pH Control Using the Nonlinear Multiple Models, Switching, and

multiple models, switching, and tuning approach are proposed in this work. For modeling, both fixed and adaptive models are considered. An experimenta...
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Ind. Eng. Chem. Res. 2000, 39, 1311-1319

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pH Control Using the Nonlinear Multiple Models, Switching, and Tuning Approach Mahmoud Reza Pishvaie and Mohammad Shahrokhi* Chemical Engineering Department, Sharif University of Technology, Tehran, I.R. Iran

pH control is known to be a challenging problem due to its highly nonlinear nature. Several control strategies are proposed for controlling pH processes among which is the strong acid equivalent scheme. Although this technique is robust to modeling error, the controller performance is degraded if the titration curve undergoes large changes. To overcome this problem, two control schemes that are based on the strong acid equivalent strategy and that use the multiple models, switching, and tuning approach are proposed in this work. For modeling, both fixed and adaptive models are considered. An experimental setup is used to evaluate the performance of the proposed scheme. The effectiveness of the proposed schemes are demonstrated via computer simulations and experimental results. Introduction pH control is recognized as a classic and difficult control problem due to its severe nonlinearity, as reflected in the titration curve of the process stream. On the other hand, pH control has found many applications in different industries. It is well established in the literature that the conventional proportional integral derivative (PID) controller has a poor performance in controlling pH processes. Many papers on modeling and control of pH have appeared in the literature. McAvoy et al.1 developed a dynamic model for a single-acidsingle-base pH process from the first principles, namely, material balances and chemical equilibrium. This method is extended to the general pH processes by Gustafsson and Waller.2 If the process is viewed as a linear timevarying system, linear adaptive control can be used. The works of Gupta and Coughanowr,3 Buchholt and Kummel,4 Palancar et al.,5 and Mahuli et al.6 are some examples of where a linear adaptive controller was used for the pH-neutralization process. A different way to cope with the nonlinear characteristics of the pH processes is by using nonlinear control methods. The nonlinear controller may be adaptive or nonadaptive. Goodwin et al.7 proposed a linear and nonlinear adaptive controller on the basis of the difference between hydrogen and hydroxyl ion concentrations. Gustafsson and Waller2 and Gustafsson8 introduced the concept of a reaction invariant (RI), and on the basis of that, they proposed a nonlinear adaptive controller and used it for controlling the multicomponent pH process. Later on, Gustafsson and Waller9 proposed a nonlinear adaptive control scheme using a model which is composed of the total ion concentrations and the dissociation constants of fictitious weak acids. Wright and Kravaris10 reduced the RI pH process model to a first-order state equation and introduced the concept of strong acid equivalent (SAE). Their control strategy resulted in a good performance and is robust to modeling error. However, the controller shows poor performance when the titration curve undergoes large changes. Lee et al.11 proposed an * To whom correspondence should be addressed. Telephone: 0119821-6005819. Fax: 0119821-6012983. E-mail: [email protected].

automatic tuning method using relay feedback to improve the performance of the SAE controller. Sung and Lee12 used the set point change method to identify the titration curve and compensate for the nonlinearities and time-varying properties of the pH process. Both aforementioned techniques use an automatic tuning procedure which needs excitation of the plant. The excitation is performed manually and it is assumed that loads are not changed during the excitation period. Therefore, these control strategies are not suitable for the cases where the titration curve undergoes changes very frequently. Some researchers have proposed different techniques for on-line identification of the titration curve. To name a few, the works of Nortcliffe and Love,13 Sung et al.,14 and Nichols and Sinhaa15 can be mentioned. The drawback of these methods is the requirement of extra measurement and equipment. Another approach for handling the pH control is using intelligent algorithms, like artificial neural networks or fuzzy theory. The works of Palancar et al.16 and Ylen and Jutila17 can be classified in this category. In this study, two control schemes, based on a control objective, similar to SAE, are proposed. To compensate for variation of the titration curve, the multiple models, switching and tuning (MMST) technique is used. The use of multiple models is not new in control theory. In fact, multiple Kalman filters were proposed in the 1970s by Lainiotis18 and Athanes et al.19 to improve the accuracy of the states estimates in control problems. Schott and Bequette20 used a bank of adaptive models to control a polymerization reactor. Banerjee et al.21 proposed a nonlinear controller based on a scheduling multiple model. In these schemes only a linear combination of controllers is used and switching is not considered. Further, no stability results for these systems have been reported. In recent years, switching is found to be important in the context of adaptive stabilization22,23 and identification of highly nonlinear systems.24 In this work, two control schemes that are based on the SAE approach and that use the multiple models, switching, and tuning technique are proposed. The paper is organized as follows: First, modeling of the pH process and synthesis of the titration curve are

10.1021/ie990412k CCC: $19.00 © 2000 American Chemical Society Published on Web 04/07/2000

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Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000

C1 - x1(t) C2 - x2(t) Cn - xn(t) ) ) ‚‚‚ ) C1 - R1 C2 - R2 Cn - Rn

(4)

Wright and Kravaris10 reduced eq 3 into the following equation,

V

reviewed and then the MMST approach is discussed. Second, the controller design is considered. Finally, the effectiveness of the proposed schemes are demonstrated by computer simulations and experimental results. pH Processes Modeling The dynamics of pH in a continuous stirred tank reactor (CSTR) was first developed by McAvoy et al.1 Gustafsson and Waller2 extended the model to a multicomponent pH process using the concept of the RI. This general model is unobservable and uncontrollable. Wright and Kravaris10 rigorously derived a minimalorder model which has the same input-output behavior as the original detailed model. This reduced-order model has two advantages. First, it facilitates controller synthesis for pH processes, and second, it is written explicitly with respect to the titration curve of the process stream. Consider the neutralization tank shown in Figure 1. Assuming constant tank volume and perfect mixing, the standard ion balances and electroneutrality conditions along with chemical equilibrium relations yield the detailed nonlinear state space model of the process as given below:

dxi ) F(Ci - xi) + (Ri - xi)u, dt

(5)

where

Figure 1. Dynamic pH process.

V

dX ) u - (F + u)X dt

i ) 1, ..., n (1)

n

ai(pH)xi + A(pH) ) 0 ∑ i)1

(2)

where A(pH) ) 10-pH - 10pH-14 and xi is the total concentration of the ith species in the effluent stream. V and F denote the volume of the reactor and process stream flow rate and are assumed to be constant. Ri and Ci are the total ion concentrations of the ith species in the titrating stream and process stream, respectively. The flow rate of the titrating stream is denoted by u and ai(pH)’s are functions of pH and the dissociation constants. In this study we have used the model developed by Wright and Kravaris10 and therefore it is discussed below very briefly. Under the assumption that Ri and Ci are not changing with time, from eq 1 we have

V d[(Ci - xi)/(Ci - Ri)]/dt ) u - (F + u)(Ci - xi)/(Ci - Ri) (3) for each i. This implies that if the system is initially at steady state, then

n

X)

Ci - xi Ci - Ri

A(pH) + )

ai(pH)Ci ∑ i)1

n

(6)

ai(pH)(Ci - Ri) ∑ i)1

If the inverse of the titration curve is denoted by T(pH), it can be related to X(pH) by the following equation:

X(pH) )

T(pH) 1 + T(pH)

(7)

The details of the above modeling are given in the Wright and Kravaris’ paper.10 On the basis of eq 5 and by introduction of the SAE n strategy as ∑i)1 ai(pHsp)xi, Wright and Kravaris10 proposed a control scheme. Note that X is related to SAE by a linear algebraic equation. Their control strategy shows a good performance and robustness to the modeling error. However, if the titration curve undergoes large changes, the performance of their scheme will be degraded. To compensate for these changes, we consider the MMST approach, which is discussed in the next section. Multiple Models, Switching, and Tuning for pH Processes For many industrial processes, the input-output characteristics are changed when the environment has changed. If a single model is used to identify the process, it will have to adapt itself to the new condition before appropriate control action is taken. Such adaptation is usually slow and results in a large transient error. One way to handle this problem is to use the multiple models, switching, and tuning approach.25 For the pH processes, the main disturbances are the variations of the feed compositions which cause the titration curve to undergo large changes. To avoid degradation of controller performance, the controller parameters should be changed accordingly. In the multiple model approach, instead of using a single model, a bank of models are used to account for all possible plant operating conditions. The criteria for selecting the best model is usually obtained by minimization of an objective function. If the jth model prediction error is defined as below,

ˆ j(t) ejj(t) ) pH(t) - pH

j ) 1, ‚‚‚, M

(8)

where pH ˆ j(t) is the estimated value of pH obtained from the jth titration curve, then the following objective function can be used for selecting the appropriate model:25

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∫0texp(-γ(t - τ))ejj2(τ) dτ

Jj(t) ) λejj2(t) + β

(9)

In the above equation λ > 0, β, and γ g 0 are the design (switching) parameters. The recursive form of eq 9 is given below,

Jj,k ) exp(-γT)Jj,k-1 +

λ{ejj,k2

-

ejj,k-12}

2

+ Tβejj,k (10)

Figure 2. Nonlinear control structure using measurement of pH and on-line estimation of X.

where T is the sampling time and subscript k denotes the kth time interval. To avoid fast switching, a hysteresis algorithm is used in the minimization of the objective function.23 If model number j is being used at sampling time k, and Jm,k ) min{Ji,k}, then the model number j will be retained if Jj,k e Jm,k + δ, and switched to number m, otherwise. Here, δ > 0 is the hysteresis constant. In what follows the procedure of calculation pH ˆ j,k is described. By integrating eq 5 between two sample times and assuming F, V, and u are constant during this period, we have

X(pHk) ) exp(-T/τk-1′)X(pHk-1) + τk-1′ 1(1 - exp(-T/τk-1′)) (11) τ

(

)

where τ ) V/F and τk-1′ ) V/(F + uk-1). Considering M models, eq 11 becomes

Xj(pHk) ) exp(-T/τk-1′)Xj(pHk-1) + τk-1′ 1(1 - exp(-T/τk-1′)) j ) 1, ..., M (12) τ

(

)

ˆ k) can Having uk-1, pHk-1, and pHk, the value of Xj(pH be obtained for each model from eq 12 and consequently the estimated value of pH at sampling time k, that is, pH ˆ j,k. The model switching algorithm can be summarized as follows. First, a set of models (titration curves) is prepared by performing experiments or by parametrization of the titration curve and determining the model parameters through experimental data. Second, at each ˆ k) is obtained from eq 12 for each sampling time Xj(pH ˆ k) and the model set, model (j ) 1, ..., M). Using Xj(pH ˆ j(k), pH ˆ j(k) can be obtained for each model. Having pH the model error ejj(k), (j ) 1, ..., M) can be calculated from eq 8 and consequently the value of the cost function Jj,k is obtained through eq 10 for each model. The model which has the minimum Jj,k is chosen as an appropriate model. Comment. Most multiple model techniques are applied to dynamic input-output models. However, in this study the MMST approach is used for estimating the titration curve which is given by a nonlinear algebraic equation. Control Algorithm The control strategy is based on the SAE technique.10 Equation 5 is used for controller design, and eq 7 is used for the estimation of X via pH measurement. If the titration curve of the process stream does not change very frequently, an accurate estimate of X will be obtained, leading to effective control of the pH process. A schematic diagram of the method is shown in Figure 2. Once X has been estimated on-line, all that remains is to design the controller. For most industrial applications the titration stream is much more concentrated than the process stream (u , F), and therefore, eq 5

Figure 3. Flowchart of the fixed MMST algorithm.

can be written as

V

dX ) u - FX dt

(13)

The discrete form of the eq 13 is as follows,

Xk ) exp(-T/τp)Xk-1 + Kp(1 - exp(-T/τp))uk-1 (14) where τp ) V/F and Kp ) 1/F. On the basis of eq 14 and using the pole-placement technique, the PI controller equation in the velocity form with its parameters is given below,

uk ) uk-1 + ∆uss + (KC + TKI)ek - KCek-1 (15) where

h (pHsp,k) - X(pHsp,k)}, ∆uss ) F{X ek ) X ˆ (pHsp) - X ˆ (pHk) KC )

exp(-T/τp) - p1p2 Kp(1 - exp(-T/τp))

, KI )

1 -(p1 + p2) + p1p2 TKp(1 - exp(-T/τp))

p1 and p2 are the desired poles. X h and X are the current (after switching) and former (before switching) titration curves, respectively. X ˆ (pH) is the estimation of X obtained from the most appropriate titration curve recognized by the switching module. Note that when switching occurs, the term ∆uss in eq 15 is being changed and results in improving the controller’s performance. A flowchart of the proposed algorithm is shown in Figure 3. If it is necessary to take the bilinearity into account, the modified PI control law will be given as shown below:

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Figure 4. Flowchart of the adaptive MMST algorithm.

uk )

1-X ˆ (pHk-1) ∆uss uk-1 + + 1-X ˆ (pHk) 1-X ˆ (pHk) (KC + TKI)ek - KCek-1 1-X ˆ (pHk)

(16)

Switching and Tuning with Adaptive Models The first step in obtaining an adaptive scheme is to formulate the problem in a parametric fashion. Because the severe nonlinearity of the pH process is reflected in the titration curve, we may parametrize the titration curve of the influent stream. However, this is a modeling issue, and any simple parametrization of a titration curve can be used. In this work, each titration curve is modeled by the following convex linearly parametrized relation:

XR. With this strategy, a reduction in the number of fixed models is obtained for the same level of performance. The flowchart of the proposed scheme, which uses a combination of fixed and adaptive models, is shown in Figure 4. In the following, the adaptation of XR(pH) between two switching occurrences is considered. Writing eq 17 at time (k - 1)T and kT gives N

XR(pHk) )

(1 - wi,R,k-1)XLi (pHk)} (18) N

XR(pHk-1) )

N

Xj(pH) )

∑ i)1

A similar approach was proposed by Gulaian and Lee.26 L In the eq 17, XH i and Xi (pH) are the base titration curves with known species and concentrations (maximum and minimum). One can use M titration curves with N-dimensional parameter vectors to account for variations of the process titration curve. M and N should be large enough to cover the whole range of pH changes. One way of reducing M is using adaptive model XR, instead of fixed models {Xj} which is considered next.25 For an adaptive version we have M fixed models with their corresponding fixed wij’s and one additional model with variable wR which can change continuously. At each sampling time, wR is updated until model switching occurs. In this case wR is reinitialized to wij of the selected model and adaptation continues. The reinitialization is used to speed up the convergence of wR. This is precisely what is accomplished by using switching to determine good initial conditions for the adaptation of

{wi,R,k-1XH ∑ i (pHk-1) + i)1 (1 - wi,R,k-1)XLi (pHk-1)} (19)

L {wi,j XH i (pH) + (1 - wi,j)Xi (pH)}

j ) 1, ..., M (17)

{wi,R,k-1XH ∑ i (pHk) + i)1

In the above equations, it has been assumed that parameters do not change between two sampling times. By substituting eqs 18 and 19 into eq 12, we have T θR,k-1 νR,k ) ΦR,k

(20)

where N

νR,k )

∑ i)1

N

(

1-

T ΦR,k

XH ∑ i (pHk-1) + i)1

XLi (pHk) + exp(-T/τk-1′)

) [φ1,k ‚‚‚ φi,k ‚‚‚ φN,k],

)

τk-1′ τ

(1 - exp(-T/τk-1′))

T ) [w1,R,k ‚‚‚ wi,R,k ‚‚‚ wN,R,k] θR,k L φi,k ) XH i (pHk) - Xi (pHk) L exp(-T/τk-1′){XH i (pHk-1) - Xi (pHk-1)}

Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000 1315 Table 1. Parameter Values Used in the Simulation Examples F ) 200 mL/min umin ) 0 mL/min umax ) 130 mL/min

V ) 1700 mL T ) 0.1 min R ) 0.03 N

uss,acidic ) 8.27 mL/min uss,no minal ) 38.95 mL/min uss,bufferic ) 116.48 mL/min

concentration, ×104 M (acidic solution)

concentration, ×104 M (nominal solution)

concentration, ×104 M (buffer solution)

10 0 0 0 1 1 1

0 0 10 15 20 15 0

0 25 25 25 25 25 100

strong acid biprotic weak acid (pKs ) 2, 8) monoprotic weak acid (pK ) 3) monoprotic weak acid (pK ) 4) monoprotic weak acid (pK ) 5) monoprotic weak acid (pK ) 6) monoprotic weak acid (pK ) 7)

Figure 5. Nominal and perturbed X(pH)’s used for simulation.

If θˆ denotes the estimated value of θ, then the identification error is given by T ejR,k ) νR,k - ΦR,k θˆ R,k-1

Figure 6. Closed-loop responses of the nonadaptive SEA and MMST schemes for changing to an acidic solution.

(21)

Any recursive identification method such as a recursive least-squares or gradient algorithm can be used for updating θˆ R,k-1. Computer Simulation To demonstrate the effectiveness of the proposed scheme, computer simulations were carried out for a synthetic and fairly complex system. A system consisting of a strong acid, a biprotic weak acid (with pKs 2 and 8), and seven monoprotic weak acids with pK values of 3, 4, 5, 6, and 7, being neutralized by sodium hydroxide, was selected for simulation purposes. The parameters and initial steady-state values of the system are given in Table 1. At time 10, the inlet concentrations are changed. The nominal titration curve and its variations before and after the disturbances are shown in Figure 5. The closed-loop responses of the nonadaptive SAE controller and proposed MMST approach for the first disturbance which is changing from a nominal solution to an acidic one are shown in Figure 6. The closed-loop poles are assigned to 0.9 and the nominal titration curve is used for the estimation of X for the SAE controller (see Figure 5.). As can be seen, the performance of the proposed scheme is superior. The second disturbance is changing feed concentrations in the reverse direction. The closed-loop responses of the two aforementioned controllers with the same tunings are shown in Figure 7. As can be seen, the nonadaptive

Figure 7. Closed-loop responses of the nonadaptive SEA and MMST schemes for changing to a buffer solution.

SAE is very sluggish. It should be mentioned that a better performance of the MMST scheme has been obtained at the expense of a greater computational effort. The corresponding model numbers and switching patterns of the two MMST schemes are shown in Figure 8. To support the MMST approach, 104 models are used to account for titration curve variations. To create the models, the following procedure is used. The minimum and maximum concentrations over all species are selected. The difference between these concentrations is divided into q equal intervals. q groups of models are

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Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000 Table 3. Parameter Values Used in the Experimental Studies F ) 185 mL/min uss ≈ 50 mL/min umin ) 0.0 mL/min umax ) 145 mL/min

acetic acid

V ) 1720 mL T ) 0.1 min R ) 0.03 N pKa ) 4.78

CL ) 1.00 × 10-4 M CH ) 2.50 × 10-3 M

Cd,decrease, M

Cnominal, M

Cd,increase, M

10-4

10-4

2.19 × 10-3

1.09 ×

8.75 ×

until the maximum concentration is reached. For the example considered for the simulation study, n is 7, q is 15, and the details are given in Table 2. The parameters of switching for the MMST scheme were λ ) 50, β ) 10, and γ ) 0.1 and the hysteresis threshold δ set to 2.2. Experimental Evaluation Figure 8. Switching patterns for changing to an acidic solution (upper) and changing to a buffer solution (lower). Table 2. Models Used in the Simulation Examples characteristic model no. concentration ×103

model no.

characteristic concentration ×103

1 2, ..., 8 9, ..., 15 16, ..., 22 23, ..., 29 30, ..., 36 37, ..., 41 42, ..., 48

49, ..., 55 56, ..., 62 63, ..., 69 70, ..., 76 77, ..., 83 84, ..., 90 91, ..., 97 98, ..., 104

8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

generated as described next. For group one, the concentration of each species is set to a minimum concentration and the concentrations of the other species are set to zero. In this way, n models are generated for group one. Next, one increment is added to the minimum concentration and the same procedure is repeated to generate the models of group two. This is continued

Figure 9. Schematic diagram of the experimental system.

To evaluate the performance of the proposed schemes, a bench-scale pH setup is used. A schematic diagram of the system is shown in Figure 9. The process consists of acid and base streams, both being fed into the neutralization process tank. The effluent pH is a measured variable. The base flow rate is regulated by a motorized control valve, while the acid stream is controlled manually. The level of liquid in the process tank is kept constant by an overflow. The process is monitored and controlled by an IBM-compatible PC through an interface card. The main unmeasured load considered for this study is a variation of the feed composition. The feed stream is a diluted acetic acid and caustic soda is used as the titrating agent. The disturbances are produced by an injection of pure acetic acid into the process stream reservoir to increase the feed concentration and an injection of water to decrease the feed concentration. The parameters of the experimental system are summarized in Table 3. and the nominal

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Figure 10. Nominal and perturbed X(pH)’s of the experimental run.

Figure 11. Closed-loop response of the nonadaptive SAE controller for increasing feed concentration.

Figure 13. Closed-loop response of the fixed MMST controller for increasing feed concentration.

Figure 14. Closed-loop response of the fixed MMST controller for decreasing feed concentration.

The poles were placed at 0.9 to avoid control valve saturation. To examine both fixed and adaptive MMST approaches, the bank of models should be produced. However, it may be done by a simple parametrization of titration curves. For this simple process, the titration curve can be modeled by the following monoparametric equation:

X(pH) ) wXH(pH) + (1 - w)XL(pH)

Figure 12. Closed-loop response of the nonadaptive SAE controller for decreasing feed concentration.

titration curves along with the disturbances are shown in Figure 10. The closed responses of the nonadaptive SAE controller for load rejection are shown in Figures 11 and 12.

(22)

To have a fairly complete model set, the parameter of w is varied from 0 to 1, whereas w ) 0 corresponds to model no. 1 or dilute base titration curve (XL(pH)) with zero concentration of acetic acid. w ) 1 corresponds to the last model in the database, the highly concentrated base titration curve (XH(pH)). For the fixed MMST approach, 51 models are used, which are generated by varying w from 0.00 to 1.00, by increments of 0.02. For the adaptive version, 11 models (with increments of 0.1 for w) were used. The closed-loop responses of the fixed MMST approach are shown in Figures 13 and 14. The controller parameters were the same as those for the nonadaptive SAE approach used for simulation and the parameters

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Conclusions In this work, control of the pH process which has highly nonlinear character is considered. Performances of most controllers proposed in the literature are degraded, if the titration curve undergoes large changes. By use of the multiple models, switching, and tuning approach, these variations are taken into account. On the basis of a performance index for model selection and the use of a SAE approach, two controller schemes are proposed. Both fixed and adaptive multiple models are considered. For the fixed models, the number of titration curves will be larger. For the adaptive multiple models, the number of titration curves is decreased, which reduces the computation efforts needed for each model, but on the other hand, computation for model identification is increased. The effectiveness of the proposed schemes are shown by computer simulations and experimental runs. Figure 15. Closed-loop response of the adaptive MMST controller for increasing feed concentration.

Figure 16. Closed-loop response of the adaptive MMST controller for decreasing feed concentration. Table 4. IAE for Three Controllers Used for the Experimental Runs controller nonadaptive SAE fixed MMST adaptive MMST

concentration increase concentration decrease 5.4725 2.0288 1.5801

6.9802 1.0149 0.7547

of switching were λ ) 50, β ) 10, γ ) 1.0, and the hysteresis threshold equal to 0.2. The closed-loop responses of adaptive MMST are shown in Figures 15 and 16. Adaptation of parameter w was done using the least-squares method with covariance resetting. To compare the performances of the proposed schemes, the integrals of absolute error (IAE) are summarized in Table 4. Regarding the pH sensor delay, it should be mentioned that the sensor used for the experiments has a time constant of 2 s which is less than the sampling period of 6 s used for the experimental study. In the simulation study sensor delay was considered (not shown). As is expected, increasing the delay degrades the controller’s performance. However, one can make the system more robust by detuning the controller.

Nomenclature A ) anion of acid A(pH) ) term depending on pH in a general titration curve ai(pH) ) pH factor, function of pH that appears as a coefficient of the ith ionic total concentration Ci ) total ion concentration of the ith species in the process stream, gmol/L CL ) the minimum concentration of acetic acid to produce a lower base titration curve CH ) the maximum concentration of acetic acid to produce an upper base titration curve ejj ) jth model prediction error e ) feedback error eR ) identification error F ) flow rate of the process stream, mL/min Jj ) cost function of the jth model [H+] ) hydrogen ion (proton) concentration in an effluent stream, gmol/L Kai ) ith dissociation constant of the acid Kbi ) ith dissociation constant of the base KC, KI ) parameters of the digital PI controller M ) gmol/L M ) number of models n ) number of species N ) number of parameters p1, p2 ) z-domain closed-loop poles pH ) -log[H+] pKai ) -log Kai pKbi ) -log Kbi t ) time, min T ) sampling period T(pH) ) inverse of standard titration curve u ) control variable, flow rate of the titrating stream, mL/ min V ) volume of the CSTR, mL xi ) total ion concentration of the ith species in the effluent stream, gmol/L X ) reduced nonlinear state of the system, T(pH)/(1 + T(pH)) Greek Symbols Ri ) total ion concentration of the ith species in the titrating stream, gmol/L τ ) time constant, min λ ) switching parameter, coefficient of the current error β ) switching parameter, coefficient of the cumulative history of error γ ) switching parameter, forgetting factor δ ) switching parameter, hysteresis threshold

Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000 1319 Subscripts d ) disturbance j ) index to model number R ) recursive sp ) set point value ss ) steady-state value Superscripts H ) high L ) low ∧ ) estimation

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Received for review June 10, 1999 Revised manuscript received December 21, 1999 Accepted January 11, 2000 IE990412K