Two-Point CompositionTemperature Control of Binary Distillation

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Ind. Eng. Chem. Res. 2006, 45, 9010-9023

Two-Point Composition-Temperature Control of Binary Distillation Columns Eduardo Castellanos-Sahagu´ n, Jose´ Alvarez-Ramı´rez, and Jesu´ s Alvarez* UniVersidad Auto´ noma Metropolitana, Unidad Iztapalapa, Departamento de Ingenierıá de Procesos e Hidra´ ulica, Apdo. Postal 55534, 09340, Me´ xico D. F., Mexico

Although the two-input-two-output (2I-2O) dual composition (DC) and two-point temperature (TPT) distillation control problems have been studied extensively, the development of two-input-four-output (2I4O) composition-temperature (CT) control schemes, driven by two composition and two temperature measurements, lags far behind. The difficulty of the 2I-4O CT control problem is due to the lack of systematic means to screen the large number of combined structural-algorithmic design degrees of freedom. With contradictory results, the problem has been recently addressed with linear and nonlinear approaches. Based on a constructive control framework, in this work, a rather simple solution to the combined 2I-4O CT control problem is obtained. The resulting scheme consists of linear PI components and static interaction compensators, has a systematic construction procedure with reduced model dependency, and has easy-to-apply tuning guidelines. The proposed approach is applied to three representative examples through simulations, yielding behaviors that improve those of previous linear and nonlinear cascade control schemes. 1. Introduction The production of many intermediate and final products in the chemical and petrochemical industries is heavily dependent on energy-intensive distillation columns,1 and their efficient (minimum energy consumption) operation requires the regulation of distillate and bottoms compositions.2 The related dual composition (DC) control problem has been studied extensively with linear3,4 and nonlinear control schemes.5 In high-purity columns with poor input-to-composition output sensitivity and measurement dead times, these control schemes may exhibit sluggish responses, because the control actions occur after the entire composition profile has been upset by disturbances. To overcome this problem, a cascade of composition-to-temperature (CT) control schemes have been used.6 Although the one-point control case (i.e., the regulation of one product composition on the basis of one concentration-temperature pair) can be reasonably handled with conventional linear cascade control designs6 and temperature sensor location criteria,7 the two-input-four output (2I-4O) control case (i.e., the regulation of both product composition on the basis of two concentration-temperature pairs) is considerably more complex, and there are few reported studies on the matter.8-10 Basically, the 2I-4O CT control problem has been addressed with the following cascade (i.e., with only primary-to-secondary interaction) schemes: on the basis of two C measurements, a pair of linear decentralized PI8 or I9 loops calculate the T setpoint values for a temperature control component, either a pair of linear-decentralized PI temperature loops8,9 or an estimatorbased nonlinear multiple input-multiple output (MIMO) controller.10 According to two of these studies, with the same application example,8,10 the 2I-4O cascade linear scheme must not be used8 for highly interactive columns, and the nonlinear controller10 successfully handles the problem. In a study with a different case example,9 an ad hoc 2I-4O scheme was implemented by applying to each section a 1I-2O linear cascade component. These results cause two basic combined structuralalgorithmic questions to be applied: (i) Are the decentralized primary structure and the C-to-T component cascade intercon* To whom correspondence should be addressed: Fax: 52 (55) 58044900. E-mail address: [email protected].

nection generic features of the distillation column class, or are they just the simplest structural choices given the lack of a systematic means to screen the large number of combined structural-algorithmic design degrees of freedom? and (ii) when and why should PI or I primary components be employed? In the chemical process systems engineering field, the choice of control structure is known to be equally important as (or more important than) the choice of control law,11,12 and the joint control structure-law design problem is regarded as an important and open research subject.13 From the aforementioned abundance of separate 2I-2O DC and two-point temperature (TPT) control schemes against the scarcity of 2I-4O CT control studies in distillation columns, and the contradictory features of the CT cascade control results, one is led to conclude that the design of the 2I-4O CT control scheme does not amount to just putting together previous DC and TPT schemes. Rigorously speaking, the 2I-4O CT structure-algorithm control problem must be addressed, given the set of all possible ways to interconnect the C and T components, as well as the ways of setting the input-output structure of each (C or T) component. In this work, the 2I-4O CT structure-algorithm control for continuous binary distillation columns is addressed within a constructive control framework, by exploiting fundamental connections between optimality, passivity, and robustness,14,15 in conjunction with the characteristics of the particular systemcontrol scheme pair under consideration. This approach provides a rather simple solution to the 2I-4O CT structure-algorithm control problem with (i) linear PI components; (ii) two possible C component structures, depending on the magnitude of the C measurement uncertainty; (iii) two possible T component (decentralized or one-way decoupling) structures, regardless of the C component structure; and (iv) a systematic construction with easy-to-apply tuning guidelines. The approach is applied to three representative examples through simulations, yielding behaviors that improve those of previous linear and nonlinear cascade control schemes. 2. Control Problem Consider the N-tray binary distillation column depicted in Figure 1, where a binary mixture with molar flow F and composition cF is fed at tray nF, yielding effluent flows B and

10.1021/ie060290d CCC: $33.50 © 2006 American Chemical Society Published on Web 11/22/2006

Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006 9011

h T ) HT(cj). The superindex T denotes matrix υ j ), ψ h c ) Hc(cj), ψ transposition, and the notation (‚h) denotes the nominal steadystate value of (‚). Our problem consists of designing a linear 2I-4O CT controller that, driven by two compositions and two temperatures (one per section18), regulates the two effluent compositions at their prescribed set points. We are interested in developing a combined structure-algorithm control design methodology that includes (i) structural decisions, given the set of all possible ways to interconnect the C and T components, as well as the ways of setting the input-output structure of each (C or T) component; and (ii) a systematic PI component-based control law construction and gain tuning procedure. The proposed control scheme must be put in perspective with existing linear8,9 and nonlinear10 C-to-T cascade techniques. 3. Control Model Figure 1. Schematic of a distillation column.

D, with compositions cB and cD, respectively. The control inputs are the vapor flow rate, V (which is proportional to the reboiler heat duty, Q), and the reflux rate, R. The objective consists of regulating the effluent composition pair (cB, cD), by controlling the temperatures Ts and Te at trays s and e (to be determined), in the stripping and rectifying sections, respectively. From standard assumptions16,17 (constant pressure, equilibrium in all trays, perfect level control, equimolal overflows), the column dynamics are given by

c˘ i )

L(mi+1)∆+ci - V∆-E(ci) + δi,nF F(cF - ci)

In this section, the adjustable-structure linear model that underlies the proposed control design is obtained from nonlinear constructive14,15 and industrial feedforward-feedback17 control ideas, given the model-control design inseparability principle.12 3.1. Nonlinear Model. Because model passivity is necessary for control robustness,14,15 the dynamical order of the column dynamics described by eq 1 must be reduced in such a way that the T measurements have relative degrees that are equal to one. This is accomplished by assuming that the holdup dynamics are in the quasi-steady-state (QSS) regime, which is a standard assumption in distillation modeling19 and control16 studies. In deviation variable form, the resulting reduced-order model is given by

xj ) f(x, d, u), yc ) hc(x), yT ) hT(x)

mi (for 0 e i e N - 1) +

c˘ N )

-

R∆ cN - V ∆ E(cN) mN

c˘ N+1 )

where x ) c - cj, yc ) ψc - ψ h c, yT ) ψT - ψ h T, u ) υ - υ j, d )δ-δ h , υ ) [V,R]T, δ ) [F, cF]T, f(x, d, u) ) Fc[cj + x,R(d,u),δ h + d,υ j + u], Fm(m,δ h + d,υ j + u) ) 0 f m ) R(d, u), and

V[E(cN) - cN+1] mD

m˘ i ) L(mi+1) - L(mi) + δi,nFF

(for 1 e i e N - 1)

m˘ N ) R - L(mN) where ∆+ci ) ci+1 - ci, ∆-E(ci) ) E(ci) - E(ci-1), E(c-1) ) c0, cN+1 ) cD, cB ) c0, Ts ) β(cs), Te ) β(ce), ψB ) ln(c0), and ψN ) ln(1 - cN). δi,nF is Kronecker’s delta, and ci (or mi) is the mole fraction (or holdup) of light component at the ith tray; E, β, and L are the nonlinear liquid-vapor equilibrium, bubble point, and hydraulic functions, respectively. The logarithmic C measurements (ψB, ψN) are used to obtain controllers that are more linear and robust,17 and cN is chosen as the regulated output to have a robust passive control structure,5 in the understanding that regulating cN is equivalent to regulating cD, because their steady-state values are bijectively related via the equilibrium function [i.e., cjN ) E-1(cjD)].5 In compact vector notation, the column dynamics are written as follows:

(2)

hc(x) ) Hc(cj + x) - Hc(cj) hT(x) ) HT(cj + x) - HT(cj) The term R denotes the static hold-up solution m of the equation Fm(m,δ,υ) ) 0. The model described by eq 2 is passive, with respect to the input-output u-yc (or u-yT) pair, because (i) y˘ c (or y˘ T) is dependent explicitly on u; and (ii) the corresponding zero dynamics, i.e., in perfect material balance control with yc (or yT) ) 0, are stable. 3.2. Linear Model. According to the constructive control approach, the application of the reduced model with yc (or yT) should lead to a robust two-point C (or T) nonlinear statefeedback controller, and its implementation would require a nonlinear state estimator that is based on the nonlinear model described by eq 2. To circumvent this problem, let us rewrite the nonlinear model (eq 2) in parametric (bc, bT) form:20

y˘ c ) Acu + bc

(3a)

y˘ T ) ATu + bT

(3b)

c˘ ) Fc(c, m, δ, υ), m˘ ) Fm(m, δ, υ ), ψc ) Hc(c), ψT ) HT(c) (1a and 1b)

x˘ I ) fI(xI, yc, yT, d, u)

(3c)

bc ) γc(yc, xI, d, u)

(4a)

where c ) (c0, ..., cN+1)T, m ) (m1, ..., mN)T, δ ) (F, cF)T, υ ) (V,R)T, ψc ) (ψB, ψN)T, ψT ) (Ts, Te)T, Hc(c) ) [ln(c0), ln(1 cN)]T, HT(c) ) [β(cs), β(ce)]T, 0 ) Fc(cj, m j , δ˙ , υ j ), 0 ) Fm(m j, δ h,

bT ) γT(yT, xI, d, u)

(4b)

where

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Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006

xI ) (cI - cjI)T cI ) [c1, ..., cs-1, cs+1, ..., ce-1, ce+1, ..., cN-1, cN+1]

θe :)

T

yT ) T - T h

[

-p θ θ A h T ≈ -ps θs θs e e e

T ) [Ts, Te]T γc(yc, xI, d, u) ) [∂xhc]fc(yc, xI, d, u) - Acu γT(yT, xI, d, u) ) [∂xhT]fT(yT, xI, d, u) - ATu Ac and AT are nonsingular (possibly diagonal or triangular) approximations of the related static interaction matrixes (A h c, A h T); that is,

h c ) G cΠ Ac ≈ A

(5a)

Gc ) diag(λ0,λN)

(5b)

AT ) A h T ≈ G TΠ

(5c)

GT ) diag(σs,σe)

(5d)

-p 1 Π ) -ps e 1

(5e)

[

]

xI is the vector with the deviated compositions in the trays without measurements, and bc (or bT) is a synthetic load input disturbance generated by the map γc (or γT). The entries of Gc, GT, and Π are given by

ps :)

(R h +F h) V h

pe :)

R h V h

∆ cj0 cj0m j0

-∆ cjN

y˘ c ) Gcµ + bc

(6a)

b˙ c ≈ 0

(6b)

y˘ T ) GTµ + bT

(6c)

b˙ T ≈ 0

(6d)

µ ) Pu

P ∈ SΠ

(6e)

(1 - cjN)m jN

σs :)

β′(cjs)∆+cjs m js

σe :)

β′(cje)∆+cje m je

β′(cj) )

where µ is a synthetic control input, Gc and GT are diagonal matrixes, eq 6b (or eq 6d) is a standard assumption to draw filter-based estimates of signal derivatives,23 P is any nonsingular structural approximation of the full matrix Π, and SΠ is the set of admissible (squared, triangular, and diagonal) nonsingular structures:

dβ(cj) dcj

SΠ ) {S,D1,D2,O11,O12,O21,O22}

where ps (or pe) is the nominal operating line slope of the stripping (or rectifying) section in a McCabe-Thiele diagram,21 λ0 (or λN) is the nominal logarithmic composition gradient-toholdup quotient in the reboiler (or the Nth tray), and σs (or σe) is the nominal temperature gradient-to-holdup quotient at the sth (or eth) tray in the stripping (or rectifying) column section. h s (or ∆+T h e), In terms of tray-to-tray temperature gradients ∆+T the constant σs (or σe) can be approximated by θs (or θe), and the matrix A h T can be approximated in terms of θs and θe:

θs )

(5h)

From the preceding observations, the linear adjustablestructure model follows:

+

λN :)

]

(5g)

The reduced nonlinear model (described by eq 3) has three interconnected subsystems: (i) a linear dynamic one (eqs 3a and b) that is driven, in a purely integral manner, by the control u and load disturbance inputs bc and bT, (ii) a nonlinear dynamical one (eq 3c) that describes the internal dynamics, and (iii) a nonlinear static one (eq 4) that sets the load inputs (bc and bT). Because the load input bc (or bT) is, time-wise, uniquely determined by the pair (y˘ c, u) [or (y˘ T, u)] [see eqs 3a and 3b], bc (or bT) is observable from the input-output pair (u, yc) [or (u, yT)], without needing the map γc (or γT).22 Thus, for control design purposes, bc and bT can be considered to be known load disturbances, and the nonlinear (static and dynamic) subsystems (eqs 3c and 4) can be dropped from the model (see eqs 3-4). The representation of matrixes Ac and AT in product form (eq 5) displays a fundamental structural property of the 2I-4O control structure: Ac and AT are linearly dependent. In our previous separate 2I-2O DC3 and TPT18 designs, the product representation was not needed, because the C-T structural interconnection was not an issue.

+

λ0 :)

∆+T he ≈ σe m je

∆+T hs ≈ σs m js

(5f)

where

[ ] [ ] [ ] [ ]

-p 1 S ) -ps e 1 -ps 0 0 D2 ) -p e D1 )

0 1 1 0

0 1 O11 ) -p e 1

(7)


[ [ [

-p 0 O12 ) -ps e 1 -ps O21 ) 0 -p O22 ) -ps e

1 1 1 0

In a similar way, enforce the decentralized temperature tracking dynamics,

] ] ]

y˘ T ) y˘ *T - KT(yT - y*T) where

KT ) diag(ωs,ωe)

The preceding control model (eq 6) consists of two linear subsystemssone dynamical-decentralized (eqs 6a-d) subsystem and one static subsystem (eq 6e) with adjustable structures and these features, as we shall see in the next section, will be incorporated into the control design. 4. Control Structure In this section, the CT control structure design problem is addressed within a feedforward-feedback (FF-FB) control framework, and the point of departure for the development, in Section 5, of the measurement-driven controller is set. 4.1. Feedforward-Feedback Controller. Let us recall a well-known paradigm among industrial control practitioners: “the most efficient way of controlling a difficult process is a combined feedforward-feedback system,”16,17 where the feedforward part performs most of the disturbance rejection task and the feedback component compensates for the errors of the feedforward model. Recall the observability of the load disturbances bc and bT, and assume that they are available for control; enforce the closed-loop decentralized composition dynamics,

y˘ c ) -Kcyc

(8)

where

Kc ) diag(ωB,ωN) on the model described by eq 6a, solve for u in the resulting equation pair, and obtain the virtual C controller:

u* ) P-1µ* -1

µ* ) -Gc (Kcyc + bc)

(9a) (9b)

where ωB (or ωN) is the bottom (or Nth tray) composition loop gain. The substitution of this controller into the temperature dynamics (eq 6c) yields the C regulation controller,

y˘ *T ) -Γ(Kcyc + bc) + bT

(10a)

where

y*T(0) ) y*To that generates the T set-point signal, y*T. Γ is an approximation of the diagonal matrix Λ, i.e.,

Λ :) GTGc-1 ) diag(σs/λ0,σe/λN) ≈ Γ ) diag (πs,πe)

(11)

(10b)

πs )

θs λ0

(10c)

πe )

θe λN

(10d)

on the model described by eq 6b; then, solve for u in the resulting equation pair and obtain the T tracking controller:

µ ) GT-1[y˘ *T - KT(yT - y*T) - bT]

(12a)

u ) P-1µ

(12b)

where ωs (or ωe) is the stripping (or rectifying) temperature loop gain. The combination of the C regulation and T-tracking controllers (eqs 10-12) yields the two-point feedforward- feedback (FF-FB) C-T cascade controller,

y˘ *T ) -Γ(Kcyc + bc) + bT

(13a)

µ ) -GT-1[Γ (Kcyc + bc) + KT(yT - y*T)]

(13b)

u ) P-1µ

(13c)

with adjustable structure P in the seven-member structure set SΠ (eq 7). This controller has two components in cascade interconnection: (i) the C controller (eq 13a), which, driven by the composition measurement yc and the loads bc and bT, yields the time-varying temperature set point y*T, and (ii) the T controller (eqs 13b and 13c), which, driven by the temperature measurement yT, the set point set point y*T, and the load bc, yields the control u. 4.2. Control Structure Selection. According to the preceding FF-FB cascade control derivation (eq 13): (i) the dynamic temperature set-point generator (eq 13a) is naturally decentralized, regardless of the structure P chosen for the static component (eq 13c) of the CT controller (eq 13), because the matrix Γ ≈ Λ is diagonal; and (ii) the structure choice P ∈ SΠ for the CT controller (eq 13) amounts to the choice of the control structure P for the T control component (eqs 13b and 13c). From eq 5, it follows that (i) Gc (in eqs 5, 10, and 13) is nonsingular and becomes ill-conditioned with the increase of product purities, and (ii) GT (in eqs 10 and 13) is nonsingular if there is one T sensor per section; and (iii) the best conditioning of GT is attained when the sensors are placed in the largest trayto-tray T change locations. From the relative gain array (RGA24) criterion for integrating processes,25 the consideration of only the principal diagonal decentralized (D1) and one-way decoupling (O12) T-control structures (eq 7) follows. The preceding analysis leads us to the next structural results. (i) The C control component (eq 13a) has an inherent closedloop decentralized structure, and (ii) the CT control (described by eq 13) structure search must be performed over two candidate structures, namely, a decentralized C component (eq 13a) with either a decentralized or (bottom-to-top) one-way decoupling T component (eq 13b); and (iii) the T component (eq 13a) performs most of the disturbance rejection task and decentralizes the structure of the C component. The latter comment explains the application of decentralized primary loops in the previous linear8 and nonlinear10 two-point cascade controllers, and the second comment is in agreement with the idea of the wave model-based nonlinear control approach:10 the regulation of the

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Figure 2. Block diagram for the closed-loop column with the C-T controller (as described by eqs 16).

wave propagation fronts, one per column section, at the stagnation points where the operating line and equilibrium curve slopes coincide. Summarizing, there are only two candidate CT control structures: (i) a decentralized C component (eq 13a) with a decentralized (with P ) D1 in eq 7) T component (eq 13b), and (ii) a decentralized C component (eq 13a) with a one-way decoupling (with P ) O12 in eq 7) T component (eqs 13b and 13c). In other words, the combined component design approach has considerably reduced the 49-structure search problem that must be tackled when the direct combination of previous separate DC3 and TPT18 control designs approach is pursued. 5. Control Law Having resolved the control structure design part, in this section, the related measurement-driven control law design problem is addressed. 5.1. Measurement-Driven CT Controller. From the decentralized and observability features of the system described by eq 4, the loads (bc and bT) can be quickly estimated with the set of decentralized reduced-order observers:26

w˘ c ) -Kco(wc + Kcoyc + Gcµ)

(14a)

bˆ c ) wc + Kcoyc, wc(0) ) wco, Kco ) diag(ωco,ωco) (14b) w˘ T ) -KTo (wT + KTo yT + GTµ) bˆ T ) wT + KTo yT, wT(0) ) wTo, KTo ) diag(ωTo ,ωTo )

(15a) (15b)

where ωco (or ωTo ) is the observer gain of the C (or T) load. The combination of these observers with the CT controller (eq 13) yields the measurement-driVen CT controller:

w˘ c ) -Kco(wc + Kcoyc + Gcµ)

(16a)

bˆ c ) wc + Kcoyc

(16b)

y˘ *T ) -ΓKcyc + bˆ

(16c)

bˆ ) bˆ T - Γbˆ c

(16d)

w˘ T ) -KTo (wT + KTo yT + GTµ)

(16e)

bˆ T ) wT + KTo yT

(16f)

u ) P-1µ

(16g)

-1

µ ) -GT [ΓKcyc + KT(yT - y*T) + bˆ T - bˆ ] P ∈ {D1,O12} (16h) with two possible structures (D1 and O12 in eq 7). When the observer dynamics are sufficiently faster than the closed-loop dynamics, the load estimation errors quickly vanish, and the CT controller (with load estimates bˆ c and bˆ T; see eq 16) practically recovers the behavior of the underlying CT controller (see eq 13) (with actual loads bc and bT). Because of the observer dynamics, the one-way (C-to-T) strict cascade interaction feature of the controller (eq 13) holds only at low frequencies, and, at high frequencies, there is two-way interaction between the C and T components. This high-frequency interaction feature must be carefully taken into consideration in the tuning and functioning assessment stages, to prevent excessive measurement error propagation and excitation of the fast holdup dynamics. These dynamical interaction features are depicted in Figure 2. If, in the CT controller (eq 16), the T-tracking controller gains are turned off (i.e., ωso ) ωs ) ωe ) 0), the adjustable structure version,

µB ) -kB(yB + τB-1

∫0t yB dζ)

(17a)

µN ) -kN(yN + τN-1

∫0t yN dζ)

(17b)


u ) P-1Gc-1µ

(17c)

µ ) [µB,µN]′

(17d)

where

kB ) ωTo + ωB 1 1 τB ) T + ω ωo B kN ) ωTo + ωN τN )

1 1 + ωco ωN

of the fixed-structure two-way decoupling DC control design3 is obtained. On the other hand, if the C component (eqs 16ad) of the CT controller (eqs 16) is dropped (i.e., ωco ) ωB ) ωN ) 0), and the set point y*T is calculated by a FF component (eqs 18a and 18b), the adjustable-structure TPT tracking controller,18

y˘ s*) κs[φs(yF) - y˘ s*]

(18a)

y˘ e* ) κe[φe(yF) - y˘ *e]

(18b)

µs ) y˘ s*- ks[(ys - y*s) + τs-1

∫0t (ys - y*s) dζ]

(18c)

µe ) y˘ *e - ke[(ye - y˘ *e) + τe-1

∫0t (ye - y˘ *e) dζ]

(18d)

u ) P-1GT-1(µs, µe)′ ks ) ωTo + ωs, τs )

(18e and 18f)

1 1 + , ke ) ωTo + ωe, ωTo ωs τe )

1 1 + ωTo ωe

is obtained, with [φs(yF), φe(yF)] being the temperature set-point static dependency on the feed temperature measurement (yF). In Laplace representation, the CT controller (described by eqs 16) is written as follows: For the C component,

[()(

)

τB-1 θs 1 Y*s(s) ) 2 k 1+ YB(s) + λ0 B s s + φss + ksτs-1

( [()(

φs +

) ]

ksτs-1 Ys(s) (19a) s

)

θe τN-1 1 Y*e(s) ) 2 k 1+ YN(s) + λN N s s + φes + keτe-1

For the T component,

(

) (

(

) ]

keτe-1 φe + Ye(s) s

τs-1 Ms(s) ) sY*s(s) - ks 1 + [Ys(s) - Y*s(s)], s τe-1 [Ye(s) - Y*e(s)] (19b) Me(s) ) sY*e(s) - ke 1 + s

)

U(s) ) P-1[Ms(s), Me(s)]′ where Y* a (a ) s, e, B, N) and M(z) denotes the Laplace

transform of ya and µ, respectively, and

( x ( x

) )

φs )

ks 1+ 2

1-

4 ksτs

(ksτs < 4)

φe )

ke 1+ 2

1-

4 keτe

(keτe < 4)

According to these expressions, the CT controller (described by eqs 19) consists of P, PI, and second-order systems interlaced in a nontrivial manner, signifying that the CT controller (described by eqs 16 or 19) can be hardly drawn by just putting together (separately designed) DC and TPT components. The presence of P and I components in the T tracking controller (see eqs 16d-g) is in agreement with earlier studies,9 and this differs from schemes with only P components in the T component.8 The presence of integral actions in the T component endows the CT controller with improved disturbance rejection and set-point tracking capabilities.27 If the C measurement signals contain errors larger than the closed-loop composition trend changes, the combination of the high-frequency unmodeled dynamics with the two-way highfrequency interaction between the C and T control components of the CT controller (described by eqs 16) may yield a degraded closed-loop behavior. In this case, the CT controller must be redesigned according to the procedure presented in the next section. 5.2. Cascade Control. Poor sensitivity of the composition outputs, instrument errors, measurement dead times, and combinations of them manifest themselves as measurement errors that are significant, in comparison to the effluent composition changes. The combination of these errors with the aforementioned high-frequency interaction between the C and T components of the CT controller (described by eqs 16) may yield degraded closed-loop behavior. To address this problem, let us redesign the CT controller as follows: (i) recall that the T-to-C high-frequency interaction is due to the presence of the term bˆ in the C component (eqs 16a-d), (ii) for this interaction to cease (i.e., bˆ ≈ 0), the C component must be tuned sufficiently slow so that the closed-loop response (yc, yT) response behaves in slow-varying regime,15 yielding the following implications:

y˘ c ) Gcµ + bc ≈ 0, ˘yT ) GTµ + bT ≈ 0 f bc ≈ Γ-1bT f bˆ ≈ b ) bT - Γbc ≈ 0(20a-d) and (iii) the substitution of bˆ ) 0 in the CT controller (described by eqs 16) yields the cascade C-T controller:

y*T ) -ΓKc yc

(primary controller)

(21a)

w˘ T ) -Kso(wT + KsoyT + GTµ), bˆ T ) wT + KsoyT (secondary controller) (21b) u ) P-1µ, µ ) GT-1[ΓKc yc - KT(yT - y*T) - bˆ T], P ∈ {D1,O12,S} (21c) with the decentralized primary C component decoupled from the secondary T component, and without the C load filter (see eqs 16a and 16b). Although the CT controller (described by eqs 16) has a C-to-T cascade interconnection at low frequencies and two-way C-T interconnection at high frequencies, the

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Figure 3. Block diagram for the closed-loop column with the cascade controller (as described by eqs 21).

preceding controller (eq 21) has a C-to-T cascade interconnection at all frequencies. (See Figure 3.) In PI form, the cascade controller (described by eqs 21) is written as follows:

y˘ s*) -ωBπsyB

(primary controller)

y˘ *e ) -ωNπeyN µs ) y˘ s*- ks[(ys - y˘ s*) + τs-1

(22a) (22b)

∫0t (ys - y˘ *s) dζ] (secondary controller) (22c)

µe ) y˘ *e - ke[(ye - y˘ *e) + τe-1

∫0t (ye - y˘ *e) dζ]

µ ) [µs,µe]′, u ) P-1µ, P ∈ {D1,O12}

(22d) (22e)

The primary C loops (i) only have I elements (see eqs 22a and 22b), which is consistent with the recommendation of using a low-gain integral control to regulate systems with significant model uncertainties, including dead times,28,29 and (ii) ensure the C effluents regulation, despite significant measurement uncertainties. The secondary controller (i) has T tracking loops (see eqs 22c and 22d), with P and I elements, and a static interaction compensator (eq 22e) that can be set with decentralized decoupling (D1) or one-way decoupling (O12); (ii) quickly performs most of the disturbance rejection tasks; and (iii) decentralizes the primary C loops. If the cascade controller (described by eq 21) is set with a decentralized T controller (with P ) D1), a previous control structure is obtained,8 with two important differences: (i) there, PI primary and P secondary loops were used, and here, PI primary and secondary loops are used, and (ii) as we shall see, in the section on application examples, the use of PI secondary T components overcomes the loop conflicts and degraded behavior. 5.3. Tuning and Implementation. The construction of the CT controller (see eq 21 or eq 16) requires only six static parameters (ps, pe, λ0, λN, θs, θe; see eq 5), which bear clear physical meaning and can be easily obtained from plant data and/or simulation packages. From standard singular perturbation theory arguments15 and conventional tuning rules for SISO loops,16,17 given the behavior-limiting presence of the fast unmodeled dynamics and of the error propagation analysis

performed in the previous sections, the next conventional-like tuning guidelines for our two-input four-measurement column case are as follows: (1) Draw estimates of (i) the open-loop dominant frequency ν, either from open-loop plant tests or from simulations, and (ii) the holdup characteristic frequency, νh (for example, νh ) 4/τh).4 (2) Set the C gains (ωB, ωN) at the open-loop dominant frequency ν, and set the T gains (ωs, ωe) three times faster; that is, ωB ) ωN :) ωc ≈ ν, ωs ) ωe :) ωT ) 3ν. (3) Set the CT (described by eqs 16) [or cascade (described by eqs 21)] control scheme with the load observer gains at least three times slower than the holdup characteristic frequency νh, i.e., ωTo ) ωco (or ωTo ) :) ωo ) νh/3. (4) Make a small set-point or load change, and then increase the filter gain ωo until (because of the excitation of the highfrequency dynamics) the response becomes oscillatory at the ultimate gain ω*o, and then back off to ωo ) ω* o/3 or slower. (5) Gradually increase the T component gain ωT until the response becomes oscillatory at the ultimate gain ω, and then back off to ωT ) ω* T/3 or slower. (6) Gradually increase the C component gains (ωB and ωN), one at a time, until an adequate tradeoff between C regulation and control effort is attained, detuning ωc and/or ωo if necessary. Preferably, a preliminary, simulation-based tuning and functioning assessment should be performed before actual implementation, to minimize actual column excitation. 5.4. Concluding Remarks. After the key control structure design issue was resolved in terms of only two candidate structures, the preceding control design led to a systematic construction procedure with conventional-type tuning guidelines. Depending on the particular column system, two component interaction modes were drawn: the CT controller (described by eqs 16) with C-to-T component cascade interaction at low frequencies, two-way component interaction at high frequencies, moderate model-measurement errors, and fast closed-loop response; and (ii) the cascade CT controller (described by eqs 21) with C-to-T component interaction at all frequencies, significant modeling-measurement errors (e.g., composition measurement dead times), and slower closed-loop response. Because of its construction-tuning simplicity, and its antireset windup capability,30 the observer-controller realizations in IMC form (see eqs 16 and 21) are more convenient for

Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006 9017 Table 1. Steady-State Data for the Studied Columns column system

benzene-toluene column benzene-toluene

methanol-water column methanol-water

thermodynamic model

Raoult’s law (vapor pressure from ref 34) 1 0.5 0.98 0.02 10 18

Wilson’s equation (ref 32; enthalpy calcalation from ref 33; vapor pressure from ref 34) 1 0.5 0.99 0.01 3 12

0.60 1.226 0.726 10.0 10.0 13.59 9.66 0.13

0.6 1.127 0.670 10.0 10.0 14.79 9.98 0.15

pressure, atm feed composition, cF distillate composition, cD bottoms composition, cB feed tray nF, (counting from below) number of trays (not including reboiler and condenser), N F (kmol/min) V/F R/F condenser holdup (kmol) reboiler holdup (kmol) condenser residence time (min) reboiler residence time (min) tray holdup (kmol) tray time constant (min) stripping section rectifying section control tray, stripping section (counting from below) control tray, rectifying section (counting from below) open-loop settling time (min) hydraulic settling time (min) open-loop characteristic frequency (min-1) hydraulics characteristic frequency (min-1)

0.0367 0.055 4 14 120 0.8 0.033 5.000

implementation purposes. Finally, it must be noted that performing a robust output feedback control design on the basis of the rather simple linear-observable model (eq 6), with reduced model dependency and decentralized dynamical features, is equivalent to performing the same task with a considerably more complex and model-dependent nonlinear passive model (eq 2) via the standard constructive interlaced control-observer designs.14 6. Application Examples To test the proposed methodology through numerical simulations, three distillation columns are considered: (i) an 18-tray benzene-toluene column, (ii) a 12-tray methanol-water column with nonideal thermodynamics, and (iii) a 39-tray column with constant relative volatility, which will be referenced as distillation column A.8 The resulting behaviors are described in sections 6.1, 6.2, and 6.3, respectively. The column characteristics are summarized in Table 1. These examples have been chosen for comparison with previous techniques and testing over a wide range of operation situation purposes. The first two columns have been studied using a two-point temperature tracking controller with set-point compensation.18 The third column has been studied with linear8 and nonlinear10 cascade controllers, meaning that a viz a viz behavior comparison can be done. Following Shinskey’s recommendation,31 the comparison of response times will be performed in terms of natural (open-loop) settling time units (Nstu). 6.1. Benzene-Toluene Column. Following the temperature sensor location guidelines of section 4, the temperature measurements were placed in the 4th and 14th trays of the stripping and rectifying sections, respectively. The column, with a natural settling time of 120 min, was subjected to the next sequence of input step disturbances: (i) at time to ) 0, the feed rate (F) increases from 10 mol/s to 12 mol/s, (ii) at t1, F decreases from 12 mol/s to 8 mol/s, (iii) at t2, the feed composition increases from 0.5 to 0.6, and finally (iv) at t3 ) the feed composition

column A (from ref 8) constant relative volatility: R ) 1.5 Raoult’s law (vapor pressure from ref 10) 1 0.5 0.99 0.01 20 39 1 3.206 2.706 0.5 0.5 0.16 0.13 0.5

0.0367 0.055 1

13

4

24

160 0.60 0.025 6.667

0.063

200 1.5 0.020 2.667

changes from 0.60 to 0.50. In the tests without measurement dead time, (t1, t2, t3) ) (120, 240, 360) min, and in the tests with composition measurement dead-time (of 2 min), (t1, t2, t3) ) (240, 480, 720) min. 6.1.1. C-T Controller (Described by eqs 16). The two (decentralized and one-way decoupling, denoted as D1 and O12, respectively) structures of the C-T controller (described by eqs 16) were tested without measurement dead times. The application of the tuning guidelines with the one-way decoupling structure (O12) yielded the gain set (ωo, ωs, ωe, ωB, ωN) ) (2.0, 0.2, 0.2, 0.05, 0.2) min-1, and the same gains were applied to the decentralized structure (D1). As it can be shown in Figure 4, the two control structures regulate both effluent compositions within ∼60 min (approximately half of the open-loop natural settling time, NSTU), and the one-way decoupling structure yields a better (less-oscillatory) response than its decentralized counterpart. It must be noted that the response of the CT control scheme (described by eqs 16) with one-way decoupling T control structure (O12), in the presence of holdup dynamics, is similar to that obtained with linear8 (or nonlinear10) cascade control with (or without) holdup dynamics. Because of the limiting performance that is caused by the presence of the holdup dynamics, a fair comparison between the linear and nonlinear schemes cannot be drawn. 6.1.2. Cascade CT Controller (Described by eqs 21 or 22). The composition measurements had dead times, and systematic actuator errors (νV and νR), in a worst-case combination4 (νV ) -νR), were included:

uV ) uˆ V(1 + νV), uR ) uˆ R(1 + νR), νV ) -νR ) 0.2 (23) where the pair (uV, uR) [or (uˆ V, uˆ R)] is the applied (and/or computed) control, νV (and/or νR) is the vapor (or reflux) error. The cascade controller (see eqs 21 or 22) was implemented with the two admissible secondary T (O12, D1) structures. The application of the tuning guidelines (see section 5.3) yielded

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Figure 4. Closed-loop response of a benzene-toluene column to a sequence of disturbances in the absence of composition measurement dead times.

Figure 5. Closed-loop response of a benzene-toluene column to a sequence of disturbances in the presence of composition measurement dead times.

the following gain set: (ωo, ωs ) ωe, ωB, ωN) ) (2.0, 0.16, 0.0133, 0.08) min-1. Figure 5 presents the resulting behaviors, showing that the two schemes required 80-120 min (i.e., 0.671.0Nstu) to regulate both effluent compositions with reasonable control efforts (i.e., the proposed cascade scheme (described by eqs 21) yields a closed-loop regulation response that is ∼2 times faster than that obtained, in the presence of composition measurement dead times,9 by applying, to each column section, a I primary-PI secondary control design. 6.1.3. Comparison with the C Control Scheme.3 Figure 6 (or Figure 7) presents the column responses with the CT (eqs 16), the CT cascade (eqs 21) (with one-way decoupling structure

O12), and the previously reported C controllers,3 in the absence (or presence) of measurement delays. In each case, the controller was tuned according to the guidelines given in section 5.3. As expected, detuning must be performed to handle the dead times properly. According to Figures 6 and 7: (1) In the absence of measurement dead times, the CT controller (described by eqs 16) yields the (quickest and lessoscillatory) best behavior with the bottom-distillate response time pair (τB, τD) ) (0.4, 0.2) (in Nstu), followed by the C controller3 with (τB, τD) ) (0.4, 0.6), and by the CT cascade controller (eqs 21) with (τB, τD) ) (1, 0.9). In other words, in the passage from the C1 to the CT control scheme (eqs 16), the


Figure 6. Comparison of closed-loop responses of a benzene-toluene column with different controllers in the absence of composition measurement dead times.

Figure 7. Comparison of closed-loop responses of a benzene-toluene column with different controllers in the presence of composition measurement dead times.

bottoms response time remains the same, and the distillate responds 3 times faster. (2) In the presence of composition measurement dead times, the CT cascade controller (eqs 21) yields the (quickest) best behavior with (τB, τD) ) (0.8, 0.7), followed by the C controller3 with (τB, τD) ) (1.6, 2-2.4), and by the CT controller (eqs 16) with (τB, τD) ) (1-1.5, 3). In other words, in the passage from the C controller1 to the CT cascade control scheme (eqs 21), the bottoms (or distillate) response time becomes 2 (or 3) times faster.

These results corroborate the theoretical developments presented in sections 5.1 and 5.2: in the absence (or presence) of significant measurement errors, the proposed CT (described by eqs 16) (or CT cascade (described by eqs 21)) controller must be used. 6.2. Methanol-Water Column. 6.2.1. CT Controller (Described by eqs 16). The 12-tray methanol-water column does not satisfy the equimolal overflow assumption, i.e., an energy balance per tray must be taken into consideration. The model18 included the vapor-liquid equilibrium with activity

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Figure 8. Closed-loop response of methanol-water column to a sequence of disturbances in the absence of composition measurement dead times.

coefficients (Wilson’s equation32) and enthalpy calculations.33 The application of the sensor location criterion (section 4.2) yielded the temperature measurements should be placed in the 1st and 4th trays. This system has an Nstu value of 160 min. The column was subjected to the following sequence of input step disturbances: (i) at t ) 0, the feed rate (F) increases from 10 mol/s to 11 mol/s; (ii) at t ) 300 min, F decreases to 9 mol/s; (iii) at t ) 600 min, the feed composition changes from 0.5 to 0.60; and, finally, (iv) at t ) 900 min, the feed composition decreases to 0.40. Three different feed disturbances were included along with the previous disturbance sequence: (i) the feed is a saturated liquid; (ii) the feed is a subcooled liquid, 10 °C below its saturation point; and (iii) the feed is subcooled 20 °C below its saturation point. The one-way decoupling structure O12 was selected to illustrate the resulting behaviors. From the application of the tuning guidelines, the set (ωo, ωs ) ωe, ωB, ωN) ) (4.0, 0.4, 0.2, 0.2) min-1 follows. The resulting behavior is shown in Figure 8, showing that (i) in the absence of composition measurement dead times, both effluent compositions are regulated within ∼40 min, i.e., 0.25Nstu, with reasonable control efforts, and (ii) the feed enthalpy disturbances are well compensated. 6.2.2. Cascade CT Controller (eqs 21 or 22). The column was subjected to the same (feed rate, feed composition, and enthalpy) disturbances of the previous case, including input actuator errors (eqs 23) and composition measurement dead times of 1 min. The one-way decoupling structure O12 was selected to illustrate the resulting behaviors. The application of the tuning guidelines yielded (ωo, ωs ) ωe, ωB, ωN) ) (1.48, 0.1333, 0.0125, 0.375) min-1. The resulting behavior is shown in Figure 9. As this figure shows, the cascade controller (described by eqs 21) behaves rather well in the combined presence of feed rate, composition, and enthalpy disturbances, as well as composition measurement dead times: both effluent purities are steered to their set points in ∼100 min, halving the regulations times obtained with a previous I primary-PI secondary controller.9 6.2.3. Comparison with T Control Scheme.14 In Figure 10, the proposed CT cascade (described by eqs 21) and the

previously reported TPT14 controllers are compared in the presence of composition measurement dead times (1 min), with a reasonable (2 °C feed subcooling) error in the feedforward component of the TPT controller. As expected, (i) the TPT controller exhibits some offset, and (ii) the use of composition measurements in the CT cascade controller (described by eq 21) eliminates the offset, and (iii) the two transient responses are similar. 6.3. Comparison with Previous Cascade Techniques. For comparison with previous two-point linear8 and nonlinear10 cascade schemes associated with the same column application example, Morari and Zafiriou’s30 distillation column A (with Nstu ) 200 min) was chosen as the third application example, in the understanding that the column has been studied with linear8 and nonlinear10 two-point cascade controllers, with and without composition measurement delay, respectively. In both cases, the closed-loop tests were performed in the presence of holdup dynamics. Accordingly, here, the holdup dynamics was included in the closed-loop tests, with the earlier reported hydraulic parameters.8 To permit comparison with the nonlinear wave-model based cascade controller,10 the column was subjected to a step disturbance from 0.50 to 0.20 in feed composition. Because the tests were performed without holdup dynamics, here, the simulations were conducted with and without such dynamics, with actuator errors (see eq 23), and with the one-way decoupling structure O12 for the T component. In the case with (or without) holdup dynamics, the application of the tuning guidelines yielded (ωo, ωs ) ωe, ωB, ωN) ) (4, 0.20, 0.05, 0.05) min-1 (or (ωo, ωs ) ωe, ωB, ωN) ) (40, 0.40, 0.067, 0.067) min-1). As expected, the performance-limiting feature of the hold-up dynamics manifests itself in the slower values of former control gains. The responses of the proposed CT controller (described by eqs 16) are shown in Figure 11: (i) in the presence (or absence) of holdup dynamics, the compositions are regulated within ∼80 min (0.40Nstu); (ii) in the test without holdup dynamics, the control performance is overestimated; and (iii) the test with holdup dynamics yields a response time that is


Figure 9. Closed-loop response of methanol-water column to a sequence of disturbances in the presence of composition measurement dead times.

Figure 10. Comparison of closed-loop responses of a methanol-water column with different controllers in the presence of composition measurement dead times.

similar to that obtained with a considerably more-complex nonlinear cascade controller in the absence of holdup dynamics.10 Finally, the column was subjected to a 50% step change in feed rate, as was done in the work of Wolff et al.,8 and the composition measurements were subjected to dead times of 6 min. In this case, the cascade C-T controller (described by eqs 21) was applied with the gains (ωo, ωs ) ωe, ωB, ωN) ) (1.6, 0.222, 0.02, 0.02) min-1 drawn from the tuning guidelines, and the resulting behavior is presented in Figure 12: (i) both effluent purities are regulated within ∼70-100 min (∼0.35-0.50Nstu),

with reasonably smooth control efforts; and (ii) these composition responses are twice as fast as those obtained with an optimally tuned linear-decentralized controller with PI primary and P secondary loops.8 Summarizing, the functioning of the proposed CT control design methodology has been tested with three different case examples, in the presence of measurement, actuator, and (holdup dynamics, nonideal thermodynamics) modeling, as well as (feed rate, composition, temperature, and enthalpy) load disturbances. With a more systematic and simpler design, the proposed control approach yields a behavior that equals or outperforms those

9022


Although separate two-input-two-output (2I-2O) dual composition (DC) and two-point temperature (TPT) control designs can be obtained as special cases of the proposed 2I-4O CT control design, the other way around is a considerably moredifficult or intractable task. In other words, the CT control design becomes unduly complex when the task is pursued by just combining separate DC and TPT designs. The control structure assessment and the control law construction require approximations of six static parameters that have clear physical meaning and can be easily drawn either from plant data or package-based simulations. The approach was tested with a set of three representative case examples, including measurement and actuator errors, measurement dead time, and (feed rate and composition, as well as thermodynamic variables) step-load disturbances, showing a behavior that improved that determined with previous linear and nonlinear controllers. Acknowledgment Figure 11. Closed-loop response of column A to a step disturbance in feed composition in the absence of measurement dead times.

The authors gratefully acknowledge the support from the Mexican National Council for Research and Technology (through CONACyT Scholarship No. 70832). Literature Cited

Figure 12. Closed-loop response of column A to a step disturbance in feed rate in the presence of measurement dead times.

obtained with previous nonlinear10 2I-4O cascade controllers, as well as linear dual composition,3 two-point temperature,14 and linear8,9 cascade schemes. 7. Conclusions The combined structure-algorithm design problem for twoinput-four-output (2I-4O) continuous binary distillation columns has been addressed within a constructive interlaced control-observer design framework, yielding a rather simple solution for the control structure search problem: (i) the C component is always decentralized; (ii) there are two candidate static compensation structures for the T component; (iii) there is C-to-T component cascade interaction at low closed-loop frequencies, and two-way interaction between those components at high-frequencies; and (iv) the CT high-frequency two-way interaction can be applied only in the case of moderate modelmeasurement errors in the input-output model behavior. The resulting control schemes (i) consist of linear PI components and static interaction compensators, (ii) have systematic construction procedure with reduced model dependency, and (iii) have easy-to-apply tuning guidelines.

(1) Humphrey, J. L.; Seibert, A. F.; Koort, R. A. Separation Technologies: Advances and Priorities; DOE Contract AC07-90ID12920, February 1991. (2) Luyben, W. L. Steady-state energy conservation aspects of distillation column control design. Ind. Eng. Chem. Fundam. 1975, 14 (4), 321-325. (3) Castellanos-Sahaguń, E.; Alvarez, J. Synthesis of two-point controllers for binary distillation columns. Chem. Eng. Commun. 2006, 193, 206232. (4) Skogestad, S. Dynamics and control of distillation columns: A critical survey. Model., Identif. Control 1997, 18 (3), 177-217. (5) Castro, R.; Alvarez, Ja.; Alvarez, Jo. Nonlinear disturbance decoupling control of a binary distillation column. Automatica 1990, 26 (3), 567572. (6) Fuentes, C.; Luyben, W. L. Control of high-purity distillation columns. Ind. Eng. Chem. Process Des. DeV. 1983, 22, 361-366. (7) Tolliver, T. L.; McCune, L. C. Finding the optimum temperature control trays for distillation columns. InTech 1980, 27 (9), 75-80. (8) Wolff, E. A.; Skogestad, S. Temperature cascade control of distillation columns. Ind. Eng. Chem. Res. 1966, 35, 475-484. (9) Alvarez-Ramı´rez, J.; Monroy-Loperena, R.; Alvarez, J. Backstepping design of composition cascade control for distillation columns. AIChE J. 2002, 48 (8), 1705-1718. (10) Shin, J.; Seo, H.; Han, M.; Park, S. A nonlinear profile observer using tray temperatures for high-purity binary distillation columns. Chem. Eng. Sci. 2000, 55, 807-816. (11) Stephanopoulos, G. Chemical Process Control: An Introduction to Theory and Practice; Prentice Hall: Englewood Cliffs, NJ, 1984. (12) Skelton, R. E., Owens, D. H., Eds. Model Error Concepts and Compensation; Pergamon Press: New York, 1986. (13) Seferlis, P., Georgiadis, M. C., Eds. The Integration of Process Design and Control; Computer-Aided Chemical Engineering 17; Elsevier: Amsterdam and Boston, 2004. (14) Krstic´, M.; Kanellakopoulos, I.; Kokotovic´, P. Nonlinear and AdaptiVe Control Design; Wiley: New York, 1995. (15) Sepulchre, R.; Jankovic´, M.; Kokotovic´, P. V. ConstructiVe Nonlinear Control; Communications and Control Engineering Series; Springer-Verlag: London, New York, 1997. (16) Luyben, W. L. Process Modeling, Simulation, and Control for Chemical Engineers, 2nd Edition; McGraw-Hill: New York, 1990. (17) Shinskey, F. G. Process-Control Systems: Application, Design, Adjustment, 3rd Edition; McGraw-Hill: New York, 1988. (18) Castellanos-Sahaguń, E.; Alvarez-Ramı´rez, J.; Alvarez, J. Twopoint temperature control and structure design for binary distillation columns. Ind. Eng. Chem. Res. 2005, 44, 142-152. (19) Levy, R. E.; Foss, A. S.; Grens, E. A., II. Response modes of a binary distillation column. Ind. Eng. Chem. Fundam. 1969, 8 (4), 765776.

Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006 9023 (20) Apostol, T. M. Calculus; Wiley: New York, 1969. (21) McCabe, W. L.; Thiele, E. W. Graphical Design of Fractionating Columns. Ind. Eng. Chem. 1925, 17 (6), 605-611. (22) Hermann, R.; Krener, A. J. Nonlinear controllability and observability. IEEE Trans. Autom. Control 1977, AC-22, 728-740. (23) Papoulis, A. Probability, Random Variables and Stochastic Processes; McGraw-Hill: New York, 1965. (24) Bristol, E. On a new measure of interaction for multivariable process control. IEEE Trans. Autom. Control 1966, 11, 133-134. (25) Woolverton, P. F. How to use relative gain analysis in systems with integrating variables. InTech 1980, 27 (9), 63-65. (26) Stefani, R. T.; Savant, C. J.; Shahian, B.; Hostetter, G. H. Design of Feedback Control Systems, 3rd Edition; Saunders College Publishing: Boston, 1994. (27) Krishnaswamy, P. R.; Rangaiah, G. P. Role of secondary integral action in cascade control. Trans. Inst. Chem. Eng. 1992, 70-A, 149-152. (28) Logemann, H.; Townley, S. Adaptive integral control of time-delay systems. IEE Proc. Control Theory Appl. 1997, 144 (6), 531-536.

(29) Lunze, J. Robust MultiVariable Feedback Control; Prentice Hall International Series in Systems and Control Engineering; Prentice Hall: Englewood Cliffs, NJ, 1989. (30) Morari, M.; Zafiriou, E. Robust Process Control; Prentice Hall: Englewood Cliffs, NJ, 1989. (31) Shinskey, F. G. Process control: As taught vs as practiced. Ind. Eng. Chem. Res. 2002, 41 (6), 3745-3750. (32) Walas, S. M. Phase Equilibria in Chemical Engineering; Butterworth: Boston, 1985. (33) Treybal, R. E. Mass Transfer Operations, 3rd Edition; McGrawHill: New York, 1980. (34) Patel, P. M.; Schorr, G. R.; Schah, P. N.; Yaws, C. Chem. Eng. 1976, 11, 159.

ReceiVed for reView March 10, 2006 ReVised manuscript receiVed September 29, 2006 Accepted September 29, 2006 IE060290D

Two-Point CompositionTemperature Control of Binary Distillation

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