Seasonal Model Based Control of Processes with Recycle Dynamics

It was found that, when a positive feedback arrangement is used to represent the linear dynamics of systems with recycle, the recycle-path time consta...
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Ind. Eng. Chem. Res. 2001, 40, 1633-1640

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PROCESS DESIGN AND CONTROL Seasonal Model Based Control of Processes with Recycle Dynamics K. Ezra Kwok,*,† Michael Chong-Ping,† and Guy A. Dumont‡ Department of Chemical & Biological Engineering, 2216 Main Mall, and Pulp and Paper Center, 2385 East Mall, The University of British Columbia, Vancouver BC, V6T 1Z4 Canada

Plants with recycle systems have complex dynamics and can present challenging control problems. In this paper, the dynamics of recycle systems were examined by analyzing a twostage heating process with recycle. It was found that, when a positive feedback arrangement is used to represent the linear dynamics of systems with recycle, the recycle-path time constant and recycle fraction must be carefully included. Also, the overall recycle dynamics can be represented by seasonal time-series models. The overall transfer function for the recycle system might be nonrational, thus complicating controller design. Three methods of applying modelbased predictive control to recycle processes were compared. The first method was a Taylor series expansion used to linearize the process model, the second method used a seasonal timeseries model to represent the process, and the third method was a recycle compensator that removed the effect of recycle dynamics. Although all three methods gave good control, practical considerations determine the method that should be used. 1. Introduction Recycle streams are widely used in the chemical industry, typically to return energy (e.g., heat) or mass (e.g., unreacted chemicals) to the process. Although recycling is economically advantageous, it causes interactions that change the process dynamics and can complicate controller design. Although interactions can be reduced using large surge tanks, this solution is expensive and presents some safety issues as well. Recycle systems are equivalent to systems with positive feedback.1,2 For feedback systems, a criterion for stability is that the recycle-loop gain be less than one. Studies have shown that the presence of recycle increases the overall time constant and the sensitivity to disturbances;3 the process might also exhibit underdamped behavior, severe overshoots, and inverse responses.4 Moreover, it has been shown that, if the recycle time delay is larger than the forward-path time constant,2 some disturbance frequencies might resonate throughout the recycle loop, and there might be a stepwise trend to steady state.5 Recycle systems also present challenging problems for control personnel. Poor control structures might cause large flow-rate changes for small input changes; this occurrence is known as the “snowball” effect.6 Also, the overall linear transfer function for the recycle system is often nonrational; thus, direct controller design is not straightforward.7 The transfer function has to be modified for use in model-based control algorithms. This paper revisits the basic properties of recycle systems. The analysis is performed using a two-stage * Author to whom all correspondence should be addressed. E-mail: [email protected]. Tel.: 604 822 3238. Fax: 604 822 6003. † Department of Chemical & Biological Engineering. ‡ Pulp and Paper Center.

Figure 1. Two-stage heating process

heating system as a model. Recycle control was studied using two first-order-plus-deadtime models. Three different methods were also used to derive a controller model for advanced control. The derived models were used to apply model-based predictive control to a generic recycle process. 2. Recycle Analysis 2.1. Recycle Process. The process studied was based on a pilot-scale two-stage heating system.8 Figure 1 shows a schematic of the process. The first stage is a continuous stirred-tank heater (CSTH). Cold water enters the tank from the top and is heated by the steam coil inside the tank. The warm water from the CSTH is split into product and recycle streams. The recycle stream is sent to a plate heat exchanger (second-stage steam heating) and then returned to the tank. The following assumptions are made for the mass and energy balance of the system: (1) The water in the tank is well mixed. (2) The physical properties of water, such

10.1021/ie0005699 CCC: $20.00 © 2001 American Chemical Society Published on Web 03/02/2001

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as density and heat capacity, are constant over the operating temperature range of this equipment. (3) Heat loss to the environment is negligible. 2.1.1. Continuous Stirred-Tank Heater. The volumetric balance of the first heating system is given by

dV ) F1 + FR - F2 dt

(1)

where V represents the volume of tank water and the subscripts 1, 2, and R under F denote inlet cold water, tank outlet warm water, and recycled warm water, respectively. Because no accumulation occurs in the plate heat exchanger, the water balance of the second stage is simply

FR ) rF2

(2)

where r represents the fraction of water being recycled to the first stage. The energy balance around the first stage is

d[FVCp(T2 - Tref)] )m ˘ 1Cp(T1 - Tref) + dt m ˘ RCp(TR - Tref) - m ˘ 2Cp(T2 - Tref) + Q1 (3) where m ˘ is the mass flow rate, Cp is the water heat capacity, F is the water density, Q1 is the energy from steam heating, and Tref is a reference temperature. Using the rule for differentiation of a product and eqs 1 and 2, one obtains

Q1 dT2 ) F1(T1 - T h 2) + rF2(TR - T h 2) - F2T ˜2 + V h dt CpF (4) where - and ∼ denote steady-state and deviation variables, respectively. If the recycle fraction r is set to zero, eq 4 will be reduced to the standard energy balance of a CSTH. Because the exit flow from the tank is fixed and the level in the tank is being regulated by manipulating F1, i.e., F1 + FR ) F2, eq 4 can be further simplified to

V h

where τ is the tank time constant. This transfer function shows that increasing the recycle fraction (r) increases the tank outlet temperature. If the recycle fraction is kept constant but the recycle temperature varies, the transfer function is Again, increasing the recycle tem-

T2(s) )

TR(s) )

Equation 5 describes the dynamic behavior of the firststage heating, which is affected by T1, r, TR, and Q1. The effect of the second-stage heating on the first stage will be via the variables r and/or TR. 2.1.2. CSTH Linear Dynamics. Consider the simple CSTH dynamics in eq 5. If the cold water inlet temperature (T1) and the recycle temperature (TR) are assumed to be constant, then eq 5 in terms of deviation variables will be reduced to

V h

Q ˜1 dT ˜2 hR - T h 1)r˜ - F h 2T ˜2 + )F h 2(T dt CpF

(6)

Q2(s)

τ2ps2 + 2ζτps + 1

(9)

V hR

Q2 dTR ) rjF2(T2 - TR) + dt CpF Q2 dTR ) rj(T2 - TR) + dt λ

(10)

where Q2 is the total heat supply to the plate heat exchanger. Its transfer function is

TR(s) )

1 λ-1 T2(s) + Q (s) τRs + 1 τ Rs + 1 2

(11)

where τR ) (VR/rF2). 2.2. Overall Model Dynamics. Because both transfer functions in eqs 8 and 11 do not involve any linearization except r, which is assumed to be constant, the overall recycle dynamics can be obtained by combining the two equations. The overall model is

T2(s) )

λ-1 Q (s) + (τs + 1)(τRs + 1) - rj 2 λ-1(τRs + 1)

The Laplace transform of eq 6 is

Q (s) (12) (τs + 1)(τRs + 1) - rj 1

-1

T2(s) )

K(τas + 1)e-τds

Although no disturbance model was reported in Khan’s9 work, one would assume that the disturbance model, such as one due to inlet temperature change T2, would generally behave like eq 9 as well. For the sake of simplicity, the plate heat exchanger is assumed to behave like a first-order model, i.e.

τR Q1 (5) CpF

(8)

perature will raise the final tank outlet temperature. Although both linear models in eqs 7 and 8 show a positive gain in the tank outlet temperature (T2) for increasing heat supply (Q1), they do not account for the subsequent increase in TR due to increased Q1, which will eventually affect T2. Also, if the heat supply to stage two remains the same, increasing r will lower the recycle temperature TR. This implies that the linear model gain between r and T2 is over estimated. Likewise, the linear model gain between TR and T2 in eq 8 is underestimated because increasing TR will eventually raise T2 to a higher value. Moreover, the transfer functions do not show the inherent coupling between r and TR, which are inversely related for a constant heat supply. As a result, one can conclude that processes with recycle flows will have their open-loop gains altered. 2.1.3. Plate Heat Exchanger Dynamics. According to Khan,9 the dynamic characteristics of a counter-current plate heat exchanger can be best represented by a leadlag second-order overdamped model with delay.

dT2 h 2) + rF h 2(TR - T h 2) )F h 2(1 - r)(T1 - T dt F h 2T ˜2 +

(CpFF h 2)-1 jr TR(s) + Q (s) τs + 1 τs + 1 1

T hR - T h1 h 2) (CpFF r(s) + τs + 1 τs +1

Q1(s)

(7)

The following observations can be made from this

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T2(s) )

λ-1 Q (s) + (τs + 1)(τRs + 1) - 1 2 λ-1(τRs + 1)

Q (s) (14) (τs + 1)(τRs + 1) - 1 1 where τR ) VR/F2. The structure is the same as that in eq 12, except in that case, τR ) VR/rF2, and the recycle fraction appears explicitly in the denominator. Because the recycle fraction should be included to properly represent the system, an alternate general representation is shown in Figure 4.10 This structure explicitly includes the effect of the recycle fraction and is the one used from this point onward. 2.3. Recycle Systems with Delays. Thus far, no delay is considered in the analysis. If delays are present in both the forward path (θF) and the recycle path (θR), the final transfer function will become

Figure 2. General recycle configuration

Figure 3. Two-stage system block diagram

T2(s) )

transfer function: (1) The steady-state gains of both models are the same even though the second-stage heating only heats a fraction of the flow from the first stage. This can be explained by the fact that the whole recycle system needs the same amount of energy to increase T2 regardless of whether the energy comes from Q1 or Q2. (2) From eq 12, the gain of the overall system is found to be λ-1/(1 - rj), which is greater than the individual gain of λ-1 without recycle. Therefore, one can conclude that the overall gain with recycle is always magnified by a factor of 1/(1 - rj). (3) The response of T2 due to Q1 is much faster because of the direct effect of Q1 on the outlet temperature. In the extreme when there is no recycle, the transfer function between T2 and Q1 reduces to a simple first-order model. On the other hand, the heating dynamics due to Q2 follow a secondorder trend because the heat energy changes the recycle temperature TR, which in turn alters T2. (4) The recycle dynamics are introduced by the recycle fraction rj in the characteristic equation, which affects the stability and damping of the overall system. The damping factor is

ζ)

τ + τR 2xττR(1 - rj)

(13)

If ζ is assumed to be greater than 1, solving for r gives rj > -(τ - τR)2/4ττR . This result is always true, as 0 < r < 1. This means that any recycle system composed of two first-order models will always be overdamped. The degree of damping depends on all time constants and the recycle fraction. (5) A common way of representing a recycling system is by arranging the forward-path and recycle-path transfer functions in a positive feedback configuration, as shown in Figure 2. This common practice is inappropriate because it does not take into the account the recycle fraction. All recycle systems must include a recycling fraction, as shown in Figure 3. Otherwise, the parameters in the individual transfer functions would have to be compromised to obtain the correct overall recycling behavior. For example, if the overall recycle transfer function is formed from the general configuration in Figure 2, the final transfer function, assuming zero time delay, will be

λ-1e-(θF+θR)sQ2(s) (τs + 1)(τRs + 1) - rje-(θF+θR)s

+

λ-1(τRs + 1)e-θFsQ1(s) (τs + 1)(τRs + 1) - rje-(θF+θR)s

(15)

According to Hugo,7 it is difficult to find the poles of the above system unless the denominator exponential term is approximated. However, the system can be analyzed in the frequency domain to assess stability.2,10,11 Alternatively, the overall dynamics can be pictured via the discrete-time equivalent of eq 15. Assuming a sampling time of Ts

T2(k) )

(b1q-1 + b2q-2)Q2(k - dF - dR) 1 + a1q-1 + a2q-2 + a3q-dF-dR

+

(c1q-1 + c2q-2)Q1(k - dF) 1 + a1q-1 + a2q-2 + a3q-dF-dR

(16)

where dF ) θF/Ts and dR ) θR/Ts. This discrete transfer function is very similar to a seasonal time-series model.12 A seasonal time series is one that shows periodic behavior (with period P). That is, similarities in the series occur after P basic time intervals. For example, the totals of international travelers over a 3-year period show two seasonal patterns. Travel is the highest in the late summer months and the second highest in early spring for each year. The general structure of a seasonal time-series model is

y(t) ) y(t - 1) + y(t - 2) + y(t - P) + y(t - P - 1) + u(t - d - 1) + u(t - d - 2) + u(t - P - d) + u(t - d - P - 1) (17) where d represents a time delay. Thus y(t) is based on sequential past values and values that are P basic time intervals in the past. 2.4. Recycle Trends. An extensive simulation study on the recycle system configuration in Figure 4 has been performed using two first-order-plus-deadtime models. The study included all possible combinations of time delays and time constants in both the forward- and the recycle-path transfer functions. It was found that the recycle dynamics can be categorized into the three types

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(1 - r)GF X ) ) U 1 - rGFGR (1 - r)KF(τRs + 1)e-θFs τFτRs2 + (τF + τR)s + 1 - rKFKRe-(θF+θR)s

The deadtime term in the denominator makes it difficult to use the transfer function for control analysis or the design of model-based controllers. Although various delay approximations such as the Pade´ approximation can be used, there is no guarantee of stability.7 3.1. Taylor Series Approximation. To remove the deadtime term in the denominator, a Taylor series expansion has been used to obtain an approximation for the overall transfer function in eq 18.14 A similar approach was recently published by Hugo.7 In the Taylor series expansion, the differentiation was performed with respect to e-(θF+θR)s, and the expansion was performed around e-(θF+θR)s ) 0. For example, a thirdorder Taylor series expansion of eq 18 is shown below

Figure 4. Modified recycle block system

(1 - r)KF(τRs + 1)e-θFs τFτRs2 + (τF + τR)s + 1 - rKFKRe-(θF+θR)s (1 - r)KFe-θFs

Figure 5. Recycle step responses

(τFs + 1) of trends shown in Figure 5 and described in the following: (1) Trend 1 is the most basic trend and is similar to first-order dynamics with an extended rise time due to a multiple time-constant effect from recycling. The condition for this to happen is when the forward-path time constant is dominant (i.e., τF > τR, θF, θR). (2) Trend 2 is like having a very short rise time but a very long time to steady state. This is because of a very dominant time constant in the recycle path (i.e., τR > τF, θF, θR). (3) Trend 3 is a stepwise increase to steady state caused by a dominant time delay in the recycle path (i.e., θR > τF, θF, τR). This response is the one usually associated with recycle systems and is the closest to seasonal models in time-series analysis. From a control viewpoint, this trend is the one that requires special consideration as the other two can be better approximated by first-order transfer functions with large time constants.

3. Recycle Control Three different methods for handling systems with recycle streams are presented in the following sections.13 They are all based on the block diagram structure shown in Figure 4. Both GF and GR are assumed to be first-order processes with deadtime

GF )

KFe-θFs τFs + 1

GR )

(18)

KRe-θRs τRs + 1

where K, τ, and θ represent the gain, time constant, and time delay of the transfer function, respectively. According to the block diagram in Figure 4, the resulting overall transfer function is

+

)

r(1 - r)KF2KRe-(2θF+θR)s (τFs + 1)2(τRs + 1)

r2(1 - r)KF3KR2e-(3θF+2θR)s 3

2

(τFs + 1) (τRs + 1)

+

+ O(e-4θds) (19)

The approximation shifts the denominator deadtime terms to the numerator, allowing the expression to be easily used as a controller model. The overall transfer function becomes a combination of the rational forwardpath transfer function, the forward-path transfer function plus one complete forward/recycle path, and the forward-path transfer function plus two complete forward/recycle paths. The accuracy will improve if more recycle loops are included, but the overall function will become more complex. In fact, even using up to the third term makes the function quite complicated. On the other hand, using only two terms causes a significant modelplant mismatch at the lower frequency that deteriorates the controller performance. 3.2. Recycle Compensator. The previous approach attempts to deal with the recycle dynamics while eliminating the irrational term in the denominator for better controller design. Another method is to eliminate the recycle dynamics completely from the overall system by designing a recycle compensator.5,15 This approach allows the controller to be designed for the forward path alone. The compensator representation is shown in Figure 6. In the figure, GF and GR are the units in the forward and recycle paths, respectively. GK is the recycle compensator, and GC is the controller. The open-loop transfer function without recycle is

X ) GFu

(20)

The open-loop transfer function of the system with recycle and compensator is

X)

GFu 1 - GFGR + GFGK

(21)

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Figure 6. Recycle compensator block system

By imposing the condition that eqs 20 and 21 be equal

GF u )

GFu 1 - GFGR + GFGK

n2

J(n1, n2, nu, λ, γ) )

1 ) 1 - GFGR + GFGK

nu

G K ) GR

(22)

Thus, the controller is designed for the forward path alone and is implemented with the recycle compensator. The performance of the compensator is affected by the plant models. The sensitivity of the controller to model plant-mismatch has been studied by Scali and coworkers.5,16-18 3.3. Seasonal Time-Series Models. An alternative to the previous two approaches is to use a seasonal timeseries model to represent the process. As shown in section 2.3, the overall transfer function for a recycle system can be represented by a seasonal time-series model. This can be further confirmed by discretizing the individual transfer functions (GF and GR) and deriving the overall transfer function for Figure 4. The discretized forward and recycle transfer functions are

GF(z) )

1 - aFq-1

GR(z) )

bRq-1-dR 1 - aRq-1

The resulting approximate overall model has the following time-series structure:

G)

∑ γ(j)[y(k + j) - w(k + j)]2 +

j)n1

GFGK ) GFGR

bFq-1-dF

prediction-error-squared term and a weighted-controlsquared term19 and is shown in eq 24.

(1 - r)(bFq-1-dF + bFaRq-2-dF) 1 + (aF + aR)q-1 + aFaRq-2 + rbFbRq-2-dF-dR (23)

where dF ) θF/Ts and dR ) θR/Ts. The presence of the term rbFbRq-2-dF-dR in the denominator indicates a seasonal model structure. It will be shown in the simulation section that this form is readily applied to model-based controller design such as generalized predictive control (GPC). 4. Simulation Results Two closed-loop simulations using GPC were used to compare the different controller models. The GPC control law is derived from the minimization of a receding-horizon quadratic cost function comprising a

λ(j)[∆u(k + j - 1)]2 ∑ j)1

(24)

where n1 is the minimum output prediction horizon; n2 is the maximum output prediction horizon; nu is the control horizon such that ∆u(t + j) ) 0, ∀j g nu; and λ(j) is a control weighting sequence. In these simulations, the forward and recycle processes are as follows:

GF(s) ) GR(s) )

e-10s 5s + 1

The recycle fraction (r) was set to 0.5, and the sampling time (Ts) was 2. In the first example, the controller was based on the true process model. The second example used an approximate model obtained from system identification so that the overall closed loop includes some model-plant mismatch. The GPC controllers were designed using default tuning parameters that have been found to work well in practice, that is, n1 ) b, n2 ) b + 10, and nu ) 1, where b is the process delay divided by the sampling time. The observer polynomial used in the disturbance model was C ) 1. 4.1. Simulation 1: True Process Model for Control. The GPC tuning parameters used were n1 ) 5, n2 ) 15, and nu ) 1. No controller weighting (λ) was used for the seasonal model or compensator approach. However, because the Taylor series approach is only an approximation and some model-plant mismatch must exist, a factor of λ ) 15 was needed to stabilize the closed-loop system. As a result, the performance is slower than those of the compensator and the seasonal model approaches. Figure 7 shows the closed-loop results. The bumps in the controller output are caused by the second and third terms of the approximation affecting the response at periodic intervals. The recycle compensator provides the best control, as the effects of the recycle have been completely removed and the controller is handling only the forward-path process. The seasonal model provides almost as good control; however, it still has to handle the recycle effect, which degrades its performance.

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Figure 8. Case 1, θFm ) 5, θRm ) 5

Figure 7. Model-based control using the true model Table 1. Modeled Delays Studied modeled delay case

θFm

θRm

figure

1 2 3 4

5 15 15 5

5 15 5 15

8 9 10 11

4.2. Simulation 2: Identified Process Model for Control. In this simulation, the controller models used for the forward- and recycle-path processes were

e-θF s 3s + 1 m

GFm )

e-θR s 3s + 1 m

GRm )

The errors in the time delay were assumed to be (50% of the true delay (θF ) θR ) 10). Four cases, shown in Table 1, were studied. The seasonal model used in the controller was built using the models for GFm and GRm and the combined model, as shown in section 3.3. The resulting simulations are shown in Figures 8-11. Because of the modelplant mismatch, the output prediction horizon n2 was extended to 35, whereas n1 remained at 5. The GPC control weighting λ was 20 for the compensator and seasonal model approaches. For the Taylor series approach, λ had to be increased to roughly 500 (the large λ weighting was required to obtain stable performance). As a result, the Taylor series approach was the slowest for all cases. For cases 1 and 2, the seasonal model and compensator gave similar performances. In case 3, depending on the control performance criteria, either the compensator or seasonal model approach was better. However, in case 4, the seasonal model outperformed the compensator method. Different model-plant mismatch scenarios were also studied. For the majority of the cases, the

Figure 9. Case 2, θFm ) 15, θRm ) 15

seasonal model gave performance comparable to that of the compensator. In some instances, either the compensator method or the seasonal model did better. 5. Implementation Issues Although all three methods combined with GPC give stable control, there are practical factors that should

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it. Accuracy can be improved with a higher-order model (i.e., more terms). In this work, a third-order approximation was used in eq 19, each term was discretized separately, and then the terms were combined to form the overall model. The application of higherorder models becomes more complex, and the controller becomes more sensitive and difficult to tune. As the compensator GPC controller was based on a first-order model and the seasonal model GPC controller was based on the structure shown in eq 23, the controller implementation was simple. Both the seasonal model and the recycle compensator gave good control. In the absence of model-plant mismatch, the recycle compensator approach is the best as it completely removes the recycle dynamics and allows the controller to handle only the forward-path dynamics. Because the compensator requires the identification of two models, the forward- and the recycle-path models, there will be a greater possibility of model-plant mismatch. As a result, complete removal of the recycle path will be unlikely, and the compensator benefits will decrease. Also, because of operating constraints, the independent identification of the forward-path and recycle-path dynamics might not be practical or possible. On the other hand, the seasonal model has the advantage that it might only require the identification of one overall model without isolation of the forward- and recycle-path processes. Figure 10. Case 3, θFm ) 15, θRm ) 5

6. Conclusions The analysis of the recycle system found that the overall gain of a recycle system is magnified by a factor of (1 - r)-1, where r is the recycle fraction. It was also found that the overall recycle dynamics can be represented by seasonal time-series models. In addition, three different methods of applying model-based predictive control to recycle processes were compared. The Taylor series approximation is a good method for handling the problem of denominator deadtime terms but is difficult to implement and tune. The recycle compensator is an elegant solution for minimizing the effect of recycle dynamics and gives very good control when no modelplant mismatch exists. The seasonal model approach is an alternative to the recycle compensator. It is more practical and provides performance comparable to that of the compensator approach, which requires independent identification of the forward- and recycle-path systems. Literature Cited

Figure 11. Case 4, θFm ) 5, θRm ) 15

be considered. First, the Taylor series approximation used by Hugo7 will always contain some degree of model-plant mismatch. The approximated model gain is always lower than the real process gain, and the resulting controller tends to be more aggressive. Therefore, more controller weighting was needed to stabilize the closed-loop process. In practice, one would prefer to overestimate the model gain instead of underestimating

(1) Gilliland, E. R.; Gould, L. A.; Boyle, T. J. Dynamics Effects of Material Recycle. In Joint Automatic Control Conference, June 24-26, 1964, Stanford, CA; American Automatic Control Council: Evanston, IL, 1964; pp 140-146. (2) Denn, M. M.; Lavie, R. Dynamics of Plants with Recycle. Chem. Eng. J. (London) 1982, 24, 55-59. (3) Verykios, X. E.; Luyben, W. L. Steady-State Sensitivity and Dynamics of a Reactor/Distillation Column System with Recycle. ISA Trans. 1978, 17 (2), 31-41 or 49. (4) Jacobsen, E. W. Effect of Recycle on Plant Zero Dynamics. Comput. Chem. Eng. 1997, 21, S279-S284. (5) Scali, C.; Ferrari, F. Control of Systems with Recycle by Means of Compensators. Comput. Chem. Eng. 1997, 21, S267S272. (6) Luyben, W. L. Snowball Effects in Reactor/Separator Processes with Recycle. Ind. Eng. Chem. Res. 1994, 33 (2), 299-305. (7) Hugo, A. J.; Taylor, P. A.; Wright, J. D. Approximate Dynamic Models for Recycle Systems. Ind. Eng. Chem. Res. 1996, 35 (2), 485-487.

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(8) Kwok, K. E. Dynamics of a two-stage heat exchanger recycle system. IASTED International Conference, Control and Applications, The International Association of Science and Technology for Development, Aug 12-14, 1998, Honolulu, HI. (9) Khan, A.; Baker, N.; Wardle, A. The dynamic characteristics of a countercurrent plate heat exchanger. Int. J. Heat Mass Transfer 1988, 31 (6), 1269-1278. (10) Knezevich, P.; Chong Ping, M.; Kwok, K. E.; Dumont, G. Simulation and Control of Recycle Systems. Canadian Conference on Electrical and Computer EngineeringsEngineering Solutions for the Next Millennium, University of Alberta, Edmonton, Alberta, Canada, May 9-12, 1999. (11) Ohbayashi, S.; Shimizu, K.; Matsubara, M. Dynamics of the Flash Fermentor System with Recycle. Ind. Eng. Chem. Res. 1989, 28 (8), 1202-1210. (12) Box, G. E. P.; Jenkins, G. M.; Reinsel, G. C. Time Series Analysis: Forecasting and Control, 3rd ed.; Prentice-Hall: Englewood Cliffs, NJ, 1994; pp 333-335, 338. (13) Kwok, K. E.; Chong Ping, M.; Dumont, G. A. Model based control of recycle process. IASTED International Conference, Control and Applications, The International Association of Science and Technology for Development, Banff, Alberta, Canada, July 25-30, 1999. (14) Marshall, J. E. Control of Time Delay Systems; Peter Peregrinus Ltd.: New York, 1979; pp 33-43.

(15) Taiwo, O. The Design of Robust Control Systems for Plants with Recycle. Int. J. Control 1986, 46, 671-678. (16) Scali, C.; Antonelli, R. Performance of different regulators for plants with recycle. Comput. Chem. Eng. 1995, 19, S409-S414. (17) Scali, C.; Antonelli, R. Issues in the Control of Systems with Recycle. In Preprints of the 13th World Congress: International Federation of Automatic Control (IFAC), San Francisco, CA, June 30-July 5, 1996; American Automatic Control Council: Evanston, IL, 1996; pp 91-96. (18) Scali, C.; Ferrari, F. Performance of control systems based on recycle compensators in integrated plants. J. Process Control 1999, 9, 425-437. (19) Clarke, D.; Mohtadi, C.; Tuffs, P. Generalised Predictive Control. Part I. The Basic Algorithm. Automatica 1987, 23 (2), 137-148.

Received for review June 12, 2000 Revised manuscript received September 13, 2000 Accepted January 12, 2001 IE0005699