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High-purity control of internal thermally coupled distillation columns based on nonlinear wave model
Journal of Process Control 21 (2011) 920–926
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Journal of Process Control journal homepage: www.elsevier.com/locate/jprocont
High-purity control of internal thermally coupled distillation columns based on nonlinear wave model Xinggao Liu a,∗ , Yexiang Zhou a , Lin Cong a , Feng Ding b,∗ a b
State Key Laboratory of Industrial Control Technology, Control Department, Zhejiang University, Zheda Road 38, Hangzhou 310027, China Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), School of IoT Engineering, Jiangnan University, Wuxi 214122, China
a r t i c l e
i n f o
Article history: Received 27 August 2010 Received in revised form 7 March 2011 Accepted 8 March 2011 Available online 8 April 2011 Keywords: Internal thermally coupled distillation column High purity control Nonlinear wave model Generic model control
a b s t r a c t Internal thermally coupled distillation column (ITCDIC) is a frontier of energy saving distillation researches, which is a great improvement on conventional distillation column (CDIC). However its high degree thermal coupling makes the control design a bottleneck problem, where data-driven model leads to obvious mismatch with the real plant in the high-purity control processes, and a first-principle model which is comprised of complex mass balance relations and thermally coupled relations could not be directly used as control model for the bad online computing efficiency. In the present work, wave theory is extended to the control design of ITCDIC with variable molar flow rates, where a general nonlinear wave model of ITCDIC processes based on the profile trial function of the component concentration distribution is proposed firstly; combined with the thermally coupled relations, a novel wave model based generic model controller (WGMC) of ITCDIC processes is developed. The benzene–toluene system for ITCDIC is studied as illustration, where WGMC is compared with another generic model controller based on a data-driven model (TGMC) and an internal model controller (IMC). In the servo control and regulatory control, WGMC exhibits the greatest performances. Detailed research results confirm the efficiency of the proposed wave model and the advantage of the proposed WGMC control strategy. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Distillation column is a key part in the separation processes. Since it consumes large amount of energy, any improvement of its energy efficiency could produce a significant impact on the economic profit in the industry. ITCDIC is an excellent improvement on conventional distillation columns (CDIC), where reboiler and condenser are no longer needed, and energy recovery is realized by transferring heat from the rectifying section to the stripping section [1–10]. Though the large energy saving potential of ITCDIC has been confirmed by many researchers, high degree of thermal coupling between the rectifying section and the stripping section brings complex influences on the dynamic behaviors in ITCDIC [11–14]. The strong nonlinearity and high sensitivity to operation conditions pose a huge challenge on its control design [15–18]. The tight control of ITCDIC not only desires an elegant control strategy but also requires an efficient control model. Traditional approximating models have great difficulty in describing the abundant complex nonlinear dynamics of ITCDIC. Linear models are often valid only in a small neighborhood of the
∗ Corresponding authors. Tel.: +86 571 87951071. E-mail addresses: [email protected] (X. Liu), [email protected] (F. Ding). 0959-1524/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jprocont.2011.03.002
nominal operating point [19] and cannot describe the distinct nonlinear behavior of ITCDIC. It hardly works in the high-purity control processes [20–22]. Data-driven models are popular alternatives to linear models, for they are easily obtained with simple constructions. Based on data-driven models, many control designs of ITCDIC have been carried out. Nakaiwa [23] proposed an internal model control (IMC) strategy, compared with multi-loop proportional, integral, and derivative control (M-PID). IMC attains better control performance when set-point transfers. However it is extremely sensitive to changes in the operating conditions, and its regulatory performance drastically deteriorates. M-PID can overcome external disturbances fairly well, but it does not deal effectively with the interactive nature of ITCDIC. Though modified IMC, adaptive predictive M-PID [24,25] make some improvement on the above problems, the performance of the controller still degrade greatly when used for high purity control. The main reason is that severe model mismatch between the formulated model and the real plant occurred when operation conditions change. Firstprinciple model [26] is proved to be a most effective one to capture the rich nonlinear dynamics. It is usually used to analyze the distinct behavior of a complex process. Due to the high degree of thermally coupled relations, the first-principle model of ITCDIC is expressed as a complex structure, which makes it hardly realized as a control model for its bad online computing efficiency.
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Table 1 Nomenclature. a, b F H kr , ks L n Pvp Pr Ps P q Q t T UA V X Xˆ Xr min , Xr max Xs min , Xs max Y Zf ˛ S1 S2 f j
[–] [kmol/h] [kmol] [–] [kmol/h] [–] [MPa] [MPa] [MPa] [MPa] [–] [kW] [h] [K] [W/K] [kmol/h] [–] [–] [–] [–] [–] [–] [–] [J/mol] [–] [–] [–] [–]
Coefficient of the Antoine equation Feed Stage holdups in each stage Characterize the tangent of the inflection points S1 and S2 Liquid flow rate Stage number Vapor saturated pressure Pressure of rectifying section Pressure of stripping section Representation of either Pr or Ps Feed thermal condition Energy required Time Absolute temperature Heat transfer rate in each stage Vapor flow rate Mole fraction of liquid The estimation of liquid mole fraction The asymptotic limits of the rectifying section The asymptotic limits in the stripping section Mole fraction of vapor Mole fraction of feed Relative volatility Latent heat Position of wave profile in the rectifying section Position of wave profiles in the stripping section Feed stage Stage number (counted from the top to the bottom)
In the research of CDICs, a novel nonlinear wave theory is employed to describe the distinct behavior in the separation processes. Systems with distributed parameters like distillation columns often exhibit dynamic phenomena which resembles traveling wave [27,28]. Luyben [29,30] pioneered a study of the propagation of temperature and composition profiles in high-purity distillation columns. Later Marquardt [31,32] derived expressions for the wave propagation velocity and the trial function of the wave profile in distillation columns from differential material balances. Hwang [33–35] developed a distributed wave theory for general counter-flow separation processes. The model was used to capture the features of the propagation, reflection, superposition and self-sharpening behaviors of component concentration waves in distillation columns. The latest research by Hankins [36,37] extends the wave theory to the distillation columns with variable molar flow rates. And detailed analysis of partial coherence of the wave propagation in the columns is also carried out. Based on the nonlinear wave model many advanced controllers are carried out for the control of CDIC. Han and Park [38,39] proposed a novel generic model control of CDIC, where a full order observer is designed to evaluate the wave position online. Shin [40] further improved the above generic model control by using the wave profile of tray temperatures. Group led by Doyle III [41,42] developed an IMC controller based on a wave model and a Kalman filter was utilized to update the parameter values. Group led by Henson [43–45] concentrated on the low order wave modeling of cryogenic distillation columns and high purity distillation columns. They also proposed a model predictive control strategy based on the nonlinear wave model of nitrogen purification columns. Wave theory is also extended to multi-component distillation columns and batch reactive distillation columns [46,47]. However due to the variable molar flow rate and distinct wave traveling characteristics, wave modeling of ITCDIC is much more difficult than CDICs’, and as far as we know, few publication is concentrated on wave modeling of ITCDIC or the controller design based on wave model. In the present work, the wave velocity formula of ITCDIC with variable molar flow rates is firstly derived based on the trial function of wave profile. Combined with the mass balance relations and thermally coupled relations, a novel generic model control
strategy is developed based on the proposed nonlinear wave model of ITCDIC (WGMC). In the illustration of benzene–toluene system, WGMC is compared with another generic model controller based on a data-driven model (TGMC) [48–52] and an internal model controller (IMC) [25]. WGMC shows excellent performances in the servo control and regulatory control. The overshoot and stabilizing time are significantly decreased by WGMC. Analysis of integral absolute error (IAE) and integral square error (ISE) further confirms the significant advantage of WGMC. Detailed results of comparison between WGMC and TGMC obviously show the efficiency of the proposed wave model, and the results of comparison between TGMC and IMC prove the advantage of generic model control strategy. The nomenclature for the model and controller is given in Table 1. 2. Nonlinear wave model of ITCDIC Wave profile is a description of a monotonic variation of an independent variable like temperature distribution and component concentration distribution [18]. In ITCDIC, since the monotonicity of the temperature distribution in the rectifying section and stripping section opposites to each other, which is much different from those of CDIC, the component concentration distribution is chosen as a representation of the wave profile, which exhibits two characteristics: (1) asymptotic properties at the top and bottom of the columns; (2) existence of a inflection point in the profile of each section. A novel trial function is introduced to describe the wave profiles in ITCDIC as follows: Xˆ j = Xr
min
+
Xr max − Xr min , 1 + exp(−kr (j − S1 ))
j = 1, 2, . . . , f − 1
(1)
Xˆ j = Xs
min
+
Xs max − Xs min , 1 + exp(−ks (j − S2 ))
j = f, f + 1, . . . , n
(2)
where Xˆ j denotes the estimation of liquid mole fraction; Xr min and Xr max denote the asymptotic limits of the rectifying section when the profiles extend to a infinite distance; Xs min and Xs max denote the asymptotic limits in the stripping section. Xr max and Xs min approximate to the liquid mole fraction at the top stage and
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the bottom stage respectively. kr , ks characterize the tangent of the inflection points S1 and S2 (kr , ks do not equal the tangents). The inflection point S1 , S2 are also the representation of the profile position in the rectifying section and stripping section respectively. Given the material balance relations: H
dXj dt
If Laplace transform of the equation above is taken, the resulting transfer function becomes: 2s + 1 y = 2 2 y∗ s + 2s + 1
j = 2, . . . , n − 1 and j = / f
(3)
Based on the trial function of Eqs. (1) and (2), a new wave velocity combined with thermal coupled relations of ITCDIC could be derived. The relationship between wave velocity and the mass balance is established by the derivatives of Eqs. (1) and (2) as follows: dXˆ j dt dXˆ j dt
=
((dkr /dt)(S1 − j) + kr (dS1 /dt))(Xr
=
max
− Xˆ j )(Xˆ j − Xr Xr
(dks /dt)(S2 − j) + ks (dS2 /dt) (Xs
max
ˆ min ) − (dXr max /dt)(Xj min − Xr
− Xˆ j )(Xˆ j − Xs Xs
where = 1/ k2 and = k1 /2 k2 . The above system gives the similar plots to the classic second-order response showing the normalized response of the system y/y* with normalized time t/ and as a parameter. Three types of trajectories can occur for this second order equation depending on the values of k1 and k2 : under damped ( < 1), critically damped ( = 1), and over damped ( > 1). Choose the right and to give the desired shape of response and “appropriate” timing of response. Then can be calculated k1 and k2 using the above equations: = 1/ k2 , = k1 /2 k2 .
= Vj+1 Yj+1 − Vj Yj + Lj−1 Xj−1 − Lj Xj ,
− Xr
min ) − (dXr min /dt)(Xr max
− Xˆ j )
(4)
max
ˆ min ) − (dXs max /dt)(Xj min − Xs
− Xs
min ) − (dXs min /dt)(Xs max
− Xˆ j )
(5)
max
Substitute dXˆ j /dt for the dXj /dt in Eq. (3), the wave velocity is derived for the rectifying section and the stripping section as follows: (1/H)(−V1 Y1 + Vf Yf − Lf −1 Xf −1 ) − (Xr
dS1 = dt
max
− Xj ))/(Xr
max
− Xr
f −1 j=1
min ) +
f −1
−
j=1
f −1 j=1
(kr (Xr
(1/H)(−Vf Yf + Lf −1 Xf −1 + FZf − Ln Xn ) −
n
− dS2 = dt
j=f
((dXs
min /dt)(Xs max
((dXr
− Xj ))/(Xs −
[((Xr
max
j=f
max
j=f
max
((dXs
− Xs
(ks (Xs
n
j=f
3. Generic model controller design for ITCDIC Consider a process described by the following state equations: x˙ = f (x, u, d, t)
(8)
ym = g(x)
(9)
where state vector x(t) ∈ Rn , manipulated input u(t) ∈ Rm , process output y(t) ∈ Rp , disturbance input d ∈ Rl ; in general, f and g are nonlinear functions. The generic model control is an optimal control approach by forcing the derivative of the process output equal to a reference rate. The reference rate is a proportional integral (PI) type function as follows:
(y∗ − y) dt
(10)
(1/H)(−V1 Y1 + Vf Yf − Lf −1 Xf −1 ) − (Xr
max
− Xr
min ) +
f −1 j=1
[((Xr
max
f −1 j=1
−K11 (S1∗ − S1 ) − K12
j=1
f −1 j=1
((dXr
min /dt)
min )](dkr /dt)(S1
− j)
(6)
min )
− Xs
− Xj )(Xj − Xs
min )
min ))/(Xs max
− Xs
− Xs
min )](dks /dt)(S2
− j)
min )
(7)
Two representative concentration positions (S1 , S2 ) in the rectifying section and the stripping section are chosen to be two state variables for the GMC formulation. When S1 , S2 deviate from the set points S1∗ , S2∗ , the changing rate of S1 , S2 should be controlled to return towards the desired steady states and the process should have zero offset. GMC formulation has following forms: dS1 = K11 (S1∗ − S1 ) + K12 dt dS2 = K21 (S2∗ − S2 ) + K22 dt
t
(S1∗ − S1 ) dt
(11)
(S2∗ − S2 ) dt
(12)
0
t
0
where subscript 1 and 2 denote rectifying section and striping section respectively; K11 , K12 , K21 , K22 are tuning constants. In WGMC, K11 = 15, K12 = 100, K21 = 7, K22 = 100. GMC formulation presents a perfect control process that the wave model should be controlled to follow. Connect Eq. (11) with Eq. (6), and connect Eq. (12) with Eq. (7), two equations for the rectifying section and stripping section are obtained as follows: − Xr
min ))/(Xr max max
− Xr
min ))/(Xs max
max
min ) −
min ))/(Xr max
− Xr
− Xj )(Xj − Xr
− Xr
min ) −
min )](dkr /dt)(S1
min ))/(Xr max
− Xr
f −1 j=1
((dXr
min /dt)(Xr max
− Xj ))/
− j) min )
t
(S1∗ − S1 ) dt = 0 0
(kr (Xr
− Xr
min ))/(Xs max
max /dt)(Xj
− Xj )(Xj − Xr
−
((dXr
f −1
[((Xs
− Xr
min ))/(Xr max
− Xs
− Xj )(Xj − Xs
where dS1 /dt and dS2 /dt denote the travel velocity of the entire profile in rectifying section and stripping section.
y˙ = k1 (y∗ − y) + k2
min ))/(Xr max
min ))/(Xr max
max /dt)(Xj
min ) + max
− Xr
− Xj )(Xj − Xr
− Xj )(Xj − Xr
n
n
max /dt)(Xj
(13)
X. Liu et al. / Journal of Process Control 21 (2011) 920–926
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Table 2 Operating conditions of the ITCDIC studied. Stage number Feed stage Feed flow rate [kmol/h] Feed composition (benzene) (toluene) Liquid holdup in reboiler [kmol]
40 21 100 0.5 0.5 0
(1/H)(−Vf Yf + Lf −1 Xf −1 + FZf − Ln Xn ) − (Xs
max
− Xs
min ) +
n
j=f
[((Xs
max
n
−K21 (S2∗ − S2 ) − K22
((dXs
− Xj )(Xj − Xs
n
−
j=f
j=f
Feed thermal condition Pressure of stripping section [MPa] Heat transfer rate in each stage [W/K] Latent heat of vaporization [kJ/kmol] Liquid holdup in each stage [kmol] Liquid holdup in reflux drum [kmol]
max /dt)(Xj
min ))/(Xs max
(ks (Xs
max
− Xs
− Xs
− Xj )(Xj − Xs
− Xs
min ))/(Xs max
min )](dks /dt)(S2 min ))/(Xs max
min ) −
n j=f
((dXs
0.501 0.3387 9803 30001.1 1.5 0
min /dt)(Xs max
− Xj ))/
− j)
− Xs
min )
t
(S2∗ − S2 ) dt = 0
(14)
0
Combine Eqs. (13) and (14) with thermal transferring relations:
Qj = UAb
1 a − ln{pr /[Xj + (1 − Xj )/˛]}
1 − a − ln{ps /[Xj+f −1 + (1 − Xj+f −1 )/˛]} j = 1, 2, . . . f − 1
,
Ln = Fq
(17) f −1
k
=
⎢ ⎣
0.155s + 6.13e−4 0.013 6s + 2.52e−6 2 2 −4 −5 s + 2.27e s + 5.68e s + 4.25e−4 s + 1.39e−3 0.013 5s − 4.43e−6 0.155s + 2.15e−3 2 −4 −5 2 s + 2.27e s + 5.68e s + 4.25e−4 s + 1.39e−3
×
(Pr − Ps )
⎤ ⎥ ⎦
q
4.1. Servo control (18)
k=1
Vf = V1 + Lf −1
⎡
(15)
(16)
Q
Y (1) X(n)
V1 = F(1 − q)
Lf −1 =
nonlinearity, logarithmic scaling has been carried out. Second, to take the inverse response into account, second- order model is adopted as follows:
(19)
The manipulating variables (the thermal condition q and the pressure of the rectifying section Pr ) could be solved out from the Eqs. (13)–(19). 4. Case study
Fig. 1 shows the control results of the three control strategies, when the set-point of the component concentration of the top product (Y1 ) increases from 0.998995 to 0.999100, and the setpoint of the component concentration of the bottom product (Xn ) decreases from 0.00299680 to 0.00290000. It can be seen obviously that WGMC shows a quicker tracking speed than the other two control strategies with less oscillation and less offset. In the rectifying section, WGMC takes less than 0.5 h to stabilize Y1 to the new setpoint whereas the other two take about 1 h. In the stripping section,
A 40-stage ideal ITCDIC with product purities of 99.9 mol% (1000 ppm, high purity) is studied as an illustrative example, where binary mixture, benzene–toluene is separated. Detailed operating conditions are shown in Table 2. Three control strategies are established to implement the high-purity control of the component concentration of the two ends. The GMC controller based on the proposed nonlinear wave model (WGMC) presents the greatest performances. Compared with the GMC controller based on the traditional data-driven nonlinear model (TGMC) and the internal model control (IMC) of ITCDIC, WGMC not only works excellent in the servo control, but also presents the superior capability of disturbance suppression. Results not only prove the advantages of the proposed WGMC control strategy but also further confirm the efficiency of the proposed wave model. The model used in the TGMC is an ARX model as follows: X(n)(K + 1) = 1.0095X(n)(K) + 0.002X(n)(K − 1) + 1.322e−4 q(K) −2.964e−4 Pr (K) Y (1)(K + 1) = 1.0104Y (1)(K) − 0.0042Y (1)(K − 1) + 2.16e−5 q(K) −4.455e−5 Pr (K) The values of the tuning constants K11 , K12 , K21 , K22 in TGMC are: K11 = 20, K12 = 25, K21 = 20, K22 = 25. The IMC control scheme adopted here has been added some modifications for better performance. First, to overcome the strong
Fig. 1. Servo control in ITCDIC.
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Table 3 Integral absolute errors (IAE) [10−6 ]. Control strategy
Servo F + 10% Zf + 10% Ps + 10%
WGMC
TGMC
IMC
Top
Bottom
Top
Bottom
Top
Bottom
11.983 0.928726 2.2337 1.06997
11.562 0.592488 2.73501 0.446747
27.293 35.5766 8.82238 59.7337
20.684 4.3396 6.22053 7.8966
29.927 57.684 8.9946 98.262
21.236 27.408 35.9465 52.061
Table 4 Integral square errors (ISE) [10−11 ]. Control strategy
Servo F + 10% Zf + 10% Ps + 10%
WGMC
TGMC
IMC
Top
Bottom
Top
Bottom
Top
Bottom
64.912 0.376300 1.7662 0.48509
52.788 0.106778 1.92805 0.10746
179.29 281.7797 11.7189 750
100.07 3.84343 7.67238 11.612
208.71 699.619 12.71 1937.8
99.712 151.778 278.326 527.284
WGMC also saves about 1 h. TGMC shows little advantage in this servo control simulation. When the set-point of Y1 increases to an even higher set-point 0.999200, TGMC fails to work as it used to be for the serious mismatch of its model, however WGMC could still work with fast stabilizing speed and little offset for the high accuracy of wave model. Tables 3 and 4 show the integrated absolute error and integrated square error of the control processes. TGMC has a slight improvement compared to IMC. However WGMC has large improvement seen from the data of IAE and ISE. The value of IAE and ISE of TGMC and IMC are more than two times of that of WGMC, revealing the efficiency of the proposed wave model and WGMC control strategy.
4.2. Regulatory control The main disturbances of ITCDIC, i.e. the feed flow rate F, the feed mole fraction Zf and the pressure of the stripping section Ps are investigated. In the high-purity control of ITCDIC, the suppression of Zf is always a challenge to the controller design for the reason that the traditional simplified models of ITCDIC are difficult to characterize the dynamical processes caused by the disturbance of Zf . In fact, Zf is a most common disturbance in the separation processes. As shown in Figs. 2–4, nonlinear wave model is able to depict dynamics of the column accurately, hence WGMC presents superior performance in the suppression of the above disturbances. The original set-points of Y1 and Xn are set on 0.998995 and 0.00299680 respectively. Fig. 2 presents the comparison of the three regulatory controllers under 10% step disturbance of F. It is obvious that WGMC has the strongest suppression capability of the feed flow rate disturbance. Compared with the drastic overshoot of TGMC and IMC, the overshoot of WGMC is tiny enough to be ignored. IMC is difficult to stabilize the system, especially in the stripping section. TGMC improves the control of stripping section efficiently though worse than WGMC, and it makes little improvement on the control of rectifying section. Tables 3 and 4 list the related IAE and ISE error statistics. It shows that IAE of WGMC in the rectifying section is only 2.6% of that of TGMC, and in the stripping section the rate is about 13.7%; ISE of WGMC is only 1.34‰ of that of TGMC in the rectifying section, about 2.8% in the stripping section. WGMC shows huge advantage in the comparison. And TGMC also shows its obvious improvement on the control of the stripping section. Fig. 3 presents the comparison of the three regulatory controllers under 10% step disturbance of the feed component
concentration Zf . It can be seen obviously that TGMC and IMC could no longer suppress the disturbance in the rectifying section effectively, though TGMC presents acceptable control in the stripping section, and IMC could stabilize the system after a long time with terrible overshoot. WGMC works excellently both in the rectifying section and stripping section. The system returns to the original steady state after about 0.7 h in the rectifying section and about 1 h in the stripping section. The advantage in the rectifying section of WGMC is obvious, and TGMC makes little improvement. From Tables 3 and 4, in the stripping section, IAE of WGMC is about a quarter of that of TGMC, ISE of WGMC is about 15.1% of that of TGMC. The advantage of WGMC over TGMC is still obvious. When the system is disturbed by the 10% step change of the pressure of the stripping section Ps , the comparison results are much like the analysis of the disturbance from F. The tiny overshoot and quick stabilization of WGMC is obvious in Fig. 4. Though TGMC and IMC could suppress the disturbance of Ps , they take much longer time to stabilize the system with large overshoot. In the rectifying section, IAE of WGMC is 1.8% of that of TGMC; ISE of WGMC is only 0.65‰ of that of TGMC. In the stripping section, IAE of WGMC is 5.7% of that of TGMC; ISE of WGMC is 9.25‰ of that of TGMC.
Fig. 2. 10% step disturbance of F.
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the mismatch between the formulated model and the real plant. In the test of illustrating ITCDIC of benzene–toluene system, the overshoot, offset, and stabilizing time of WGMC is much smaller than TGMC and IMC. WGMC performs special advantage in the regulatory control, where the influence on the system from the disturbances like F, Zf and Ps is tiny. In contrast, the system controlled by TGMC and IMC is disturbed with large overshoot and long stabilizing time. Comparison of the evaluating parameters like IAE and ISE of the three control strategies firmly proves the accuracy and validity of the proposed wave model and further confirm the excellent performances of the proposed WGMC control strategy. Acknowledgments This work is supported by National Natural Science Foundation of China (Grant 50876093), International Cooperation and Exchange Project of Science and Technology Department of Zhejiang Province (Grant 2009C34008), National High Technology Research and Development Program (863, Grant 2006AA05Z226) and Zhejiang Provincial Natural Science Foundation for Distinguished Young Scientists (Grant R4100133), and their supports are thereby acknowledged. Fig. 3. 10% step disturbance of Zf .
References
Fig. 4. 10% step disturbance of Ps .
The above research results show that WGMC is a great control strategy compared with TGMC and IMC. And detailed analysis shows the efficiency of the proposed nonlinear wave model and the advantage of WGMC control strategy obviously. 5. Conclusion In this work, nonlinear wave theory is extended to the GMC controller design for the high-purity control of ITCDIC, where a novel nonlinear wave model with variable molar flow rates is proposed firstly, and the further development of the GMC controller is carried out based on the proposed wave model. Since the high accuracy of wave model and the strong robust performance of GMC, WGMC shows great advantage in servo control and regulatory control. Compared with TGMC and IMC, WGMC effectively eliminate
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Contents lists available at ScienceDirect
Journal of Process Control journal homepage: www.elsevier.com/locate/jprocont
High-purity control of internal thermally coupled distillation columns based on nonlinear wave model Xinggao Liu a,∗ , Yexiang Zhou a , Lin Cong a , Feng Ding b,∗ a b
State Key Laboratory of Industrial Control Technology, Control Department, Zhejiang University, Zheda Road 38, Hangzhou 310027, China Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), School of IoT Engineering, Jiangnan University, Wuxi 214122, China
a r t i c l e
i n f o
Article history: Received 27 August 2010 Received in revised form 7 March 2011 Accepted 8 March 2011 Available online 8 April 2011 Keywords: Internal thermally coupled distillation column High purity control Nonlinear wave model Generic model control
a b s t r a c t Internal thermally coupled distillation column (ITCDIC) is a frontier of energy saving distillation researches, which is a great improvement on conventional distillation column (CDIC). However its high degree thermal coupling makes the control design a bottleneck problem, where data-driven model leads to obvious mismatch with the real plant in the high-purity control processes, and a first-principle model which is comprised of complex mass balance relations and thermally coupled relations could not be directly used as control model for the bad online computing efficiency. In the present work, wave theory is extended to the control design of ITCDIC with variable molar flow rates, where a general nonlinear wave model of ITCDIC processes based on the profile trial function of the component concentration distribution is proposed firstly; combined with the thermally coupled relations, a novel wave model based generic model controller (WGMC) of ITCDIC processes is developed. The benzene–toluene system for ITCDIC is studied as illustration, where WGMC is compared with another generic model controller based on a data-driven model (TGMC) and an internal model controller (IMC). In the servo control and regulatory control, WGMC exhibits the greatest performances. Detailed research results confirm the efficiency of the proposed wave model and the advantage of the proposed WGMC control strategy. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Distillation column is a key part in the separation processes. Since it consumes large amount of energy, any improvement of its energy efficiency could produce a significant impact on the economic profit in the industry. ITCDIC is an excellent improvement on conventional distillation columns (CDIC), where reboiler and condenser are no longer needed, and energy recovery is realized by transferring heat from the rectifying section to the stripping section [1–10]. Though the large energy saving potential of ITCDIC has been confirmed by many researchers, high degree of thermal coupling between the rectifying section and the stripping section brings complex influences on the dynamic behaviors in ITCDIC [11–14]. The strong nonlinearity and high sensitivity to operation conditions pose a huge challenge on its control design [15–18]. The tight control of ITCDIC not only desires an elegant control strategy but also requires an efficient control model. Traditional approximating models have great difficulty in describing the abundant complex nonlinear dynamics of ITCDIC. Linear models are often valid only in a small neighborhood of the
∗ Corresponding authors. Tel.: +86 571 87951071. E-mail addresses: [email protected] (X. Liu), [email protected] (F. Ding). 0959-1524/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jprocont.2011.03.002
nominal operating point [19] and cannot describe the distinct nonlinear behavior of ITCDIC. It hardly works in the high-purity control processes [20–22]. Data-driven models are popular alternatives to linear models, for they are easily obtained with simple constructions. Based on data-driven models, many control designs of ITCDIC have been carried out. Nakaiwa [23] proposed an internal model control (IMC) strategy, compared with multi-loop proportional, integral, and derivative control (M-PID). IMC attains better control performance when set-point transfers. However it is extremely sensitive to changes in the operating conditions, and its regulatory performance drastically deteriorates. M-PID can overcome external disturbances fairly well, but it does not deal effectively with the interactive nature of ITCDIC. Though modified IMC, adaptive predictive M-PID [24,25] make some improvement on the above problems, the performance of the controller still degrade greatly when used for high purity control. The main reason is that severe model mismatch between the formulated model and the real plant occurred when operation conditions change. Firstprinciple model [26] is proved to be a most effective one to capture the rich nonlinear dynamics. It is usually used to analyze the distinct behavior of a complex process. Due to the high degree of thermally coupled relations, the first-principle model of ITCDIC is expressed as a complex structure, which makes it hardly realized as a control model for its bad online computing efficiency.
X. Liu et al. / Journal of Process Control 21 (2011) 920–926
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Table 1 Nomenclature. a, b F H kr , ks L n Pvp Pr Ps P q Q t T UA V X Xˆ Xr min , Xr max Xs min , Xs max Y Zf ˛ S1 S2 f j
[–] [kmol/h] [kmol] [–] [kmol/h] [–] [MPa] [MPa] [MPa] [MPa] [–] [kW] [h] [K] [W/K] [kmol/h] [–] [–] [–] [–] [–] [–] [–] [J/mol] [–] [–] [–] [–]
Coefficient of the Antoine equation Feed Stage holdups in each stage Characterize the tangent of the inflection points S1 and S2 Liquid flow rate Stage number Vapor saturated pressure Pressure of rectifying section Pressure of stripping section Representation of either Pr or Ps Feed thermal condition Energy required Time Absolute temperature Heat transfer rate in each stage Vapor flow rate Mole fraction of liquid The estimation of liquid mole fraction The asymptotic limits of the rectifying section The asymptotic limits in the stripping section Mole fraction of vapor Mole fraction of feed Relative volatility Latent heat Position of wave profile in the rectifying section Position of wave profiles in the stripping section Feed stage Stage number (counted from the top to the bottom)
In the research of CDICs, a novel nonlinear wave theory is employed to describe the distinct behavior in the separation processes. Systems with distributed parameters like distillation columns often exhibit dynamic phenomena which resembles traveling wave [27,28]. Luyben [29,30] pioneered a study of the propagation of temperature and composition profiles in high-purity distillation columns. Later Marquardt [31,32] derived expressions for the wave propagation velocity and the trial function of the wave profile in distillation columns from differential material balances. Hwang [33–35] developed a distributed wave theory for general counter-flow separation processes. The model was used to capture the features of the propagation, reflection, superposition and self-sharpening behaviors of component concentration waves in distillation columns. The latest research by Hankins [36,37] extends the wave theory to the distillation columns with variable molar flow rates. And detailed analysis of partial coherence of the wave propagation in the columns is also carried out. Based on the nonlinear wave model many advanced controllers are carried out for the control of CDIC. Han and Park [38,39] proposed a novel generic model control of CDIC, where a full order observer is designed to evaluate the wave position online. Shin [40] further improved the above generic model control by using the wave profile of tray temperatures. Group led by Doyle III [41,42] developed an IMC controller based on a wave model and a Kalman filter was utilized to update the parameter values. Group led by Henson [43–45] concentrated on the low order wave modeling of cryogenic distillation columns and high purity distillation columns. They also proposed a model predictive control strategy based on the nonlinear wave model of nitrogen purification columns. Wave theory is also extended to multi-component distillation columns and batch reactive distillation columns [46,47]. However due to the variable molar flow rate and distinct wave traveling characteristics, wave modeling of ITCDIC is much more difficult than CDICs’, and as far as we know, few publication is concentrated on wave modeling of ITCDIC or the controller design based on wave model. In the present work, the wave velocity formula of ITCDIC with variable molar flow rates is firstly derived based on the trial function of wave profile. Combined with the mass balance relations and thermally coupled relations, a novel generic model control
strategy is developed based on the proposed nonlinear wave model of ITCDIC (WGMC). In the illustration of benzene–toluene system, WGMC is compared with another generic model controller based on a data-driven model (TGMC) [48–52] and an internal model controller (IMC) [25]. WGMC shows excellent performances in the servo control and regulatory control. The overshoot and stabilizing time are significantly decreased by WGMC. Analysis of integral absolute error (IAE) and integral square error (ISE) further confirms the significant advantage of WGMC. Detailed results of comparison between WGMC and TGMC obviously show the efficiency of the proposed wave model, and the results of comparison between TGMC and IMC prove the advantage of generic model control strategy. The nomenclature for the model and controller is given in Table 1. 2. Nonlinear wave model of ITCDIC Wave profile is a description of a monotonic variation of an independent variable like temperature distribution and component concentration distribution [18]. In ITCDIC, since the monotonicity of the temperature distribution in the rectifying section and stripping section opposites to each other, which is much different from those of CDIC, the component concentration distribution is chosen as a representation of the wave profile, which exhibits two characteristics: (1) asymptotic properties at the top and bottom of the columns; (2) existence of a inflection point in the profile of each section. A novel trial function is introduced to describe the wave profiles in ITCDIC as follows: Xˆ j = Xr
min
+
Xr max − Xr min , 1 + exp(−kr (j − S1 ))
j = 1, 2, . . . , f − 1
(1)
Xˆ j = Xs
min
+
Xs max − Xs min , 1 + exp(−ks (j − S2 ))
j = f, f + 1, . . . , n
(2)
where Xˆ j denotes the estimation of liquid mole fraction; Xr min and Xr max denote the asymptotic limits of the rectifying section when the profiles extend to a infinite distance; Xs min and Xs max denote the asymptotic limits in the stripping section. Xr max and Xs min approximate to the liquid mole fraction at the top stage and
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X. Liu et al. / Journal of Process Control 21 (2011) 920–926
the bottom stage respectively. kr , ks characterize the tangent of the inflection points S1 and S2 (kr , ks do not equal the tangents). The inflection point S1 , S2 are also the representation of the profile position in the rectifying section and stripping section respectively. Given the material balance relations: H
dXj dt
If Laplace transform of the equation above is taken, the resulting transfer function becomes: 2s + 1 y = 2 2 y∗ s + 2s + 1
j = 2, . . . , n − 1 and j = / f
(3)
Based on the trial function of Eqs. (1) and (2), a new wave velocity combined with thermal coupled relations of ITCDIC could be derived. The relationship between wave velocity and the mass balance is established by the derivatives of Eqs. (1) and (2) as follows: dXˆ j dt dXˆ j dt
=
((dkr /dt)(S1 − j) + kr (dS1 /dt))(Xr
=
max
− Xˆ j )(Xˆ j − Xr Xr
(dks /dt)(S2 − j) + ks (dS2 /dt) (Xs
max
ˆ min ) − (dXr max /dt)(Xj min − Xr
− Xˆ j )(Xˆ j − Xs Xs
where = 1/ k2 and = k1 /2 k2 . The above system gives the similar plots to the classic second-order response showing the normalized response of the system y/y* with normalized time t/ and as a parameter. Three types of trajectories can occur for this second order equation depending on the values of k1 and k2 : under damped ( < 1), critically damped ( = 1), and over damped ( > 1). Choose the right and to give the desired shape of response and “appropriate” timing of response. Then can be calculated k1 and k2 using the above equations: = 1/ k2 , = k1 /2 k2 .
= Vj+1 Yj+1 − Vj Yj + Lj−1 Xj−1 − Lj Xj ,
− Xr
min ) − (dXr min /dt)(Xr max
− Xˆ j )
(4)
max
ˆ min ) − (dXs max /dt)(Xj min − Xs
− Xs
min ) − (dXs min /dt)(Xs max
− Xˆ j )
(5)
max
Substitute dXˆ j /dt for the dXj /dt in Eq. (3), the wave velocity is derived for the rectifying section and the stripping section as follows: (1/H)(−V1 Y1 + Vf Yf − Lf −1 Xf −1 ) − (Xr
dS1 = dt
max
− Xj ))/(Xr
max
− Xr
f −1 j=1
min ) +
f −1
−
j=1
f −1 j=1
(kr (Xr
(1/H)(−Vf Yf + Lf −1 Xf −1 + FZf − Ln Xn ) −
n
− dS2 = dt
j=f
((dXs
min /dt)(Xs max
((dXr
− Xj ))/(Xs −
[((Xr
max
j=f
max
j=f
max
((dXs
− Xs
(ks (Xs
n
j=f
3. Generic model controller design for ITCDIC Consider a process described by the following state equations: x˙ = f (x, u, d, t)
(8)
ym = g(x)
(9)
where state vector x(t) ∈ Rn , manipulated input u(t) ∈ Rm , process output y(t) ∈ Rp , disturbance input d ∈ Rl ; in general, f and g are nonlinear functions. The generic model control is an optimal control approach by forcing the derivative of the process output equal to a reference rate. The reference rate is a proportional integral (PI) type function as follows:
(y∗ − y) dt
(10)
(1/H)(−V1 Y1 + Vf Yf − Lf −1 Xf −1 ) − (Xr
max
− Xr
min ) +
f −1 j=1
[((Xr
max
f −1 j=1
−K11 (S1∗ − S1 ) − K12
j=1
f −1 j=1
((dXr
min /dt)
min )](dkr /dt)(S1
− j)
(6)
min )
− Xs
− Xj )(Xj − Xs
min )
min ))/(Xs max
− Xs
− Xs
min )](dks /dt)(S2
− j)
min )
(7)
Two representative concentration positions (S1 , S2 ) in the rectifying section and the stripping section are chosen to be two state variables for the GMC formulation. When S1 , S2 deviate from the set points S1∗ , S2∗ , the changing rate of S1 , S2 should be controlled to return towards the desired steady states and the process should have zero offset. GMC formulation has following forms: dS1 = K11 (S1∗ − S1 ) + K12 dt dS2 = K21 (S2∗ − S2 ) + K22 dt
t
(S1∗ − S1 ) dt
(11)
(S2∗ − S2 ) dt
(12)
0
t
0
where subscript 1 and 2 denote rectifying section and striping section respectively; K11 , K12 , K21 , K22 are tuning constants. In WGMC, K11 = 15, K12 = 100, K21 = 7, K22 = 100. GMC formulation presents a perfect control process that the wave model should be controlled to follow. Connect Eq. (11) with Eq. (6), and connect Eq. (12) with Eq. (7), two equations for the rectifying section and stripping section are obtained as follows: − Xr
min ))/(Xr max max
− Xr
min ))/(Xs max
max
min ) −
min ))/(Xr max
− Xr
− Xj )(Xj − Xr
− Xr
min ) −
min )](dkr /dt)(S1
min ))/(Xr max
− Xr
f −1 j=1
((dXr
min /dt)(Xr max
− Xj ))/
− j) min )
t
(S1∗ − S1 ) dt = 0 0
(kr (Xr
− Xr
min ))/(Xs max
max /dt)(Xj
− Xj )(Xj − Xr
−
((dXr
f −1
[((Xs
− Xr
min ))/(Xr max
− Xs
− Xj )(Xj − Xs
where dS1 /dt and dS2 /dt denote the travel velocity of the entire profile in rectifying section and stripping section.
y˙ = k1 (y∗ − y) + k2
min ))/(Xr max
min ))/(Xr max
max /dt)(Xj
min ) + max
− Xr
− Xj )(Xj − Xr
− Xj )(Xj − Xr
n
n
max /dt)(Xj
(13)
X. Liu et al. / Journal of Process Control 21 (2011) 920–926
923
Table 2 Operating conditions of the ITCDIC studied. Stage number Feed stage Feed flow rate [kmol/h] Feed composition (benzene) (toluene) Liquid holdup in reboiler [kmol]
40 21 100 0.5 0.5 0
(1/H)(−Vf Yf + Lf −1 Xf −1 + FZf − Ln Xn ) − (Xs
max
− Xs
min ) +
n
j=f
[((Xs
max
n
−K21 (S2∗ − S2 ) − K22
((dXs
− Xj )(Xj − Xs
n
−
j=f
j=f
Feed thermal condition Pressure of stripping section [MPa] Heat transfer rate in each stage [W/K] Latent heat of vaporization [kJ/kmol] Liquid holdup in each stage [kmol] Liquid holdup in reflux drum [kmol]
max /dt)(Xj
min ))/(Xs max
(ks (Xs
max
− Xs
− Xs
− Xj )(Xj − Xs
− Xs
min ))/(Xs max
min )](dks /dt)(S2 min ))/(Xs max
min ) −
n j=f
((dXs
0.501 0.3387 9803 30001.1 1.5 0
min /dt)(Xs max
− Xj ))/
− j)
− Xs
min )
t
(S2∗ − S2 ) dt = 0
(14)
0
Combine Eqs. (13) and (14) with thermal transferring relations:
Qj = UAb
1 a − ln{pr /[Xj + (1 − Xj )/˛]}
1 − a − ln{ps /[Xj+f −1 + (1 − Xj+f −1 )/˛]} j = 1, 2, . . . f − 1
,
Ln = Fq
(17) f −1
k
=
⎢ ⎣
0.155s + 6.13e−4 0.013 6s + 2.52e−6 2 2 −4 −5 s + 2.27e s + 5.68e s + 4.25e−4 s + 1.39e−3 0.013 5s − 4.43e−6 0.155s + 2.15e−3 2 −4 −5 2 s + 2.27e s + 5.68e s + 4.25e−4 s + 1.39e−3
×
(Pr − Ps )
⎤ ⎥ ⎦
q
4.1. Servo control (18)
k=1
Vf = V1 + Lf −1
⎡
(15)
(16)
Q
Y (1) X(n)
V1 = F(1 − q)
Lf −1 =
nonlinearity, logarithmic scaling has been carried out. Second, to take the inverse response into account, second- order model is adopted as follows:
(19)
The manipulating variables (the thermal condition q and the pressure of the rectifying section Pr ) could be solved out from the Eqs. (13)–(19). 4. Case study
Fig. 1 shows the control results of the three control strategies, when the set-point of the component concentration of the top product (Y1 ) increases from 0.998995 to 0.999100, and the setpoint of the component concentration of the bottom product (Xn ) decreases from 0.00299680 to 0.00290000. It can be seen obviously that WGMC shows a quicker tracking speed than the other two control strategies with less oscillation and less offset. In the rectifying section, WGMC takes less than 0.5 h to stabilize Y1 to the new setpoint whereas the other two take about 1 h. In the stripping section,
A 40-stage ideal ITCDIC with product purities of 99.9 mol% (1000 ppm, high purity) is studied as an illustrative example, where binary mixture, benzene–toluene is separated. Detailed operating conditions are shown in Table 2. Three control strategies are established to implement the high-purity control of the component concentration of the two ends. The GMC controller based on the proposed nonlinear wave model (WGMC) presents the greatest performances. Compared with the GMC controller based on the traditional data-driven nonlinear model (TGMC) and the internal model control (IMC) of ITCDIC, WGMC not only works excellent in the servo control, but also presents the superior capability of disturbance suppression. Results not only prove the advantages of the proposed WGMC control strategy but also further confirm the efficiency of the proposed wave model. The model used in the TGMC is an ARX model as follows: X(n)(K + 1) = 1.0095X(n)(K) + 0.002X(n)(K − 1) + 1.322e−4 q(K) −2.964e−4 Pr (K) Y (1)(K + 1) = 1.0104Y (1)(K) − 0.0042Y (1)(K − 1) + 2.16e−5 q(K) −4.455e−5 Pr (K) The values of the tuning constants K11 , K12 , K21 , K22 in TGMC are: K11 = 20, K12 = 25, K21 = 20, K22 = 25. The IMC control scheme adopted here has been added some modifications for better performance. First, to overcome the strong
Fig. 1. Servo control in ITCDIC.
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Table 3 Integral absolute errors (IAE) [10−6 ]. Control strategy
Servo F + 10% Zf + 10% Ps + 10%
WGMC
TGMC
IMC
Top
Bottom
Top
Bottom
Top
Bottom
11.983 0.928726 2.2337 1.06997
11.562 0.592488 2.73501 0.446747
27.293 35.5766 8.82238 59.7337
20.684 4.3396 6.22053 7.8966
29.927 57.684 8.9946 98.262
21.236 27.408 35.9465 52.061
Table 4 Integral square errors (ISE) [10−11 ]. Control strategy
Servo F + 10% Zf + 10% Ps + 10%
WGMC
TGMC
IMC
Top
Bottom
Top
Bottom
Top
Bottom
64.912 0.376300 1.7662 0.48509
52.788 0.106778 1.92805 0.10746
179.29 281.7797 11.7189 750
100.07 3.84343 7.67238 11.612
208.71 699.619 12.71 1937.8
99.712 151.778 278.326 527.284
WGMC also saves about 1 h. TGMC shows little advantage in this servo control simulation. When the set-point of Y1 increases to an even higher set-point 0.999200, TGMC fails to work as it used to be for the serious mismatch of its model, however WGMC could still work with fast stabilizing speed and little offset for the high accuracy of wave model. Tables 3 and 4 show the integrated absolute error and integrated square error of the control processes. TGMC has a slight improvement compared to IMC. However WGMC has large improvement seen from the data of IAE and ISE. The value of IAE and ISE of TGMC and IMC are more than two times of that of WGMC, revealing the efficiency of the proposed wave model and WGMC control strategy.
4.2. Regulatory control The main disturbances of ITCDIC, i.e. the feed flow rate F, the feed mole fraction Zf and the pressure of the stripping section Ps are investigated. In the high-purity control of ITCDIC, the suppression of Zf is always a challenge to the controller design for the reason that the traditional simplified models of ITCDIC are difficult to characterize the dynamical processes caused by the disturbance of Zf . In fact, Zf is a most common disturbance in the separation processes. As shown in Figs. 2–4, nonlinear wave model is able to depict dynamics of the column accurately, hence WGMC presents superior performance in the suppression of the above disturbances. The original set-points of Y1 and Xn are set on 0.998995 and 0.00299680 respectively. Fig. 2 presents the comparison of the three regulatory controllers under 10% step disturbance of F. It is obvious that WGMC has the strongest suppression capability of the feed flow rate disturbance. Compared with the drastic overshoot of TGMC and IMC, the overshoot of WGMC is tiny enough to be ignored. IMC is difficult to stabilize the system, especially in the stripping section. TGMC improves the control of stripping section efficiently though worse than WGMC, and it makes little improvement on the control of rectifying section. Tables 3 and 4 list the related IAE and ISE error statistics. It shows that IAE of WGMC in the rectifying section is only 2.6% of that of TGMC, and in the stripping section the rate is about 13.7%; ISE of WGMC is only 1.34‰ of that of TGMC in the rectifying section, about 2.8% in the stripping section. WGMC shows huge advantage in the comparison. And TGMC also shows its obvious improvement on the control of the stripping section. Fig. 3 presents the comparison of the three regulatory controllers under 10% step disturbance of the feed component
concentration Zf . It can be seen obviously that TGMC and IMC could no longer suppress the disturbance in the rectifying section effectively, though TGMC presents acceptable control in the stripping section, and IMC could stabilize the system after a long time with terrible overshoot. WGMC works excellently both in the rectifying section and stripping section. The system returns to the original steady state after about 0.7 h in the rectifying section and about 1 h in the stripping section. The advantage in the rectifying section of WGMC is obvious, and TGMC makes little improvement. From Tables 3 and 4, in the stripping section, IAE of WGMC is about a quarter of that of TGMC, ISE of WGMC is about 15.1% of that of TGMC. The advantage of WGMC over TGMC is still obvious. When the system is disturbed by the 10% step change of the pressure of the stripping section Ps , the comparison results are much like the analysis of the disturbance from F. The tiny overshoot and quick stabilization of WGMC is obvious in Fig. 4. Though TGMC and IMC could suppress the disturbance of Ps , they take much longer time to stabilize the system with large overshoot. In the rectifying section, IAE of WGMC is 1.8% of that of TGMC; ISE of WGMC is only 0.65‰ of that of TGMC. In the stripping section, IAE of WGMC is 5.7% of that of TGMC; ISE of WGMC is 9.25‰ of that of TGMC.
Fig. 2. 10% step disturbance of F.
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the mismatch between the formulated model and the real plant. In the test of illustrating ITCDIC of benzene–toluene system, the overshoot, offset, and stabilizing time of WGMC is much smaller than TGMC and IMC. WGMC performs special advantage in the regulatory control, where the influence on the system from the disturbances like F, Zf and Ps is tiny. In contrast, the system controlled by TGMC and IMC is disturbed with large overshoot and long stabilizing time. Comparison of the evaluating parameters like IAE and ISE of the three control strategies firmly proves the accuracy and validity of the proposed wave model and further confirm the excellent performances of the proposed WGMC control strategy. Acknowledgments This work is supported by National Natural Science Foundation of China (Grant 50876093), International Cooperation and Exchange Project of Science and Technology Department of Zhejiang Province (Grant 2009C34008), National High Technology Research and Development Program (863, Grant 2006AA05Z226) and Zhejiang Provincial Natural Science Foundation for Distinguished Young Scientists (Grant R4100133), and their supports are thereby acknowledged. Fig. 3. 10% step disturbance of Zf .
References
Fig. 4. 10% step disturbance of Ps .
The above research results show that WGMC is a great control strategy compared with TGMC and IMC. And detailed analysis shows the efficiency of the proposed nonlinear wave model and the advantage of WGMC control strategy obviously. 5. Conclusion In this work, nonlinear wave theory is extended to the GMC controller design for the high-purity control of ITCDIC, where a novel nonlinear wave model with variable molar flow rates is proposed firstly, and the further development of the GMC controller is carried out based on the proposed wave model. Since the high accuracy of wave model and the strong robust performance of GMC, WGMC shows great advantage in servo control and regulatory control. Compared with TGMC and IMC, WGMC effectively eliminate
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