Dual Composition Control in a Middle-Vessel Batch Distillation Column

Ind. Eng. Chem. Res. 2001, 40, 4377-4390

4377

PROCESS DESIGN AND CONTROL Dual Composition Control in a Middle-Vessel Batch Distillation Column Rosendo Monroy-Loperena*,† and Jose Alvarez-Ramirez‡ Simulacio´ n Molecular, Instituto Mexicano del Petro´ leo, Eje Central La´ zaro Ca´ rdenas 152, 07730 Distrito Federal, Me´ xico, and Departamento de Ingenierı´a de Procesos e Hidra´ ulica, Universidad Auto´ noma Metropolitana-Iztapalapa, Apartado Postal 55-534, Me´ xico D.F., 09340 Me´ xico

The dual composition control problem in a middle-vessel batch distillation column is studied in this work. The control objective is to regulate the overhead purity by manipulating the reflux flow rate and the bottoms product composition by manipulating the boilup flow rate. By using estimators of the input-output modeling error, output-feedback compensators are designed that are shown to be equivalent to a PID controller with an antireset windup structure for the overhead purity regulator and a PI controller with an antireset windup structure for the bottoms composition regulator. Tuning issues are discussed, showing that they have straightforward physical meanings. The ability of the controllers to regulate the overhead product composition and the bottoms product composition in the face of unknown nonlinearities and sampled/delayed measurements is shown via numerical simulations on a dynamical model. 1. Introduction Batch distillation is a flexible process that has become widely used during the past decade. The main reason for its increased use is that production amounts are usually small, with minimum raw material inventories, which often results in an economic incentive.1 Batch distillation operation is designed via optimal control techniques where a prescribed profit function is maximized. This results in operation with a time-varying reflux ratio that is implemented in open-loop mode, which suffers from the drawback of lack of guaranteed robustness against model/plant mismatches. In principle, such a drawback can be reduced via closed-loop implementation of the optimal profiles, although this is still an open issue.2 Middle-vessel continuous and batch column control configurations have been discussed by Barolo et al.,3 Farschman and Diwekar,4 Phimister and Seider,5 and Barolo and Papini.6 All emphasize the importance of the middle vessel in decoupling the composition controllers. Nevertheless, it should be pointed out that there is a basic dynamic difference between continuous and batch processes. Specifically, whereas continuous processes display steady-state operation, batch processes display sustained (i.e., nonvanishing) dynamics. That is, batch process operation evolves along (nontrivial) trajectories, rather than about steady-state operating points. This fact seriously limits the use of traditional control designs and tuning based on local linearizations. This justifies the application of more advanced control techniques to the regulation of the operation of batch processes, including distillation columns. * Corresponding author. E-mail: [email protected]. Tel.: +52-5333-8105. Fax: +52-5333-6239. † Instituto Mexicano del Petro ´ leo. ‡ Universidad Auto ´ noma Metropolitana-Iztapalapa.

During the past decade, the feedback linearization control technique has been successfully used to address some practical control problems, such as the control of nonlinear fermentation processes,7 polymerization process,8,9 and pH neutralization processes.10 Successful applications to nonminimum-phase nonlinear systems have been reported.11-13 The important practical case of output feedback regulation has also been addressed,14,15 where the idea is to use some kind of state estimation within an input-output feedback linearization framework. In the special case of batch distillation, AlvarezRamirez et al.16 proposed a one-point proportionalintegral-derivative (PID) control configuration based on an observer structure, and Monroy-Loperena and Alvarez-Ramirez17 extended this feedback one-point control to a reactive batch distillation column. In the spirit of these papers, our work addresses several aspects of output feedback control of middle-vessel batch distillation (MVBD) processes by extending the onepoint structure used in batch rectifiers or strippers16 to a dual composition feedback control of a MVBD column (two-point structure). As pointed out above, the extension from single control to dual control is not straightforward, as different control configurations result for the top and bottom composition control. Because the profitability of the process is closely related to the product compositions, the control objective is to track prescribed product compositions via manipulations of the reflux rate and vapor boilup rate. An L-V configuration (also called an energy-balance configuration) is assumed, where the reflux rate is used to control the overhead purity, and the vapor boilup rate is used to control the bottoms composition. (In practice, however, the heat supply to the reboiler is manipulated rather than the vapor boilup.) Although the L-V configuration might not be the best one from the point of view of coupling between control loops,14 it is a

10.1021/ie0005405 CCC: $20.00 © 2001 American Chemical Society Published on Web 08/30/2001

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Ind. Eng. Chem. Res., Vol. 40, No. 20, 2001

mixtures in our case studies are similar and their heats of vaporization do not differ very much, an energy balance for the column is not performed. Also, the constant molar overflow assumption is used, but the model takes into account the dynamics of the molar holdups on each tray, and the internal liquid rate on each stage is determined by means of the linearized version of the Francis weir formula. The numerical solution involves the calculation of the liquid-vapor equilibrium outside of the integration. That is, the equilibrium temperature and the composition of the vapor phase are calculated for a given composition of the liquid phase and pressure (bubblepoint calculation). With this approach, only the massbalance equation must be integrated. 3. Ideal Control Design

Figure 1. Schematic diagram of the middle-vessel batch distillation column.

commonly used control structure for dual-composition control,15 because it is simple to implement and easy to understand and, therefore, easily accepted among operators. The main interests are (i) establishing an output feedback strategy with guaranteed tracking properties despite strong uncertainties in the dynamics of the MVBD process and (ii) demonstrating via numerical examples the robustness of the proposed control. The control designs are based on an approximate model of the composition dynamics of the products and make use of reduced-order observers to estimate the modeling error. Input-output linearizing feedbacks are proposed where the estimated modeling errors are included to achieve robust tracking of the product composition references. Applying the results of the relative gain array analysis done by Farschman and Diwekar4 that the interactions between the two loops for a MVBD are mostly negligible, we develop our control analysis considering uncoupled control loops, and we show that the resulting controllers have the structure of a PID controller with an antireset windup scheme for the regulation of the overhead purity and a PI controller with an antireset windup scheme for the regulation of the bottoms composition. The controllers’ performances are tested using a simulation model.

Given a desired output trajectory, we derive a stable inverse-dynamics feedback control law assuming complete knowledge of the process dynamics. Of course, this is not a realistic assumption, as process dynamics (e.g., thermodynamics, internal flow rates, etc.) are highly uncertain. However, we use this approach as a methodological step toward our robust control design in section 4. Distillate Composition Control. We consider first the problem of regulating the distillate concentration of a desired component about any physically realizable trajectory, say ωD(t), by manipulating the reflux rate R. The desired trajectory ωD(t) can arise, e.g., from an optimal control problem with an economic index. Consider the case of a system with the overhead purity xi,D as the regulated variable. Let us take Ei,j(xj) as the vapor-liquid equilibrium relationship (see eq A.14). Let eD ) xi,D - ωD be the regulation or tracking error. We can compute the time derivative of the regulation error to obtain

dωD deD )+ν dt dt

where ν ) V(Ei,1(x1) - xi,D)/HD. Therefore, the relative degree is not 1 because the control input does not affect directly the time derivative deD/dt. We can go farther by computing the second time derivative of the regulation error

d2eD dt2

)-

d2ωD dt2

+ Φ + ΨR

A schematic diagram of a MVBD column is presented in Figure 1. The ordinary differential equations describing the process dynamics for an N-stage MVBD are described in the Appendix. The model is based on the assumption of negligible vapor holdup, theoretical trays, perfect mixing on trays, constant operating pressure, total condensation with no subcooling, and adiabatic operation. Because the components involved in the

(2)

where def

Φ ) (V/HD)[(dEi,1(x1)/dxi,1)Ω - ν] def

2. Process Model

(1)

Ω ) V(yi,2 - yi,1)/H1 - L1xi,1/HD def

Ψ ) [V(HDH1)-1][dEi,1(x1)/dxi,1]

(3) (4) (5)

Note that, now, in eq 2, the control input R directly affects the dynamics of the controlled distillation concentration via the second-order derivative, and therefore, the relative degree is 2. Suppose that the following stable error trajectory description18-20

Ind. Eng. Chem. Res., Vol. 40, No. 20, 2001 4379

d2eD dt2

+ 2ξcDτcD-1

deD + τcD-2eD ) 0 dt

(6)

is the prescribed behavior for the regulation error. In eq 6, τcD and ξcD are the closed-loop time constant and the damping factor, respectively. From eq 2, the control law leading to the stable closed-loop behavior in eq 6 is id

[

R ) -Φ +

d2ωD dt2

+ 2ξcDτcD-1 2ξcDτcD-1

dωD + τcD-2ωD dt

]

dxi,D - τcD-2xi,D /Ψ (7) dt

The control input Rid is well-defined provided that Ψ(t) * 0 for t g 0. The main feature of the feedback function in eq 7 is that it leads to asymptotic convergence of the tracking error to zero within a mean operating time τcD. In this way, tuning of the controller in eq 7 can be easily performed just by choosing the closed-loop time constant τcD and the damping coefficient ξcD. To avoid unrealistic situations due to hard input bounds, a saturated version of eq 7 is proposed

{

Rmin if Rid < Rmin id id if Rmin e Rid e Rmax (8) Rid s ) Sat[R ] ) R Rmax if Rid > Rmax Bottoms Product Composition Control. Now consider the problem of regulating the bottoms product concentration of a desired component about any physically realizable trajectory, say ωB(t), by manipulating the vapor boilup rate V′. Consider the case where the concentration xi,B is the regulated variable. Let eB def xi,B - ωB be the regula) tion error. Computing the time derivative of the regulation error, we obtain

dωB dxi,B deB )+ dt dt dt

(9)

We can rewrite eq 9 using the dynamics for xi,B (see the Appendix)

dωB deB )+ Λ + ΘV′ dt dt

(10)

where def

Λ ) L′NHB-1 (xi,N - xi,B) def

Θ ) - HB-1(yi,B - xi,B)

(11) (12)

[

V′id ) -Λ +

]

dωB + τcB-1ωB - τcB-1xi,B /Θ (14) dt

Note that, under regular operating conditions, Θ(t) * 0 for all t g 0. Hence, the bottoms concentration is controllable via manipulations of the boilup rate V′. Again, the main feature of the feedback function in eq 14 is that it leads to asymptotic convergence of the tracking error to zero within a mean operating time τcB, and tuning of the controller in eq 14 can be easily done just by choosing the closed-loop time constant τcB. A drawback of this feedback function is that arbitrarily large control actions might be required to attain the control objective. In fact, the control input is subjected to physical saturation. In this way, the saturated version of the feedback function in eq 14 is

{

V′min if V′,id < V′min ,id ,id if V′min e V′,id e V′max (15) V′,id s ) Sat[V′ ] ) V′ V′max if V′,id > V′max Some Issues about the Ideal Control Law. Equations 2 and 10 show that the relative order of xi,D with respect to R is 2 and that the relative order of xi,B with respect to V′ is 1. The relative degree 2 of the R-xi,D map means that the control changes of R have to be transmitted to the first stage and then to the condenser, to act on the dynamics of xi,D. On the other hand, the relative degree 1 of the V′-xi,B map means that the control changes of V′ act directly on the dynamics of xi,B. It must be pointed out that, although the this paper focuses on the L-V configuration, the control design procedure can be extended to other control configurations. Numerical Simulations with the Theoretical Control Laws in Equations 8 and 15. We have carried out several numerical simulations for the separation of propylene oxide (1)-acetone (2) using the MVBD column described in Table 1. For this purpose, we used the thermodynamic data from Gmeling et al.21 First, Figure 2 shows the dynamics of the uncontrolled column for the distillate, middle-vessel, and bottom compositions at total reflux for several values of the ratio between the vapor flow rate leaving and entering the middle vessel, which is denoted q′ ) V/V′. For this purpose, the initial charge to the middle vessel was distributed throughout the column in such a way that the plate holdup is 1 mol on each tray, 10 mol for the top drum, and 10 mol for the reboiler, with the remaining charge in the middle vessel. The vapor rate in the rectification section of the column was fixed at 10 mol h-1, Table 1. Model and Systems Characteristics binary

Suppose that the following stable error trajectory description

deB + τcB-1eB ) 0 dt

(13)

is the prescribed behavior for the regulation error. In eq 13, τcB is the closed-loop time constant. From eq 10, the control law leading to the stable closed-loop behavior in eq 13 is

system

ternary

propylene oxide (1) n-pentane (1) n-hexane (2) acetone (2) n-heptane (3) thermodynamic model UNIQUAC ideal nominal tray holdup, mol 1 1 reflux drum holdup, mol 10 10 reboiler holdup, mol 10 10 total feed charge, mol 100 250 nominal feed composition 0.7/0.3 0.2/0.6/0.2 tray hydraulic time constant, h 0.001 0.001 number of ideal rectifying trays 5 10 number of ideal stripping trays 5 10

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the stripping section for small values of q′ and by the rectifying section for large values of q′. This behavior can be taken into account by any control design. Assume that the control objective was to simultaneously regulate the distillate composition and the bottoms composition to given constant purities for several values of q′. The reflux flow rate was computed with the feedback control law in eq 8 and the boilup rate with the feedback control law in eq 15, until the middle-vessel holdup reach 1 mol. The control parameters were chosen as ξcD ) 1, τcD ) 0.05 h, and τcB ) 0.25 h, with 0.1 e Rid/V e 1.0 and 0.1 e V′,id/LN e 1.0. Figure 4 shows the dynamics of the computed internal ratios and product purities. For this case, the starting point was the total reflux steady state. After about 1 h, the product purities were maintained at the reference values until the end of the process. Note that, because we started at the total reflux state, the rectifying control attained its lower limit until the distillate composition reached the desired goal. 4. Robust Control Design

Figure 2. Time evolution of the middle-vessel batch distillation column at total reflux.

The aim of this section is to develop a control structure with guaranteed robust stabilization capabilities. An on-line computation of eqs 8 and 15 would require measurements of vapor and liquid compositions, which are expensive. Moreover, implementation of the control laws in eqs 8 and 15 would require perfect knowledge of thermodynamics (vapor-liquid equilibria), which is not possible in practice. In our design, we assume that only concentration measurements are available for the product purities. These measurements can be obtained either from a gas chromatograph or from an extended Luenberger observer (see the interesting work by Quintero-Marmol et al.22 on this topic). Distillate Composition Control. As in the ideal control design, let us begin with the problem of regulating the distillate concentration by manipulating the reflux rate. The dynamics of the distillate composition can be written as

d2xi,D dt2

) Φ + ΨR

(16)

It is noted that the function Φ involves a set of quite complex and uncertain functions. As a worst-case control design, assume that the function Φ is unknown. On the other hand, the function Ψ involves the gradient of the vapor-liquid equilibrium relationship dE(x1)/dxi,1 > 0. Let Ψ ˜ be an estimate of Ψ, which can be taken as ref Ψ ˜ ) [V ˜ (H ˜ DH ˜ 1)-1][dE(xD)/dxi,D ]

Figure 3. Time evolution of the uncontrolled middle-vessel batch distillation column.

and the vapor rate in the stripping section of the column was calculated from the q′ relation. Note that, for small values of q′, the dynamics of the MVBD is faster than that for large values of q′. In Figure 3, we present the time evolution of the same MVBD column with the external reflux ratio (R/D) set to 2.2 and the external boilup ratio (V′/B) set to 20.0. Here, we can appreciate that the behavior of the MVBD column is great affected by the selection of q′ as the dynamics is dominated by

(17)

following the next considerations. We can obtain estimates of the vapor flow rate V and the holdups HD and H1 from the nominal operating conditions. These variables change slightly during the separation phase. On the other hand, the term dE(x1)/dxi,1 is time-varying, but we can use the approximation dE(x1)/dxi,1 ≈ dE(xD)/ dxi,D. A further approximation can be taken as dE(xD)/ ref . dxi,D ≈ dE(xD)/dxi,D Introducing the modeling error function

η ) Φ + (Ψ - Ψ ˜ )R eq 16 can be written as

(18)


Figure 4. Time evolution of the controlled middle-vessel batch distillation column with the theoretical control laws.

d2xi,D dt2

)η+Ψ ˜R

(19)

dη j /dt ) τeD-2(dxi,D/dt - zj)

Following the ideas described in section 3, the theoretical feedback control leading to the desired closed-loop behavior in eq 6 is

(

RT ) -η +

d2ωD dt

2

+ 2ξcDτcD-1 2ξcDτcD-1

dzj/dt ) η j+Ψ ˜ R + 2ξeDτeD-1(dxi,D/dt - zj)

dωD + τcD-2ωD dt

)

where ξeD and τeD are, respectively, the estimation damping factor and the estimation time constant. To implement eq 21, we define the variables q1 ) zj j - τeD-2xi,D. Then, the estimates 2ξeDτeD-1xi,D and q2 ) η zj and η j are computed from the following equations

dq1/dt ) η j+Ψ ˜ R - 2ξeDτeD-1zj

dxi,D - τcD-2xi,D /Ψ ˜ (20) dt

This feedback control cannot be implemented just as it is, because the modeling error η and the time derivative dxi,D/dt are not available for feedback. An alternative is to use estimates of η and dxi,D/dt. To this end, we introduce the variable z ) dxi,D/dt. Let zj and η j be estimates of z and η, respectively. The following estimator is proposed

(21)

dq2/dt ) - τeD-2zj

(22)

where

zj ) q1 + 2ξeDτeD-1xi,D η j ) q2 + τeD-2xi,D

(23)

The initial conditions for eq 22 can be chosen as follows. Because the signals z(t) and η(t) are unknown,

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take q1(0) ) -2ξeτe-1xi,D(0) and q2(0) ) -τe-2xi,D(0). In this way, the practical feedback control is given by

(

RP ) -η j+

d2ωD dt

2

+ 2ξcDτcD-1

dωD + τcD-2ωD dt

dxi,B ) Λ + ΘV′ dt

As a worst-case control design, assume that the function Λ is unknown. Let Θ ˜ be an estimate of Θ, which can be taken as

)

˜ (24) 2ξcDτcD-1zj - τcD-2xi,D /Ψ such that the saturated version becomes

RPsat

{

P

Rmin if R < Rmin ) Sat[R ] ) RP if Rmin e RP e Rmax Rmax if RP > Rmax P

(25)

In this way, the reflux ratio policy is computed online with the feedback function in eq 25 and the estimators in eqs 22 and 23. This control law has an interesting structure. It can be interpreted as a PIDlike control law with an antireset windup (ARW) scheme.23,24 In fact, after straightforward algebraic manipulations, it can be concluded that the feedback control in eqs 22-25 can be written as

˜ -1 RP ) Ψ

d2ωD dt2

- CPID(s) F(s)er - GARW(s)(RP - RPsat) (26)

The first term is a feed-forward term to account for the second-order nature of the map R-xi,D. This term takes into account the dynamics of the desired trajectory ωD(t). The second term is the regulatory part, which tries to take the regulation error to zero, where CPID(s) is a classical PID controller with control gain, integral, and derivative time constants given by

ξcDτcD + ξeDτeD ˜ -1 KcD ) Ψ ξeDτeDτcD2 + ξcDτcDτeD2 τDD )

τcD2 + 4ξeDξcDτeDτcD + τeD2 2(ξcDτcD + ξeDτeD)

τID ) 2(ξcDτcD + ξeDτeD)

τf )

2(ξeDτcD + ξcDτeD)

(27)

(28)

The last term corrects the integral action when the control input is saturated, where GARW(s) is the ARW P operator acting on the “saturation error” RP - Rsat given by

GARW(s) )

τcD - 2ξcDτeD2s s[τeD2τcDs + 2τeD(ξeDτcD + ξcDτeD)]

Θ ˜ ) -H ˜ B-1Γ

(31)

following the next considerations. We can obtain an estimate of the holdup HB from the nominal operating conditions. On the other hand, the function Θ involves a composition gradient (yi,B - xi,B) that can be assumed to be almost constant over the separation process, and this gradient is denoted by the constant Γ. Furthermore, ref ref - xi,B ). Γ can be approximated as Γ ) (yi,B - xi,B) ≈ (yi,B Introducing the modeling error function

µ ) Λ + (Θ - Θ ˜ )V′

(32)

eq 30 can be written as

dxi,B )µ+Θ ˜ V′ dt

(33)

The theoretical feedback control leading to desired closed-loop behavior in eq 13 is

(

V′T ) -µ +

)

dωB + τcB-1ωB - τcB-1xi,B /Θ ˜ (34) dt

From the input-output representation in eq 33, it is possible to show that the dynamics of the modeling error signal µ(t) can be reconstructed from measurements of the regulated output xi,B and the control input V′. In ˜ (t) V′(t) provides evidence of a fact, µ(t) ) dxi,B/dt - Θ kind of strong observability25 of the modeling error signal µ(t). This property can be exploited to propose ˜ V′ an observer-based estimator. Let θm ) dxi,B/dt - Θ be the equivalent measured signal. Note that θm(t) ≡ µ(t), t g 0. To estimate the signal µ(t), we propose the reduced-order observer

dµ˜ ˜ V′ - µ˜ ) ) τeB-1(θm - µ˜ ) ) τeB-1(dxi,B/dt - Θ dt

F(s) is a first-order low-pass filter [i.e., F(s) ) (τfs + 1)-1] to make causal the PID control24 with a filter time constant given by

τcDτeD

(30)

(29)

Bottoms Product Composition Control. Now consider the bottoms product concentration regulation problem. The dynamics of the bottom product composition can be written as

(35)

where τeB > 0 is the estimation time constant. A realization of the former estimator is giving by introdef

ducing the variable w ) τeµ˜ - eB. Then, we have

dw dωB ) -Θ ˜ V′ - τe-1(w - eB) dt dt

(36)

where the modeling error estimate is given as µ˜ ) τe-1(w + eB). In this way, the practical feedback control is given by

(

V′P ) -µ˜ +

)

dωB + τcB-1ωB - τcB-1xi,B /Θ ˜ (37) dt

such that the saturated version becomes

{

P V′min if V′min < V′min P P P if V′min e V′P e V′max V′sat ) Sat[V′ ] ) V′ V′max if V′P > V′max

(38)


Figure 5. Time evolution of the controlled middle-vessel batch distillation column for two different values of the estimation time constant.

The resulting control law is composed of the feedback function in eq 38 and the first-order estimator in eq 36 and depends only on the known function Θ ˜ and the measured signals eB(t) and V′(t). As in the case of distillate composition control, this control law has an interesting structure. It can be interpreted as a PI-like control law with an antireset windup (ARW) scheme.23,24 In fact, after straightforward algebraic manipulations, it can be concluded that the feedback control in eqs 35-38 can be written as

V′P ) Θ ˜ -1

dωB - CPI(s)er - GARW(s)(V′P - V′satP) (39) dt

The first term represents the feed-forward term to account for the first-order nature of the map V′-xi,B. The second term is the regulatory part that tries to take the regulation error to zero, where CPI(s) is a classical PI controller with control gain and integral time constant given by

KcB ) Θ ˜ -1(τeB-1 + τcB-1)

τIB ) τeBτcB(τeB-1 + τcB-1)

(40)

The third term corrects the integral action when the control input is saturated, where GARW(s) is the ARW P given operator acting on the saturation error V′ - V′sat by

GARW(s) ) τe-1

(41)

The set {τcB, τeB} uniquely defines the control gain and integral time of a PI controller. This {τcB, τeB} parametrization of the PI controller has the enormous advantage that both time constants τcB and τeB have straightforward physical meanings. On the other hand, a dead time in the input channel imposes serious limitations on the achievable closedloop performance.24 Following some internal model control ideas (see Morari and Zafiriou24), and given an upper bound for the dead time, the estimation and closed-loop time constants for the proposed control scheme should be set at values no smaller that the deadtime upper bound. This tuning guideline is used in the numerical simulations presented below.

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Figure 6. Time evolution of the controlled middle-vessel batch distillation column with the robust control laws.

Comments on the Robust Control Design. It is not surprising that the proposed controllers resulted in a PI compensator for the bottom product and a PID compensator for the top product. From internal model control arguments (Morari and Zafiriou24), such control configurations are expected because of the relative degrees of the maps R-xi,D and V′-xi,B. The real advantage of the proposed PI/PID control configuration over standard PI controllers can be summarized as follows: (1) These configurations have easy design and tuning methodologies in the face of timevarying operating conditions. Most standard PI/PID controllers are designed on the basis of a linear model about a constant operating point. (2) The PI/PID controllers are endowed with an antireset windup scheme to cope with control input saturations. (3) (Tuning Guidelines) The above-described PID and PI controllers for the distillate and bottom compositions, respectively, can be tuned easily in view on their estimator-based configurations. In the first step, the closed-loop time constants τcD and τcB should be chosen. We suggest fixing τcD and τcB at values no larger than 0.1 times the expected batch operating time. The underlying idea is

to achieve convergence to the prescribed setpoint as fast as possible. In the second step, the estimation time constants τeD and τeB should be chosen. This can be done by noting that, as τeD and τeB go to zero, the corresponding robust controllers converge to the ideal ones (see section 3). We suggest choosing τeD and τeB as no larger than about 0.5 times τcD and τcB. The idea behind this tuning guideline is that the estimation procedure should be faster than the nominal closed-loop dynamics. The tuning of the PI and PID control gains is particularly easy to carry out in view to the fact that, up to the point where the influence of nonmodeled dynamics and measurement noise is no longer negligible, τeB and τeD influence the control performance inversely. In this way, the minimum allowable values of τeB and τeD are limited by the underlying measurement noise and unmodeled (e.g., actuator) dynamics. In practical situations, τeB and τeD must be chosen no smaller than the dominant period of the measurement noise and actuator dynamics. 5. Numerical Simulations In this section, the functioning of the proposed control scheme is illustrated via numerical simulations.


Figure 7. Time evolution of the controlled middle-vessel batch distillation column with input delay time and multiple feed charges.

Binary System. We have carried out several numerical simulations to illustrate the performance of the proposed robust control laws. To rework the binary example of Table 1, used to show the behavior of the theoretical control law, the closed-loop damping factor and time constant, ξcD, τcD, and τcB, have been chosen as in the above case. The estimation damping factor has been chosen as ξe ) 1. From the nominal design parameters and the vapor-liquid equilibrium relation˜ )ship, we have chosen Ψ ˜ ) 0.0238 mol h-1 and Θ -1 0.0043 mol . Figure 5 shows the time evolution of the internal ratios and product purities for ωD ) 0.95 and ωB ) 0.05 for two different sets of the estimation time constants {τeD, τeB} for the case when q′ ) 0.5. For comparison, the ideal behavior under the theoretical feedback controls is also shown. It is noted that the smaller the estimation time constants, the closer the behavior to the ideal. This implies that, in principle, the ideal behavior under the theoretical feedback controls in eqs 8 and 15 could be achieved as τeD f 0 and τeB f 0. Of course, this is not possible in the presence of delay times and unmodeled (actuator) dynamics.

Figure 6 presents the behavior of the proposed robust control laws for different values of q′ using the same tuning parameters as above, with τeD ) 0.01 h and τeB ) 0.005 h. It is clear that the proposed robust control laws do not depend of the value of q′ to regulate the product purities. In this way, the proposed controller is not fragile in the face of variations of the process parameter q′. Let us simulate a more realistic situation, where the distillate and bottoms product compositions are available from sampled and delayed measurements. Furthermore, consider a multibatch process with negative and positive 5 mol % changes in the charge composition that are applied every time the middle-vessel holdup reaches 10% of the original charge. Specifically, the first charge (t ) 0) has 40% propylene oxide, the second charge has 35% propylene oxide, the third charge has 40% propylene oxide, the fourth charge has 45% propylene oxide, the fifth charge has 40% propylene oxide, and so on. Figure 7 presents the dynamical behavior of the controls inputs and the regulated outputs for τeD ) 0.5 h and τeB ) 1.5 h for a sample and delay of 5 min.

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Figure 8. Time evolution of the controlled middle-vessel batch distillation column with the robust control laws for a ternary mixture.

In this case, the column was charged with the initial composition of the charge and began to work at total reflux during the first 2 h; after that, the process entered the production phase. Notice that the behavior of the controllers is affected by the sampled/delayed measurements, and better behavior is observed when the modeling error estimators have enough information to become stabilized. In general, however, the feedback controllers are able to regulate and reject the exogenous perturbations induced by the change of the charge composition. It is clear that the disturbance rejection capabilities are induced by the modeling error estimation technique used in the development of the control laws. Multicomponent System. To illustrate the performance of the proposed control laws in multicomponent systems, we have selected a ternary mixture composed of n-pentane (1), n-hexane (2), and n-heptane (3). The model and system characteristics are described in Table 1. The objective of the separation is to maintain the composition of the more volatile component at 0.95 mol fraction in the top product and to maintain the intermediate component composition at 0.05 mol fraction in

the bottom product. Figure 8 shows the dynamics of the controlled MVBD. For this purpose, the initial charge to the middle vessel was distributed throughout the column in such a way that the plate holdup was 1 mol on each tray, 10 mol for the top drum, and 10 mol for the reboiler, with the remaining charge in the middle vessel. The vapor rate in the rectification section of the column was fixed at 10 mol h-1 and q′ at 0.5. The control parameters were chosen as ξcD ) 1, τcD ) 0.10 h, τcB ) 0.25 h, τeD ) 0.025 h, and τeB ) 0.001 h, with 0.6 e R/V e 1.0 and 0.6 e V′/LN e 1.0. From the nominal design parameters and the vapor-liquid equilibrium relation˜ ) -0.0070 ship, we chose Ψ ˜ ) 0.0985 mol h-1 and Θ mol-1. Notice that the performance of the controllers is very good, as in the binary case. Both products are kept at the desired setpoints, and the internal reflux profiles are smooth. Now consider the case where the above column has a 0.05-h dead time because of composition measurements. Figure 9 shows the dynamics under this situation, without any parameter being changed. It is noted that the internal reflux profiles display an oscillatory be-


Figure 9. Time evolution of the controlled middle-vessel batch distillation column with input delay time for a ternary mixture.

havior during the first hour of the process, as the dynamic of the process is faster than the estimation with this delay. To avoid this situation, we can fix the estimation time constants to larger values, but this would affect the performance of the process. We can conclude that, under the presence of dead times, a tradeoff between the dynamics of the process due to the process design and the controller performance is present. Measurement noise is always present in practical applications. Consider that the input control signals are subject to noise. We take the ternary system column without changing any control parameter. Because the observer-based PI/PID control configurations are of a high-gain nature, measurement noise can affect adversely the performance of the control input. Figure 10 shows the behavior of the control inputs r and r′ when the measured compositions are contaminated with (3% Gaussian noise. As expected, the behavior of the bottom controller is better than the behavior of the top controller, because the input signal for the top controller has a span larger than that for the bottom controller. Notice that the control inputs display an oscillatory behavior, induced by the amplification of the measurement noise

in the estimation of the modeling error signals η and µ. However, it is shown that the proposed control laws provide good performance in the tracking of the error to preserve the imposed trajectories. It should be stressed that, because the underlying batch process is of a time-varying nature (i.e., nonvanishing signal tracking), a simple PI/PID controller generally does not suffice to face the adverse effects induced by measurement noise. In such a case, a higher-order controller would perform better than PI/PID controllers. The design of high-order controllers to confront measurement noise and compensate for large measurement delays is under study and will be reported elsewhere. Conclusions In this work, we address the dual composition control of a MVBD problem. Because batch processes impose nonstationary control problems, the composition control problem is posed as an inverse dynamics problem. The new idea that we proposed for robust stabilization is based on modeling error compensation techniques and consists of interpreting the modeling error signal as a

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Figure 10. Time evolution of the manipulated variables when the measured compositions are contaminated with (3% Gaussian noise.

new state whose dynamics can be reconstructed from available measurements. In this way, we used observerbased estimators that were used in saturated versions of the input-output feedback functions. The resulting controllers are shown to be equivalent to a standard PID controller with an antireset windup structure for the regulation of the overhead purity and a standard PI controller with an antireset windup structure for the regulation of the bottoms product composition. An advantage of the proposed controllers is their simple structure. In fact, their tuning parameters have straightforward interpretations and uniquely define the controller parameters. This conclusion is important because PID and PI compensators are the most widely used control strategies in the actual chemical industry. For simplicity in presentation, all control computations and numerical simulations were based on a distillate model with constant vapor and liquid flow rates. This is not a serious drawback because, as Cui et al.26 have demonstrated, such a modeling approach is able to provide a close description of the middle-vessel distillation process. Finally, our numerical simulations

have shown that measurement dead times and noise can limit the performance of the proposed PID/PI control configuration. This is because batch distillation displays sustained dynamics with finite operation times. In principle, these control performance problems can be reduced if secondary temperature measurements are used. In this case, a cascade control design must be confronted. The cascade control design for batch distillation deserves in-depth study. Research in this particular problem is under progress and will be presented elsewhere. Nomenclature E ) vapor-liquid equilibrium relationship e ) tracking error H ) holdup L ) liquid flow rate q′ ) ratio between the vapor flow rate leaving and entering the middle vessel R ) reflux rate r ) internal reflux ratio

Ind. Eng. Chem. Res., Vol. 40, No. 20, 2001 4389 r′ ) internal boilup ratio t ) time V ) vapor flow rate x ) liquid mole fraction y ) vapor mole fraction ξ ) damping factor τ ) time constant ω ) trajectory

dHN ) L′N-1 - L′N dt

dHkxi,k ) L′N-1xi,N-1 + V′yi,B - L′Nxi,N - V′yi,N (A.12) dt Reboiler (subscript B). For i ) 1,..., C - 1

dHBxi,B ) L′N(xi,N + xi,B) - V′(yi,B - xi,B) dt

8. Appendix: A Dynamical Model of Middle-Vessel Batch Distillation A basic model of the process dynamics is described here. The internal flow rates of the liquid are calculated by means of the Francis weir formula

Lj ) Lj,0 +

(A.11)

Hj - H0 τL

where Lj,0 is the reference value of the internal liquid flow rate. Hj and H0 are the actual and reference molar holdups on tray j, respectively, and τL is the tray hydraulic time constant. The energy balances are not included in the model; therefore, the vapor rate is constant inside the column. Other assumptions are ideal trays, well-mixed trays, boiling feed, total condensation with no subcooling, and negligible heat losses. C denotes the number of components. Reflux Drum (subscript D). For i ) 1,..., C - 1

dxi,D V ) (y - xi,D) dt HD i,D

(A.2)

Top Tray (subscript 1). For i ) 1,..., C - 1

dH1 ) R - L1 dt

(A.3)

dH1xi,1 ) Rxi,d + Vyi,2 - L1xi,1 - Vyi,1 dt

(A.4)

Rectifying Trays (suscript j). For i ) 1,..., C - 1

dHj ) Lj-1 - Lj dt

(A.5)

dHjxi,j ) Lj-1xi,j-1 + Vyi,j+1 - Ljxi,j - Vyi,j (A.6) dt Middle Vessel (subscript M). For i ) 1,..., C - 1

dHM ) LM-1 + V′ - LM - V dt

(A.7)

dHMxi,M ) LM-1xi,M-1 + V′yi,M+1 - LMxi,M - Vyi,M dt (A.8) Stripping Trays (subscript k). For i ) 1,..., C - 1

dHk ) L′k-1 - L′k dt dHkxi,k ) L′k-1xi,k-1 + V′yi,k+1 - L′kxi,k - V′yi,k dt

The vapor and liquid compositions are related by the equilibrium relationship

yi,j ) Ei,j(xj) (A.1)

(A.9) (A.10)

Bottom Tray (subscript N). For i ) 1,..., C - 1

(A.13)

(A.14)

where xj ) (x1,j, ..., xc,j)T. This expression can be computed using any thermodynamic method. Literature Cited (1) Diwekar, U. M. Batch Distillation: Simulation, Optimal Design and Control; Taylor and Francis International Publishers: Washington, D.C., 1995. (2) Edgar, T. F. Control of unconventional processes. J. Process Control 1996, 6, 99. (3) Barolo, M.; Guarise, G. B.; Ribon, N.; Rienzi, S. A.; Trotta, A.; Macchietto, S. Some issues in the design and operation of batch distillation column with a middle vessel. Comput. Chem. Eng. 1996, 20, S37. (4) Farschman, C. A.; Diwekar, U. Dual composition control in a novel batch distillation column. Ind. Eng. Chem. Res. 1998, 37, 89. (5) Phimister, J. R.; Seider, W. D. Distillate-Bottoms Control of Middle-Vessel Distillation Columns. Ind. Eng. Chem. Res. 2000, 39, 1840. (6) Barolo, M.; Papini, C. A. Improving dual composition control in continuous distillation by a novel column design. AIChE J. 2000, 46, 146. (7) Henson, M. A.; Seborg, D. E. Nonlinear control strategies for continuous fermenters. Chem. Eng. Sci. 1992, 47, 821. (8) Schork, F. J.; Deshpande, P. B.; Leffew, K. W. Control of Polymerization Reactors; Marcel Dekker: New York, 1993. (9) Soroush, M.; Kravaris, C. Nonlinear control of a polymerization CSTR with singular characteristic matrix. AIChE J. 1994, 40, 6. (10) Henson, M. A.; Seborg, D. E. Adaptive nonlinear control of a pH neutralization process. IEEE Trans. Control Syst. Technol. 1994, 2, 169. (11) Kravaris, C.; Daoutidis, P. Nonlinear state feedback control of second-order non-minimum-phase nonlinear systems. Comput. Chem. Eng. 1990, 49, 439. (12) Wright, R. A.; Kravaris, C. Nonminimum-phase compensation for nonlinear processes. AIChE J. 1992, 38, 26. (13) Kravaris, C.; Daoutidis, P.; Wright, M. A. Output feedback control of nonminimum-phase nonlinear processes. Chem. Eng. Sci. 1994, 49, 2107. (14) Shinskey, F. G. Distillation Control; McGraw-Hill: New York, 1984. (15) Ha¨ggblom, K. E.; Waller, K. V. Control Structures, Consistency and Transformations, Practical Distillation Control; Luyben, W. L., Ed.; Van Nostrand Reinhold: New York, 1992. (16) Alvarez-Ramirez, J.; Monroy-Loperena, R.; Cervantes, I.; Morales, A. A novel proportional-integral-derivative control configuration with application to the control of batch distillation. Ind. Eng. Chem. Res. 2000, 39, 432. (17) Monroy-Loperena, R.; Alvarez-Ramirez, J. Output-feedback control of reactive batch distillation columns. Ind. Eng. Chem. Res. 1998, 37, 378. (18) Bartusiak, R. D.; Georgakis, C.; Reilly, M. J. Nonlinear feedforward/feedback control structures designed by reference systems synthesis. Chem. Eng. Sci. 1989, 44, 1837. (19) McLellan, P. J.; Harris, T. J.; Bacon, D. W. Error trajectory descriptions of nonlinear controller designs. Chem. Eng. Sci. 1990, 45, 3017.

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(20) Lee, P. L.; Sullivan, G. R. Generic model control (GMC). Comput. Chem. Eng. 1988, 12, 573. (21) Gmehling, J., Onken, U., Arlt, W., Eds. Vapor-Liquid Equilibrium Data Collection. Aldehydes and Ketones Ethers; DECHEMA Chemistry Data Series; DECHEMA: Frankfurt, Federal Republic of Germany, 1979; Vol. I, Parts 3 and 4. (22) Quintero-Marmol, E.; Luyben, W. L.; Georgakis, C. Application of an extended Luenberger observer to the control of multicomponent batch distillation. Ind. Eng. Chem. Res. 1991, 30, 1870. (23) Kothare, M. V.; Campo, P. J.; Morari, M.; Nett, N. N. A unified framework for the study of anti-windup designs. Automatica 1994, 30, 1869.

(24) Morari, E.; Zafiriou, E. Robust Process Control; Prentice Hall: New York, 1989. (25) Diop, S.; Fliess, M. On nonlinear observability. In Proceedings of the 1st European Control Conference; Herme`s: Hamburg, Germany, 1991; Vol. 1, 152. (26) Cui, X.; Yang, Z.; Shao, H.; Qu, H. Batch distillation in a column with a cold middle vessel for heat-sensitive compounds. Ind. Eng. Chem. Res. 2001, 40, 879.

Received for review June 1, 2000 Revised manuscript received May 4, 2001 Accepted July 2, 2001 IE0005405

Dual Composition Control in a Middle-Vessel Batch Distillation Column

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