISA), 1993.25. Ogata K. IngeniernHa de control moderno, Prentice-Hall: Englewood Cli!s, NJ, 1993; 634}641.26. Astrom KJ, Hagglund T. PID Controllers: ¹heory, Design and ¹unning (2nd edn), Instrument Society of America

(ISA), 1995.27. Smith CL, Corripio AB. Controller tuning from simple process models. Instrumentation ¹echnology 1975; 39}44.28. Lopez AM, Miller JA, Smith CL, Murrill PW. Tunning controllers with error-integral criteria. Instrumentation

¹echnology 1967; 57}62.29. Chien I, Fruenhauf PS. Considerer IMC tunning to improve performance. Chemical Engineer Progress 1991; 33}44.30. Morari M, Za"riou E. Robust Process Control, Prentice-Hall: NY, 1989.31. Rivera DE, Morari M, Skogestad S. Internal model control: PID controller design. Industrial and Engineering

Chemistry, Process Design and Development 1986; 25:252}265.32. Rojo F. ComparacioH n de estrategias para el control de pH, Electrical Engineering Thesis, Santiago: Electrical

Engineering Dept., University of Chile, 1997.



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