Abstract

CONTROL AND STABILITY ANALYSIS OF THE GMC ALGORITHM APPLIED TO pH SYSTEMS

J.T. Manzi¹ and D. Odloak²

¹Department of Nuclear Energy, Federal University Federal of Pernambuco, Rua Prof. Luiz Freire 1000, 50740-540 ,Recife - PE - Brazsil

²Department of Chemical Engineering, University of São Paulo, C. P. 61548, 05424-970, São Paulo - Brazil email: odloak@usp.br

(Received: February 7, 1998; Accepted: May 11, 1998)

Abstract - This paper deals with the control of the neutralization processes of the strong acid-strong base and the weak acid-strong base systems using the Generic Model Control (GMC) algorithm. The control strategy is applied to a pilot plant where hydrochloric acid-sodium hydroxide and acetic acid-sodium hydroxide systems are neutralized. The GMC algorithm includes in the controller structure a nonlinear model of the process in the controller structure. The paper also focuses the provides a stability analysis of the controller for some of the uncertainties involved in the system. The rResults indicate that the controller stabilizes the system for a large range of uncertainties, but the performance may deteriorate when the system is submitted to large disturbances.

Keywords: GMC algorithm, control strategy, stability analysis, neutralization processes.

INTRODUCTION

The pH control plays an important role in the chemical, biotechnological and wastewater treatment processes. The pH control structure must satisfy the requirements of process specifications and environmental regulations which haves become increasingly more rigorous. The fundamental neutralization process fundamental has been well established as the electro-chemical in nature and the developed concepts developed have provided a comprehensive approach of to the phenomena. However, the pH control is still recognized as a challenging problem from the practical point of viewstandpoint. Much effort has been addressed directed towards development of an adequate modeling, as well as a control structure which would be appropriate for to treating the nonlinear system. The major difficulty arises from the strong nonlinearity reflected in the S- shaped gain curve of the system. Significanttive progress has been obtained in the development of advanced control methods applied to the pH system, as reported by McAvoy et al. (1972), Gustafsson and Waller (1983) and Williams et al. (1990). However, in practice, the conventional PI or PID control strategy is still implemented in the majority of the applications. Recently, Manzi et al. (1997) presented the experimental results of the model-based Generic Model Control (GMC) pH control algorithm for the strong acid-strong base system applied to a pilot plant. It was shown that the GMC algorithm clearly produces a better performance than that produced by the conventional PI algorithm.

The An analysis of GMCthe stability conditions of GMC for the process-model mismatch has been reported in the literature (Lee et al., 1989). Dunia and Edgar (1996) have shown that the closed -loop may become unstable, due to the model uncertainty, when the controller includes a continuous linear model of the system. Orava and Niemi (1974) carried outpresented the a stability analysis of the pH process with the aid of Lyapunov’s methods. They also developed a specificparticular control law and showed that a small mismatch in the process-model can result in an unstable system. One of the goals of this paper is to investigate the stability conditions of the system using the approach of the structured singular value approach for the model-based GMC structure applied to the strong acid-strong base and weak acid-strong base systems. Some experimental results obtained from a pilot plant are also included here. The pilot plant of the neutralization process is represented in Figure 1. The reactor inlet stream consists of polluted industrial water (e.g.,: carbonate and phosphate salts in addition tobesides hydrochloric and acetic acid). A disturbance can be introduced by manipulation of the flow of strong acid (HCl-0.2N) or weak acid (-0.2N). The tTitration stream flow (-1N) is then determined by the control law to hold maintain the pH of the outlet stream at the desired set -point. The iInlet feed flow is controlled by manipulation of the rotation speed of the centrifugal pump. The pH values of the inlet and outlet streams of the reactor are measured.

The dynamic models presented here are based on the a methodology developed by McAvoy et al.(1972) and on the concepts of reaction invariants introduced by Waller and Mäkilä (1981). Thisese approachmodels It resultss in a model-based nonlinear transformation. The chosen pH control strategy chosen uses the GMC algorithm developed by Lee and Sullivan (1988). Such a controller is particularly appropriate to for treatment of the nonlinearities of the system in addition to besides of the clarificationty of the ideas involved aiming at the in industrial application.

MODEL FOR THE pH-NEUTRALIZATION PROCESS

Modeling the Hydrochloric Acid-sodium Hydroxide System

Consider the strong acid-strong base system described by the following equations:

(1)

with (2)

(3)

The principal difficulty in working with mass balances with chemical reactions consists in the need to include reaction rates. Use of the concept of reaction invariants (Waller and Mäkilä, 1981) allows the utilization of mass balances without employing reaction rates in the balance equations. The system can be adequately described in the following form:

(4)

where is the matrix of stoichiometric coefficients and s is the vector of chemical symbols. For the hydrochloric acid-sodium hydroxide system, these variables are given below:

We can define vector , whose components are the concentrations of chemical species involved in the system. It is given by

where stands for the molar concentration of component . Vector can be decomposed by the following linear transformation:

(5)

where q and z are the vectors of reaction invariants and reaction variants, respectively.Considering the equation above, it is clear that

(6)

According to Waller and Mäkilä (1981), matrix D can be calculated by equation 7 as follows:

(7)

where I is the identity matrix and W₁ and W₂ correspond to the partitioning of W:

Here W₁ is a nonsingular matrix and represents a base for the reaction invariants. Thus, for the hydrochloric acid-sodium hydroxide system, the partitioning considered above results in

, ,

Consequently, for this system, the vector of reaction invariants q is given by

From vector q, the first component is suitable for the development of a dynamic model for the purpose of controlling the system and for carrying out the stability analysis. Since the combination of invariants is also an invariant, it can be shown that and [Cl^-] are also reaction invariants. Since we know that q(1) for the pure neutral water is equal to zero, it should remain equal to zero after the addition of strong acid and strong base. Hence the titration curve can be written as follows:

(8)

Derivation of the following balance equations, taking into account the reaction invariants of the system, is also straightforward:

(9)

(10)

where V_R is reactor volume, relates to the inlet stream and relates to the titration stream. Combining equations 9 and 10 results in:

(11)

The term can be calculated from the measured pH of the feed stream, and it is given by:

(12)

Equations 9 to 12 describe the structure of the model to be used for control of the hydrochloric acid-sodium hydroxide system.

Modeling the Acetic Acid-strong Base System

The acetic acid-sodium hydroxide system can be described by the following reactions:

with (13)

with (14)

(15)

where HA and A^- represent the acetic acid and the acetate ion, respectively.

Using the same results and notations as those in the preceding section, an analogous procedure can be developed and vector p, matrices W₁, W₂ and D can be found as follows:

, ,

As a consequence, vector q of the reaction invariants is now given by:

From the above set of reaction invariants, q(2) is particularly suitable for the purpose of either modeling or controlling the system. Since is also a reaction invariant, we can derive the following balance equations:

(16)

(17)

where , and

(18)

From equations 16 and 17 and using the electroneutrality condition expressed by q(1) being an invariant, the following equations are obtained:

(19)

(20)

Substituting equation 20 into equation 19 at the steady state, the titration curve is obtained and can be expressed by:

(21)

Equations 16 to 20 describe the dynamic behavior of the acetic acid-sodium hydroxide system.

THE CONTROL ALGORITHM

Consider now the general nonlinear model

(22)

(23)

where x is the state vector, u the manipulated input, d the disturbance and y the controlled output of the system. In the classical control structure, the trajectory of the controlled output is compared with a reference signal and the controller minimizes the difference between the two signals. In the Generic Model Control strategy (Lee and Sullivan, 1988), the time derivative of the model output is compared to the time derivative of the output of a reference system. The following structure is produced:

(24)

where

(25)

There are several ways to choose the reference system (Bartusiak et al., 1989). For the pilot plant studied here, the reference trajectory is represented by the following equation:

(26)

where and are the controller parameters that are tuned to produce a specified closed-loop response and is the set point.

Combining equations 24 and 26 under the assumption that

(27)

results in the GMC algorithm

(28)

Control input u is calculated by solving equation 28 together with model equations 22 and 23.

Control of the Hydrochloric Acid-sodium Hydroxide System

The model of the neutralization system represented by equation 11 can be written in the following form:

(29)

where

, and

Equation 8 can be used to calculate the pH corresponding to a given . It should also be noted that system output is equal to the state of the system . Thus, applying equation 28 to the

system model represented by equation 29, we obtain the following expression for the manipulated input of the system:

(30)

Control of the Acetic Acid-sodium Hydroxide System

For the system considered here, equation 19 governs dynamic behavior and can be rewritten in the following form:

(31)

where is given by equation 20, and .

From equations 28 and 31 we can find the control law for the acetic acid-sodium hydroxide system which is expressed by

(32)

PI Algorithm Based on the GMC Strategy

Since we use the PI algorithm in the experimental tests, it is interesting to derive the PI control law as a particular reference system adopted by the GMC strategy. For this purpose consider the linearized form of equation 29:

(33)

where and correspond to the steady-state values of variables F_t and x, respectively. Equation 33 can be expressed in terms of the following deviation variables:

, and

and results in

If the reference system corresponding to the PI strategy is given by

(34)

where is the set point of the state deviation variable, then using equation 27 we can obtain the following PI algorithm:

EXPERIMENTAL RESULTS AND DISCUSSION

The systems composed of hydrochloric acid-sodium hydroxide and acetic acid-sodium hydroxide are considered for the experimental tests and for the stability studies. The feed stream to be neutralized consists of industrial water with pH equal to 7.3. Hydrochloric acid or acetic acid is added to the feed stream to generate the inlet stream to the neutralization reactor. The process parameters are flow rate , reactor volume and sampling period . The parameters of the GMC algorithm are and for the hydrochloric acid-sodium hydroxide system, while the parameters for the acetic acid-sodium hydroxide system are and . The tuning procedures of Ziegler and Nichols (1942) and Chien et al. (1952) lead to unsatisfactory performance of the control strategy. Consequently, parameters k₁ and k₂ are found to be the most adequate in terms of speed of response and stability after exhaustive experimental tests.

Pulse disturbances are introduced into the systems by changing the pH of the feed stream from 7.3 to approximately 4.6 and back to 7.3 after

3 min. The disturbance introduced into the system is shown in Figure 2, and the performance of the GMC controller is compared to that of the PI controller, with both controllers manipulating the flow of the strong base solution. The set point of the pH of the outlet stream remains equal to 7 for all the experiments. Figures 2 and 3 show the trends of the manipulated variable and the system output. Observing the responses of each system for the two control strategies, we conclude that the performance of the GMC controller is considerably better than the performance of the PI controller for the particular disturbance considered in this experiment. In both cases the system is remarkably less sensitive to the disturbance when the GMC strategy is used. It is also interesting to note that the GMC controller provides a more aggressive control action than does the PI controller. At this point it should be observed that our assumption that the systems contain exclusively the species considered in the model is not completely true. It is possible that small amounts of unknown weak acid salts (e.g., calcium carbonate or phosphate) are also present in the feed stream. This explains why the pH of the acid-free water is 7.3 instead of 7, as expected. The weak acid contamination results in a robustness problem in the GMC strategy since control action is calculated based on the response of the system model. This aspect causes a structural model mismatch between plant and model. It is beyond the scope of this paper to analyze this kind of problem. In the next section we consider another robustness problem that is present in our pilot plant and is related to uncertainties in input and output measurements.

STABILITY ANALYSIS OF GMC FOR THE NEUTRALIZATION PROCESS

The study performed in this section is motivated by the practical observation that performance of the GMC controller deteriorates when the disturbance represented by the pH of the feed stream suffers a step change from the steady-state value of 7.3 to a value below 4.5. The response of the hydrochloric acid-sodium hydroxide system is represented in Figure 4. Apparently the system approaches its stability limit since any further reduction in the pH of the feed stream results in lack of stability.

Figure 2: Responses of the hydrochloric acid-sodium hydroxide system tofor the GMC and PI control strategies.

Figure 4: Response of the hydrochloric acid-sodium hydroxide system tofor the GMC control strategy by changing the pH of the feed stream from 7.3 to 4.6.

The analysis here is based on the structured singular value reported by Packhard and Doyle (1993) and the small gain theorem (Zames, 1966), which are used to perform the robustness stability analysis for a class of uncertainties. In this section our discussion on of stability analysis will be focused on the same systems considered in experimental tests runs.

Analysis of Hhydrochloric Acid-Sodium Hydroxide System

For the purpose of the stability analysis, the following differential set of equations is considered:

(35)

(36)

(37)

where and represent the state of the model system model and the state of the plant, respectively, relates to the integral term of the reference system and is the set -point of the state variable. Applying the GMC algorithm to the system above we can derive the following control law:

(38)

Substituting equation (38) into equations (35) and (36) result inleads to:

(39)

(40)

Now Cconsider now the following definitions:

and (41)

where and reflect the uncertainties in the state of the system state and disturbance measurement, respectively. Then equation (40) can be written as follows:

(42)

Reorganizing equations (37), (39) and (42) in matrix form, results in:

(43)

For the analysis of the stability of this system, two fundamental issues are considered, namely:

1- The Small Gain Theorem (Zames, 1966)

2- Definition and properties of the structured singular value (Packhard and Doyle, 1993)

Definition of :

For

is defined by

and if there is no such that , then

where is a block diagonal matrix. is the set of all matrices that are associated to three basic characteristics: the total number of blocks and, the type and the dimensions of each block. These characteristics are defined for a particular problem. The can be interpreted as a measure of the smallest structured that causes instability in the system. The norm of this perturbation is given by , as shown in the theorem of the small gain.

From equation (43) we obtain the disturbed transition matrix of the system:

(44)

It is obvious that the transition matrix is of the form where can be considered as a structured uncertainty. If all the eigenvalues of matrix are located on the left half of the complex -plane, then for any , where and is any real number. Thusen, for the purposess of the stability analysis of the system, the problem is to determine the uncertainty with minimal norms such that:

(45)

It is straightforward to show that equation (45) is equivalent to:

or

where

(46)

Consequently, the stability limit corresponds to the calculation of forw , where w . For the hydrochloric acid-sodium hydroxide system underin study, we consider the same tuning parameters of GMC and the same process parameters reported in section Experimental Results and Discussion. In this case matrices and are given by:

The eigenvalues of matrix are: - 0.55, - 50,0 and - 20.0. Consequently, if the system in closed -loop with the GMC algorithm is stable. Now can be calculated to estimate the smallest uncertainty which makes the closed -loop unstable. The minimum uncertainty is shown in Figure 5 where is plotted as a function of the frequency .

In Figure 5 it can be observed that the minimum value of is 0.88. As stated previously, this number expresses the norm of the smallest structured that causes instability to the system. Using equation 41, the system will be locally stable if the following condition is satisfied:

(47)

It is obvious that global stability will occur if the system is stable for each pair corresponding to a possible operational condition of the system. At this point it is necessary to define what we mean by a possible operational condition for the neutralization system. It is reasonable to assume that the true pH of the system for all conceivable practical conditions will be in the range of . These limits can be considered to be quite wide, since for the industrial neutralization process the pH normally remains between 4 and 10. Then, if we prove that the system is stable for such a conservative range, it will certainly remain stable for all practical conditions. The corresponding range for the process state variable is . For the model used in the GMC algorithm, the constraints on the calculated state variable are in the same range as those on . This can be done easily since if the result is outside the assumed range, we can always round up the calculated state to the nearest limit. Assuming these limits on the state variables, the following relation holds true:

(48)

Figure 5: Structured singular value [] for the hydrochloric acid-sodium hydroxide system.

It can be observed that equation (48) is inside the stability range of the system represented by equationrelation (ratio ) (47). Consequently, we can state that GMC will stabilize the system for all conceivable practical situations as long as for both plant and model we have for both plant and model. So, there is still no explanation for the poor behavior of the system observed in Figure 4. To verify the stabilizing properties of GMC when applied to a system such as the pilot plant studied in this paper, we present several simulations with the hydrochloric acid-sodium hydroxide system. Uncertainties are assumed to exist in the system and the responses of the closed loop withto with GMC are observed. Figure 6 shows the trends of the pH of the reactor output stream and the pH calculated by the model used by the GMC algorithm for a step disturbance on the pH of the feed stream. At instance the pH of the feed is changed from 7 to 4. In this case it is assumed that there is a 5% multiplicative uncertainty () in the manipulated variable and an additive uncertainty () of 0.5 in the measurement of the reactor pH, that is:

(49)

From a practical point of view these uncertainties can be considered to be as very large. The disturbance introduced into the system is also much larger than the disturbance normally expected in the nan industrial system. In spite of these extreme conditions, we observe that the system remains stable as predicted by the stability analysis and the control strategy produces an acceptable performance. However, Figure 7 shows the results of another experiment where it is considered an additional uncertainty in the measurement (= 0.5) of the pH of the feed stream, which is the measured disturbance of the system is considered:

It is clear that the performance in this case is not adequate and that the system response is that similar to the pilot plant response shown in Figure 4. However, upon inspecting Figure 7, we observe that during the simulation period we have while the stability range is . This means that the closed- -loop performance can deteriorate, even when the system is at a large distance from the stability limit. In other words, knowing the stability limit of the system and imposing a an stability margin is of little help in the design of the nonlinear controller. Also, it is not a specificparticular uncertainty that causes the deterioration in of performance. This can be in seening in Figure 8, where we have the uncertainty in the manipulated input is associated to the with uncertainty in the disturbance measurement, while the controller performs well.

Figure 6: Response of hydrochloric acid-sodium hydroxide system withwith to GMC and uncertainties in the manipulated input and in the measurement of the output. The pHpH of the feed stream was changed from 7 to 4.

Figure 7: Response of the hydrochloric acid-sodium hydroxide system withtowith GMC and uncertainties in the input, and in the measurement of the output and in the measurement of the disturbance. The pHpH of the feed stream was changed from 7 to 4.

Analysis of Acetic Acid-Sodium Hydroxide System

Here we repeat the study performed in the section Control of the Hydrochloric acid-sodium hydroxide system() now considering now the acetic acid-sodium hydroxide system. As stated previously, From what has been said before, equation (31) describes the dynamic of the acetic acid-sodium hydroxide system. Thus,en for the purpose of the stability analysis, the following set of equations should be considered:

(50)

(51)

(52)

where is the set -point of the state, , and is given by the following nonlinear transformation:

(53)

Here and represent the state of the system model and plant, respectively. is related to the integral term of the reference system. The previously derived GMC control law previously derived (eq. 32) can be rewritten as:

(54)

Substituting equation (20) into equations (17) and (18), results in:

(55)

(56)

Let us now define the system uncertainty as follows:

(57)

in which represents the uncertainty in the system state of the system.

Using the same strategy developed in section (Control of the Hydrochloric acid-sodium hydroxide system), the following system of differential equations can be obtained:

(58)

Consequently, the disturbed transition matrix of the system is given by:

(59)

Based on the same procedures and definitions established in the last section, the stability analysis will be performed for the weak acid-strong base system under in study. We consider the same tuning parameters offor GMC and the same process parameters as those reported by Manzi et al. (1997). In this case matrices and are given by:

Since the eigenvalues of matrix A (, and ) indicate that the nominal system in closed loop with GMC is stable, we can calculate and then estimate the minimum uncertainty which makes the system unstable. Figure 9 10 shows as function of the frequency w .

Observe in Figure 9 that the minimal value of is 0.658. This number means the minimum norm of that causes instability to the system. The system will be locally stable if the following condition is satisfied:

(60)

Figure 9: Structured singular value [] for the acetic acid-sodium hydroxide system.

Again we can assume that the (overall) global stability will occur if the system is stable for any pair corresponding to a possible operational condition of the system. Following the same procedure as that of the hydrochloric acid-sodium hydroxide system, we will establish the practical operational conditions for the acetic acid-sodium hydroxide system. Initially we assume that the true pH of the system will be in the range: range. These limits are quite exaggerated, as we mentioned previouslybefore. Consequently, if we show that the system is stable for the assumed range, then it will certainly remain stable for all practical situations. The corresponding range for the system state variable is . For the model used in the GMC algorithm, the constraints of calculated state constraints of can also be constrained into are also the same range of . Considering are also these limits on the state variables and assuming a conservative value for equal to , then the condition(ratio) relation below is valid:

(61)

We can observe that the limits of the (ratio) conditionrelation (61) are outside the stability range of the system established by (ratio) relationcondition (60) and we cannot guarantee that GMC will stabilize the system underin the assumed conditions. However, if the practical operational conditions are assumed to remain within stay in a narrower range, given by which corresponds to the state variable being inside the limits , then the following condition(ratio) relation holds true:

(62)

The new range defined by equation(ratio) relation (62) is inside the stability limits established by equation(ratio) relation (60). Hence the GMC algorithm will stabilize the system whenever the true pH of the system and the pH calculated using the neutralization model remain within the range: range. It should be emphasized highlighted that the considerations onofon the structured singular value and the stability limits are true for a specific set of tuning parameters that produce a specified closed- loop response.

Now we present several simulations to verify the stabilizing properties of the GMC when applied to anthe acetic acid-sodium hydroxide system such as the pilot plant studied here. The responses of the closed -loop are observed considering the presence of uncertainties in the system. Figure 10 shows the behavior of the pH of the reactor output stream and the pH calculated by the model used by in the GMC algorithm for a step disturbance on the pH of the feed stream. At instance 0.15 min the pH of the feed stream is disturbed from 7 to 4. It is also assumed that there is a 5% additive uncertainty in the manipulated variable and an additive uncertainty of 0.5 in the measurement of the reactor pH. We observe that for these conditions the GMC controller remains stable, as predicted by the stability analysis, and it showshas a good performance. However the performance of the GMC controller deteriorates, if under the same conditions, an additional 5% uncertainty is considered in the pH measurement of the feed stream. This is shown in Figure 11. It should be emphasized that it is not solely the presence of this new uncertainty that causes the loss inof performance, since as shown in Figure 12, the performance is considerably better if the disturbance goes from 7 to 4.2, instead of from 7 to 4. Also, there is almost a complete failure of the control system when the disturbance goes from 7 to 3 and there is a 5% uncertainty in its measurement. As shown in Figure 13, in this case for most of the time we have , which is well inside the stability range of the system, but the performance is very poor. A similar situation occurs when the pHpH of the feed is changed moved from 7 to 1, with uncertainties in the input and output measurements. In this case and the system is still stable, but as shown in fig. 14, there is no performance.

Figure 14: System response of the acetic acid-sodium hydroxide system withtowith GMC and uncertainties in the manipulated input and controlled output. The pH of the feed stream was changed from 7.0 to 1.0.

CONCLUSIONS

In this paper we presented the results of the modeling and implementation of the a GMC controller in to a pilot plant of the waste water neutralization process. The modeling and controller were developed for the strong -acid--strong base and weak acid-strong base systems. The pPlant experiments were performed for the hydrochloric acid-sodium hydroxide and acetic acid-sodium hydroxide systems. The pPerformance of the GMC algorithm was compared to that of the conventional PI algorithm and it showed a superior performance superior to that of the latterthan this controller which is usually adopted for this kind of system. It was also presented aA stability analysis was also presented for both neutralization systems. TheThe A stability analysis was performed using the structured singular value as well as the small gain theorem. It was demonstrated that the GMC control strategy is stable for a very large range of practical operational conditions of for the considered systems considered. However, it was observed that the performance of GMC is very sensitive to large disturbances when uncertainties are present in the measurements of the system. BesidesIn addition, the study reveals that performance may deteriorate underin conditions far from the stability limit of the system. The practical experiments and the simulations show that sensitivity increases when the pHpH of the feed stream falls below 4.5. If in the industrial system this situation occurshappens frequently, then another kind of controller should be considered in the design stage of the neutralization system.

NOMENCLATURE

A_d Disturbed transition matrix

ACT Auxiliary concentration (see equation following eq. 17)

D Transformation matrix

d Disturbance

Flow rate of the feed stream

Flow rate of the titration stream

I Identity matrix

k₁, k₂ GMC control parameters

k_c Proportional action for PI controller

Equilibrium constant of the weak acid

Dissociation constant of the water

L Transformation matrix

p Vector of the concentrations of chemical species

P Transformation matrix

q Vector of reaction invariants

s Vector of chemical symbols

t_I Integral action for PI controller

T Transformation matrix (equation. 5) and sampling period

u Manipulated input

V_R Volume of the neutralization reactor

W Stoichiometric coefficients matrix

W₁,W₂ Partitioning matrices

x Model state

y Model output

Greek letters

Uncertainty in the system state of the system for weak acid-strong base system

Uncertainty in the system state of the system for strong acid-strong base system

Uncertainty in in the disturbance the measurement of disturbance

Additive uncertainty in the manipulated variable

Additive uncertainty in the measurement of the pH

Maximum singular value

m Structured singular value

Subscripts

e Inlet stream

t Titration stream

T Transpose matrix or vector

s Steady -state

p Plant

ref. Reference system

* Set point

Abbreviations

GMC Generic Model Control

PI or PID Proportional-Integral-Derivative feedback control

ACKNOWLEDGMENTS

Support for this work was provided by the Fundação de Amparo à Pesquisa de São Paulo (FAPESP) under grant 95/6686-1 and from the a Krupp Ffellowship administered by the Alexander von Humboldt Foundation. The discussions held with Dr. O. U. Langer (DECHEMA e.V.) are gratefully acknowledged.

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