Recycle Effect Index: A Measure to Aid in Control System Design for

Ind. Eng. Chem. Res. 2004, 43, 1499-1511

1499

Recycle Effect Index: A Measure to Aid in Control System Design for Recycle Processes† S. Lakshminarayanan,*,‡ K. Onodera,§ and G. M. Madhukar‡ Department of Chemical & Environmental Engineering, National University of Singapore, Singapore 117576, and Mitsubishi Chemical Corporation, Mizushima Plant, Okayama 712-8054, Japan

Systems with material and/or energy recycles are quite common in the chemical industry. Modeling and DCS-level controller design are quite challenging for such processes because separation of the overall process response into a direct (forward-path) response and an indirect (recycle-path) response is often difficult. Even when this is possible, one needs to know whether a feedback-only control structure is sufficient or an advanced control structure that employs a recycle compensator is necessary. Using control loop performance assessment concepts, this paper proposes a quantitative measuresthe recycle effect index (REI)sto help determine whether a simple feedback control system can provide the desired quality of control. Introduction Owing to economic pressures in the increasingly competitive global markets, there has been a relentless drive toward tighter process integration via material and/or energy recycles. The behavior of plants comprising material and/or energy recycles can be quite different from the behavior of the component units. The presence of recycle streams introduces either positive or negative feedback structures into the system,1-3 leading to phenomena such as extremely slow responses, limit cycles, and even instability. For multivariable systems, using the concept of partial feedback, Jacobsen4 showed that the recycle paths can move both the zeros and the poles of the transfer functions between the inputs and outputs that are not part of the recycling loop. Luyben5 showed that a steady-state phenomenon called the “snowball effect” occurs for systems with recycle streams specifically for certain control structure configurations. Several research articles that highlight control strategies for a variety of processes with recycles have been published by Luyben and co-workers6 (also see the references cited therein). Trierweiler et al.7 showed instability at low frequencies can be controlled. Samyudia et al.8 proposed a gap metric method to analyze the effect of recycle dynamics on closed-loop stability and performance. There is ample evidence in the published literature that the presence of recycles can adversely alter both the steady-state and dynamic characteristics of the process compared to the case where fresh material and energy streams are employed. In physiological systems,9 the recycle (actually a negative feedback mechanism) can, in fact, have a benign effect on the system. The nature of integration in chemical processes makes recycles quite problematic from operational or control points of view, and this article addresses issues relevant to such chemical processes. † A preliminary version of this paper was presented at the DYCOPS-6 meeting held at Cheju Island, Korea, Jun 3-6, 2001. * To whom correspondence should be addressed. E-mail: [email protected].: (65) 68748484. Fax: (65) 67791936. ‡ National University of Singapore. § Mitsubishi Chemical Corporation.

The model for a process with recycle is often highorder and complicated. It is not possible to represent the process dynamics of recycle systems using a standard first-order plus dead time (FOPDT) element representation. If a first-principles-based process model is available, it is possible to separate the effect of a manipulated variable on a controlled variable into a direct-effect (forward-path) component and an indirecteffect (recycle-path) component. The forward-path and recycle-path models can be used to design recycle compensators to alleviate the harmful effects of the recycle (Scali and Ferrari10). The issue of identification of process models from plant step response data and open-/closed-loop time series data was addressed very recently by Lakshminarayanan and Takada11 and Lakshminarayanan et al.12 Once the forward-path and recycle-path models are available, the control engineer must quantify the strength of the recycle and determine whether an advanced control structure such as a recycle compensator is needed. If a recycle compensator is deemed important, then it can be implemented, and a feedback controller can be designed for the compensated plant. The feedback controller can then be tuned using the forward-path model only. We consider the case where it is desired to control a single output y using a single manipulated variable u of a recycle process using control functions available at the distributed control system (DCS) level. For this system, this paper proposes and provides a measure of the effect of the recycle using concepts from the minimum-variance benchmarking of control loop performance. This measure, termed the recycle effect index, can help determine the control structure necessary for the process in question. This paper is organized as follows: The recycle process structure is first introduced, and a description of the recycle effect index (REI) is provided. Theoretical expression for the REI and methods for its practical use are then outlined. Several industrial and simulation examples are considered to reinforce the theoretical concepts discussed in this paper. Recycle Process Structure A process with recycle can be modeled using a forward component G and a recycle component Gr. Two struc-

10.1021/ie030533r CCC: $27.50 © 2004 American Chemical Society Published on Web 02/24/2004

1500 Ind. Eng. Chem. Res., Vol. 43, No. 6, 2004

Figure 2. Closed-loop system with feedback and recycle compensation.

Figure 1. Structure 1 (top) and 2 (bottom) representations of a recycle process.

tures are frequently used to describe processes with recycles. These are represented as structure 1 and structure 2 (see Figure 1). The prescribed control strategy is to compensate the detrimental effects of the recycle using a recycle compensator element and to tune the feedback controller on the basis of the forward model G. In the context of SISO systems, the recycle compensator can be developed as follows: Consider the structure 1 representation of a process with recycle. We can represent the output y as a function of the manipulated variable u and the disturbance d as

y)

Gd G u+ d 1 - GGr 1 - GGr

(1)

To alleviate the negative effects of the recycle, a recycle compensator can be introduced as part of the control system. The aim in introducing the recycle compensator element is to achieve an open-loop relationship between y and the process inputs (u and d) as in eq 2

y ) Gu + Gdd

(2)

If this is done, then a feedback controller can be designed for the (hopefully) simpler process G rather than the complicated process G/(1 - GGr). With the ideal recycle compensator in place, the feedback controller will “see and act” on a simpler process that has been compensated for the potentially negative effects of the recycle. The feedback controller for the compensated plant can be a simple controller (such as PI) because of the “simple” process it has to control. If a PID-type controller is used on the uncompensated “overall” process, performance will likely be compromised. For the structure 1 representation of the recycle process, the ideal recycle compensator is given by GRC ) -Gr. Figure 2 shows the overall control system that combines feedback compensation with recycle compensation. The recycle compensator, denoted by RC, can be designed once the model of the recycle-path is available. The feedback controller, C, can be designed using the knowledge of the forward-path model G. The recycle

compensator can be considered to be a feed-forward controller that acts on internally generated disturbances. Although implementation of the recycle compensator might be easy, any plant/model mismatch could have implications on the stability of the closed-loop system. Also, to keep things simple, the recycle compensator should be implemented only when it is absolutely necessary. Thus, it would be worthwhile to obtain a measure of the performance degradation that would result from completely neglecting the recycle effects, i.e., from designing a feedback controller based only on the forward model G. Control Loop Performance and Recycle Effect Index Control loop performance is usually assessed by subjecting the system to a step change in set point or a disturbance value and computing measures such as the integral of the absolute value of the error (IAE), maximum deviation, settling time, and decay ratio. This method is useful when experiments or set-point changes can be made periodically on the loops. A more practical approach is to be able to determine control loop performance from routine operating data. Minimum-variance control is the best possible control in the sense that no controller can achieve a lower variance for the process output. Its implementation is not desired in practice because its control action can be extremely aggressive and also because of its low tolerance to modeling errors. However, such a control model does provide a convenient bound on achievable performance with which the performance of other controllers can be compared.13-15 An overview of the available methods for control loop performance assessment is available in Qin.16 The minimum-variance benchmark possibly requires the least process information (only knowledge of the time delay, d) and places the lowest demand on data (routine closed-loop data is sufficient) and is therefore most suited for the first-cut analysis of control loop performance. We now extend the minimum-variance benchmarking ideas to the domain of systems with recycle. We assume to have designed a minimum-variance controller on the basis of the forward-path model G. With no recycle effects, this controller will deliver minimum-variance performance, i.e., the autocorrelation function of the process variable y will be identically zero for all lags beyond the process delay. However, when this minimumvariance controller is implemented on a process with

Ind. Eng. Chem. Res., Vol. 43, No. 6, 2004 1501

When the set point is constant (zero in terms of deviation variables), we can derive the relationship

yt )

N at 1 + q-dTf(Q - TR)

(4)

N can be expanded as

N ) F + q-dR Figure 3. Block diagram of the closed-loop system.

Substituting this in the expression for yt, we obtain

recycle, the controller performance will deviate from the minimum-variance performance depending on the nature of the recycle-path transfer function. Let σMV,RC2 denote the output variance in the presence of the recycle, and let σMV,NRC2 denote the output variance when there is no recycle. The subscript MV in both σMV,RC2 and σMV,NRC2 denote the fact that the controller employed is a minimum-variance controller based on the forward-path model alone. It is obvious that σMV,RC2 g σMV,NRC2. We can therefore define a performance degradation measure (recycle effect index) φ as

φ)1-

σMV,NRC2 σMV,RC2

(5)

(3)

φ can take any value between 0 and 1. A value of φ close to 0 implies that the effect of the recycle is lower, and a value of φ close to 1 means that σMV,RC2 is large, implying that the effect of the recycle is quite strong. A value of φ close to 1 would indicate that that the use of an advanced control scheme such as a recycle compensator might be necessary should a high control performance be desired. If the implementation of the recycle compensator is not possible in the above case, it is advisable that a detuned feedback controller be implemented to avoid potential instability (this will be accompanied by a performance loss). The detuned feedback controller can then be based on the overall process G/(1 - GGr). A value of φ close to 0 indicates the futility of employing a recycle compensator. Because φ is calculated by assuming that a minimum-variance controller is controlling the process, it cannot be estimated by the analysis of plant operating data (unless a minimum-variance controller designed from the forward model G is in place). The approach considered here devises a formula relating φ to the parameters of the forward and recycle paths. This procedure is described in the following section. Computation of the Recycle Effect Index Consider the plant (under feedback control) shown in the block diagram of Figure 3. The forward-path transfer function is denoted as q-dTf (the deadtime of d samples has been explicitly factored out, and hence, Tf is the delay-free part of the transfer function) and the recycle path is denoted by TR. Q is the controller. N is the disturbance or load-transfer function. yt is the process output, ut is the manipulated variable, ysp,t is the set-point value for y, and at is random noise sequence with zero mean and variance σa2. Notice that this corresponds to the structure 1 representation of the recycle process.

yt ) )

F + q-dR at 1 + q-dTf(Q - TR) F[1 + q-dTf(Q - TR)] + q-d[R - FTf(Q - TR)] at 1 + q-dTf (Q - TR)

q-d(R - FTf Q + FTfTR) ) Fat + at 1 + q-dTf (Q - TR)

(6)

The first part of eq 6, Fat, denotes the controllerinvariant term that is independent of the control law Q. The second part is dependent on the controller and can be manipulated by a proper choice of the controller Q. If Q is designed as in the equation

Q ) TR +

R FTf

(7)

then the second term becomes identically zero, and the lower bound on the output error variance can be achieved. Q is then the minimum-variance controller for the overall (forward path + recycle path) process. When the minimum-variance controller is designed on the basis of the forward-path model alone (by neglecting the recycle transfer function TR), Q is given by

Q)

R FTf

(8)

With Q designed as in eq 8, the expression for yt becomes

yt,MVFP ) Fat +

q-dFTfTR at R 1 + q-d - q-dTfTR F

(

)

(9)

(the subscript MVFP denotes that this output can be expected when there is a minimum-variance controller based on the forward path only acting on a process with recycle). When TR is zero, one obtains the minimumvariance performance, as expected

yt,MV ) Fat ) F0at + F1at-1 + ‚‚‚ + Fd-1at-d+1

(10)

This gives σMV,NRC2 ) (F02 + F12 + ‚‚‚ + Fd-12)σa2. As remarked earlier F0, F1, ..., Fd-1 are all independent of the control law and can be estimated from routine


operating data. The second term in the expression for yt,MVFP can be denoted as

et )

[(

]

FTfTR at-d ) R 1 + q-d - q-dTfTR F (L0 + L1q-1 + L2q-2 + ‚‚‚)at-d (11)

)

This implies that σe2 ) (L02 + L12 + L22 + ‚‚‚)σa2. Using the expression for yt,MVFP, we can write

σMV,RC2 ) σMV,NRC2 + (L02 + L12 + L22 + ‚‚‚)σa2 ) (F02 + F12 + ‚‚‚ + Fd-12 + L02 + L12 + L22 + ‚‚‚)σa2 (12) This results in

φ)1-

σMV,NRC2 σMV,RC2

)

L02 + L12 + L22 + ‚‚‚ F02 + F12 + ‚‚‚ + Fd-12 + L02 + L12 + L22 + ‚‚‚

(13)

The above equation implies that, with knowledge of the forward-path model, the recycle-path model, and the disturbance model N, one can compute the recycle effect index. From the expressions derived above, one can see that φ depends on (i) the product of the forward-path and recycle-path gains, (ii) the nature of the disturbance Nat [we have assumed that at is a white noise sequence and, therefore, that the time constant of N (equivalently the pole location of N) will have a bearing on φ], (iii) the time constants of the forward- and recycle-path elements (to keep things simple and practical, we assume that the forward- and recycle-path transfer functions can be adequately described by first-order plus deadtime elements), and (iv) the delays associated with the forward-path and recycle-path elements. The challenge is to determine the conditions under which the recycle effect index is under a certain threshold value (say, φ < 0.25) so that one need not bother with advanced control structures such as the recycle compensator. Two possible options exist. The first option is to perform numerous simulations spanning several combinations of G, Gr, and N (all assumed to be first-order plus time delay elements). In each case, a minimum-variance controller is designed on the basis of the forward-path element only, and the recycle effect index is calculated. Data mining procedures can then be used to determine “rules”, i.e., the conditions under which the recycle compensator will or will not be needed. A better alternative to the rule generation procedure is to use a simulation-based approach. Using symbolic computing, a minimum-variance controller can be designed utilizing the noise model (FOPDT element) and the forward-path model. With the designed minimumvariance controller and the identified process model, the closed-loop system can be simulated. Using this approach, the REI is computed for different values of the noise model pole location, leading to an REI plot for the

recycle process. The noise transfer function is assumed to be of the form

N)

1 1 - az-1

(14)

with z ) a as the pole location. Note that, when a ) 0, we have N ) 1 and Nat ) at, which is white noise. When a ) 1, we have N ) 1/(1 - z-1), which makes Nat equal to integrated white noise (random walk). These extremes generally represent the types of stochastic disturbances that an industrial loop experiences. The REI plot therefore portrays the loss in control performance due to ignoring the recycle dynamics as a function of the noise characteristics that are likely to affect the process. We point out here that REI plots can be constructed for any noise characteristics and the recycle process need not conform to any particular structure. Examples of using the REI are demonstrated in the next section. Chodavarapu and Zheng17 present guidelines for tuning feedback PID controllers for processes with recycles. They adopt the structure 2 representation of the recycle system. Using a fixed model for the forwardpath element (first-order plus time delay) and considering various values for the parameters of the recyclepath transfer function (gain, time constant, and time delay), they provide heuristics for designing PID-type controllers. Although their results are interesting and potentially useful, they provide heuristics for only a portion of the parameter space. Also, they consider a first-order plus dead time (FOPDT) representation for G/r . Approximation of Gr by a FOPDT is much more reasonable than approximating G/r . The heuristics developed by Chodavarapu and Zheng17 indicate that the ratio of the recycle time constant to the forward-path time constant (τ/r /τf) holds the key to the tuning of the feedback controller. The analytical expressions derived in this work (which are based on a stochastic setting) indicate that the recycle effect index (φ) depends on other quantities, such as the gains and time delays of the forward path and recycle path, and the disturbance dynamics in addition to the forward-path and recyclepath time constants. Remark 1. Theoretically, it is possible to envision systems in which the disturbances affect the output after the recycle. In this case, the expression for the output y is

y)

G u + Gdd 1 - GGr

(15)

If the recycle compensator GRC ) -Gr is employed, the “compensated” plant will have the relationship

y ) Gu + Gd(1 - GGr)d

(16)

This is in contrast to the relationship that was obtained for the structure considered in this paper, i.e., y ) Gu + Gdd. This means that the disturbance rejection behaviors will be quite different for these two structures. If the noise dynamics were to have an effect outside the recycle loop, one can show that eq 4 will become

yt )

N(1 - TfTR) -d

1 + q Tf(Q - TR)

at )

N* at 1 + q Tf (Q - TR) (17) -d


One can now express N* as N* ) F* + q-dR* and repeat the steps followed in our manuscript and derive the recycle effect index for this case. F* will depend on N as well as Tf and TR. One can assume a structure for N, say, N ) 1/(1 - az-1) with a ranging from 0 to 1, and arrive at the REI plot. It must be pointed out that a vast proportion of industrial processes fall under the structure considered in Figure 3. Therefore, in this paper, we consider only the more industrially relevant structure, in which the disturbances affect the process upstream of the recycle. Remark 2. The information one seeks places certain demands on the quality of the data needed. If one wants to benchmark the current control performance against the minimum-variance performance, routine operating data plus knowledge of the process delay are sufficient. If the maximum performance that can be achieved with PI/PID controllers is sought, routine operating data plus a process model (Ko and Edgar18) or experimental closedloop data (Agrawal and Lakshminarayanan19) will be needed. In this work, we are not interested in knowing how well the current controller is doing relative to minimum-variance performance. This would be easy to quantify with only routine operating data and knowledge of process delay. We are rather interested in answering the following questions: (i) What is the control strategy to be adopted (feedback vs feedback plus recycle compensation) to achieve the maximum control loop performance? (ii) If only feedback control is chosen, what should be the basis for controller tuning (i.e., forward-path model alone or overall model)? Analysis of routine operating data will not be able to provide answers to these questions. One needs experimental data to identify the relevant models (Tf and TR) and construct the REI plot. This will determine the responses to the two questions posed above. Case Studies The first four examples described here primarily deal with systems that involve energy recycles. Three of these examples deal with analysis of industrial process systems. Examples 5 and 6 deal with processes that involve material recycles. It must be pointed out that, as long as the relevant models are available, it should not matter whether material and/or energy recycles are involved. However, obtaining models for concentrationrelated variables from plant data is difficult because of the relatively “slow” measurement of compositions. Developments in multirate system identification theory and process instrumentation are expected to make the construction of models for composition variables easier in the future. Example 1: Simulated Feed-Effluent Heat Exchanger, Furnace, Reactor System. We consider a process in which the fuel supply to a furnace is manipulated to control the temperature of the stream leaving the furnace. The stream leaving the furnace enters a reactor in which a simple homogeneous reaction takes place. The effluent stream from the reactor heats the fresh feed coming into the system in a feed-effluent heat exchanger before leaving the system. The “warm” fresh feed is further heated in the furnace. The schematic of the system, the models employed, and the operating conditions are specified in Chodavarapu and Zheng.17 The forward-path and recycle-path models for this process (structure 1) are given by G ) 252/(2.6s + 1) and Gr ) 0.6K/[252(25s + 1)]. Here, K denotes the

Figure 4. REI plots for example 1.

reactor gain, which changes with operating conditions. The recycle-path gain and, consequently, the gain of GGr change with the operating conditions. Chodavarapu and Zheng17 consider three specific values of K (1, 1.5, and 3). When K ) 1 or 1.5, the product of the forward-path and recycle-path gains is less than 1, so the system is open-loop stable. For values of K ) 1 and 1.5, the REI plots are shown in Figure 4 (top). The REI values are quite small, indicating that the feedback controller can be designed on the basis of the forward-path model alone (ignoring the recycle dynamics). This also means that this feedback controller can be tuned very aggressively. Indeed, Chodavarapu and Zheng17 advocate the use of aggressive tuning for these values of K. Note that a significant separation exists in the time scales between the forward-path (fast) and recycle-path (slow) dynamics. On the basis of this separation (especially when the value of K is small), one would expect the recycle effect to be less significant. This is also indicated by our analysis. When K ) 3, the system is open-loop unstable. The REI plot shown in the bottom plot of Figure 4 indicates that an aggressive feedback controller based on the forward-path model alone can be designed and implemented if the expected disturbances are “structured” (e.g., integrated white noise) rather than being random. Chodavarapu and Zheng17 also show that an aggressive feedback controller based on the forward-path dynamics is suited for set-point tracking when K ) 3. If whitenoise-type disturbances are expected to affect the process, it is not appropriate to design an aggressive feedback controller on the basis of the forward-path model. The recycle-path dynamics must be taken into consideration for the design of the control system. In summary, good control loop performance can be ensured (for all disturbance types) if a recycle compensator plus feedback controller is implemented when K ) 3. When a perfect recycle compensator is employed, it is very easy to control the compensated plant, which is a simple firstorder process. It is conceivable that a very aggressive feedback controller can now be employed in conjunction with the recycle compensator. In fact, an “aggressive” PI controller designed using the direct synthesis method, Gc(z-1) ) (0.008 - 0.005z-1)/(1 - z-1), performed good stochastic regulation (for all types of disturbances, η ≈


Figure 5. Stochastic regulation and set-point tracking performance with aggressive PI controllers designed for the forward-path process in example 1. Subplots a and b are for the white noise disturbance case, and subplots c and d are for the integrated white noise disturbance case.

0.8), set-point tracking, and rejection of step-type disturbances on the recycle-compensated process. What happens if one neglects the recycle-path dynamics and designs aggressive PI controllers (by maximizing η) based on the forward-path model only that are then implemented on the process? This situation was investigated by considering two extremes for the noise: “white” noise and “integrated white” noise. For the white noise case, the aggressive feedback controller turned out to be a pure integrator, i.e., Gc(z-1) ) (1.852 × 10-6)/(1 - z-1). When this controller was implemented on the process, the stochastic regulation and the servo performance are as indicated in Figure 5 (a and b, respectively). The system is unstable, as was clearly indicated in Figure 4. For the integrated white noise case, the optimal PI controller based only on the forward path turned out to be Gc(z-1) ) (0.0122 - 0.0086 z-1)/(1 - z-1). When this controller was implemented on the overall process, the stochastic regulation and servo performances are as indicated in Figure 5c and d, respectively. The control loop performance index based on the data shown in Figure 5c turns out to be 0.999 (almost minimum-variance performance, as predicted in the bottom subplot of Figure 4 for the integrated white noise case). The set-point tracking is also good (quick response, less overshoot, and no offset). We also proceeded to investigate the consequences of designing an aggressive PI controller based on the overall process model. With the assumption that the disturbances were white, then the best PI controller for the process was found to be an integral controller with Gc1(z-1) ) (3.134 × 10-4)/(1 - z-1). PI-type controllers were not feasible in this case. With this controller, the disturbance rejection and set-point tracking response for a step-type excitation initiated at sample 10 are as illustrated in Figure 6 (subplots a and b, respectively).

If we assume the integrated white noise disturbance scenario, the “best” PI controller was determined to be Gc2(z-1) ) (0.0121 - 0.0084z-1)/(1 - z-1), which is practically the same as that obtained by neglecting the recycle-path dynamics. This shows that, for an integrated white noise disturbance, the recycle path is as good as nonexistent. With this controller, the disturbance rejection and set-point tracking responses are as shown in Figure 6c and d, respectively. It is easy to see that the responses with the controller designed for the integrated white noise disturbance case provides better performance for this type of input (as would be expected). The choice between controllers Gc1 and Gc2 really depends on the types of disturbances this loop is likely to experience. Gc1 is to be employed if the disturbances are expected to be of white noise type. If “random walk” or step-type disturbances are expected, controller Gc2 should be employed. As seen earlier, if a recycle compensator plus feedback controller strategy is employed, one can ensure high performance against all kinds of disturbances as well as good set-point tracking. These results are indicative of the soundness of the tuning guidelines offered by the REI plot in controlling processes with recycle. Example 2: Industrial Quench Column. A quench column immediately follows the primary fractionator in the ethylene plant. Cracked hydrocarbon gas containing water is fed to the quench column after the heavy components are separated in the primary fractionator. This cracked gas is cooled by water (quench water) in upper and lower packed sections. In the bottom section, cooled cracked gas condenses and is separated into gasoline and water. The bottom water, used as the heating medium in several process reboilers is at about 80 °C. After heat recovery in several process reboilers, cooled water is recycled to the quench column as an


Figure 6. Disturbance regulation and set-point tracking performance with aggressive PI controllers designed for the overall process in example 1. Subplots a and b are for the white noise disturbance case, and subplots c and d are for the integrated white noise disturbance case.

Figure 7. Process diagram of the quench column.

upper reflux (30 °C) and a lower reflux (60 °C). The process flow diagram is depicted in Figure 7. In the current system, the bottom temperature controller manipulates the bypass flow rate of the reflux cooler (E-121). This control loop is called TC105. To design a feedback PID controller for TC105, Emoto et al.20 conducted an open-loop step test on the process. The data collected from the step test are shown in Figure 8. The top subplot shows the output response, and the bottom subplot indicates the step change in the manipulated variable. It is apparent that the response

comprises of two time scalessthe output response shows an immediate jump and then slowly approaches the steady state. Using these data, the following models were obtained for the forward and recycle paths, respectively.

G)

0.0777e-s , 1.25s + 1

Gr )

6.2173e-11s 1.44s + 1

(18)

The model was rigorously cross-validated using multiple fresh data sets.


Figure 8. Step test of the quench column.

Figure 10. Schematic for reactor-preheater system.

Figure 9. Recycle effect index plot for the quench column system.

The recycle effect index plot for the quench column is given in Figure 9. It can be seen that the largest value of φ is about 0.085, which is quite small. Hence, we expect that a recycle compensator will not be needed and that the feedback controller could be tuned on the basis of the forward-path model alone. In fact, Emoto et al.20 did obtain good closed-loop control of the TC105 loop using a feedback-only control structure and PID controller settings based on the forward-path model. Example 3: Reactor-Preheater System at a Solvent Manufacturing Plant. The schematic of the system that is part of a solvent manufacturing plant at MCC’s Mizushima petrochemical complex is shown in Figure 10. The reactor effluent goes into a separator whose overhead product (temperature v) is used to preheat the feed coming into the process unit. The feed is maintained at a constant temperature. To maintain the reactor entrance temperature (y) constant, an electric heater is used. The heat supplied by this heater element serves as the manipulated variable u. The temperature of the stream entering the heater is denoted as w. The process was operating on manual mode, and it was desired to achieve closed-loop control that manipulates u to keep y at a specific value. The flow rate of the feed stream is the main process disturbance. Identification performed using open-loop plant data resulted in the following transfer functions

Figure 11. Recycle effect index plot for the reactor-preheater system.

(the model was accepted after cross-validation with new data sets)

y(s) 0.2013 ) u(s) 4.93s + 1

(19)

y(s) 4.5238 ) 4.93s +1 w(s)

(20)

w(s) 0.9478 ) y(s) 20.25s + 1

(21)

Assembling these models in terms of the structure 1 representation, we obtain

G(s) )

0.2012 , 4.93s + 1

Gr(s) )

4.5238 20.25s + 1

(22)

The REI plot for this system is shown in Figure 11. The results indicate that the maximum value of the REI is 0.12, implying that the recycle path will not significantly deteriorate the performance of the closed-loop system if the feedback controller is designed and tuned on the


Figure 12. Schematic of the industrial reactor system with multiple recycles.

basis of the forward-path model G alone. A recycle compensator would not be required. This recommendation was implemented in the plant, i.e., the loop was closed using a PID controller tuned on the basis of the forward-path model. The performance of the controller has been extraordinarily good: the control loop performance index (η) based on the minimum-variance benchmark is about 0.93 for this loop (η ) 1 is the theoretical optimum) over a period of 3 years following our controller implementation. Example 4: Industrial Reactor System with Multiple Recycles. A schematic of the process unit under consideration is shown in Figure 12. This unit is part of a larger plant and has a significant impact on the profitable operation of the plant. Two reactors (R1 and R2) are used to convert a feedstock comprising mainly species A into a product comprising mainly species B. Many undesired byproducts might also be produced in the system. The operational goal is to select

Figure 13. Model fit (top) and model validation (bottom) for Ti1.

operating conditions such that as much of A as possible is converted into B. Advanced control schemes implement these chosen operating conditions using a hierarchical control structure. In this study, we aim at improving the control of the first reactor inlet temperature (Ti1), which exhibited oscillations and deviations of unacceptable levels. The available manipulated variable is the bypass stream flow (F1). The inlet temperature for reactor 2 (Ti2) was manipulated using F4. As shown in Figure 12, there are two feed-effluent preheaters (H1 and H2) designed for efficient use of energy. Closed-loop experimental data were used to identify the models. The model fit (top) and cross-validation results (bottom) are shown in Figure 13. The plant data are shown as continuous lines, and the model predictions are shown as broken lines. In this identification effort, the input variables were F1, F4, T1, and T2, and Ti1 was the output variable. The inclusion of T1 and T2 in the list of input variables for identification helps to quantify the forward-path and recycle-path dynamics separately, as demonstrated by Lakshminarayanan and Takada.11 A similar modeling effort was made for Ti2 using available open-loop data. The fit and cross validation for Ti2 are shown in Figure 14. Again, as for Ti1, a good model is indicated for Ti2 as well. With all of the relevant models (forward path and recycle path) now available, the REI plots could be constructed. The identified models for this system are not explicitly provided here for confidentiality reasons. During plant operation, it was noticed that when the profiles of the manipulated variables (F1 and F4) showed a high correlation, the control performances of the two loops were poor. This similarity in the time series of the two manipulated variables appeared too often. With this in mind, the REI plot for Ti1 was generated by varying the level of correlation between F1 and F4. Figure 15 shows that, as long as the correlation between F1 and F4 is low or moderate, the REI values are less than 0.1 even for an integrated white noise disturbance. When the correlation becomes high, the REI can reach a value


Figure 14. Model fit (top left) and model validation (middle and bottom) for Ti2. Figure 16. Performance of control loops before and after tuning.

Figure 15. REI plots for different levels of correlation between F1 and F4.

of about 0.35 and can be considered to be in the “border zone”. This led us to conclude that designing the controllers on the basis of the forward-path models alone (pure effect of F1 on Ti1 and pure effect of F4 on Ti2) would lead to very satisfactory performance of the controllers in regulating the stochastic disturbances affecting the system. There is no need to design the controllers on the basis of the overall process models (direct plus recycle effects) in this case; in fact, such a strategy could lead to poor regulatory performance of the control system.

With this design, the control performance improved, as shown in Figure 16. The y-axis values are not shown here for confidentiality reasons. The first and third subplots show the trajectories of the controlled variables (Ti1 and Ti2) along with their set-point values. The second and fourth subplots show the movement of the manipulated variables F1 and F4. It can be seen that the control performance has increased dramatically after tuning and the manipulated variable moves are also more acceptable. During long-term monitoring, it was seen that high correlations between F1 and F4 rarely occurred and the good performances of the loops continue to this day. This improvement was achieved through identification of the forward-path and recyclepath dynamics and through the use of REI to determine the appropriate control strategy. Example 5: Simulation of Two CSTR System with Recycle. The system we consider next consists of two reactors in series with a recycle stream from the outlet of the second reactor to the inlet of the first reactor. The two reactors are assumed to be well-mixed systems in which first-order irreversible reactions not accompanied by any heat effects occur. The levels and flow rates are constant; we are concerned only with composition effects. The control objective is to maintain the reactor 1 outlet composition, y1, at a specified level by manipulating the feed composition u1. The main disturbance is the composition d of the wild stream entering the first reactor and the composition u2 of the feed stream entering the second reactor. The schematic of the system is shown in Figure 17, and a description


Figure 17. Schematic of the simulated two CSTR with recycle system. Table 1. Variables and Parameters for the Two CSTR with Recycle System parameter feed flow rate into reactor 1 feed flow rate into reactor 2 recycle flow rate disturbance stream flow rate product removal rate from reactor 1 product removal rate from reactor 2 composition of stream F1 composition of stream F2 composition of disturbance stream reactor 1 outlet composition reactor 2 outlet composition recycle stream composition at the entrance of reactor 1 volume of reactor 1 volume of reactor 2 measurement delay in composition sensors recycle delay (outlet of reactor 2 to inlet of reactor 1) kinetic rate constant (reactor 1) kinetic rate constant (reactor 2)

symbol

value

F1 F2 R Fd Fp1 Fp2 u1 u2 d y1 y2 y2

m3/min

1 0.5 m3/min 10 m3/min 0.5 m3/min 1 m3/min 1 m3/min 2 kmol/min 3 kmol/min 1 kmol/min 1 kmol/min 1 kmol/min 1 kmol/min

V1 V2 θm θr

1 m3 10 m3 1 min 2 min

k1 k2

1 min-1 0.1 min-1

of the variables and values of the operating parameters is provided in Table 1. The ordinary differential equations describing the system can be found in Scali and Ferrari.10 As noted in Table 1, we assume that the reactor outlet composition measurement is perfect but available after a delay of 1 min. The composition measurements for the inlet streams (u1, u2, and d) are assumed to be instantaneously available. The delay between measurements y2 and y/2 is 2 min (this is the time taken for the material to move from the reactor 2 exit to the inlet of reactor 1). We further assume that we do not have access to measurements y2 and y/2. Using a two-step identification strategy, Lakshminarayanan and Takada11 derived the forward- and recycle-path models using open-loop time series data as

0.0769e-0.9s G) , 0.1396s + 1

8.1169e-0.8s Gr ) 0.7003s + 1

The REI plot for this system is shown in Figure 18. The results indicate that the REI gets larger as the pole location of the noise model moves toward the unit circle (i.e., with integrated white noise). The largest value of REI is about 0.16, implying that the recycle dynamics can be neglected for the purposes of control design. Example 6: Reactor-Separator System with Recycle. We now consider a reactor-separator system as discussed in Papadourakis et al.21 The process schematic is shown in Figure 19. The system comprises

Figure 18. REI plot for the simulated two CSTR with recycle system.

Figure 19. Schematic of the reactor-separator with recycle system.

an isothermal CSTR in which a simple the first-order reaction A f B takes place. The reactor effluent stream (assumed to be an ideal AB mixture with constant relative volatility) is fed to a distillation column. The distillate, rich in unconverted A, is recycled back to the reactor. Product B is withdrawn from the bottom of the column. All flow rates are assumed to be constant. Details of the operating conditions are given in Papadourakis et al.21 They derived the following model relating the feed composition to the reactor product composition

Direct Effect:

GR1 )

xF(s) xO(s)

)

0.2580 1.3498s + 1

Direct Effect: GCD ) Direct Effect:

xD(s) xF(s)

)

GR2 )

1.4494(0.3684s + 1)e-0.2912s (1.9624s + 1)(0.43256s + 1)

xF(s) xD(s)

)

0.2810 1.3498s + 1

Overall Effect: xF(s) GR1 ) ) GO ) xO(s) 1 - GR2GCD

GR1

( )

1 - GR1GCD

GR2 GR1

In terms of the structure 1 representation of recycle


Figure 20. REI plot for the simulated reactor-separator with recycle system.

processes, we can write the forward- and recycle-path models as

G)

1.5786(0.3684s + 1)e-0.2912s 0.2580 , Gr ) 1.3498s + 1 (1.9624s + 1)(0.43256s + 1)

Gr can be well approximated by the following FOPDT transfer function

Gr )

1.5786e-0.2912s (2.0266s + 1)

Using G and the FOPDT approximation of Gr, the REI plot was constructed. The REI values are shown as model 1 in Figure 20. The REI values are small, indicating that a recycle compensator is not needed. For illustration purposes, the REI values were also computed for cases in which the recycle-path gain was taken to be 2 and 3 times the actual value. These results are plotted as REI curves labeled model 2 and model 3, respectively. Even when the recycle-path gain is 2 times the base value, the REI is small, and a recycle compensator is not recommended. However, when the recyclepath gain increases to 3 times the base value, we see that, for certain noise model pole locations, the closedloop system is unstable when a minimum-variance controller is designed on the basis of the forward-path model alone. These points show an REI value of 1. As the noise term Nat goes from pure white noise to integrated white noise, the REI value decreases to a very small value. For this hypothetical process, this indicates that, when certain types of noise affect the process, it is better not to aggressively tune the controller based on the forward-path model alone. For the situation represented by model 3, the control strategy will be the same as that employed in example 1 for the case K ) 3. Conclusions A measure termed the recycle effect index (REI) has been developed to quantify the loss in control loop performance encountered when the feedback controller is based on the forward-path model alone for systems with recycles. Because a theoretically optimal feedback controller is used as a benchmark, the REI can be used

to judge whether a more advanced control structure (e.g., a control structure employing a recycle compensator) is needed. If the REI value is significant (say, 0.25 or more), then the use of a recycle compensator can be considered. The theory was demonstrated on three simulation examples and three industrial case studies in which the REI measure helped to identify the best control structure. It must be remembered that the REI is calculated on the basis of the identified models: plant/ model mismatch can have a significant effect on the computed REI. With models that have been well validated with many fresh data sets (as was done in all of the industrial case studies discussed here), the REI measure can be used with confidence. In all three industrial case studies considered, through proper system identification and computation of the REI, we were able to determine that (a) a recycle compensator is not needed and (b) a high-performance feedback controller could be designed on the basis of the forwardpath model alone for the regulation of stochastic-type disturbances. Without this clear insight, the industrial systems considered were either being operated in the manual mode (reactor with preheater system) or were poorly controlled because of inappropriate tuning (quench column temperature loop and reactor with multiple recycles). With the concepts described in this paper, excellent control performance was obtained for all three industrial systems. The REI analysis of these industrial systems provided the needed confidence to design and implement a feedback controller on the basis of the forward-path dynamics alone. Examples 5 and 6 showed that the ideas presented here are equally applicable to systems with material recycle as long as the models for the forward path and recycle path are available. Acknowledgment The authors extend their sincere appreciation to the two anonymous referees for making some interesting suggestions and comments during the review process. Their remarks have, undoubtedly, contributed to the utility of this article. Symbols C ) feedback controller (continuous domain) F ) polynomial (in powers of q) depicting the controllerinvariant part in the moving average model for the process output F0, ..., Fd-1 ) coefficients of the F polynomial G ) forward-path transfer function (continuous domain) Gd ) disturbance transfer function (continuous domain) GRC ) transfer function of the recycle compensator Gr ) recycle-path transfer function (continuous domain, structure 1) G/r ) recycle-path transfer function (continuous domain, structure 2) L0, L1, ... ) coefficients of the controller-variant part of the moving average model for process output N ) disturbance/noise transfer function (discrete domain) Q ) feedback controller (discrete domain) Tf ) forward-path transfer function (delay-free part, discrete domain) TR ) recycle-path transfer function (discrete domain) at ) white noise sequence d ) process disturbance; also, sample delays in forwardpath model q ) backward difference operator u ) manipulated variable

Ind. Eng. Chem. Res., Vol. 43, No. 6, 2004 1511 y ) controlled variable ysp ) set point of the controlled variable Greek Symbols φ ) recycle effect index η ) control loop performance index σ ) standard deviation τf ) time constant of the forward-path model τr ) time constant of the recycle-path model (structure 1) τ/r ) time constant of the recycle-path model (structure 2)

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Received for review June 30, 2003 Revised manuscript received November 22, 2003 Accepted January 14, 2004 IE030533R

Recycle Effect Index: A Measure to Aid in Control System Design for

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