Robustness of a Recycle Compensator - American Chemical Society

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Ind. Eng. Chem. Res. 2006, 45, 1924-1934

PROCESS DESIGN AND CONTROL Control of a Process with Recycle: Robustness of a Recycle Compensator Etienne Tremblay, Andre´ Desbiens,* and Andre´ Pomerleau LOOP (Laboratoire d’obserVation et d’optimisation des proce´ de´ s), Department of Electrical and Computer Engineering, UniVersite´ LaVal, Que´ bec City, Que´ bec, Canada G1K 7P4

Several studies on the dynamics of processes with recycle are reported in the literature. A complete summary of the effects of the recycle, along with a detailed analysis of the disturbances behavior when matter or energy is recycled, is presented in the first part of the paper. The control of processes with recycle is then discussed, including the benefits of adding a recycle compensator to a controller. Representing the system equations with Bode plots shows that the performance of a PI (proportional and integral) controller with a recycle compensator is usually better than the use of a PI controller alone. A systematic analysis of the sensitivity of the model demonstrates that this is true, even for important model errors in the recycle path. This work suggests that the negative result of recycle addition on the process dynamics can be overcome when the recycle effects are anticipated even though the recycle path model is not error free. 1. Introduction Recycling of material or energy is very common. In chemical plants, for example, environmental and economical constraints force the reuse of the majority of reagents. An example from the mineral industry is the ore that must often be ground several times in order to obtain the desired particle size distribution. The fact that typical plants nowadays recycle both matter and energy has justified many studies on systems with recycle. Before 1990, Denn and Lavie1 had already shown that a recycle stream can greatly increase the time constants of a process, while Taiwo2 had proposed the use of the recycle compensator in the regulator design. However, developments on process control with recycle only emerged a few years later with the works of Luyben,3-6 using transfer function models to show the effects of various parameters of the individual units in a process with recycle. Luyben considered a process consisting of a reactor equipped with a distillation column and searched for the control structure that best reduced the “snowball effect”, i.e., a large increase of the recycle flow rates caused by a small increase of the fresh feed. Scali and Ferrari7 later detailed the recycle compensator of Taiwo.2 They analyzed the recycle compensator, mentionned its benefits, and applied it to a few benchmarks. The goal of the recycle compensator, a part of the regulator, is to reduce or eliminate the effects of the recycle, allowing for the design of a regulator independent of the recycle. The recycle compensator suppresses the negative consequences of the recycle like the slowing of the overall process. Scali and Ferrari8 also studied the robustness of the recycle compensator for applications within the time domain. According to their results, the recycle compensator presents several benefits but requires that a model of the recycle path be identified. This problem was addressed by Lakshminarayanan et al.9 The present paper compares and completes the more recent works and focuses especially on the robustness of the recycle * To whom correspondence should be addressed. Tel.: 1-418-6262131 ext. 3408. Fax: 1-418-656-3159. E-mail: [email protected].

compensator. The impacts of disturbances or model errors are studied in the frequency domain. The effects of the recycle will first be summarized using firstorder transfer functions with delay. This paper will demonstrate the snowball effect and both the increase of time constants and the impact of disturbances described by Luyben,3-6 as well as the stair effect first explained by Hugo et al.10 Bode plots of output versus input transfer functions and Bode plots of output versus disturbances transfer functions will also be constructed, which constitutes a more in-depth study of the work originally presented by Knezevich et al.11 The design of two regulators for first-order plus time delay systems will then be described. The first controller (proportional and integral (PI)) will be calculated from a process model that ignores the internal structure of the process, i.e., without special consideration for the recycling. The second controller will make use of the recycle compensator designed by Taiwo2 in addition to the PI controller. An analysis of the robustness of the regulator with the recycle compensator is also included, as suggested in a recent paper by Samyudia et al.12 Scali and Ferrari8 have looked at only a few examples in the time domain to analyze the robustness of a regulator with a recycle compensator. This paper will explicitly develop the equations of sensitivity and the expression of the output as a function of the disturbances, which will be accompanied by the related Bode plots. Finally, the performance of both regulators will be compared to determine the circumstances that dictate the use of the recycle compensator. 2. First-Order Process with Recycle A first-order process with recycle can be represented in many ways. In every case, it is composed of a transfer function for the direct path (often a reactor or a transformation unit) and a transfer function for the recycle path. The representations given by Figures 1 and 2 will be used in this paper. In these figures, Gd(s) and Gd2(s) are the transfer functions of the direct path, Gr(s) and Gr2(s) are the transfer functions of the recycle path,

10.1021/ie050643t CCC: $33.50 © 2006 American Chemical Society Published on Web 02/14/2006

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2.1. Steady-State Gains and Snowball Effect. All the authors agree that the gains of Gd(s) and Gr(s) are very important in processes with recycle. For instance, Luyben3 has shown that the poles of the transfer functions are changed with Kr. More recently, Lakshminararayanan et al.13 have shown that the “recycle effect index”, a measure of the recycle effect, was directly related to the product of Kr and Kp. According to the denominator of eqs 2-4, the poles of the process depend on KrKp. Furthermore, the steady-state gain of the plant is given by:

Figure 1. Process representation.

Hyu(0) ) Figure 2. Process representation with a recycle fraction.

and U(s) and Y(s) are, respectively, the input and output. The signals Dindice(s) are disturbances. Figure 2 introduces the recycle fraction a. This representation is frequent for systems with a transformation unit coupled with a separator (e.g., a reactor with a distillation column). For a system using flows, a is the fraction of the flow at the output of the transformation unit that is sent back as an input to the transformation unit. The variable a will be used in the following sections. The representation of Figure 2 is equivalent to the system of Figure 1 if Gr(s) ) aGr2(s)/(1 - a), Gd(s) ) (1 a)Gd2(s), Di(s) ) Di2(s), Do(s) ) (1 - a)Do2(s), and Dr(s) ) (1 - a)Dr2(s)/a. If Gd(s) and Gr(s) are both stable first-order systems plus time delay,

Gd(s) )

Kpe-θps Kre-θrs Gr(s) ) τps + 1 τrs + 1

(1)

the transfer functions of the plant then become

(τrs + 1)Kpe-θps Y(s) ) U(s) (τps + 1)(τrs + 1) - KpKre-(θp+θr)s

(2)

(3)

KpKre-(θp+θr)s Y(s) ) Dr(s) (τps + 1)(τrs + 1) - KpKre-(θp+θr)s

(4)

The transfer functions will be identified as follows:

Hyu(s) )

Y2(s) Y(s) ; Hy2u(s) ) U(s) U(s)

Hyr(s) )

Y(s) Y(s) ; Hydi(s) ) R(s) Di(s)

Hydo(s) )

Y(s) Y(s) ; Hydr(s) ) Do(s) Dr(s)

Hydo2(s) )

Y(s) Y(s) ; Hydr2(s) ) Do2(s) Dr2(s)

In the previous equations, R(s) is the set point for Y(s). For simplicity, the s variable is often omitted.

(5)

Because of the importance of the gains Kr and Kp, two different processes will be analyzed in the following sections. The gains of the first process will be Kp ) 1 - a and Kr ) a/(1 - a) (KrKp ) a; a is between 0 and 1 to make physical sense), characteristic of a process with flows where the overall process gain is always 1 (Hyu(0) ) 1) because of the law of conservation of matter. The gains of the second process will be Kp ) 1 and 0 < Kr < 1, typical of a generic process that exhibits a large increase of gain as Kr approaches 1 (Hyu(0) ) 1/(1 - Kr)). Analyzing the gain of Hy2u causes the snowball effect to appear:

Hy2u(0) )

KpKr 1 - KrKp

(6)

The steady-state gain of Hy2u can be very large if KrKp is close to a value of 1. For a process with flows and a recycle fraction of 0.8, for instance (a ) KrKp ) 0.8), a variation of 1 at the process input will increase the recycle flow by 4. This is the snowball effect introduced by Luyben.14 2.2. Poles. All the transfer functions of the process have the same poles. When there are no delays, the values of these poles (F) are

F)

(τrs + 1)(τps + 1) Y(s) ) Do(s) (τps + 1)(τrs + 1) - KpKre-(θp+θr)s

Kp 1 - KrKp

-(τr + τp) ( x(τr + τp)2 - 4τrτp(1 - KrKp) 2τrτp

(7)

Several conclusions can be drawn from eq 7: (i) When the KrKp product is negative, the poles of the process are always negative. They might become complex if the absolute value of KrKp is large (-KrKp > (τr + τp)2/(4τrτp) - 1). When the KrKp product is negative, the process with recycle is always stable and damped oscillations may appear if KrKp is large. (ii) When the KrKp product is positive and close to 0, the poles of the system will be close to -1/τp and -1/τr. In other words, if the recycle is low, the process will have the same dynamics as the dynamics of its individual units in series. (iii) When the KrKp product is between 0 and 1, the poles move from -1/τp and -1/τr to 0 and -(τr + τp)/τrτp. The pole moving toward the origin slows the process down. Hence, as the recycle increases, the effect of the disturbances on the output will last longer. (iv) When the KrKp product is 1, the process is integrating because there is a pole at 0. It is unstable for KrKp > 1 because there is a positive pole. The scope of the current paper will be limited to KrKp between 0 and 1. KrKp < 0 will not be studied because it is equivalent to the study of a process with negative feedback. KrKp g 1 will not be investigated because such an unstable process is rare. Note: For Hyu(s), the pole close to -1/τr is almost canceled by the zero at -1/τr whenever the KrKp product is close to 0.

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Figure 3. Frequency responses of the process without a controller for increasing values of KrKp.

This is not true for the disturbances. Furthermore, the phase hump observed by Knezevich et al.11 appears only for Hyu(s). Indeed, the zero at -1/τr in Hyu(s) is found between the two poles of the process, creating a phase hump. These poles tend toward 0 and -2/τr when KrKp increases (in ref 11: τr ) τp). 2.3. Delays and Stair Effect. Using x ) e-(θp+θr)s and expanding Hyu(s) into a Taylor series around x ) 0 allows for a better understanding of the effects of delays on the process dynamics. The goal of this method, presented by Hugo et al.,10 was to eliminate the presence of dead times in the denominator in order to allow application of advanced control algorithms. However, the steady-state error caused by the approximation (Error ) Kp(KpKr)n/(1 - KpKr) where n is the approximation order) and the complexity of the resulting controllers render the concept difficult to apply.15 Thus, its only purpose will be to gain a better understanding of the effects of the delays:

Hyu )

Kp2Kre-(2θp+θr)s Kpe-θps + ‚‚‚ + τps + 1 (τ s + 1)2(τ s + 1) p r

(8)

As observed by Hugo et al.,10 every term of eq 8 is an additional recycling of the input into Gd(s). Writing Hyu(s) this way also suggests the following: (i) The delay within the recycle path will have few effects if KrKp , 1. If there is almost no recycle, delay in the recycle path has less impact on the process. (ii) If a delay is large compared to the time constants, the contribution of every term of the Taylor series will appear with

a delay after the contribution of the previous term, defined as the stair effect. The height of the steps is a function of KrKp, and the width of the steps is a function of the delays. 2.4. Bode Plots Versus KrKp. The frequency responses of the plant with recycle have been plotted for different values of KrKp. Two relationships between Kr and Kp have been used: (1) Kp ) 1 and 0 < Kr < 1 and (2) Kp ) 1 - a, Kr ) a/(1 a), and 0 < a < 1. The curves of Figure 3 are for delays and time constants equal to 1. In addition to confirming some of the previous results (poles and gains), the curves display more information on the behavior of the disturbances in a process with recycle as a function of the process gains: (i) The cutoff frequencies of Hyu(s), Hydo(s), and Hydr(s) decrease when KrKp increases: the process is slowed. (ii) For a process with a recycle fraction, the steady-state gains of all the transfer functions are always 1, for any KrKp. For a process with Kp ) 1, the steady-state gain of Hyu(s) increases with an increase in Kr. The gain at low frequency increases. In Figure 3, as well as in several other figures, Hydo(s) and Hydr(s) are plotted. To obtain the magnitudes of Hydo2(s) and Hydr2(s), one must multiply the magnitudes of Hydo(s) and Hydr(s), respectively, by (1 - a) and (1 - a)/a. For processes with a recycle fraction, an increase of recycle does not change the steadystate gains related to Y(s). However, for some processes such as a processes with Kp ) 1, an increase of recycle changes the steady-state gains as explained in Section 2.1:


Figure 4. Effects of the delays on the process in the frequency domain.

Steady-State Gains and Snowball Effect. (iii) For processes with Kp ) 1, the gain of the disturbances is increased at low frequencies as KrKp increases. Disturbances such as steps will be amplified by the recycle. (iv) For processes with a recycle fraction, the gain of the disturbances is reduced at high frequencies as KrKp increases. The presence of large recycle flows and large time constants reduces the impact of fast disturbances. 2.4.1. Effects of Delays in the Frequency Domain. In the time domain, the delays create a stair effect. This section will describe the effects of the delays in the frequency domain. According to eq 2, θp and θr have the same effects on the poles of the process. Furthermore, the delays will change the poles of the process periodically (e-jw(θp+θr) is periodic in the frequency domain). At low frequencies, KrKpe-jw(θp+θr) is close to KrKp and the delays have almost no effect on the process dynamics. At high frequencies, KrKpe-jw(θp+θr) is small compared to jwτp or jwτr. Thus, the delays change the poles of the system mainly at midfrequencies. For significant pole changes, KrKp and (θp + θr)/(τr + τp) must be large enough, approximately above 0.5 and 1, respectively. Since the delays are in the denominator of eq 2, they should increase the phase of the process at midfrequencies. Finally, since θp is in the numerator of eq 2, a decrease of the phase of the process is expected as θp increases. Figure 4 summarizes all these effects. The values of the process parameters are the following: Kr ) 4, Kp ) 0.2, τr ) 1, and τp ) 1.

Figure 5. Process with controller C.

3. Control of a First-Order Process with Recycle The control of a process with recycle will be investigated in the following sections. First, a PI controller will be used. Then, a recycle compensator will be added to the PI controller. Finally, both regulators will be compared. Two values of KrKp will be used. The first value of KrKp will be 0.2, representing a process with low recycle. The second value of KrKp will be 0.8, characteristic of a process with high recycle. The PI controller (C(s)) and the recycle compensator (K(s)) will be

Kc(τis + 1) Kke-θks C(s) ) K(s) ) τis τks + 1

(9)

3.1. Feedback Controller without a Recycle Compensator. It is possible to identify the process independently of its internal structure. A controller can then be built, based on the process identification. The present section describes the performance of such controllers. As shown in Figure 5, the controller is a standard feedback controller.

Figure 6. Comparison of the process and its model as a function of KrKp. The delays and time constants are all equal to 1.

To identify the plant dynamics Y(s)/U(s), a step has been applied to the input. From the step responses, the model parameters have been identified using a least-squares algorithm. The PI controller integral time τi is set equal to the plant time constant, and the PI controller proportional gain Kc is adjusted to get a phase margin of 60°. Figure 6 compares the model output with the process output, while Table 1 presents the model (Pm) and controller parameters. This table is built for a process with a recycle fraction. For a process with Kp ) 1 and for the same values of KrKp, the resulting models and controllers are the same, except for the gains: the gain of the model is multiplied by 1/(1 - KrKp), and the gain of the PI controller is multiplied by (1 - KrKp).

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Table 1. Models and PI Controllers Kr ) 0.25; Kp ) 0.8

Pm ) C)

e-θs 2s + 1

2s + 1 3s

Kr ) 4; Kp ) 0.2

Pm ) C)

e-θs 17s + 1

17s + 1 18s

As seen in Figure 6, the process is adequately approximated by a first-order model. This is explained by the fact that Hyu has a dominant time constant while its second pole is relatively close to a zero. If a stair effect occurs, identifying a single transfer function may be not adequate. One solution would be to record the step response and to use it directly for the design of a DMC (dynamic matrix control) controller.16 3.1.1. Performance of Controller C, Frequency Domain. Y(s) as a function of the set point or disturbances is

C(s)Gd(s) Y(s) ) R(s) 1 + C(s)Gd(s) - Gr(s)Gd(s)

(10)

Gd(s) Y(s) ) Di(s) 1 + C(s)Gd(s) - Gr(s)Gd(s)

(11)

Y(s) 1 ) Do(s) 1 + C(s)Gd(s) - R(s)GrGd(s)

(12)

Gr(s)Gd(s) Y(s) ) Dr(s) 1 + C(s)Gd(s) - Gr(s)Gd(s)

(13)

Substituting the transfer functions of eqs 1 and 9 in eq 10 yields the following equation:

KcKp(s + 1)(τrs + 1)e-θps Hyr ) ψ1

(14)

where

ψ1 ) (s)(τps + 1)(τrs + 1) + KcKp(s + 1)(τrs + 1)e-θps KrKp(s)e-(θps+θrs) (15) With this equation, and the similar one for the disturbances, it is possible to evaluate the performance of the regulator in the frequency domain. Figure 7 depicts the frequency responses for two types of processes: a process where Kp ) 1 and a process with a recycle fraction. These curves will be used to compare the performance of a controller with and without a recycle compensator in section 3.3. For all the curves, τr ) τp ) θr ) θp ) 1. The following observations can be made from Figure 7: (i) The cutoff frequency of Hyr(s) decreases when KrKp increases if the controller C(s) is always tuned the same way. A PI controller is unable to counterbalance the deceleration caused by the recycle if the phase margin is preserved. (ii) The gain of Hydo(s) at low frequencies increases when KrKp increases. However, it does not change at high frequencies. This suggests that the gain of Hydo2(s) decreases at high frequencies when KrKp increases. (iii) The gain of Hydi(s) at low frequencies increases when KrKp increases. However, it decreases at high frequencies for the process with a recycle fraction. (iv) The gain of Hydr(s) increases significantly at low frequencies when KrKp increases.

It only slightly increases at high frequencies. Note that the curve of Hydr2(s) (not shown) is only shifted toward the low frequencies when KrKp increases. (v) As for the process without a controller, all disturbances are recycled and the overall gain of the disturbances would increase at all frequencies as KrKp increases. However, large flows and large time constants counterbalance this increase at high frequencies by reducing fast transitions of process variables. The PI controller does not change this result because it is tuned according to a unique process model that includes all process parts. It ignores the relationships between the disturbances and the process. 3.2. Feedback Controller with a Recycle Compensator. The design of a controller with a recycle compensator was initially proposed by Taiwo.2 This design, illustrated in Figure 8, shows that the controller is split in two parts. First, C(s) is designed according to Gd(s), independently of the recycle. Then, the recycle compensator (K(s)) is designed from the model of Gr(s), neglecting the transformation unit, Gd(s). The theoretical value of K(s) in Figure 8 is K(s) ) Gr(s). The basic theory of the recycle compensator is not detailed here. As shown by Scali and Ferrari7 and Kwok et al.,15 the use of a recycle compensator with any kind of controller C(s) (PI, IMC (internal model control), predictive, etc.) is often advantageous. For instance, the addition of the recycle compensator to a controller of any type allows better disturbance rejection for similar set point responses. Appendix A mathematically demonstrates the previous statement when Gd(s) and the overall process are invertible like the processes used by Scali and Ferrari.7 Similar conclusions about the recycle compensator will be reached for noninvertible processes (e.g., processes with delay) in the following paragraphs. In addition to discussing the benefits of the recycle compensator, the robustness of a controller with a recycle compensator will be detailed, as discussed by Samyudia et al.12 or Cheng et al.17 3.2.1. Performance of the Controller C with a Recycle Compensator, Frequency Domain. The equations of the system with a recycle compensator are

C(s)Gd(s) Y(s) ) R(s) 1 + C(s)Gd(s) + (K(s) - Gr(s))Gd(s)

(16)

Gd(s) Y(s) ) Di(s) 1 + C(s)Gd(s) + (K(s) - Gr(s))Gd(s)

(17)

Y(s) 1 ) Do(s) 1 + C(s)Gd(s) + (K(s) - Gr(s))Gd(s)

(18)

Gr(s)Gd(s) Y(s) ) Dr(s) 1 + C(s)Gd(s) + (K(s) - Gr(s))Gd(s)

(19)

The sensitivities to model errors are

∂Hyr/Hyr 1 ) ∂Gd/Gd 1 + CGd + (K - Gr)Gd

(20)

GrGd ∂Hyr/Hyr ) ∂Gr/Gr 1 + CGd + (K - Gr)Gd

(21)

Equation 20 reveals that the sensitivity to errors on Gd(s) is identical to Y(s)/Do(s) (eq 18), while eq 21, the sensitivity to errors on Gr(s), is identical to Y(s)/Dr(s) (eq 19). These


Figure 7. Frequency responses of the process with C(s) for increasing values of KrKp.

observations will be used to analyze the robustness of the controller with a recycle compensator. Combining eqs 1 and 9 into eq 16 yields

Hyr )

KcKp(s + 1)(τks + 1)(τrs + 1)e-θps ψ2

(22)

where

ψ2 ) (s)(τks + 1)(τps + 1)(τrs + 1) + KcKp(s + 1)(τks + 1)(τrs + 1)e-θps + KkKp(s)(τrs + 1)e-(θps+θks) - KrKp(s)(τks + 1)e-(θps+θrs) (23) As for the controller without a compensator, this transfer function, and the similar transfer functions for the disturbances, is used to build Figure 9, which includes the Bode plots of

Figure 8. Process with controller C and a recycle compensator.

equations linking Y(s) to the set point and the disturbances for two types of processes: the process where Kp ) 1 and the process with a recycle fraction. Once again, these curves will be used for the evaluation of the performances of the controller with the compensator in section 3.3. For all the curves, τr ) τp ) θr ) θp ) 1, K(s) ) Gr(s), and C(s) ) 0.5(s + 1)/s.

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Figure 9. Frequency responses of the process with C(s) and a recycle compensator for increasing values of KrKp.

The following observations can be made from Figure 9: (i) Hyr(s) is unaffected by the recycle when the model of Gr(s) is perfect. K(s) being equal to Gr(s) eliminates all the recycle effects. (ii) The gains of all the disturbances (Hydi(s), Hydo2(s), and Hydr2(s)) decrease when KrKp increases for a process with a recycle fraction. Recycled flows are anticipated by the recycle compensator. (iii) The gains of Hydi(s) and Hydo(s) are unaffected by the recycle for a process with Kp ) 1. The recycle compensator eliminates these disturbances. However, the gain of Hydr(s) shows a small increase at all frequencies when KrKp increases. This appears because the effects of a disturbance before the recycle path are measured only after going through the whole process dynamics. (iv) Because Hydo(s) is unaffected by the value of KrKp when the model of Gr(s) is perfect, the

sensitivity to model error of Gd(s) is unaffected by the value of KrKp. (v) Because the gain of Hydr(s) increases when KrKp increases, the sensitivity to model error of Gr(s) increases when the recycle increases. However, the increase is small. Since the gain of Hydr(s) is small (at least for low and high frequencies), the errors in the model of Gr(s) have little impact on the transfer functions when KrKp is small. When KrKp is close to 1, errors in the model of Gr(s) in the midfrequencies may cause large variations of the system behavior. To depict more precisely the sensitivity of the control scheme with a recycle compensator to errors in the identification of the recycle component Gr, 600 uniformly distributed random systems have been simulated (300 systems with small KrKp and 300 systems with large KrKp). They are defined as follows:


Figure 10. Frequency responses of the process with C(s) and a recycle compensator for various modeling errors on Gr.

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Figure 11. Time responses of the process with C(s) and a recycle compensator for various modeling errors on Gr.

Small KrKp

Large KrKp

K(s) )

0.2e-s s+1

Kr ∈ [0.16, 0.24]

(24) (25)

K(s) )

0.8e-s s+1

Kr ∈ [0.64, 0.96]

(26) (27)


For all simulations, the following parameters were used:

θr ∈ [0.8, 1.2]

(28)

τr ∈ [0.8, 1.2]

(29)

C(s) )

0.5(s + 1) s

(30)

1e-s s+1

(31)

Gd(s) )

Figures 10 and 11 respectively show the frequency and time responses of the systems. The design is therefore robust to 20% error on the estimated Gr parameters. Obviously, the sensitivity increases with the amount of recycling. (i) Analysis of the denominator of eq 22 (ψ2) reveals that an error in the estimation of Kr within physical limits (KrKp < 1) will have a small effect over the poles of the process because Kk cancels out part of Kr. (Kr - Kk)Kp must be close to 1 to significantly move the system poles (assuming that other parameters are perfectly known). (ii) Analysis of the denominator of eq 22 (ψ2) reveals that an error (absolute error) in the estimation of θr can significantly modify the poles of the system when KrKp or KkKp is close to 1 and if θr is large enough as compared to the time constants of the process. In the worst case, an error in the estimation of θr may lead to instability. For instance, for Kr ) 0.8, τr ) 1, and K(s) defined by eq 26, estimating θr larger than 3.1 when the real plant value is 1 renders the system unstable. Note that this information is also found in the frequency plot of Hydr(s). As stated above, this plot reveals that the system is sensitive to errors in the Gr(s) model at midfrequencies. Since the delays change the process frequency responses at midfrequency (section 2.4.1), they indeed may cause instability. (iii) An error in the estimation of τr will have a very small impact on the transfer functions of the system. For instance, for Kr ) 0.8, θr ) 1, and K(s) defined by eq 26, the system remains stable for any positive value of τr. This is explained by the fact that this modeling error does not increase the gain of Hydr(s) at midfrequency. 3.3. Comparison of the Controllers. Using Figures 7 and 9 and the previous equations allows a comparison of the performance of the two controllers studied: (i) The performance of a PI controller with a recycle compensator is always better than the performance of a PI controller alone for a first-order plus time delay systems: the set point is reached earlier, the disturbances are more attenuated, and the effects of model errors are reduced. (ii) The PI controller alone is nearly as good as the PI controller with a recycle compensator when KrKp is small. (iii) When KrKp increases, the gain of the disturbances generally increases for the PI controller alone. The inverse occurs for the controller with a recycle compensator. (iv) Even with errors in the model of Gr(s) (except for θr), the performance of the PI controller with a compensator is better than for the PI controller alone. 4. Conclusion This paper has shown that the negative effects of recycle on the dynamics of some processes can be counterbalanced with the design of a suitable controller. The recycle compensator eliminates the negative effects of the recycle on the process dynamics, and it is robust to model mismatches.

A review of the effect of recycle on a first-order process with delay was presented. An analysis of the process poles has shown that the process is slowed as KrKp goes from 0 to 1. Therefore, the KrKp product has an important role in processes with recycle. It also determines the gain of the transfer function of Y2(s)/ U(s). Whenever KrKp is close to 1, this gain is very high, characteristic of the snowball effect. The relationship between Kr and Kp fixes the gains. For instance, if Kr ) a/(1 - a) and Kp ) a, as is the case for a system with a separator when considering the flows, the gain of Hyu is 1. Thus, an increase of KrKp or Kr does not always increase the gain of Hyu even though this is the case when Kp is set to 1. Moreover, it has been shown that the delays of a process with recycle may produce a stair effect if they are large enough compared to the time constants. The performance of a PI controller was then analyzed for a process with recycle. These performances deteriorate when the recycle increases. Indeed, the deceleration of the process yields a poor controller design: the set point tracking is gradually slower. The disturbances are also rejected more slowly. Finally, the robustness of the recycle compensator toward the model errors and the disturbance rejection performance were systematically analyzed. First, it was demonstrated that the sensitivity to the errors in model of Gr(s) had the same representation as Hydr(s). This fact was used to analyze the sensitivity to errors of a model starting from Hydr(s). It was then shown that the performance of the controller with a recycle compensator usually remained constant, or slightly increased in some cases, when the recycle increased. Without modeling errors, the performance is the same as for a PI controller used on a process without recycle. Errors in model of Gr(s) have little impact on the transfer functions of the system when the recycle is low. They change the behavior of the system more when the recycle is high. However, even with high recycle and with important model errors of Kr (within physical limits) or τr, the controller with a compensator of recirculation remains stable. It also exhibits better disturbance rejection and a faster time response to a set point change than the use of a PI controller alone. Unfortunately, the recycle compensator may be sensitive to estimation errors of θr. Appendix A. Disturbance Rejection Analysis This appendix demonstrates that the addition of a recycle compensator to a feedback controller improves the rejection of disturbances when Gd(s) and Gd(s)/(1 - Gr(s)Gd(s)) are invertible. First, assume that only the following feedback controller is used:

CwithoutK(s) )

1 - Gr(s)Gd(s) τisGd(s)

(32)

where τi is a design parameter. Assume now that a recycle compensator is added to the feedback controller, as depicted in Figure 8. The same set point response for both cases can be achieved if

K(s) ) Gr(s)

(33)

CwithK(s) ) 1/(τisGd(s))

(34)

Equations 10 and 16 show that the responses, with or without the recycle compensator, to a set point step are indeed the

1934


same: Hyr ) 1/(τis + 1). However, the disturbance dynamics are not:

YwithoutK(s) Di(s)

)

τis τ s 1 - Gr(s)Gd(s) i + 1 Gd(s)

τis ) Gd(s) τis + 1 Di(s)

YwithK(s)

(35)

(36)

The 1 - Gr(s)Gd(s) term included in CwithoutK(s) appears in the denominator of all the transfer functions of the disturbances (eq 35 is an example). Therefore, the dynamics of the process with recycle appears in the overall dynamics. The dynamics of the overall process is often much slower than the dynamics of the individual units of the process (see section 2.2). With a recycle compensator and with no model error, Gr(s) disappears from the denominator of all the transfer functions of the disturbances (eq 36 is an example). Thus, the dynamics of the disturbances will be determined by the dynamics of the individual units of the process (Gd(s) or Gr(s); see ref 18) and τi. For the same step response, the disturbance rejection is faster when a recycle compensator is added to the feedback controller if the model of the process is invertible. Literature Cited (1) Denn, M. M.; Lavie, R. Dynamics of Plants with Recycle. Chem. Eng. J. 1982, 24, 55. (2) Taiwo, O. The design of robust control system for plants with recycle. Int. J. Control 1986, 43, 671. (3) Luyben, W. L. Dynamic and Control of Recycle System. 1. Simple Open-Loop and Closed-Loop Systems. Ind. Eng. Chem. Res. 1993, 32, 466.

(4) Luyben, W. L. Dynamic and Control of Recycle System. 2. Comparison of Alternative Process Design. Ind. Eng. Chem. Res. 1993, 32, 476. (5) Luyben, W. L. Dynamic and Control of Recycle System. 3. Comparison of Alternative Process Design. Ind. Eng. Chem. Res. 1993, 32, 1142. (6) Luyben, W. L. Dynamic and Control of Recycle System. 4. Comparison of Alternative Process Design. Ind. Eng. Chem. Res. 1993, 32, 1154. (7) Scali, C.; Ferrari, F. Performance of different regulators for plants with recycle. Comput. Chem. Eng. 1995, 19, S409. (8) Scali, C.; Ferrari, F. Performance of control systems based on recycle compensators in integrated plants. J. Process Control 1999, 9, 425. (9) Lakshminarayanan, S.; Takada, H. Empirical modelling and control of processes with recycle: some insights via case studies. Chem. Eng. Sci. 2001, 56, 3327. (10) Hugo, A. J.; Taylor, P. A.; Wright, J. D. Approximate dynamic models for recycle systems. Ind. Eng. Chem. Res. 1996, 35, 485. (11) Knezevich, P.; Chong-Ping, M.; Kwok, K. E. Simulation and Control of recycle systems. Can. Conf. Electr. Comput. Eng. 1999, 3, 1604. (12) Samyudia, Y.; Kadiman, K. Control design for recycled, multi unit processes. J. Process Control 2003, 13, 311. (13) Lakshminarayanan, S.; Onodera, K.; Madhukar, G. M. Recycle Effect Index: A mesure to Aid in Control System Design for Processes with Recycle. Ind. Eng. Chem. Res. 2004, 43, 1499. (14) Luyben, W. L. Snowball Effects in Reactor/Separator Processes with Recycle. Ind. Eng. Chem. Res. 1994, 33, 299. (15) Kwok, K. E.; Chong-Ping, M.; Dumont, G. A. Seasonnal Modelbased control of recycle processes. Ind. Eng. Chem. Res. 2001, 40, 1633. (16) Cutler, C. R.; Ramaker, B. L. Dynamic Matrix ControlsA Computer Control Algorithm. Proceedings of the Joint American Control Conference, San Francisco, CA, 1980. (17) Cheng, Y. C.; Yu, C. C. Effects of Process Design on recycle Dynamics and Its Implication to Control Structure Selection. Ind. Eng. Chem. Res. 2003, 42, 4348. (18) When K(s) ) Gr(s), Gd(s) determines the dynamics of all transfer functions and Gr(s) only determines the dynamics of Hydr.

ReceiVed for reView June 2, 2005 ReVised manuscript receiVed January 6, 2006 Accepted January 18, 2006 IE050643T

Robustness of a Recycle Compensator - American Chemical Society

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