RESEARCH NOTE pubs.acs.org/IECR
Iterative Feedback Tuning of PID Controllers for Reactive Distillation Processes: a Comparison with Relay Feedback Tuning Steffen Sommer,*,† Peter M€uller,† and Achim Kienle†,‡ † ‡
Otto von Guericke University, Magdeburg, Germany Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany ABSTRACT: The iterative feedback tuning approach is used to design decentralized PID controllers for an ideal two-product reactive distillation column. The method is compared with relay feedback tuning including TyreusLuyben tuning rules. A control structure where two tray temperatures are the controlled variables and the feeds are the manipulated variables is considered. It is shown that iterative feedback tuning gives superior performance, compared to standard relay feedback.
1. INTRODUCTION Reactive distillation (RD) processes integrate chemical reactions and distillation into the same process unit.1 This has several positive implications (for instance, a reduction of investment and operational costs and potential improvement of performance). However, the dynamic behavior of reactive distillation columns (RDCs) is quite complex, which complicates controller design. Suitable control strategies are needed. The first investigations on reactive distillation column control was published in ref 2. An overview about RD control is given in refs 1 and 3. Autotuning methods for controller design are extremely popular in industry, because they are model-free methods and can be implemented in process control systems as an automated design procedure. In this context, the PID controller is often used, because a wide range of control problems can be solved with this simple and effective controller.4 A first model-free tuning method for PID controllers was developed by Ziegler and Nichols.5 Related methods followed. They are summarized, for example, in refs 68. Typical autotuning methods include relay feedback (RF) and iterative feedback tuning (IFT). The IFT method was originally suggested in ref 9. Surveys are presented in refs 10 and 11. The advantage of the IFT technique is that it is an optimization-based approach, where no process model and no empirical tuning rules are needed. The optimization is only done by means of experiments. In this contribution, the IFT approach is introduced for RD processes to design decentralized PID controllers, which, so far, has not been reported in the literature. Originally, IFT was developed for linear time-invariant systems. It is shown in ref 12 that IFT can also be applied successfully to nonlinear systems. Here, it is investigated if IFT is suitable for RD processes, which are highly nonlinear processes. An ideal generic two-reactant two-product RD system, such as that introduced in ref 13, is considered as a typical RDC control benchmark problem. The purities of both products are to be controlled using decentralized PID controllers. Online composition measurement is assumed in refs 1315. However, composition measurement devices are expensive, slow, and error-prone. Therefore, an inferential control approach is taken into account in this contribution, where tray temperatures are used as an indirect measure for the product purities. Inferential control r 2011 American Chemical Society
approaches are also considered in refs 1618 and an estimatorbased control approach is discussed in ref 19. The relay feedback test introduced by Åstr€om and H€agglund20 is an effective controller tuning method, especially for highly nonlinear chemical processes such as RDCs.21 An overview of this technique is given in refs 6, 7, and 22. Relay feedback tuning, especially in combination with TyreusLuyben tuning rules, is often used for chemical processes, because it provides good results. This approach was also successfully applied to the considered RDC in different publications (for example, ref 18). That is, it can be seen as a standard method for this column. For that reason, the iterative feedback tuning results are compared with those from the established relay feedback method including TyreusLuyben tuning rules.
2. PROCESS DESCRIPTION In this section, we present a model to describe the dynamic behavior of the considered generic two-reactant two-product RDC, as shown in Figure 1. The process model was adapted from refs 13, 14, and 23. The column consists of nRX = 10 reactive trays, nS = 5 stripping trays, nR = 5 rectifying trays, a reboiler, and a condenser. Thus, the total number of trays is nT = 22. The trays are numbered from bottom to the top. The reaction A þ BhC þ D
ð1Þ
occurs in every reactive tray. The volatilities Rj are related as follows: RC > RA > RB > RD
ð2Þ
Product C is the lightest of the process, and product D is the heaviest of the process. The volatilities of reactants A and B are between those of the products. Feed stream FA is located at the bottom of the reactive zone (i = 2 + nS = 7) and contains reactant A. The feed stream FB is located at the top of the reactive zone Received: March 8, 2011 Accepted: July 6, 2011 Revised: July 4, 2011 Published: July 15, 2011 9821
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Table 1. Process Parameters parameter
value
parameter
value
nT
22
P
9 bar
nS
5
RA
4
nR
5
RB
2
nRX
10
RC
8
feed stage A
7
RD
1
feed stage B
16
τ
6s
aF
0.008 kmol s1
aB
0.004 kmol s1
EF
kmol 30 000 cal/mol
EB
kmol1 40 000 cal/mol
Λ
10 000 cal/mol
ΔHV
6944 cal/mol
AVP,1
12.34
BVP,1
3862
ν
[1,1,1,1]
R
8.3145 J/(K mol)
1
Table 2. Steady-State Values parameter
value Flow Rates
VS,0 D0
0.0294 kmol/s 0.0126 kmol/s
FA,0
0.0126 kmol/s
R0
0.0340 kmol/s
B0
0.0126 kmol/s
FB,0
0.0126 kmol/s
Figure 1. Reactive distillation column (RDC).
Molar Holdup Mi,0
(i = 1 + nS + nRX = 16) and contains reactant B. Product C is withdrawn at the top, and product D is withdrawn at the bottom of the column. D is the distillate flow rate, and B is the bottom flow rate. R is the reflux flow rate, and VS is the vapor boil-up or reboiler vapor flow rate. For each tray (i = 1, ..., nT), one total mass balance and nC1 = 3 component mass balances are formulated. The kinetics of reaction 1 is modeled by the equation Ri ¼ Mi ðkF, i xi, 1 xi, 2 kB, i xi, 3 xi, 4 Þ
Backward :
kB, i ¼ aB exp
Liquid Mole Fractions—Reboiler
EB RTi
ð5Þ
Parameter xi,j denotes the liquid mole fraction. The rate of reaction Ri is zero in all nonreactive trays. An ideal vaporliquid equilibrium is included. The relations are Rj xi, j ð6Þ yi, j ¼ nC Rk xi, k
∑
0.0221
B C
0.0280 0.0000
D
0.9499
A
0.0287
B
0.0227
C
0.9486
D
0.0000
Moreover, yi,j denotes the vapor mole fraction and Ti is the tray temperature. Tray hydraulics are taken into consideration using a linearized version of the Francis weir formula: Mi Mi, 0 Li ¼ Li, 0 þ ð8Þ τ where Mi is the molar holdup, Li the liquid flow rate, and τ the hydraulic time constant. The vapor flow rate Vi changes only in the reactive section, because the heat of reaction increases vapor formation. The relations are as follows: Vi ¼ Vi1 þ VRX ðiÞ
k¼1
Ti ¼
A
Liquid Mole Fractions—Condenser
ð3Þ
with the forward and backward specific reaction rates defined as follows: EF Forward : kF, i ¼ aF exp ð4Þ RTi
1 kmol
"
BVP, 1
AVP, 1 ln R1 P=
nC
∑ k¼1
Rk xi, k
#
VRX, i ¼
ð7Þ
λ Ri ΔHV
ð9Þ ð10Þ
All process parameters and steady-state values are presented in Tables 1 and 2. 9822
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’ CONTROL OBJECTIVE AND CONTROL STRUCTURE First, we focus on the control objective. The product purities are to be kept constant during the chemical production process, particularly in the case of unforeseen disturbances. This problem is solved using a decentralized two-temperature control strategy. The feed FB is used to control one temperature in the reactive section and feed FA is used to control one temperature in the stripping section. A challenging problem is the choice of appropriate trays, where the temperatures are measured. Two conditions have to be met. First, both controllers must have direct action,17 which means that an increase in the controlled variable temperature will cause an increase in the controller output (feed flow rate). For this reason, the process steadystate gains between tray temperatures and feed flow rates should be negative. Second, the most sensitive tray temperature is paired with the corresponding feed. Temperature T3 in the stripping section and temperature T12 in the reactive section have been chosen. That choice is structurally equal to the choice in ref 18, where the same control structure is investigated. Molar holdups in the condenser and in the reboiler are controlled using proportional controllers via the distillate flow rate D and the bottom flow rate B: D ¼ D0 þ KD ðM22 M22, 0 Þ
ð11Þ
B ¼ B0 þ KB ðM1 M1, 0 Þ
ð12Þ
The gains are KD = KB = 0.5. The reflux flow rate R depends on D. The ratio of both flow rates is constant: R = 2.7 D. The remaining input VS is used for the disturbance scenario. Strictly speaking, the deviation from the operating point VS,0 is the disturbance. In addition, an antireset windup strategy is combined with the PID controllers. This action is necessary if the calculated manipulated variables FA and FB do not satisfy the following constraints:
Single-input single-output (SISO) PID controllers (eq 13, 14) with KI = KP/TI and KD = KPTD are considered. The gains of the controller transfer function (eq 13) are derived by means of optimization. Therefore, the controller is parametrized: p2 þ p3 s ð15Þ Cðs, pÞ ¼ p1 þ s where p1 = KP, p2 = KI and p3 = KD. The controller parameters can be collected in a vector, 0 1 p1 B C C n ð16Þ p¼B @ l A∈Rp pnp where np is the number of parameters. The PID controller transfer function has np = 3 parameters, which are determined by solving the following optimization problem:
’ PID AUTO-TUNING, USING ITERATIVE FEEDBACK TUNING In this section, the autotuning design method iterative feedback tuning is described. KI þ KD s s 1 þ TD sÞ ¼ KP ð1 þ TI s
CðsÞ ¼ KP þ
ð13Þ ð14Þ
ð17Þ
subject to C(s,p) stabilizing the closed loop. The cost function, Z i 1 Tf h fQy eðt, pÞg2 þ λfRu uðt, pÞg2 dt ð18Þ JðpÞ ¼ 2Tf 0 is a quadratic criterion of the control error e(t,p) and the manipulated variable u(t,p). Here, Tf is the considered time interval or the duration of the tuning experiment, respectively. Qy and Ru are the weighting filters, and λ is a weighting constant. For the sake of simplicity, Qy and Ru are set to identity and weighting is done exclusively with the scalar factor λ. In order to obtain the optimal controller parameters, the following equation must be solved: 0¼
∂JðpÞ ∂p
1 ¼ Tf
FA,mineFAeFA,max FB,mineFBeFB,max In such situations, the derivatives of the controller state variables (i.e., the inputs of the integrators) are reset. At steady state, all valves are regarded to be half open. One motivation for using the introduced control structure is the compatibility to the start-up strategy in refs 24 and 25, where also the feed is the manipulated variable. After start-up stages, the start-up control scheme can be switched to the operating point control without much effort. In this paper, one control structure is used to compare two controller design methods. The considered control structure is discussed in ref 18 and compared with other configurations.
JðpÞ
min
wrt p ∈ R np
Z 0
Tf
(
) ∂eðt, pÞ ∂uðt, pÞ þ λuðt, pÞ eðt, pÞ dt ∂p ∂p
ð19Þ
Equation 19 represents the necessary condition for optimality of the unconstrained problem described by eq 17. The solution of eq 19 can be found by the following iterative procedure: pkþ1 ¼ pk γk Hk1
∂Jðpk Þ ∂p
ð20Þ
k denotes the current iteration step. γk is a positive real scalar and represents the step size. Hk is a positive definite matrix. The easiest choice is the identity matrix: Hk = Inp. In this paper, a GaussNewton approximation of the Hessian of J(p) is used: Z ( 1 Tf ∂eðt, pkÞ ∂eðt, pkÞ T Hk ¼ Tf 0 ∂P ∂P ) ∂uðt, pkÞ ∂uðt, pkÞ T þλ dt ð21Þ ∂P ∂P which is recommended in the literature (e.g., see ref 10) and provides good convergence properties. To calculate the current parameter vector pk+1 (eq 20) at iteration step k + 1, ∂J(pk)/∂p (eq 19) and Hk (eq 21) are required. e(t) and u(t) can be measured directly. In addition, the gradients ∂e(t,p)/∂p and 9823
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the partial derivative of the manipulated variable U(s,p), with respect to the parameter vector p, is ∂Uðs, pÞ 1 ∂Cðs, pÞ ¼ Sðs, pÞEðs, pÞ ∂p Cðs, pÞ ∂p
Figure 2. Unity feedback control system.
∂u(t,p)/∂p are needed. As shown below, the gradients can be found by conducting an additional experiment. For further investigations, the unity feedback control system in Figure 2 is considered. The following summary is based on linear time-invariant SISO processes.7 However, the obtained method is applied to the nonlinear reactive distillation process in the next section. The closed-loop equations are GðsÞCðs, pÞ RðsÞ ¼ Tðs, pÞRðsÞ 1 þ GðsÞCðs, pÞ
ð22Þ
1 Eðs, pÞ ¼ RðsÞ ¼ Sðs, pÞRðsÞ 1 þ GðsÞCðs, pÞ
ð23Þ
Y ðs, pÞ ¼
Uðs, pÞ ¼
Cðs, pÞ RðsÞ ¼ Sðs, pÞCðs, pÞRðsÞ 1 þ GðsÞCðs, pÞ
ð24Þ
With ∂Eðs, pÞ ∂Eðs, pÞ ∂Cðs, pÞ ¼ ∂p ∂Cðs, pÞ ∂p
ð25Þ
and ∂Eðs, pÞ 1 ¼ Tðs, pÞEðs, pÞ ∂Cðs, pÞ Cðs, pÞ
ð26Þ
the partial derivative of the error signal E(s,p), with respect to the parameter vector p, is ∂Eðs, pÞ 1 ∂Cðs, pÞ ¼ Tðs, pÞEðs, pÞ ∂p Cðs, pÞ ∂p ¼ Ggrad ðs, pÞY 1 ðs, pÞ
ð27Þ
where Ggrad ðs, pÞ ¼
1 ∂Cðs, pÞ Cðs, pÞ ∂p
ð28Þ
and Y 1 ðs, pÞ ¼ Tðs, pÞEðs, pÞ
ð29Þ
Y1(s,p) denotes the output of the control loop (controlled variable) of an additional experiment, when E(s,p) is the reference signal. With ∂Uðs, pÞ ∂Uðs, pÞ ∂Cðs, pÞ ¼ ∂p ∂Cðs, pÞ ∂p
ð30Þ
and ∂Uðs, pÞ ¼ Sðs, pÞEðs, pÞ ∂Cðs, pÞ
ð31Þ
¼ Ggrad ðs, pÞU 1 ðs, pÞ
ð32Þ
where U 1 ðs, pÞ ¼ Sðs, pÞEðs, pÞ
ð33Þ
U1(s,p) denotes the manipulated variable of the closed loop of an additional experiment, when E(s,p) is the reference signal. In eq 28 the partial derivatives of the controller transfer function, with respect to the components of the parameter vector, are needed. For the PID controller, the individual derivatives ∂C(s,p)/ ∂pi (i = 1, ..., 3) are given as ∂Cðs, pÞ ¼ 1, ∂p1
∂Cðs, pÞ 1 ¼ , ∂p2 s
∂Cðs, pÞ ¼s ∂p3
ð34Þ
The complete experiment, which must be performed during one iteration step k, is illustrated in Figure 3 and is described below. In the first experiment, the closed loop must be operated with the reference signal r(t) (Figure 3, upper control loop). The signals e(t,pk) and u(t,pk) are recorded. In a second experiment, the closed loop must be run with the error e(t,pk) from the first experiment as a reference signal (Figure 3, lower control loop). Therefore, the gradients ∂e(t,pk)/∂p and ∂u(t,pk)/∂p are obtained and ∂J(pk)/∂p (eq 19) and Hk (eq 21) can be calculated. The IFT procedure is summarized as follows: (1) Initialization phase (k = 0): choose initial parameter vector p(k = 0), weighting constant λ, time interval Tf, and convergence tolerance ε. (2) Execution of the two experiments (Figure 3), gradient calculation: compute gradient ∂J(pk)/∂p (eq 19); if |∂J(pk)/∂p| < ε, then stop. (3) Parameter update: determine γk and Hk (eq 21), compute pk+1 (eq 20), k = k + 1, go to step 2.
’ SIMULATION AND DESIGN The decentralized SISO PID controllers of the temperature control loops FAT3 and FBT12 are designed sequentially (see Figure 4). Thus, the MIMO system is treated as a sequence of SISO systems, which provides a simple design procedure considering plant couplings. Because of negative steady-state process gains, positive feedback loops are implemented. All in all, a negative feedback system is obtained as desired. First, the relay feedback design is described. The procedure starts with step (I) (see Figure 4). The process is at steady state (operating point). The steady-state value of the manipulated variable FA,0 is increased or decreased by h (relay height) if T3 < T3,SP (e > 0) or T3 > T3,SP (e < 0). A value of h = 0.00126 kmol/s was chosen and represents 10% of the steady-state value FA,0 = 0.0126 kmol/s. A switching procedure is applied and a sustained oscillation of the temperature T3 is observed, from which the first PID controller can be configured. In this contribution, tuning rules by Tyreus and Luyben are applied, which provide good results. The tuning procedure is proceeded with step (II). The steady-state value of the manipulated variable FB,0 is increased or decreased by h = 0.00126 kmol/s if T12 < T12,SP (e > 0) 9824
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Table 3. PID Controller Parameters, Relay Feedback control loop
Kp
TI[s]
TD[s]
FAT3
0.0025
212.03
15.16
FBT12
0.0150
186.66
13.51
Table 4. IFT Iteration, Control Loop FBT12
Figure 3. Experimental design.
γ
k
Kp
TI [s]
TD [s]
J
0
0.0150
186.66
13.51
1
0.0246
160.78
34.39
1
0.0017
2
0.0266
159.26
35.90
0.1
0.0016
0.0049
Table 5. IFT Iteration, Control Loop FAT3
Figure 4. Sequential design.
or T12 > T12,SP (e < 0). After this, steps (III) and (II) are repeated alternatingly until the controller parameters converge. The final results are presented in Table 3. The IFT approach is now applied to the above-mentioned nonlinear RD process. We assume that the process is at steadystate (operating point). In the linear case, the variables u, y, u1, and y1 represent absolute values. Here, deviations from the operating point are considered for computing the controller. Compared to the standard IFT tuning procedure (see Figure 3) a modification is carried out. The control loop concerning the first experiment is not be run with a set-point change but with introducing a disturbance. This can be done because the gradient computation in the second experiment is not dependent on what signal (reference or disturbance) is used in the first experiment. An increase of 20% in the operating point value of the reboiler flow rate VS,0 is considered as an unneeded disturbance. That means the controller is constructed for a known disturbance case. During the second experiment, the control loop is only be operated with the first experiment error response as reference signal (see Figure 3). The RF controllers
γ
k
Kp
TI [s]
TD [s]
J
0
0.0025
212.03
15.16
1
0.0037
170.51
28.54
1
0.125
2
0.0039
169.56
28.85
0.1
0.122
0.246
are used as initial controllers. For this reason, the tuning procedure starts with step (II) (see Figure 4). A weighting constant of λ = 0 is chosen for the control loop FBT12. To prevent oscillations with huge amplitudes in the manipulated variable FA, a weighting constant of λ = 10 000 is chosen for the control loop FAT3. The duration of one tuning experiment is Tf = 1000 s, because after that duration, the controlled variables reach their steady-state values. For the derivation of the gradients, the initial conditions of the process and controller are assumed to be the same in all experiments. That is, the experiments start from the same consistent operating point. The iteration processes of both control loops are presented in Tables 4 and 5. In both cases, the cost function could be decreased by additional iterations but without much improvements in the control behavior. In the following, the simulation results are compared. A change of +20% in the reboiler vapor flow rate VS is considered. The temperature responses are illustrated in Figure 5. For both control loops, the disturbance is rejected with steadystate accuracy. The temperatures return to the desired steadystate values. It can be observed that both IFT controllers improve the closed-loop dynamic behavior remarkably. The corresponding product purities are presented in Figure 6. The control results regarding the bottom product purity xD can be improved. However, both responses do not reach the desired steady-state value. There is a small difference. The top product purity responses xC take a long time to reach the steadystate value. The reason is the great distance between the top of the column and the reactive section where the corresponding temperature is measured. The difference between the actual and desired steady-state values is greater when compared with the bottom product purity. In connection with the top product purity, the IFT controller can improve the results only slightly. Finally, the manipulated variables can be seen in Figure 7. The disadvantages of the mentioned two-temperature control structure slow dynamics and great inaccuracy concerning the top product purity in the disturbance case could be improved using a three-temperature control scheme. Thereby, an additional 9825
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Figure 5. Temperature responses, given a +20% change in the reboiler vapor flow rate (VS).
Figure 6. Product purities, given a +20% change in the reboiler vapor flow rate (VS).
Figure 7. Feed flow rates, given a +20% change in the reboiler vapor flow rate (VS).
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Figure 8. Temperature responses, given a 20% change in the reboiler vapor flow rate (VS).
temperature in the rectifying section is controlled via the reflux flow rate R. The test input signal of the control loop has to be chosen with care, because the controller is tuned for that signal. For the considered reactive distillation process the controller is tuned for a change of +20% in the reboiler vapor flow rate. The control behavior might change if another disturbance scenario occurs. However, it can be shown that also very good results are obtained compared to RF when a change of 20% in the reboiler vapor flow rate occurs (see Figure 8). Changes of +20% and 20% in the disturbance input are worst-case scenarios. Smaller disturbances are less critical. That is, the robustness with respect to the disturbance input is demonstrated. It is appropriate to highlight that PID controllers tuned by means of IFT are able to improve the control dynamics of reactive distillation processes compared with PID controllers tuned using relay feedback. Because of additional experiments, the tuning time with respect to the IFT method is larger, compared with RF.
’ CONCLUSION In this work, iterative feedback tuning was successfully applied to the design of decentralized PID controllers for an ideal twoproduct reactive distillation process. One control structure was used to compare the method with relay feedback including TyreusLuyben tuning rules. The control results concerning a two-temperature-control scheme were improved significantly. However, better results are obtained with much tuning effort. For further studies, the application of the IFT approach, with respect to the other reactive distillation column control structures is reasonable. According to our experience, we expect even greater benefit when IFT is applied to a highly nonideal reactive distillation process, such as methyl acetate synthesis for example. ’ AUTHOR INFORMATION Corresponding Author
*E-mail: steff
[email protected].
’ NOMENCLATURE A, B = reactants C, D = products
nC = number of components nT = total number of trays nS = number of stripping trays nRX = number of reactive trays nR = number of rectifying trays Fi = feed flow rate Li = liquid flow rate Vi = vapor flow rate R = reflux flow rate D = distillate flow rate B = bottom flow rate Mi = molar holdup xi,j = liquid mole fraction yi,j = vapor mole fraction zi,j = liquid mole fraction, feed Ri = reaction rate kF,i = forward specific reaction rate kB,i = backward specific reaction rate aF, aB = pre-exponential factors EF, EB = activation energies R = ideal gas constant Ti = absolute temperature AVP,j, BVP,j = vapor pressure constants P = pressure ΔHV = heat of vaporization C(s) = controller transfer function G(s) = process transfer function KD, KB = proportional controller gains KP, KI, KD, TI, TD = PID controller parameters r(t) = reference signal in the time domain e(t) = error signal in the time domain u(t) = manipulated variable in the time domain y(t) = controlled variable in the time domain R(s) = reference signal in the Laplace domain E(s) = error signal in the Laplace domain U(s) = manipulated variable in the Laplace domain Y(s) = controlled variable in the Laplace domain S(s) = sensitivity function T(s) = complementary sensitivity function h = relay height p = parameter vector np = number of parameters 9827
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Industrial & Engineering Chemistry Research J = cost function Tf = duration of experiment Qy, Qu = weighting filters H = positive definite matrix Greek Symbols
RA,...,D = relative volatilities νj = stoichiometric coefficients τ = hydraulic time constant Λ = heat of reaction λ = weighting constant γ = step size ε = convergence tolerance Subscripts
i = tray j = component S = stripping RX = reactive R = rectifying 0 = steady-state, operating point k = iteration SP = set point C, D = products
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