Recycle Compensator Facilitates Rapid Parameterization of

Article pubs.acs.org/IECR

Recycle Compensator Facilitates Rapid Parameterization of Proportional Integral/Derivative (PI/D) Controllers for Open-Loop Unstable Recycle Processes Ayorinde Bamimore* and Femi Taiwo Process Systems Engineering Laboratory, Department of Chemical Engineering, Obafemi Awolowo University, Ile-Ife, Nigeria S Supporting Information *

ABSTRACT: Processes with recycle are quite common in the process industries. It has been pointed out that such processes often exhibit unusual dynamics, such as open-loop instability, which makes the task of designing a model-based controller for them challenging. In this work, a design procedure for rapid parametrization of simple internal model control (IMC)-based proportional integral/derivative (PI/D) controller for open-loop unstable recycle processes is proposed. In the proposed strategy, first, the recycle process model is decomposed into the forward and recycle path. Thereafter, a perfect/approximate compensator is designed for the global process. The final controllers for the global process are then parametrized using the compensated system model and implemented on the system. It was observed that the compensator was able to restore stability to the open-loop unstable recycle process model and simplifies its model order considerably. Simulation results obtained revealed that the recycle compensated system displays better closed-loop performance, in terms of set-point tracking and disturbance rejection. Furthermore, the proposed method results in a closed-loop system that is less sensitive to disturbance and has a smoother control signal with higher gain and phase margins.

1. INTRODUCTION Processes with heat integration and material recycle are commonplace in the process industries. An economic analysis favors this type of design, because it allows efficient use of material and energy. Also, such a design usually results in a more environmentally friendly plant. However, the introduction of recycle into process plants has been known to result in poor dynamics of the global process, although the dynamics of the individual units that comprise the global plants are simple. Unusual dynamic behaviors that have been observed by researchers such as Denn and Lavie,1 Kapoor et al.,2 and Morud and Skogestad,3,4 in processes with recycle, include extremely long response time, limit cycles, oscillatory response, and even instability, as reported by Silverstein and Shinnar;5 Luyben,6 and Trierweiler et al.7 These effects result from the positive feedback nature of plants with recycle, which often acts to affect the pole location of the global process. Because of the overall complicated dynamic model of recycle processes, various methods have been proposed by researchers for controlling them. Hugo et al.8 proposed the use of moment matching for reducing the model and the subsequent use of any standard control method; del-Muro-Cuéllar et al.9 proposed the use of discrete time approximation; and Kwok et al.10 suggested the use of seasonal time-series model, while Papadourakis et al.11 proposed the use of the variable order model. All these methods have their merits and demerits. Most are not convenient to use if the overall model is open-loop unstable. In order to retain the steady-state advantage of recycle and avoid its dynamic penalty, Taiwo12−15 proposed the idea of using the recycle compensator in the control of plants with recycle. Its effectiveness has been widely supported by other researchers.16−19 Its industrial applicability has also been reported by Lakshminarayanan and Takada.20 © 2015 American Chemical Society

This paper highlights how the application of a recycle compensator can be used for rapid tuning of a proportional integral derivative (PI/D) controller for open-loop unstable recycle processes by first restoring stability to the process model for onward standard controller tuning. The open-loop instability of some recycle systems and the quasi-rational nature of their transfer functions often preclude the use of modelbased PI/D control strategy for controlling them. Hence, most researchers often resort to controller design using gain and phase margin specifications or settle for a higher-order controller. Therefore, in this paper, we propose a design procedure for the rapid parametrization of a high-performance internal model control (IMC)-based PI/D controller for openloop unstable recycle processes. In the proposed design procedure, first, the global process is decomposed into forward and recycle path transfer functions. Second, the recycle compensator is specified and applied to restore open-loop stability. Subsequently, the compensated plant model is used to parametrize the PI/D controller for the recycle process. Finally, the tuned PI/D controller, together with the designed compensator, is implemented in a recycle compensation scheme. The rest of the paper is structured as follows: in section 2, the general representation of recycle processes is presented with an emphasis on specification of recycle compensator. Illustrative examples are presented in section 3, to show the effectiveness of the proposed method. In section 4, results are discussed and then some conclusions are drawn. Received: Revised: Accepted: Published: 5115

January 15, 2015 March 27, 2015 April 19, 2015 April 19, 2015 DOI: 10.1021/acs.iecr.5b00192 Ind. Eng. Chem. Res. 2015, 54, 5115−5127

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Industrial & Engineering Chemistry Research

2. GENERAL REPRESENTATION OF RECYCLE SYSTEMS 2.1. Recycle Compensator Design. The general block diagram representation of a process with recycle including the feedback controller Gc(s) and the recycle compensator F(s) is shown in Figure 1, where y, u, d, uc, uF, and ym are controlled,

compensator in Figure 2; this is equivalent to the original process in Figure 1, without a compensator, but with the

Figure 2. Recycle process with the recycle acting upstream.

recycle acting on the process upstream. It is easily converted to Figure 1 by setting G3 = G1G3*. For single-input−single-output (SISO) systems, the equivalent global process, designated as Gglobal in Figure 1, can be represented as

G1 =

k1e−θ1s τ1s + 1

G3* =

k 3e−θ3s τ3s + 1

y(s) =

(3)

Upon applying the recycle compensator F(s) to the subsystem Gglobal, the open-loop compensated system (Gcompensated) becomes

(4)

ym (s) = G5G4G′(I + FG5G4G′)−1uc(s) + G5G4Gd′ (5)

where G′ = (I − G3)−1G1 and Gd′ = (I − G3)−1G2. If F(s) is perfect and realizable, substituting for F(s) in eqs 4 and 5 results in

ym (s) = G5G4G1u(s) + G5G4G2d(s)

(7)

(8)

k1(τ3s + 1)e−(θ1+ θ4)s (τ1s + 1)(τ3s + 1) − k1k 3e−(θ1+ θ3)s

u(s)

(τ1s + 1)(τ3s + 1)e−θ4s (τ1s + 1)(τ3s + 1) − k1k 3e−(θ1+ θ3)s

d (s ) (9)

The presence of the deadtime term in the denominator makes controller design a daunting task. For even when the time delays θ1 and θ2 are zero, it is easy to see that, when k1k3 > 1, the system becomes open-loop unstable, even though the component systems are stable. Luyben21 referred to this phenomenon as external open-loop instability. Such is the case with a heat exchanger/tubular reactor system. The application of Taiwo12−15 recycle compensator will both simplify the global process model and restore open-loop stability, thereby allowing for the use of a model-based control system design. 2.3. Proposed IMC-PI/D Design Procedure. Following the notations in Figure 1, for the purpose of controller design, the feedback controller Gc(s) sees the recycle process, given by

y(s) = G4G′(I + FG5G4G′)−1uc(s) + G4Gd′

(6)

d (s )

The global process is given by

+

y(s) = G4G1u(s) + G4G2d(s)

(1 − G1G3*)

G2 = 1

The compensator F(s), which totally cancels the detrimental effect of recycle and is known as the perfect recycle compensator, can be specified as found in the work of Taiwo:12−15

(I + FG5G4G′)−1G2d(s)

G4G2

and

(1)

(2)

(I + FG5G4G′)−1G2d(s)

u(s) +

G4 = e−θ4s

ym (s) = G5G4(I − G3)−1G1u(s) + G5G4(I − G3)−1G2d(s)

F(s) = G1−1G3G4 −1G5−1(s)

(1 − G1G3*)

Assuming that G1, G*3 , and G4 are stable transfer functions, given as

manipulated, disturbance, controller output, compensator output, and measured variables, respectively. G1 refers to the forward path process transfer function, G2 the process disturbance, G3 the recycle path process transfer function, G4 additional elements in the process path, and G5 the process sensor dynamics. The effect of manipulated variable (u) and disturbance (d) on the controlled output (y) and measured output (ym) for the open-loop global process (Gglobal) is given by y(s) = G4(I − G3)−1G1u(s) + G4(I − G3)−1G2d(s)

G4G1

y(s) =

Figure 1. Block diagram of a system consisting of a plant with recycle, recycle compensator (F), and feedback controller (Gc).

ym (s) = G5G4(I − G3)−1G1 u(s) + G5G4(I − G3)−1G2 d(s)   G R (s)

G D(s)

(10)

2.2. General Characteristics of Recycle Processes. Consider the block diagram of a recycle process without

It has been established in previous works that GR(s) usually exhibits complex and unfavorable dynamics such as oscillatory 5116

DOI: 10.1021/acs.iecr.5b00192 Ind. Eng. Chem. Res. 2015, 54, 5115−5127

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bypass flow rate (FB) around FEHE. The recycle compensator (RC), which acts like a feedforward controller, compensates for the effect of this positive heat feedback. Note that both the compensator (RC) and the feedback controller (TC) use the same measurement from the temperature transmitter (TT). Hence, there is no need for any additional sensor. Following the linear analysis in the Luyben work,22 the reactor is modeled by a first-order lag transfer function

response, limit cycles, inverse response, and even instability, which makes the design of the controller very challenging. For our situation here, we are going to consider processes where GR(s) is open-loop unstable. One of the often-overlooked advantages of the recycle compensator is that, when properly designed, it restores the open-loop stability of the global process and results in the open-loop stable compensated plant that has been described by eqs 6 and 7. Therefore, here, we propose a design procedure for rapidly tuning IMC-based PI/D controllers for such processes, which includes: Step 1: Decompose the global process into the component transfer functions G1, G2, G3, G4, and G5. Step 2: Design the recycle compensator F(s) using eq 3. If unrealizability is encountered, an approximate compensator can be designed, which often works well. Step 3: Apply the compensator to the open-loop unstable model of the recycle process denoted by eq 10 to obtain a open-loop stable process, which is given by

Gr (s) =

δTout Kr = δTin τrs + 1

The reactor inlet temperature (Tin) is related to the bypass flow rate (FB), the reactor exit temperature (Tout) and the disturbance (d) by the algebraic equation δTin = K1δTout + K 2δFB + d

Figure 4 gives the control system block diagram for the process where Gm is used to account for the dynamics of the

ym (s) = G5G4G′(I + FG5G4G′)−1 uc(s)  G K (s)

+ G5G4Gd′(I + FG5G4G′)−1G2 d(s)  G L(s)

(11)

Step 4: Parameterize the IMC-PI/D controller, Gc(s) for the process using the compensated system model, denoted as GK(s) in eq 11. If F(s) is perfect and realizable, GK(s) from eq 11 easily reduces to eq 12. Thus, Gc(s) can be parametrized using eq 12. G K (s ) =

ym (s) uc(s)

= G5G4G1

Figure 4. Block diagram of the coupled FEHE/reactor system. (12)

Step 5: Implement both the designed feedback controller Gc(s) and the compensator F(s) on the recycle process in a closed loop, according to the scheme presented in Figure 1. The IMC filter parameter (τc) can be varied to achieve the desired closed-loop response.

temperature measurement, modeled by a small third-order lag. In this figure, the selected process parameters are Kr = 4, τr = K1 = K2 = 1, τm = 0.1. According to the scheme presented in Figure 1, the component transfer functions are easily derived as

G1 = K 2 = 1

3. ILLUSTRATIVE EXAMPLES 3.1. Example 1: Coupled FEHE/Reactor Process with High Reactor Gain. This example is taken from the work of Luyben.22 It consists of a feed-effluent heat exchanger (FEHE) coupled with an adiabatic exothermic reactor (Figure 3). The heat of reaction generated in the reactor results in an increase in the inlet temperature (Tin) to an effluent temperature Tout. This heat can be recovered from the reactor effluent through the heat exchanger FEHE. The control objective of this process is to maintain the reactor inlet temperature by manipulating the

G2 = 1 G3 =

K1K r 4 = τrs + 1 s+1

G4 = 1 and G5 =

1 1 = 3 (τms + 1) (0.1s + 1)3

3.1.1. Controller Design for the Uncompensated System. The uncompensated system model GR is computed as follows, using eq 10: G R (s) =

Tin_lag(s) FB(s)

=

s+1 0.001s 4 + 0.027s 3 + 0.21s 2 + 0.1s − 3

It is easily noticeable that GR is open-loop unstable. To be able to tune an IMC-based PID controller, this transfer function is reduced to GRr by first separating the unstable pole and optimally reducing the remaining stable part using the time integral of squared error (ITSE) as a loss function to give

Figure 3. FEHE/reactor process with inlet temperature control. 5117


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Industrial & Engineering Chemistry Research G Rr (s) =

Tin_lag(s) FB(s)

1

=

−3 1

( − 3 s + 1)(0.1s + 1)

Using the IMC-PID tuning rules for unstable processes, found in the work of Rotstein and Lewin,23 the following PID controller is parametrized for the process (with the IMC filter parameter as τc = 0.5): ⎛ ⎞ 1 Gc,(a) = 7.4⎜1 + + 0.0946s⎟ ⎝ ⎠ 1.85s

An optimal PI controller, Gc(b) designed in the work of Luyben,22 was also able to stabilize the system, but with a very high overshoot and oscillation. ⎛ 1 ⎞ ⎟ Gc,(b) = 2.31⎜1 + ⎝ 0.669s ⎠

3.1.2. Recycle Compensator Design. The recycle compensator is designed using eq 3 as F=

4(0.1s + 1)3 s+1

Figure 5. Measured and manipulated variables in response to a step change in reference r introduced at t = 1.

Note that this is unrealizable, since it is improper. A realizable compensator is designed by first reducing G5 optimally to G5r = 1/(0.2714s + 1), using ITSE as the loss function. This is then subsequently used in designing F = [4(0.2714s + 1)]/(s + 1). 3.1.3. Controller Design for the Compensated System. Upon using the designed compensator on the recycle process according to eq 11, the open-loop compensated system model GK becomes G K (s) =

ym (s) uc(s)

=

s+1 0.001s + 0.027s + 0.21s 2 + 1.1856s + 1 4

3

In order to tune an IMC-based PID, GK(s) is approximated to the second-order model. GKr(s) ≅ 1/(0.0242s2 + 0.1595s + 1), using an optimal model reduction with ITSE as the loss function. Thus, the PID controller is parametrized for the compensated system with a tuning parameter chosen as τc = 0.05: ⎛ ⎞ 1 Gc,(c) = 3.19⎜1 + + 0.15s⎟ ⎝ ⎠ 0.16s

The closed-loop step responses of this process to reference and disturbance step changes as shown in Figures 5 and 6 reveal that the compensated system provide a better set-point tracking and disturbance rejection. Notice also a high overshoot in the uncompensated system. To compare the closed-loop performance, stability margins, and robustness of the two designs, the following performance metrics were computed: integrated absolute error (IAE), peak sensitivity (MS), total variation (TV) of manipulated input, gain margin (GM), and phase margin (PM). The results, as summarized in Table 1, show that the compensated system has a lower output error, is less sensitive to disturbance, has a smoother control signal, and has higher gain and phase margins. 3.2. Example 2: Coupled FEHE/Reactor Process with Reactor Exhibiting Inverse Response. The process in Example 1 is now recast with the reactor represented by the transfer function that contains a gain, a positive zero, dead time, and two equal first-order lags, as found in the work of Tyreus and Luyben.24

Figure 6. Measured and manipulated variables in response to a step change in disturbance d introduced at t = 1.

Gr (s) =

K ( −τ1s + 1)e−Ds (τ2s + 1)2

with the new process parameters chosen as K = 7, τ1 = 0.8, τ2 = 1, K1 = 0.8094, K2 = −8.946, D = 1, and τm = 0.1. According to the proposed scheme represented as Figure 1, the component transfer functions are obtained as G1 = K 2 = −8.946

G2 = G4 = 1 G3 = K1Gr = 5118

5.6658(− 0.8s + 1)e−s (s + 1)2 DOI: 10.1021/acs.iecr.5b00192 Ind. Eng. Chem. Res. 2015, 54, 5115−5127

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Industrial & Engineering Chemistry Research Table 1. Performance Indices for Example 1a Set-Point Tracking controller, Gc,(c) + compensator, F controller, Gc,(b) controller, Gc,(a)

Disturbance Rejection

MS

GM

PM

IAE

TV

IAE

TV

1.56 3.12 1.88

∞ 0.57 0.28

47.4 23.4 33.9

0.24 1.63 1.07

10.70 12.22 21.57

0.06 0.62 0.28

2.25 4.34 3.90

a

Legend of the various abbreviations used in this table: MS = peak sensitivity, GM = gain margin, PM = phase margin, IAE = integrated absolute error, TV = total variation of manipulated input.

G5 =

1 (0.1s + 1)3

3.2.1. Controller Design for the Uncompensated System. Using eq 10, a complex quasi-rational transfer function, GR is obtained for the recycle process. A Nyquist plot of GR reveals that it is open-loop unstable. The time domain step response, which is observed to be very oscillatory with increasing amplitude, also supports this observation. GR =

−8.946(s + 1)2 (0.1s + 1)3 [(s + 1)2 − 5.6658(− 0.8s + 1)e−s]

The PI controller, Gc,(b), which was designed in the work of Tyreus and Luyben,24 was used to stabilize the system but with a very oscillatory response: Gc,(b) = −0.33 +

Figure 8. Measured variable in response to a step change in disturbance d introduced at t = 5.

0.33 s

The closed-loop step responses are displayed in Figures 7 and 8.

G K (s ) =

−8.946 0.001s 3 + 0.03s 2 + 0.3s + 1

This is optimally reduced to a second-order transfer function: G Kr (s) =

−8.946 0.0425s + 0.324s + 1 2

With the IMC tuning parameter chosen as τc = 0.5, PID controller was parametrized for the compensated system, but only the proportional and the integral parts were implemented as given in Gc,(c) = −0.0725 −

A comparison of the closed-loop responses of the compensated and uncompensated system, as shown in Figures 7 and 8, reveal that the combination of PI controller plus recycle compensator provides a better set-point tracking and disturbance rejection, compared to a PI controller alone, with the response of the uncompensated system being so oscillatory with larger overshoot. This is supported by the performance metrics summarized in Table S1 in the Supporting Information. 3.3. Example 3: Coupled FEHE/Furnace/Reactor Process. This process is a modification of the one presented in Example 1. One great disadvantage of the structure presented in Figure 3 is its tendency to quench. As a result, a furnace is introduced before the reactor, which introduces an additional degree of freedom (see Figure 9). In this modified configuration, the first controller Gc1, which is the primary controller, controls Tmix by manipulating the bypass flow FB while the second controller Gc2 controls Tin by manipulating the furnace firing (QF). The various components in the coupled FEHE/furnace/ reactor process are modeled by the transfer functions

Figure 7. Measured variable in response to a step change in reference r introduced at t = 5.

3.2.2. Recycle Compensator Design. Using eq 3, the recycle compensator is designed as F (s ) =

0.2236 s

−0.6334(− 0.8s + 1)(0.1s + 1)3 e−s (s + 1)2

which is unrealizable, because it is improper. A realizable compensator is designed by first reducing G5 optimally to G5r = 1/(0.2714s + 1), which is then subsequently used in designing F(s) = [−0.6334(−0.8s + 1)(0.2714s + 1)e−s]/(s + 1)2. 3.2.3. Controller Design for the Compensated System. Applying the compensator to the global process results in an open-loop stable process model 5119


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exponential function e−0.3s using Taylor expansion, the secondary loop can be modeled by transfer function (Tin_lag(s))/(QF(s)) = GF2 × Gm = 1/[(s + 1)(0.1s + 1)3] ≅ e−0.3s/(s + 1). With a tuning parameter of τc = 0.6, the following IMC-PI controller was parametrized for the process: ⎛ 1⎞ Gc2,(a) = Gc2,(c) = 1.11⎜1 + ⎟ ⎝ s⎠

3.3.2. Controller Design for the Uncompensated Primary Loop. According to the proposed compensation scheme in Figure 1, the component transfer functions are obtained as

G1 = G X f = 1

G4 = 1 Figure 9. Schematic of the coupled FEHE/furnace/reactor process.

G3 = G F1 × Gr × G Xr =

FEHE: G Xf

δT = mix = K 2 , δFB

G Xr

G5 =

δT = mix = K1 δTout

δTin KF1 , = δTmix τF1s + 1

G F2 =

δTin KF2 = δQ F τF2s + 1

GR =

δTlag δT

=

1 (τms + 1)3

G Rr (s) =

Reactor:

Gr =

Tmix s 2 + 2s + 1 = 5 4 FB 0.001s + 0.032s + 0.351s 3 + 1.33s 2 − 0.7s − 9

To be able to tune an IMC-based PID controller, this transfer function is reduced to GRr by first separating the unstable pole and optimally reducing the remaining stable part to give

Sensor dynamics: Gm =

1 (0.1s + 1)3

Using eq 10, the open-loop transfer function of the recycle process is given by

Furnace: G F1 =

10 (s + 1)2

δTout Kr = δTin τrs + 1

Tmix −0.11 = FB ( −0.4625s + 1)(0.1s + 1)

Using the IMC-PID tuning rules for unstable process found in the work of Rotstein and Lewin,23 the following PID controller is parametrized for the process (with the IMC filter parameter as τc = 0.8):

Figure 10 gives the block diagram of the individual unit in the coupled process, together with proposed control configuration.

⎛ ⎞ 1 Gc1,(a) = 20.3⎜1 + + 0.0968s⎟ ⎝ ⎠ 3.1s

The uncompensated system model GR is open-loop unstable. The PI controller settings computed in the work of Reyes and Luyben25 are able to stabilize the system but with less satisfactory performance. ⎛ 1 ⎞⎟ Gc1,(b) = 2.56⎜1 + ⎝ 0.801s ⎠

⎛ 1 ⎞ ⎟ Gc2,(b) = 2.13⎜1 + ⎝ 1.94s ⎠

3.3.3. Recycle Compensator Design. Using eq 3, the recycle compensator is designed as Figure 10. Block diagram of the coupled FEHE/furnace/reactor process.

F (s ) =

10(0.1s + 1)3 (s + 1)2

which is unrealizable, because it is improper. A realizable compensator is designed by first reducing G5 optimally to G5r = 1/(0.2714s + 1), which is then subsequently used in designing F(s) = 10(0.2714s + 1)/(s + 1)2. 3.3.4. Controller Design for the Compensated Primary Loop (Tmix − FB). Upon using the designed compensator, the compensated model is obtained as (using eq 12)

The process parameters are K1 = K2 = 1, Kr = 5, τr = 1, KF1 = 2, τF1 = 1, KF2 = 1, τF2 = 1, τm = 0.1. Note that there are two control loops in this process: (Tmix − FB) and (Tin − QF). The compensator is applied to the primary loop (Tmix − FB). 3.3.1. Controller Design for Secondary Loop. With the measurement lag Gm = 1/(0.1s + 1)3 approximated by the 5120


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Industrial & Engineering Chemistry Research G K (s ) =

ym (s) uc(s)

=

1 0.001s + 0.03s 2 + 0.3s + 1 3

For the purpose of controller design, GK(s) is reduced to GKr(s), using an optimal model reduction to obtain 1 G Kr (s) = 2 0.0784s + 0.2576s + 1 An IMC-PID controller with a tuning parameter of τc = 0.03 was parametrized for the primary loop: ⎛ ⎞ 1 Gc1,(c) = 8.59⎜1 + + 0.3s⎟ ⎝ ⎠ 0.26s

Figure 11 gives a comparison of the closed-loop responses of the compensated and uncompensated system in response to a

Figure 12. Measured variable in response to a step change in disturbance d introduced at t = 5.

Figure 11. Measured variable in response to a step change in reference Tset in introduced at t = 5.

step change in Tset in introduced at t = 5. Note a small oscillation in Tin and a large oscillation in Tmix for the uncompensated system. In Figure 12, a step change in output disturbance d that affects Tin is introduced at t = 5; note that the compensated system provides a better disturbance rejection to this disturbance. The performance metrics that are summarized in Table S2 in the Supporting Information corroborates this assertion. 3.4. Example 4: Nonlinear Simulation of Coupled FEHE/Furnace/Reactor Process. To verify the results of linear simulation presented in the earlier examples, the recycle compensation scheme was implemented on a nonlinear model of the coupled FEHE/furnace/reactor process, since the nonlinear model is the closest to the real physical system. The details of the nonlinear modeling can be found in the work of Reyes and Luyben.25 However, we present the basic equations here. Two different flowsheets are considered here for the application of the compensation scheme: FS1 (Figure 13), which is without furnace, and FS2 (Figure 14), which is with furnace. 3.4.1. Dynamic Model of Reactor and Gas Loop. The tubular reactor used here is a distributed parameter system;

Figure 13. Control system for the coupled FEHE/reactor process without furnace.

hence, it is modeled rigorously as partial differential equations. Using the numerical method of lines, they are approximated as lumped dynamic models. The gas temperature at each of the lumps is modeled as c p,catWcat, n

dTn = Fn − 1c p, n − 1Tn − 1 − Fnc p, nTn − λ 9 Cn dt

(13)

where Wcat,n = Wcat/NR (NR is the number of reactor lumps) and the heat capacity of the catalyst is given as cp,cat = 0.837 kJ kg−1 K−1. The heat of reaction is given as λ = −23237 kJ kmol−1. It is assumed that an exothermic, gas-phase irreversible reaction A + B → C occurs in the reactor, with a reaction rate 9 Cn , which is given by the expression 9 Cn = Wcat, nkn(Tn)yAn yBn P 2 5121

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numbered starting from the cold inlet, and cp,H and cp,C are the specific heat capacities of the hot and cold gaseous mixtures, respectively, which can be calculated from the specific heats of the components, given in Table 2. An average molar density ρav Table 2. Specific Heats and Molecular Weights of Components parameter

value

−1 −1

specific heat (kJ mol K ) Cp,A Cp,B Cp,C molecular weight (kg mol−1) MWA MWB MWC

Here, kn(Tn) = αe−E/(RTn) (where α = 0.19038 kmol s−1 bar−2, E = 69 710 kJ kmol−1, and R is the universal gas constant (R = 8.314 kJ kmol−1 K−1)). The dynamic change in the mole fraction (yj,n) of each reacting species j in the gas phase in each lump, assuming a constant molar density of ρav = 1.124 kmol m−3 (at 535 K and 50 bar) is given by dyjn

Q H, n = UHAH, n(TH, n − TM, n)

Q C, n = UCAH, n(TM, n − TC, n)

c p,mWn

ρav c p,CVC, n ρav VH, n

dt

dTC, n dt

dyH, j , n dt

= FHc p,H(TH, n + 1 − TH, n) − Q H, n

(17)

= FCc p,C(TC, n − 1 − TC, n) + Q C, n

(18)

= FH, n + 1yH, j , n + 1 − FH, nyH, j , n

dTM, n dt

= Q H, n − Q C, n

(22)

where Wn is the weight of tube metal in each lump, which is equal to the total tube weight (Wtubes) divided by the number of lumps, where Wtubes = 44 500 kg, and cp,m = 0.05 kJ kg−1 K−1. 3.4.3. Dynamic Model of the Furnace. The furnace is modeled with a simple energy balance:

(16)

where F0 is the fresh feed flow rate, FNR is the reactor outlet flow, and yNR,C is the mole fraction of C in the reactor outlet flow. VTotalGas = 60.14 m3. 3.4.2. Dynamic Model of the FEHE. In the FEHE, the cold side is always a 50/50 mixture of A and B, while the hot side contains varying amount of component C, because the conditions in the reactor change dynamically with time. Assuming that both temperatures and compositions can change with time and axial position, each lump of the cold and hot sides of the exchanger is described by an energy balance and three composition balances, given by ρav c p,HVH, n

(21)

where AH,n is the heat-transfer area in each lump, which is equal to the total area divided by the number of lumps. The two heattransfer coefficients are set at UH = UC = 0.497 kJ s−1 m−2 K−1. The dynamic change of temperature in each lump of the metal is described by energy balance

(15) dt where the stoichiometric coefficients are ϑA = ϑB = −1 and ϑC = 1, and Vgas = 1.55 m3. The dynamic change in the total pressure in the gas loop is given by

dTH, n

(20)

QC,n is the rate of heat transferred from the hotter tube metal to the cold gas, given by

= Fn − 1yj , n − 1 − Fnyj , n + ϑj9 Cn

17.5(VTotalGas) dP = F0(17.5) − 35FNR yNR,C (0.08314)(500) dt

15 20 35

is used in the component balances. There is flow in and out of each lump. Pressure drops are neglected. The hot and cold side volumes are assumed to be equal. QH,n is the rate of heat transferred from the hot gas into the tube metal at temperature TM,n, given by

Figure 14. Control system for the coupled FEHE/reactor process with furnace.

Vgasρav

30 40 70

c p,CMF

dTin = Q F + Fmixc p,C(Tmix − Tin) dt

(23)

where MF = 240 kmol s, Tmix is the temperature of the blended stream, and Fmix is the flow rate of the blended stream. 3.4.4. Dynamic Model of the Separating Drum. The total molar balance on the drum is given by dMD = 35FNR yNR,C − LC dt

(24)

where LC is the liquid product C. 3.4.5. Controller Structure Selection. To stabilize the plant, the control structure used in the work of Reyes and Luyben25 was adopted for controlling the process. It has the following loops: (i) The gas loop pressure (P) is controlled by manipulating the fresh feed flow ware F0. (ii) The liquid level in the separator is controlled by manipulating the liquid flow rate.

(19)

where j = A, B, and C, and yH,j,n is the mole fraction of component j in hot-side lump n. The stage parameter n is 5122


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Industrial & Engineering Chemistry Research (iii) The recycle gas flow rate (FR) is held constant by fixing the compressor speed. (iv) The temperature of the blended stream of gas from the heat exchanger (Tmix) is controlled by manipulating the bypass flow rate (FB). (v) In the FS2 flowsheet with furnace, the reactor inlet temperature (Tin) is controlled by manipulating the furnace heat input (QF). (vi) In both FS1 and FS2, the recycle compensator (RC), which uses Tmix as an input signal, is designed to compensate for the effect of energy recycle. Figures 13 and 14 show the Process and Instrumentation Diagrams for the implementation of these control structures. 3.4.6. Solution of the Dynamic Model. For the purpose of control system implementation, a simulink model of the process is developed from the systems of differential equations describing the dynamics of the process. A 10-lump model is assumed for both the reactor and the heat exchanger. This model predicts a reactor outlet temperature of 501.5 K. Table 3 gives a comparison of the process steady states from the lumped dynamic model and the steady-state design, which shows that they are very close.

G3 = G Xr × Gr =

G4 = 1 G5 = Gm =

−1

Gc =

F0 (kmol s ) FB (kmol s−1) QF (kJ s−1) Tmix (K) Tin (K) Tout (K) area (m2)

dynamic

design

dynamic

0.24 0.184 0 460 460 500 2261

0.2327 0.1875 0 460 460 501.5 2261

0.24 0.365 1,023 445.3 460 500 1712

0.2327 0.4107 1,021 445.3 460 501.5 1712

F (s ) =

−0.0758(6s + 1)3 e−280s (51.7s + 1)(125s + 1)3

3.4.7.3. Controller Design for the Compensated Temperature Loop Model. Applying the compensator to the recycle process, the open-loop compensated system model for the temperature loop is obtained using eq 11 as G K (s ) =

Tin(s) −43.8 = FB(s) (6s + 1)3

which is optimally reduced to −43.8 G Kr (s) = (9.11s + 1)2 An IMC-based PID controller is tuned for this loop (with the tuning parameter chosen as τc = 50), but only PI is implemented as Gc1 = −0.0083 −

3.4.7. Controller Design for Flowhseet 1 (FS1). For the purpose of controllers and compensator design, transfer function models were obtained for the plant by linearizing the model at the plant’s nominal operating point. The heat exchanger is modeled by the following transfer functions:

0.0004566 s

3.4.7.4. Controller Design for the Pressure Loop. The gas loop pressure is modeled by P(s) 0.6912 = G P (s ) = F0(s) s

Tin(s) = G Xf = −43.8 FB(s)

Two 30 s lags, 1/(30s + 1)2, are assumed in the pressure loop, which is approximated to e−60s, using Taylor’s expansion. Thus,

Tin(s) 0.777 = G Xr = Tout(s) 51.7s + 1

G Pm(s) =

0.6912 0.6912e−60s ≅ 2 s s(30s + 1)

An IMC-PI controller is parametrized for this loop as follows, with the filter parameter selected as τc = 10:

The tubular reactor is modeled by Tout(s) 4.27e−280s = Gr = Tin(s) (125s + 1)3

Gc = 0.0236 +

0.00029526 s

3.4.8. Controller Design for Flowhseet 2 (FS2). In FS2, the heat exchanger is modeled by

Three 6-s lags are assumed in the temperature loops: Gm =

⎛ −0.000278 ⎞ ⎜ ⎟ ⎝ ⎠ s

3.4.7.2. Recycle Compensator Design. The recycle compensator is specified according to eq 3 as

FS2

design

1 (6s + 1)3

3.4.7.1. Controller Design for the Uncompensated Temperature Loop Model. The transfer function for the uncompensated temperature loop is quasi-rational. A linear analysis shows that it is unstable. A controller with integral action only is able to stabilize this loop.

Table 3. Design and Dynamic Model Steady States FS1

3.318e−280s (51.7s + 1)(125s + 1)3

1 (6s + 1)3

Tmix(s) = G Xf = −52.4 FB(s)

Using the compensation scheme in Figure 1, the component transfer functions are obtained as follows:

Tmix(s) 0.7 = G Xr = Tout(s) 51.7s + 1

G1 = −43.8 5123


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Figure 15. FS1 response to a step change of +5 K in reference Tset in introduced at t = 10 min. [Legend: dotted line = set point, solid line = feedback controllers + compensator, dashed line = feedback controllers alone.]

The furnace is modeled by

F (s ) =

Tin(s) 0.0143 = G F2 = Q F(s) 120.5s + 1

3.4.8.3. Controller Design for Compensated Temperature Loop (Tmix − FB). Applying the compensator to the recycle process, the open-loop compensated plant model for the temperature loop (Tmix − FB) is obtained using eq 11 as

Tin(s) 1 = G F1 = Tmix(s) 120.5s + 1

The reactor modeled by

G K (s) =

Tout(s) 4.27e−280s = Gr = Tmix(s) (125s + 1)3 1 (6s + 1)3

Using the compensation scheme in Figure 1, the component transfer functions are obtained as follows:

Gc1 = −0.0232 −

G3 = G Xr × Gr × G F1 2.99e−280s (51.7s + 1)(120.5s + 1)(125s + 1)3

G5 = Gm =

G F2 =

1 (6s + 1)3

0.0143e−18s 120.5s + 1

An IMC-PI controller is parametrized for this loop as follows, with the filter parameter selected as τc = 32:

3.4.8.1. Controller Design for Uncompensated Temperature Loop (Tmix − FB). The transfer function for the uncompensated temperature loop is quasi-rational. A linear analysis shows that it is unstable. Only a controller with integrating action is able to stabilize this loop. Gc =

0.0013 s

3.4.8.4. Controller Design for the Temperature Loop (Tin − QF). Taking into consideration the temperature lag, Gm = 1/(6s + 1)3 in the loop, which, according to Taylor’s expansion, is approximated to e−18s, the loop can be modeled by the transfer function:

G1 = −52.4

=

Tmix(s) −52.4 = FB(s) (6s + 1)3

For the purpose of IMC-PID controller parametrization, this is optimally reduced to GKr(s) = −52.4/(9.11s + 1)2. An IMC-based PID controller is tuned for this loop (with the tuning parameter chosen as τc = 15), but only PI is implemented as

Three 6 s lags are assumed in the temperature loops: Gm =

−0.057(6s + 1)3 e−280s (51.7s + 1)(120.5s + 1)(125s + 1)3

Gc = 168.53 +

1.3986 s

3.4.8.5. Controller Design for the Pressure Loop (P − F0). The same controller settings as those used in FS1 were used. 3.4.9. Simulation Results. The designed controllers were implemented on a simulink model of the process according to the proposed flowsheets in Figures 13 and 14. Simulation results as presented in Figures 15 and 16 show that the control

−0.00077 s

3.4.8.2. Recycle Compensator Design. The recycle compensator is specified according to eq 3 as 5124


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Figure 16. FS2 response to a step change of −6 K in reference Tset in introduced at t = 2 min. [Legend: dotted line = set point, solid line = feedback controllers + compensator, dashed line = feedback controllers alone.]

The Nyquist plot of GR(s) and the time response plot reveal that it is open-loop unstable. An attempt was made at designing PID controller for this process via optimization of time integral indices. The simulation results obtained showed that the process could not be stabilized using the PID controller alone. The recycle compensator is specified for this system as

scheme with feedback controllers plus recycle compensator performs better than the control scheme with feedback controllers alone. The performance indices computed in Table S3 in the Supporting Information also confirms this. It was investigated in this study whether the feedback controllers plus the recycle compensator obtained at the nominal operating point suffice in controlling the plant at all operating points. Simulation results as presented in the Supporting Information (Figures S1−S6) show that the combination of the nominal feedback controller plus the recycle compensator will adequately control the process flowsheet (FS1) for inlet temperature range of (460 − 4) K ≤ Tin ≤ (460 + 12) K and the manipulated input range of 0 kmol s−1 ≤ FB ≤ 1.2 kmol s−1. Note that this is a large enough range and it falls within the physical constraints of the process. For the process flowsheet (FS2), the limits are given as (460 − 7) K ≤ Tin ≤ (460 + 15) K, 0 kmol s−1 ≤ FB ≤ 1.2 kmol s−1, and 0 kJ s−1 ≤ QF ≤ 3000 kJ/s. 3.5. Example 5: Recycle Process with an Unstable Pole in the Forward Path. Consider a recycle system with an unstable pole in the forward path, taken from the work of Marquez-Rubio et al.26 According to the block diagram given in Figure 2, the forward and recycle path transfer functions are given as

F (s ) =

Upon applying F(s) to GR(s), it simplifies to G K (s ) =

⎛ ⎞ 1 Gc = 0.639⎜1 + + 0.86s⎟ ⎝ ⎠ 10.55s

and

4e 4s − 1

fR (s) =

5e−2s (s + 1)(0.5s + 1)

The transfer function of the overall process is given as G R (s) =

1 9.63s + 1

Figure 17 shows the comparison between our compensation scheme and the observer−predictor-based control scheme proposed in the work of Marquez-Rubio et al.26 The recycle compensation scheme outperforms their scheme for both servo and regulatory control.

and G3*(s) =

4e−2s (4s − 1)

Two-degrees-of-freedom PID controllers were parametrized for this system using the PID controller tuning rule proposed for first-order-plus delayed unstable systems by Lee et al.27 The feedback controller and the set point filter were respectively parametrized (with the IMC tuning parameter selected as τc = 1.75) as

− 2s

G1(s) =

5e−2s (s + 1)(0.5s + 1)

4. DISCUSSION AND CONCLUSION This work has proposed an IMC-based PI/D control strategy for controlling open-loop unstable recycle processes. It has been revealed through the process examples studied that the

G1 4(s + 1)(0.5s + 1)e−2s = * (1 − G1G3 ) (4s − 1)(s + 1)(0.5s + 1) − 20e−4s 5125


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can be designed and compensators can be appropriately switched as necessary. In conclusion, the implementation of the combination of a recycle compensator and a feedback controller (compared with feedback controller alone), as revealed through the computed performance indices and simulations, results in a control system that is less sensitive to disturbance and has lower output error, smoother control signal, and higher gain and phase margins.

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ASSOCIATED CONTENT

S Supporting Information *

Tables S1−S3 give performance indices for Example 2, Example 3, and flowsheets FS1 and FS2. Figures S1−S6 show the FS1 and FS2 responses to step changes in Tset in . This material is available free of charge via the Internet at http:// pubs.acs.org.

■

Figure 17. System output for a unit step change in set point introduced at t = 0 and a step change in output disturbance of −0.1 and +0.1 introduced at t = 40 and 80, respectively.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

recycle compensator restores stability to open-loop unstable plants, which is an often-overlooked advantage of recycle compensator, and also simplifies considerably the process model used for controller design, thus facilitating the design of high-performance closed-loop systems. It is noteworthy here that the proposed strategy is in the same spirit with the approach of Tan et al.,28 in which a compensator is initially applied to the unstable model of the system to restore stability, and, subsequently, the stable compensated model is used for controller design. However, unlike their scheme, our approach is more effective for systems whose instability is caused by recycle. Furthermore, the recycle compensator has general applicability to processes with recycle. This work also compared the proposed strategy with the simple PI/PID tuning rules for unstable systems proposed by Rotstein and Lewin23 and the optimal controllers designed in the work of Luyben.22,24,25 In all of the process examples considered, the simulation results obtained, together with the computed performance indices, revealed that the application of the recycle compensator has brought significant improvement in process performance. It is always expedient to know a priori when the combination of feedback controller plus recycle compensator will give a far more superior performance to feedback controller alone. Work on this has been carried out by Madhukar et al.29 The major conclusion of their work is that, if the forward path and the recycle path dynamics of the recycle process can be represented using first-order-plus time delay, and if 0.75 ≤ |kFkR| ≤ 1 with a small (τR/τF) ratio or |kFkR| ≥ 1 with a small (τR/τF) ratio, then a control system that implements both feedback controller and recycle compensator will give a superior performance to the control system with only a feedback controller, where (kF and kR) and (τR and τF) respectively denote the gains and the time constants for the forward path and recycle path process. More details about this can be found in their paper. Here, note that all the processes considered in this paper satisfied this condition, which explains the better performance that is obtained. In Example 4, the nominal feedback controllers plus nominal recycle compensator suffice in controlling the plant at the operating points of interest within the constraints of the system. However, if the nominal compensator could not give an adequate performance, more than a single recycle compensator

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ACKNOWLEDGMENTS The authors are grateful to anonymous reviewers for their useful suggestions and comments, which greatly helped in improving the quality and the presentation of this paper.

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REFERENCES

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Recycle Compensator Facilitates Rapid Parameterization of

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