Nonlinear Control of Neutralization Processes by Gain-Scheduling

Res. 1991, 30, 1561−1572). The central idea of this paper is to combine exact feedback linearization and gain-scheduling techniques to obtain a nonl...
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Ind. Eng. Chem. Res. 1996, 35, 3511-3518

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Nonlinear Control of Neutralization Processes by Gain-Scheduling Trajectory Control Karsten-U. Klatt*,† and Sebastian Engell Process Control Group, Department of Chemical Engineering, University of Dortmund, D-44221 Dortmund, Germany

A new approach to the nonlinear control of neutralization processes in continuous stirred tank reactors is presented. The controller design is based on a process model in terms of the standard titration curve of the process which was proposed by Wright and Kravaris (Wright, R. A.; Kravaris, C. Ind. Eng. Chem. Res. 1991, 30, 1561-1572). The central idea of this paper is to combine exact feedback linearization and gain-scheduling techniques to obtain a nonlinear control scheme which preserves the advantages and overcomes some of the problems of the two single concepts. In a first step, a nonlinear state feedback controller is computed by exact input/output linearization of the process model to shape the nominal closed-loop system. The required state variables are computed by an on-line simulation of the process model. This part of the controller thus is a pure nonlinear feedforward compensator for the nominal plant. To act against disturbances and model uncertainty, a nonlinear gain-scheduled reference controller is designed by approximately linearizing the process model not for a number of operating points, as in the standard gain-scheduling approach, but around the nominal trajectory generated by the feedforward controller. The feasibility of the proposed design approach is demonstrated on a real laboratory scale neutralization process where acetic acid is neutralized by sodium hydroxide. 1. Introduction The control of pH processes is widely recognized as a difficult problem in chemical process control and has received considerable attention in the literature (e.g., McAvoy et al. (1972), Shinskey (1973), Gustafsson and Waller (1983), and Wright and Kravaris (1991)). We here consider neutralization processes in a continuous stirred tank reactor (CSTR) as shown in Figure 1, where the pH-value is controlled by the flow rate u of the titrating stream. The inherent nonlinearity of the neutralization process arises from the nonlinear dependence of the pH-value on the amount of reagent which is represented by the so called standard titration curve. If this nonlinearity is severe, the process gain changes widely within the range of operation and classical linear feedback does not achieve satisfactory performance (Shinskey, 1973). The development of nonlinear modelbased control techniques, beginning in the mid-1980s (see, e.g., Bequette (1991) for a detailed review), has provided tools for handling strongly nonlinear systems. Two important approaches to the design of nonlinear control laws are exact feedback linearization (Kravaris and Kantor, 1990; Henson and Seborg, 1990) based on the differential geometric control theory (Isidori, 1989; Nijmeijer and van der Schaft, 1990) and gain-scheduling techniques based on approximate linearization of the process model and the concept of linearization families (Rugh, 1991; Wang and Rugh, 1987). The central idea of exact feedback linearization is to transform a nonlinear system into a (fully or partly) linear one by means of a coordinate transformation and/ or nonlinear feedback so that linear control techniques can be applied and stability as well as performance of the resulting control scheme can be ensured a priori for the nominal system. However, there are also some important shortcomings and limitations associated with the feedback linearization approach. One major draw†

E-mail address: [email protected].

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Figure 1. Neutralization process in a CSTR.

back of feedback linearization is that one generally needs to know the whole state of the nonlinear process to design the control structure. In general, this information is not available from measurements and the use of nonlinear observers is a difficult problem because the separation principle does not hold in the nonlinear case. Another problem is due to the fact that an exact model of the nonlinear process is generally not available. Thus, robustness to parameter uncertainty and unmodeled dynamics is required but cannot be guaranteed. This may lead to offset or deviations from the specified closed-loop dynamics. Gain-scheduling is an attempt to apply the wellknown linear control methodology directly to the control of nonlinear systems. The main idea is to select a number of operating points which cover the intended range of process operation. The linear time-invariant approximation of the process at these points is parametrized, and a linear controller is designed for each operating point, thus leading to a parametrized linear control law. Between the operating points, the parameters of the controller are then interpolated, or scheduled; this yields a global compensator. Its approximate linearization at any operating point yields the previously designed linear controller. The main problem associated with gain-scheduling is that the design is inherently local so that global stability and performance of the overall control scheme cannot be assured by design. The scheduled controller parameters are valid for constant operating points, which are steady states of the system. However, on an arbitrary transient © 1996 American Chemical Society

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trajectory the system operates more or less far from the operating point which corresponds to the actual values of the scheduling variables. Thus, the controller operates correctly only in the vicinity of a steady state; otherwise performance deterioration or even closed-loop instability may occur (Klatt, 1995). These considerations motivated the idea of gainscheduling trajectory control (GSTC) as a new process control scheme which combines exact feedback linearization and gain-scheduling techniques to make use of the advantages and reduce or even overcome some of the disadvantages of the single concepts (Klatt and Engell, 1995a). In a first step, a suitable process model is used to design a nominal nonlinear state feedback controller using exact linearization techniques to specify a nominal closed-loop trajectory. An on-line simulation of the process model forced by the output of the feedback linearizing controller is used to calculate the nominal state variables for the state feedback controller. This part of the controller, thus, is a nonlinear feedforward compensator for the nominal plant. However, in practice the process model will not exactly match the actual process dynamics, and unmodeled disturbances will be present. Therefore, a nonlinear gain-scheduling reference controller is designed whose output is added to the output of the nonlinear feedforward controller to act against disturbances and model mismatch. The gainscheduling controller is computed by approximately linearizing the process model around the nominal trajectory generated by the feedback linearizing controller and not for a number of operating points, as in the standard gain-scheduling approach. Thus, the transient dynamics is correctly represented in the gain-scheduling controller design. The linear approximation of the trajectory control loop meets the specifications of the robust linear design, and the gain-scheduling reference controller forces the process output to keep up with the nominal trajectory in order to ensure robustness of the overall nonlinear control scheme. The standard approach to the modeling of pH processes involves ion balances and chemical equilibrium relations (Wright and Kravaris, 1991; Gustafsson and Waller, 1983). The resulting model is partly unobservable and uncontrollable in a system theoretical sense and has an implicit output equation. Thus, it is not well-suited for the design of a GSTC structure. Wright and Kravaris showed that a minimal-order model, which has the same input/output behavior as the original detailed model, can be derived by using the standard titration curve of the process (Wright and Kravaris, 1991). Here, this modeling approach is applied together with the gain-scheduling trajectory control strategy to obtain a nonlinear control design concept for neutralization processes, the standard titration curve of which is known approximately. This paper is organized as follows. In the next section we explain how to get a minimal order model for neutralization processes as shown in Figure 1 in terms of their standard titration curve. Then, the basic concepts of exact input/output linearization and standard gain-scheduling are briefly reviewed. Subsequently we explain the concept of gain-scheduling trajectory control in detail and use this approach together with the proposed model to develop an overall nonlinear control scheme for neutralization processes. Finally we show the feasibility of the proposed control method by applying it to a laboratory scale neutralization reactor where acetic acid is neutralized by sodium hydroxide.

Figure 2. Standard titration curve for acetic acid (0.007 m) and sodium hydroxide (0.01 m).

2. Minimal Order Process Model in Terms of the Standard Titration Curve We consider a neutralization process as shown in Figure 1. This process is characterized by the presence of acid/base reactions which take place very quickly so that the overall system can usually be considered to be in chemical equilibrium. The pH-value of the mixture depends on the hydrogen ion concentration by

pH ) -log10[H+] When a sample of the process stream is titrated with the same agent as in the titrating stream, a standard titration curve, as shown in Figure 2, results. The standard titration curve shows the pH-value of the mixture depending on the ratio of the titrating stream u and the process stream F:

pH ) fT(u/F)

(1)

Because the standard first-principle approach for modeling this process involving material balances and chemical equilibrium relations leads to a process model which is not well-suited for model-based nonlinear control design, Wright and Kravaris derived a minimalorder model using the inverse of the standard titration curve

u IT(pH) ) (pH) ) fT-1(pH) F

(2)

to facilitate controller synthesis for neutralization processes (Wright and Kravaris, 1991). Assuming constant temperature and perfect mixing within the reactor, the process dynamics can be described by the first order model

1 dz ) [u - (F + u)z] dt VR

(3)

where the state variable z is related to the inverse of the standard titration curve by

z)

IT(pH) 1 + IT(pH)

) φ(pH)

(4)

The process output is explicitly given by

y ) pH ) φ-1(z)

(5)

The main difficulty with this modeling approach is to compute the inverse function φ-1(z). In Wright and

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Kravaris (1991) and Wright et al. (1991) this is avoided by defining an auxiliary output, the strong acid equivalent, which is directly related to the titration curve and can be computed without inverting φ. However, this only enables the control of the real process output pH in an indirect way. Therefore, we follow a different approach by differentiating (5) with respect to time

y˘ )

∂φ-1(z) dz ∂z dt

(7)

This results in a first order nonlinear differential equation

y˘ )

u - (F + u)φ(y) VR(∂φ(y)/∂y)

(8)

which explicitly describes the idealized dynamics of the neutralization process for a known standard titration curve. In general, the nonideal mixing characteristic of a real reactor and/or measurement delays increase the relative degree of the system. This has to be considered for the controller design and is approximated by an additional first order system with time constant τ. So we obtain the following nonlinear process model for neutralization processes in a CSTR:

x˘ 1 )

r ) min{k:{LgLfk-1h] * 0}

(6)

and substituting

∂φ-1(z) 1 ) ∂z ∂φ(y)/∂y

Since the Lie derivative is also a vector function, repeated use of this operation is possible. The relative degree of a system indicates the number of times the output y has to be differentiated with respect to time in order to have the input u appear explicitly. In terms of the Lie derivatives the relative degree r of the nonlinear system (10) is defined as

(1 + IT(x1))(u - IT(x1)F)

3.1. Exact Input/Output Linearization. Input/ output linearizing controllers have been designed for a variety of chemical process models. The design of an exact input/output linearizing controller for nonlinear SISO systems of the form of (10) was originally presented in Kravaris and Chung (1987) and is only sketched here. Suppose the nonlinear system (10) is minimum phase (i.e., its zero dynamics is stable) and has a well-defined relative degree r. Then the static nonlinear state feedback

u ) k(x,v) )

u(t) ) k(z(t), e(t)) (9)

where y is the measured output. This modeling approach leads to a minimal-order description of the process dynamics which only depends on the standard titration curve of the process streams. Knowledge of the particular chemical species and their dissociation constants is not required; thus, a more macroscopic description of the process dynamics results.

RL(R) ) {uS(R), xS(R), yS(R)}

The process model (9) belongs to the class of single intput/single output (SISO) analytical nonlinear systems which are affine in the input u. These systems are generally described by

(10)

where x is an n-dimensional state vector, u is the scalar manipulated input, and y is the scalar output. f and g denote smooth vector fields on Rn, and h denotes a smooth vector function on Rn. The Lie derivative of the function h(x) along the vector field f is defined as

∑

(13)

so that

f(xS(R)) + g(xS(R)) uS(R) ) 0, h(xS(R)) ) yS(R) Then, the corresponding linearization family of the nonlinear system (10) is

x3 (t) ) f(x(t)) + g(x(t)) u(t)

n ∂h(x) ∂h(x) ‚f(x) ) fi(x) ∂x i)1 ∂xi

(12)

such that the linear approximation (Taylor linearization) of the closed-loop system at each constant operating point has fixed dynamical characteristics (eigenvalues, gain margin, etc.). Here, z represents the state vector of the controller and e denotes the difference e ) w - y between a scalar exogenous input w (e.g., a reference signal) and the process output y. Suppose that (10) has a parametrized family of constant operating points

3. Controller Design

y(t) ) h(x(t))

(11)

z3 (t) ) l(z(t), e(t), u(t))

1 x˘ 2 ) (x1 - x2) τ

Lfh )

LgLfr-1h(x)

leads to a linear map from the new input v to the output: y(r) ) v, and one can impose linear feedback control on it in order to assign a specific closed-loop behavior. 3.2. Standard Gain-Scheduling. The objective of the standard gain-scheduling approach (Wang and Rugh, 1987; Rugh, 1991) is to compute a nonlinear output control law of the form

VR(∂IT(x1)/∂x1)

y ) x2 ) pH

v - Lfrh(x)

∆x3 ) A(R)∆x + B(R)∆u ∆y ) C(R)∆x

(14)

where ∆ denotes small deviations from the respective operating point. A(R), B(R), and C(R) are the corresponding parametrized Jacobians. Linear controllers for (14) are computed at each operating point to get a specified dynamical behavior of the linearized closedloop system. This leads to a parametrized linear controller:

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∆z3 ) L1(R)∆z + L2(R)∆e ∆u ) K1(R)∆z + K2(R)∆e

(15)

From this parametrized linear controller the overall nonlinear controller (12) has to be constructed such that its linear approximation at a certain operating point in RL(R) leads to the corresponding linear controller. There is no unique solution for this problem. One possibility is to replace the parameter R by a proper chosen measurable exogenous or endogenous variable β(t) of the control loop such that βS ) R holds for each operating point of RL(R). Replacing the parameter R of the parametrized linear control law (15) by the socalled scheduling variable β(t), one gets the resulting nonlinear control law:

z3 ) L1(β)[z - zS(β)] + L2(β)[e - eS(β)] u ) uS(β) + K1(β)[z - zS(β)] + K2(β)[e - eS(β)] (16) The linearization of this nonlinear controller at each operating point of RL(R) yields the corresponding linear controller from (15) if certain linearization conditions are met. These conditions result from computing the partial derivatives of (16) and comparing them with the corresponding terms of the parametrized linear control law (15) (Wang and Rugh, 1987). The major advantage of the standard gain-scheduling approach is that one can use all known techniques for linear controller design to compute a nonlinear controller for a nonlinear process in a comparatively easy way. 3.3. Gain-Scheduling Trajectory Control. Robustness against model uncertainties and unknown disturbances is a critical issue in the exact linearization approach, especially for its implementation in real processes. Usually the resulting error e ) w - y is fed to an integral part added to the feedback control law in order to achieve convergence to the reference signal. However, this slows down the closed-loop response considerably, even if there is no model mismatch or disturbance, and may lead to stability problems and wind-up effects in the case of manipulated variable constraints. Gain-scheduling for a family of operating points often leads to nonlinear controllers which show satisfactory performance even for fast transient processes, although the design approach is valid only in the immediate vicinity of the respective operating points. Between these points the controller parameters are scheduled by a projection onto the corresponding operating point, and one tries to keep the resulting error small by a proper choice of the scheduling variable (Klatt, 1995). However, on an arbitrary transient trajectory the linearized dynamics of the gain-scheduling controller are different from their steady state linearization. This may lead to bad performance or even to instability of the overall control scheme for fast transients. Therefore linearization around a specified transient trajectory seems to be more meaningful when gain-scheduled controllers are computed for systems which are not operated only in the neighborhood of steady state operating points. The transient trajectory itself has to be generated by suitable nonlinear control techniques. These considerations motivated the idea of gainscheduling trajectory control as described in Klatt and Engell (1995b). The nominal process model (9) is an open-loop stable, minimum phase and has a well-defined

Figure 3. Gain-scheduling trajectory control (overall structure).

relative degree r ) 2 within the physically meaningful range of operation for neutralization processes, so that the necessary conditions for exact input/output linearization are satisfied. Thus, in the first step we design an input/output linearizing nonlinear state feedback controller for the nonlinear process model (9) to specify a nominal transient trajectory for the pH process ensuring nominal closed-loop stability and performance. The nonlinear static state feedback controller

u˜ )

(∂IT(x1)/∂x1)(wVRτ2 + VR(x1 - x2)) τ(1 + IT(x1))

+ IT(x1)F

(17)

where

w)

-k1 (x - x2) + k0(ysp - y) τ 1

achieves a linear input/output behavior with closed-loop transfer function

k0 Y(s) ) 2 Ysp(s) s + k1s + k0

(18)

from a specified set point ysp to the controlled output y. The state variables are reconstructed by using an online simulation of the nominal process model (9) forced by the output u˜ of the nonlinear state feedback (17) as an open-loop observer. Because the process model (9) is open-loop-stable, the open-loop observer can be properly initialized by assuming that the process is in the steady state which corresponds to the actual measured output. In the case of a perfect model and vanishing disturbances, this nonlinear feedforward compensator forces the process to track the desired transient trajectory (u˜ , x˜ ) which is specified by the choice of k0 and k1. However, this nominal nonlinear controller is not able to deal with model uncertainties and/or disturbances, for example, deviations from the standard titration curve, so steady state offset and deviations from the specified nominal dynamics occur. Therefore, the resulting trajectory tracking error e ) y˜ - y is fed to a nonlinear gain-scheduling reference controller whose output uδ is added to the output u˜ of the nominal controller in order to force the process output y to keep up with the desired nominal trajectory. The resulting control structure is shown in Figure 3. It has some similarity with adaptive feedback linearization (Kabuli and Kosut, 1992) and asymptotically exact linearization (Gilles et al., 1994), but here a gain-scheduling reference controller is used to ensure robustness of the overall control scheme. The linear approximation of the neutralization process model (9) at an (arbitrary) point (u0, x0) of a transient trajectory results in a parametrized linear transfer function

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G(s) )

b (s + a)(1 + τs)

(19)

where

a)

F0 + u0 VR

b ) f(x0)

A parametrized linear PI-controller was designed such that the tracking error for the desired nominal trajectory e ) y˜ - y converges to zero with specified dynamics for the linear approximation around the nominal trajectory. Because the parameter a of the transfer function (19) changes only slightly within the standard operating range of neutralization processes whereas the process gain varies by several orders of magnitude depending on the actual pH-value within the reactor, the PIcontroller is parametrized only in the pH-value on the nominal trajectory y˜ :

(20)

Here, ∆ denotes small deviations from the nominal trajectory, and xR represents the controller state. As in the standard gain-scheduling approach, the parameter y˜ in eq 20 is replaced by an appropriate scheduling variable β so that for each point on the nominal trajectory β0 ) y˜ holds. The actual process output y is an obvious choice for the scheduling variable, because y(t) ) y˜ (t) on the nominal trajectory. This results in an overall nonlinear gain-scheduling reference controller:

x˘ R(t) ) KI(y(t))[y˜ (t) - y(t)] uδ(t) ) xR(t) + KP(y(t))[y˜ (t) - y(t)]

(21)

This nonlinear reference controller forces the process output y(t) to track the desired nominal trajectory also in the case of model uncertainties and disturbances. As e0 ) 0 and xR0 ) 0 for an arbitrary point on the nominal trajectory, the linear approximation of (21) yields the respective linear controller:

| |

| |

∂x˘ R ∂x˘ R ) 0, ) KI(y˜ ) ∂xR 0 ∂e 0 ∂uδ ∂uδ ) 1, ) KP(y˜ ) ∂xR 0 ∂e 0 Thus, the linear approximation of the trajectory control loop around the nominal trajectory has the dynamics specified by the gain-scheduling controller design. The actual value of the manipulated variable u(t) is the sum of the output of the nominal nonlinear feedback controller (17) and the gain-scheduling reference controller (21)

u(t) ) u˜ (t) + uδ(t)

The pH-value of the reactor effluent has to be adjusted without steady state offset in a range between pH 6 and pH 10 in the presence of varying process streams and deviations from the standard titration curve. The concentration of sodium hydroxide in the titrating stream is cB0 ) 0.01 mol/L. The nominal concentration of acetic acid in the process stream is cA0 ) 0.007 mol/ L; the nominal flow rate of the process stream is F ) 20 L/h. Due to the incomplete dissociation of acetic acid in water and its reaction with sodium acetate, the system behaves like a buffer solution for low pH-values. Figure 2 shows the standard titration curve of this process which represents its steady state characteristics. For this reaction system the inverse of the standard titration curve is explicitly given by

IT(pH) ) -

∆x˘ R ) KI(y˜ )∆e ∆uδ ) ∆xR + KP(y˜ )∆e

CH3COOH + NaOH ) CH3COONa + H2O

(22)

4. Experimental Results The feasibility of the proposed nonlinear control scheme for neutralization processes was tested on a laboratory scale neutralization process where acetic acid (CH3COOH) is neutralized by sodium hydroxide (NaOH) in a CSTR with a volume of VR ) 5.5 l. The stoichiometry of this reaction is described by

10-pH - 10pHKW - [cA0/(1 + 10(pKS-pH))] cB0 + 10-pH - 10pHKW

(23)

where pKS ) 4.75 and KW ) 10-14 mol2/L2 for room temperature. Because of the low concentration of the reactants and the small reaction enthalpy of the neutralization process, we can assume constant temperature within the reactor. The process control equipment has several features which are not represented in the model equations (9). The flow rate of the peristaltic pump which supplies the titrating stream is limited to the range 0 e u e 45 L/h. All control actions are performed by a PC-based process control system which has a minimal sampling period of 1 s. The flow of the titrating stream is controlled by the modulation of an impulse frequency, which leads to a quantization of the control amplitude because the frequency can assume only certain discrete values. These nonmodeled process characteristics have to be considered in the controller design. Thus, we have to choose the closed-loop dynamics in such a way that the manipulated variable constraints are met for normal operation and such that the sampling time of the process control system can be neglected. Moreover, the resulting controller has to be robust against the quantization of the control amplitude. In Klatt and Engell (1995b) we presented first experimental results for gain-scheduling trajectory control of this neutralization reactor. The additional time constant τ in (9) was assumed to be 50 s, and the parameters k0 and k1 of the nominal controller (17) were chosen to get a nominal closed-loop transfer function (18) with a double time constant at T1,2 ) 40 s. Later, the pH sensor was exchanged, and the delay was reduced to τ ) 10 s. This allows a faster nominal closedloop trajectory within the admitted range of the manipulated variable; we chose T1,2 ) 15 s. The on-line simulation of the nominal model (9) forced by u˜ is used to calculate the necessary state variables for the input/ output linearizing feedback. Because of the structural uncertainty of the process model, which in this case is caused by the nonideal mixing characteristic and the features of the process control equipment, in connection with deviations from the standard titration curve and process stream disturbances, the robustness requirements for the resulting control are demanding. We used the “symmetric optimum” tuning rule (Kessler, 1958) to specify the parameters of the linearized reference

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Figure 4. Set point step response of the neutralization process controlled with GSTC in the case of nominal values of cA0 and F.

Figure 5. Step response of the process output compared with the desired nominal trajectory.

controller (20) in terms of the respective parameters of the plant transfer function (19). The symmetric optimum tuning rule is a robust linear controller design approach which is based on frequency domain design specifications. The parametrized plant transfer function (19) has two time constant, one of which (τ) is small compared to the dominating time constant T ) 1/a. For this case, the symmetric optimum rule yields

K1 )

T 1 ) 8τKS 8τb

KP )

T 1 ) 2KS 2b

where KS ) b/a is the steady state gain. The parameter b of the transfer function (19) depends on the pH-value on the nominal trajectory. Choosing the process output y(t) ) pH(t) as the scheduling variable for the control law yields the nonlinear gain-scheduling reference controller:

x˘ R(t) )

1 [y˜ (t) - y(t)] 8b(y(t))τ

uδ(t) ) xR(t) +

1 [y˜ (t) - y(t)] 2b(y(t))

(24)

The overall control structure, which consists of the nominal exactly linearizing state feedback according to eq 17, the simulation of the process model (9), and the nonlinear gain-scheduling reference controller (24), was implemented in a commercial PC-based DCS, where the respective controller parameters were calculated on-line from the measured pH-value and the inverse of the standard titration curve of the process (23). The resulting feed of the titrating stream u is the sum of the output of the nominal controller u˜ and the output of the gain-scheduling reference controller uδ. On the basis of experimental experience, the range of the output of the nominal controller was restricted to 80% of the overall range of the manipulated variable in order to provide a sufficient range for the reference controller output uδ at any time. Figure 4 shows the closed-loop time response to a set point step sequence in the case of nominal values of the process stream inflow F ) 20 L/h and the acetic acid concentration cA0 ) 0.007 mol/L. The GSTC control structure works well even for large set point changes and leads to a uniform reaction to set point changes within the whole operating range of the process. Fur-

Figure 6. Set point step response of the neutralization process controlled with GSTC in the case of inflow disturbances.

thermore, this figure shows that the overall performance of the proposed control structure is significantly better than that obtained with a standard linear PI-controller. The parameters of the PI-controller (KP ) 0.0015, KI ) KP/100) resulted from a pole placement design for the linear approximation of the process model at the main operating point pH 7. Figure 5 presents the closed-loop response of the process output to a set point step from pH 7 to pH 9.5 compared to the nominal trajectory output pHnom specified by the nominal controller (17) and the nonlinear process model (9). Here, it can be seen that the nonlinear gain-scheduling reference controller (24) rejects the tracking error caused by the mismatch of the process model (9) and leads to good tracking of the nominal trajectory. Figure 6 shows the behavior of the closed-loop system in the case of variations of the process stream flow rate F. The gain-scheduling reference controller rejects the disturbances in all cases and forces the process output to converge to the desired pH-value. The step changes of the process stream inflow have a different effect on the behavior of the controlled neutralization reactor, depending on the set point and the direction of the step change. The gain-scheduling reference controller is able to reject all disturbances after a sufficiently short period of time but the maximum deviations of the pH-value differ much. This is due to the dramatic change of the

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steady state gain within the operating range of the process. For lower pH-values, the chemical reaction behaves as a buffer system. Thus, the process gain is very low in this operating region and considerable changes of the inflow stream produce only minor changes of the pH-value. However, the controller actions have to be very strong in this region to force the process output to track the nominal trajectory and stay close to the specified set point. In the vicinity of the neutral point of the reaction system (pH 8.3), the process gain is very high so that the process is very sensitive against inflow disturbances. On the other hand, the controller gain of the gain-scheduling reference controller has to be low in this region in order to guarantee robust stability. This leads to a major deviation of the process output from its set point in the case of inflow disturbances which drive the process toward the neutral point. Thus, the different values of the maximum deviation in the case of disturbances are caused by the process characteristics. 5. Conclusions We have presented the application of gain-scheduling trajectory control to the control of a neutralization process in a continuous stirred tank reactor. The control design is based on a model of the neutralization process in terms of its standard titration curve. This leads to a macroscopic minimal-order description of the process dynamics. The standard titration curve can be calculated explicitly from the chemical equilibrium constants in many cases; otherwise it can be determined by laboratory experiments and approximation by an analytical function. An exact feedback linearizing controller is combined with a simulation of the nonlinear process model to generate a desired nominal transient trajectory. If there are no modeling errors and disturbances, the closed-loop dynamics exactly matches the desired dynamics forced by this nonlinear feedforward control scheme and overall stability and offset-free tracking can be ensured. Of course, these assumptions do not hold for real processes, as can be seen for the experimental setup considered here. The gain-scheduling reference controller is used to achieve insensitivity against model uncertainties and disturbances and steady state accuracy. It only acts for nonvanishing reference error and forces the process output to converge to the desired nominal trajectory if the reference controller dynamics is sufficiently fast compared to the nominal dynamics. The overall stability of the closed-loop system can only be ensured in the first approximation for a neighborhood of the nominal trajectory, provided that the controller dynamics are properly chosen and initialized. The region of attraction cannot be specified in the design. Gain-scheduling trajectory control is a local design approach. However, the transient dynamics are correctly represented in the gain-scheduling controller design, which is a considerable advantage compared to standard gain-scheduling. Although global stability cannot be proven a priori, the proposed design approach makes it possible for the designer to address stability and robustness issues for practical problems in a transparent way using methods from linear control theory in the design of the gain-scheduling controller. The resulting control structure was tested on a real laboratory scale neutralization process where acetic acid is neutralized by sodium hydroxide. The proposed control scheme achieves good set point tracking and sufficiently fast disturbance rejection over the whole

range of operation. The GSTC scheme performs significantly better than a linear controller. Of course, other nonlinear control approaches, especially variants of model predictive control, e.g., extended DMC control (Hernandez and Arkun, 1990), could also be used. Research in this direction is reported in Draeger et al. (1995). There, a neutral net was trained and used as a process model in the extended DMC-scheme with comparable results. Nonlinear model predictive control schemes require the use of iterative nonlinear optimization algorithms, either of the SQP-type or implicitly as in the extended DMC approach, where a series of linear problems for which a closed-form solution exists is solved. Convergence and on-line computation therefore are issues which have to be considered very carefully. Also, stability, even in the nominal case, can only be guaranteed if the internal state variables of the plant are known and can be estimated. Rigorous proofs of stability which take the actual behavior of the iterative optimization algorithm into account (as discussed in Kreisselmeier and Birkho¨lzer (1994)) are very difficult. Our algorithm is noniterative, and nominal stability is ensured by design as well as a certain degree of robustness (i.e., for linear perturbations of the plant). Complex computations are only necessary off-line during the controller design. The two concepts used in our approach, linearizing feedforward and robust gainscheduled linear feedback control, are intuitive and transparent; thus, the chances that such a scheme is accepted by the operators at a real plant are good. Roman Symbols A, B, C ) linear system matrices a, b ) transfer function parameters c ) concentration e ) error F ) process stream inflow f(‚‚‚), g(‚‚‚), l(‚‚‚) ) vector fields fT ) standard titration curve G(s) ) transfer function h(‚‚‚), k(‚‚‚) ) vector function IT ) inverse titration curve K, L ) linear controller matrices KI, KP ) PI-controller parameters KW ) dissociation constant of water k0, k1 ) closed-loop transfer function parameters pKS ) logarithmic dissociation constant of acid RL ) family of operation points r ) relative degree s ) Laplace transform variable T ) time constant t ) time u ) input, titrating stream inflow VR ) reactor volume v, w ) exogenous inputs x ) state (vector) y ) output z ) state (vector) Greek Symbols R ) parameter β ) scheduling variable τ ) time constant φ ) nonlinear coordinate transformation Indices 0 ) initial value, point on a trajectory A ) acid

3518 Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996 B ) base R ) reference controller S ) steady state sp ) set point value

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Received for review January 18, 1996 Revised manuscript received May 30, 1996 Accepted May 30, 1996X IE960033G

X Abstract published in Advance ACS Abstracts, August 15, 1996.