ISA 1

TRANSACTIONS

ISA Transactions 37 (1998) 265±276

Simpli®ed, ideal or inverted decoupling?

E. Gagnon *, A. Pomerleau, A. Desbiens
GRAIIM (Groupe de recherche sur les applications de l'informatique aÁ l'industrie mineÂrale), Department of Electrical
and Computer Engineering, Laval University, QueÂbec, Canada, G1K 7P4

Abstract
This paper presents a comparative study of simpli®ed, ideal and inverted decoupling. The stability, robustness and
implementation of the three decoupling methods are studied. The structured singular value (SSV) is used to carry out
some comparisons. It is demonstrated that robust performance and robust stability of a nominally stable control sys-
tem are equivalent for the three decoupling methods when the controllers are tuned to obtain identical nominal per-
formance. A relation is derived between the presence of right-half plane (RHP) zeros of a process in series with its
simpli®ed decoupler and the instability of the ideal and inverted decouplers for the same process. This paper also
describes a potential implementation problem related to the particular structure of the inverted decoupling. Finally, a
recapitulative table of the main advantages and limitations of each decoupling method is presented. # 1998 Elsevier
Science Ltd. All rights reserved.
Keywords: Decoupling; Robustness; Stability; Implementation

1. Introduction Weischedel and McAvoy [2] have compared ideal

and simpli®ed decoupling methods using distilla-
The choice of a decoupling method is a rela- tion column simulators. They concluded that sim-
tively complex task since all techniques have their pli®ed decoupling is more robust than ideal
advantages and limitations. Simpli®ed decoupling decoupling. According to Waller [3], stability pro-
is by far the most popular method. Its main blems encountered by Luyben [1] with ideal
advantage is the simplicity of its elements. Ideal decoupling are explained by the fact that he used
decoupling, which is rarely used in practice, the same controller tuning for both decoupling
greatly facilitates the tuning of the controller methods. Weischedel and McAvoy [2] also kept
transfer matrix. Inverted decoupling, which is also the same controller tuning for both decoupling
rarely implemented, presents at the same time the techniques, therefore leading to the same conclu-
main advantage of both the simpli®ed and ideal sion about robustness. Following these studies,
decoupling methods. McAvoy [4] concluded that ideal decoupling is
Some authors have already compared simpli- very sensitive to modeling errors.
®ed, ideal and inverted decoupling. Luyben [1] and To evaluate control systems robustness, Arkun
et al. [5] proposed a general analysis procedure
based on the singular values. To illustrate their
* Corresponding author. Fax: +1-418-656-5343. methodology, they studied decoupling control

0019-0578/98/$Ðsee front matter # 1998 Elsevier Science Ltd. All rights reserved.
PII: S0019 -0 578(98)00023 -8
266 E. Gagnon et al./ISA Transactions 37 (1998) 265±276

systems applied to several distillation columns. structured singular value, depicts this important
The distillation columns used came from the lit- point. However, the three decoupling methods
erature [2,6,7]. They compared ideal decoupling may not have the same nominal stability. In fact, a
and simpli®ed decoupling. However, to be able to relation is established between the presence of
carry out direct analysis and comparisons with right-half plane (RHP) zeros of a process in series
results already presented, they used the same with its simpli®ed decoupler and the nominal
decouplers and controllers parameters as found in instability of the ideal and inverted decouplers for
the literature. As in the preceding studies, they the same process. A potential implementation
also concluded that ideal decoupling can be less problem with inverted decoupling, which can
robust than simpli®ed decoupling. deteriorate performance, is also explained. It is
In his book, Shinskey [8] detailed both simpli- shown, with an example, that this problem can
®ed and inverted decoupling structures. He even destabilize the control system. Finally, this
explains why the initialization problem, which paper summarizes in a simple table the main
consists in ®nding the right controller outputs advantages and limitations of each decoupling
values to allow bumpless switches between the method.
manual and automatic modes, is easier to solve
with inverted decoupling technique. He also
describes why it is much more easier to take into 2. Decoupling methods
account saturation of manipulated variables when
using inverted decoupling. Furthermore, it can be Decoupling at the input of a two input±two
added that ideal decoupling presents the same output (TITO) process P s† requires the design of
de®ciencies as simpli®ed decoupling when analyz- a transfer matrix D s†, such that P s†D s† is a
ing initialization and saturation. diagonal transfer matrix T s†:
Simpli®ed and inverted decoupling methods are  
also described by Seborg et al. [9]. Referring D11 s† D12 s†
D s† ˆ ;
respectively to Shinskey [8] and to Luyben [1], D21 s† D22 s†
Waller [3] and Weischedel and McAvoy [2], they  
concluded that inverted decoupling is appropriate P11 s† P12 s†
P s† ˆ and 1†
to take into account the saturation of manipulated P21 s† P22 s†
variables, however it is more sensitive to modeling  
errors. T11 s† 0
T s† ˆ
Recently, Wade [10] discussed implementation 0 T22 s†
issues for the inverted decoupling method. In most
commercial distributed control systems (DCS), the
PID function block has an auxiliary input called and
``feedforward input''. The feedforward input is
summed, within the PID function block, with the P s†D s† ˆ T s† 2†
output of the PID algorithm. Consequently, such
a PID function block is appropriate for direct Fig. 1 shows a decoupling control system for a
implementation of inverted decouplers, allowing, TITO process. The variables r1 and r2 are the set
without any programming, correct initialization, points, c1 and c2 are the controller outputs, u1 and
which facilitates bumpless switches between man- u2 are the manipulated variables and y1 and y2 are
ual and automatic modes. the process outputs. The controller transfer matrix
The purpose of this paper is to show that the C s† is diagonal and is de®ned as follows:
three decoupling methods present the same robust
stability and robust performance when the con-  
trollers are tuned to obtain equal closed loop C1 s† 0
C s† ˆ 3†
nominal performance. An example, which uses the 0 C2 s†
 E. Gagnon et al./ISA Transactions 37 (1998) 265±276 267

used in the literature. It consists in selecting the

decoupler as follows:
" #
1 ÿ PP1211 s†
s†
D s† ˆ 5†
ÿ PP2122 s†
s† 1

The resulting transfer matrix T s† is then:

" #
P11 s† ÿ P12Ps†P 21 s†
22 s†
0
Fig. 1. Decoupling control system of a TITO process. T s† ˆ
0 P22 s† ÿ P12Ps†P 21 s†
11 s†

Substituting Eq. (1) into Eq. (2) leads to: 6†

D s† ˆ P s†ÿ1 T s† This choice makes the realization of the decoupler

1 easy, but the diagonal transfer matrix T s†
ˆ obtained is complex since its elements are the sum
P11 s†P22 s† ÿ P12 s†P21 s† 4†
of transfer functions. Controller tuning can there-
 
P22 s†T11 s† ÿP12 s†T22 s† fore be dicult. It is then often suggested to
ÿP2 s†T11 s† P11 s†T22 s† approximate each sum by a simpler transfer func-
tion to facilitate controller tuning.

The elements P11 s†, P12 s†, P21 s† and P22 s† of 2.3. Inverted decoupling
Eq. (4), which represent the transfer functions of
the process, are supposed to be known. The only To avoid the realization problems of ideal
unknown elements are T11 s† and T22 s†. They decoupling while keeping its advantages, an inter-
represent the desired dynamics of the decoupled esting method, found in Shinskey [8], consists in
system. modifying the decoupling structure of Fig. 1.
According to this ®gure and Eq. (4), when
2.1. Ideal decoupling T11 s† ˆ P11 s† and T22 s† ˆ P22 s†, the manipu-
lated variables are:
A ®rst decoupling control design consists in  
selecting the transfer functions T11 s† and T22 s†. P11 s†P22 s†
u1 s† ˆ c1 s†
The decoupling transfer matrix D s† is then P11 s†P22 s† ÿ P12 s†P21 s†
deduced from Eq. (4). The diagonal controller   7†
elements C1 s† and C2 s† are independently and P12 s†P22 s†
ÿ c2 s†
respectively tuned based on T11 s† and T22 s†. A P11 s†P22 s† ÿ P12 s†P21 s†
logical choice for T s† is T11 s† ˆ P11 s† and
 
T22 s† ˆ P22 s†. With this choice, the same con- P21 s†P11 s†
troller tunings can be kept even if one loop is set in u2 s† ˆ ÿc1 s†
P11 s†P22 s† ÿ P12 s†P21 s†
manual mode. However, this technique, called 8†
 
``ideal decoupling'' by Luyben [1], often leads to P11 s†P22 s†
complicated D s† expressions, which can be di- ‡ c2 s†
P11 s†P22 s† ÿ P12 s†P21 s†
cult to realize.
It can easily be demonstrated (Appendix A) that
2.2. Simpli®ed decoupling these equations can be simpli®ed as follows:

A second decoupling control design, called P12 s†

u1 s† ˆ c1 s† ÿ u2 s† 9†
``simpli®ed decoupling'' by Luyben [1], is widely P11 s†
268 E. Gagnon et al./ISA Transactions 37 (1998) 265±276

P21 s† decoupling are unstable. For ideal decoupling, by

u2 s† ˆ c2 s† ÿ u1 s† 10†
P22 s† de®ning:
n11 s† n12 s†
Fig. 2 depicts a control system with the decoupling P11 s† ˆ ; P12 s† ˆ ;
structure de®ned by Eqs. (9) and (10). This repre- d11 s† d12 s†
11†
sentation is called ``inverted decoupling'' by Wade n21 s† n22 s†
P21 s† ˆ and P22 s† ˆ
[10]. It should be noted that the transfer functions d21 s† d22 s†
of the decoupler are the same as the ones used
with simpli®ed decoupling. Therefore, inverted with T11 s† ˆ P11 s† and T22 s† ˆ P22 s†, the
decoupling o€ers at the same time, the ease of denominator of each element of D s† (Eq. (4))
realization of the simpli®ed decoupling elements becomes:
and the more appropriate diagonal transfer matrix
T s† of the ideal decoupling. M s† ˆ n11 s†n22 s†d12 s†d21 s†
12†
ÿ d11 s†d22 s†n12 s†n21 s†
3. Robustness and stability
The transfer matrix D s† of ideal decoupling is
For equivalent nominal performance, the robust stable if M s† does not have any root with a posi-
performance and robust stability of nominally tive real part. With simpli®ed decoupling, sub-
stable control systems with simpli®ed, ideal and stituting Eq. (11) into Eq. (6) leads to the
inverted decoupling are identical. In fact, equiva- following expression for the numerator of each
lent nominal performance implies that the closed element of T s†:
loop transfer matrices are identical. The process
being the same in each case, the controllers trans- N s† ˆ n11 s†n22 s†d12 s†d21 s†
fer matrices in series with the decoupling transfer 13†
matrices are also identical for all three decoupling ÿ d11 s†d22 s†n12 s†n21 s†
methods. Therefore, they all present the same
robustness. However, to obtain the same nominal Since M s† ˆ N s†, when a process in series with
performance with di€erent decouplers, the con- its simpli®ed decoupler has RHP zeros (more dif-
troller cannot be the same in all cases. ®cult controller design), the ideal and inverted
decouplers for the same process would be unstable
3.1. Nominal stability (unstable system).
Therefore, the ®rst step of a robustness study is
When analyzing nominal stability, some di€er- to verify the nominal stability of the three decou-
ences can appear between the three decoupling pling methods. Fig. 3 shows a closed loop system
methods. For some processes, simpli®ed decou- where d is the process input disturbance vector.
pling is nominally stable but ideal and inverted P s† represents the transfer matrix of the nominal
process model and K s† ˆ D s†C s† represents the
controller in series with the decoupler. Nominal
stability of the closed loop system is veri®ed if and

Fig. 2. Inverted decoupling control system of a TITO process. Fig. 3. Closed loop system with process input disturbance.
 E. Gagnon et al./ISA Transactions 37 (1998) 265±276 269

only if all elements in transfer matrix F s† of the

following equation:
   
y s† r s†
ˆ F s† 14†
u s† d s†
Fig. 4. Closed loop system with multiplicative input uncer-
have all their poles in the left-half plane. The tainty.
matrix F s† is de®ned with the sensitivity function
E s† ˆ ‰I ‡ P s†K s†Šÿ1 , where I is the identity
matrix, as: where  is the structured singular value and
  HI s† ˆ ‰I ‡ K s†P s†Šÿ1 K s†P s† is the input
P s†K s†E s† E s†P s† complementary sensitivity function.
F s† ˆ 15†
K s†E s† ÿK s†E s†P s†
3.5. Robust performance

To evaluate if the closed loop system will

3.2. Nominal performance respect the desired performance even in presence
of multiplicative input uncertainty, a robust per-
Nominal performance analysis based on the formance condition must be calculated. A neces-
singular values permits to verify if the closed loop sary and sucient condition based on the
system respects the desired performance at the structured singular value [12] is:
nominal point. This condition [11,12] is: " #
ÿP s†ÿ1 H s†P s†!I s† ÿP s†ÿ1 H s†!P s†
 s†!P s†Š < 1 8!
‰E 16† 
E s†P s†!I s† E s†!P s†

where  is the maximum singular value and !P s† < 1 8! 18†

is the desired performance weight.

3.3. Input uncertainty where H s† ˆ P s†K s†‰I ‡ P s†K s†Šÿ1 is the

complementary sensitivity function.
To evaluate the robustness of a control system,
multiplicative input uncertainty can be added to 3.6. Example
the process input. Multiplicative input uncertainty
[12,13] is represented as illustrated in Fig. 4, where It is possible to compare the robustness of the
!I s† is the relative uncertainty and
I s† is the three decoupling methods using a simple process
disturbance. From this uncertainty description given by:
and the desired performance, robust stability and  4 3 
robust performance conditions can be obtained. P s† ˆ 1‡10s 1‡10s
19†
3 4
1‡10s 1‡10s
3.4. Robust stability
For this process, decoupling systems will be sym-
A robust stability condition, based on the metrical and both non-zero elements of the con-
structured singular value, allows to verify if the trollers will be identical. The nominal closed loop
closed loop system would remain stable in pre- performance required for the three systems studied
sence of multiplicative input uncertainty. The is described by a ®rst order dynamics with a time
necessary and sucient condition [12] is: constant of ®ve time units. PI controllers will be
sucient to reach the desired performance. With
‰HI s†!I s†Š < 1 8! 17† simpli®ed decoupling, the matrices D s† and C s†
270 E. Gagnon et al./ISA Transactions 37 (1998) 265±276
   
required respectively to perform decoupling and to 0:3 0 !I 0

1 s† ˆ and
I s† ˆ
obtain the desired nominal performance are: 0 0:3 0
1 s†
  22†
1 ÿ3=4
D s† ˆ and
ÿ3=4 1 It is important to precise that an equal uncertainty
" 8 1‡10s†=7 # 20† over all frequencies is not realistic since modeling
10s 0 errors are generally larger at high frequencies than
C s† ˆ 8 1‡10s†=7 at low frequencies. In order to verify the speci®-
0 10s
cations, !P s† has to be de®ned as follows:
 5s‡1 
0
For this example, an IMC-based tuning method !P s† ˆ 5s 5s‡1 23†
0 5s
[12] has been used. Any other method leading to
speci®c closed-loop dynamics could have also been Fig. 6 shows the robust stability, nominal perfor-
used. When designing ideal decoupling, the mance and robust performance conditions for the
required transfer matrices D s† and C s† are: three decoupling methods. All three decoupling
  methods lead to identical results. The nominal
16=7 ÿ12=7 performance is described by a unitary magnitude
D s† ˆ and
ÿ12=7 16=7 over the whole frequency window.
" 1‡10s†=2 # 21† Therefore, the systems exactly respect the nom-
10s 0 inal speci®cations. The robust stability magnitude
C s† ˆ 1‡10s†=2 is clearly smaller than one for all frequencies,
0 10s
indicating that the systems will remain stable in
spite of an uncertainty of 30% on each process
The two transfer functions of the inverted decou- input. The robust performance analysis shows that
pler are ÿ3/4 and the controller C s† is the same as the performance will deteriorate at low fre-
with ideal decoupling. Fig. 5 shows the poles locus quencies, but in the same way for the three systems.
of the transfer matrix [13] F s† given by Eq. (15), Hence, from a robustness point of view, the
for all three decoupling methods. Obviously, they three decoupling methods are equivalent when the
are the same in all cases. This ®gure points out controllers are tuned to obtain identical closed loop
that the three systems are nominally stable. To nominal performance. Other types of uncertainty
analyze robustness, an arbitrary uncertainty of
30% on each process input is selected:

Fig. 6. Robust stability, nominal performance and robust per-

Fig. 5. Poles locus of the closed loop transfer matrix F s† for formance conditions for simpli®ed, ideal and inverted decou-
simpli®ed, ideal and inverted decoupling. pling.
 E. Gagnon et al./ISA Transactions 37 (1998) 265±276 271

would lead to the same conclusion. In all cases, to The same diculty arises with inverted decoupling
obtain the same robustness, closed loop nominal when the PID function bloc does not have feed-
performance must also be perfectly identical. For forward input. Fig. 8 shows a decoupling control
processes with more complex dynamics (with dead system (ideal or simpli®ed) where the antireset
times for instance), it can be very dicult, even windup features of the PID blocks are used. This
impossible, to tune the controllers to obtain iden- ®gure shows that it may be dicult to use the
tical nominal performance for the three decou- antireset windup feature with ideal decoupling due
pling methods. For some processes, it may also to the impossibility, in some cases, to invert the main
happen that the elements of the decouplers cannot diagonal elements of the decoupling transfer matrix.
be perfectly realized and some di€erences may
then appear in the robustness analysis. 4.2. Manual mode

Occasionally, for maintenance or any other

4. Implementation reason, it may be necessary to operate one loop in
manual while the other one remains in automatic
When a system is nominally stable with ideal or
inverted decoupler, it is as robust as a simpli®ed
decoupled system. Since the robustness study deals
with process variations and not with controller or
decoupler variations, it is independent of the
decoupler proximity to instability. In practice,
however, it is recommended to make sure that
some variations to the controller and decoupler
parameters do not lead to instability [14]. In fact,
slight di€erences between calculated and imple-
mented parameters may appear due to the number
of signi®cant digits. Fine-tuning of the controller
and decoupler must also be possible without
destabilizing the system. Finally, an unwanted delay
can be created in one of the decoupling elements,
when implementing inverted decoupling on a TITO
process using lead-lag and delay function blocks. Fig. 7. Inverted decoupling control system with PID function
blocks which have antireset windup feature.

4.1. Antireset windup

In most commercial DCS, the PID function

block can prevent reset windup through the use of
a reset feedback input. The signal sent to the reset
feedback input should be equal to the PID output,
unless there is saturation. In this case, the reset
feedback is the saturated PID output. The anti-
reset windup feature of the PID, combined to its
feedforward input, can be used to directly take into
account the saturation of the manipulated variables
when an inverted decoupling structure is imple-
mented. This is shown for a TITO process in Fig. 7.
The antireset windup feature is more dicult to Fig. 8. Decoupling control system (ideal or simpli®ed) with
implement with simpli®ed and ideal decoupling. PID function blocks which have antireset windup feature.
272 E. Gagnon et al./ISA Transactions 37 (1998) 265±276

mode. With inverted decoupling, the system The resulting transfer matrix T s† is therefore:
remains decoupled when the manual manipulated  3 
1‡15s 0
variable is modi®ed, which is not the case for ideal T s† ˆ 3 26†
0 1‡15s
or simpli®ed decoupling.
To reach the nominal performance de®ned by a
4.3. Initialization ®rst order dynamics with a time constant of ®ve
time units, the controller is:
The decoupling control system must allow  1‡15s 
bumpless switches between manual and automatic 0
C s† ˆ 15s
1‡15s 27†
modes. Therefore, before switching from the 0 15s
manual to the automatic mode, the controller
outputs must be back calculated from the The next step in the design consists in verifying
manipulated variables in manual mode. Unfortu- nominal stability. The process is obviously stable.
nately, since the decoupling elements are often not The inverted decoupler, equivalent to D s† de®ned
static, their outputs will not instantaneously take by Eq. (4) with T11 s† ˆ P11 s† and T22 s† ˆ
the right values, unless the lead-lag and delay P22 s†, is also stable. Fig. 9 shows the poles locus
blocks can be initialized to a speci®c value. of the equivalent decoupling transfer matrix D s†.
The initialization problem can be avoided by The nominal stability of the closed loop system is
waiting the decoupler to be in steady-state before veri®ed, if the closed loop transfer matrix F s†
switching to the automatic mode. This limitation does not have any poles with a positive real part.
is not present with inverted decoupling because Fig. 10 shows the nominal stability of the closed
the input of each decoupling element is a manipu- loop system.
lated variable measurement. Fig. 11 shows the nominal performance, robust
stability and robust performance conditions for an
4.4. Unwanted delay uncertainty of 30% on each process input. The
nominal performance is respected. The system
Because of the blocks scanning sequence, when remains stable in spite of the uncertainty and the
implementing inverted decoupling for a TITO performance deterioration is more important at
process on a DCS using lead-lag and delay func- low frequencies. However, the system becomes
tion blocks, an unwanted delay can be created in unstable if the implementation of this control sys-
one of the decoupling elements. This delay of one tem with function blocks introduces a delay of one
sampling period slightly decreases decoupling sys- scan time (1 time unit in this case) in one of the
tem performance. With some processes, it can
even destabilize the control system if the calcu-
lated decoupler was close to instability, as illu-
strated by the following example.

4.5. Example

The TITO process is de®ned as follows:

 3 4 
P s† ˆ 1‡15s
4
1‡10s
3 24†
1‡10s 1‡15s

As illustrated in Fig. 2, the transfer functions of

the inverted decoupler are:

P12 s† P21 s† 4 1 ‡ 15s†

ÿ ˆÿ ˆÿ 25†
P11 s† P22 s† 3 1 ‡ 10s† Fig. 9. Poles locus of the equivalent transfer matrix D s†.
 E. Gagnon et al./ISA Transactions 37 (1998) 265±276 273

Fig. 10. Poles locus of the closed loop transfer matrix F s†. Fig. 12. Poles locus of the closed loop transfer matrix F s†.

Fig. 11. Robust stability, nominal performance and robust Fig. 13. Poles locus of the equivalent transfer matrix D s†.
performance conditions.

decoupler elements. Fig. 12 shows this instability can subtract a delay of one scan time in one cross-
by using a Pade approximation for the unwanted element of the inverted decoupling structure at the
delay added to the closed loop transfer matrix implementation time. However, the designer must
F s†. Fig. 13, which shows the poles locus of the be careful to the order in which the chains of blocs
equivalent transfer matrix D s† of the inverted are executed in the control system, to really com-
decoupler with the unwanted delay, indicates that pensate for the arti®cial delay and to avoid the
the instability comes from the decoupler. ampli®cation of the problem.
It is therefore very important to take into
account a possible delay introduced by the blocks
scanning sequence. To avoid the introduction of 5. Summary table
the delay, inverted decoupler could be imple-
mented using a unique multivariable block (possi- Table 1 summarizes the advantages and dis-
bly using a state-space representation). Also, if the advantages of each decoupling method. The
inverted decoupling cross-elements require dead robustness is not discussed since it is identical for
times due to the delays of the process, the designer all three decoupling techniques.
274 E. Gagnon et al./ISA Transactions 37 (1998) 265±276

Table 1
Advantages and disadvantages of each decoupling method

Decoupling methods characteristics Simplified Ideal Inverted

decoupling decoupling decoupling

When one loop is in manual mode, dynamics of the remaining NO YES YES
loop is unchanged
Decoupling elements do not contain a sum of transfer functions YES NO YES
Transfer matrix of the decoupler in series with the process does NO YES YES
not contain sums of transfer functions
When loops are switched from manual to automatic mode, NO NO YES
decoupling system initialization is simple
Saturation of the manipulated variables is easily taken into NO NO YES
account with a PID function block having an antireset
feature and a feedforward input
Implementation with lead-lag and delay function blocks may YES YES NO
not decrease performance
A feedforward input to the PID function block facilitates the NO NO YES
decoupling system implementation
The antireset windup of the PID function block can be used YES NO YES
without inverting a transfer function

6. Conclusion [3] K.V.T. Waller (Toijala), Decoupling in distillation,

AIChE Journal 20 (3) (1974) 592±594.
[4] T.J. McAvoy, Interaction Analysis: Principles and Appli-
This paper gives some guidelines for the selec- cations. Instrument Society of America, Research Triangle
tion of a decoupler. It is shown that robust per- Park, NC, 1983.
formance and robust stability of nominally stable [5] Y. Arkun, B. Manouslouthakis, A. Palazoglu, Robustness
control systems, using simpli®ed, ideal and inver- analysis of process control systems. A case study of
ted techniques, are the same if the controllers are decoupling control in distillation, I&EC Process Design
and Development 23 (1) (1984) 93±101.
tuned to obtain the same nominal performance. [6] K.V.T., Toijala (Waller), K.C., Fagervik, A digital simu-
Therefore, the selection of one of the three meth- lation study of two-point feedback control of distillation
ods must not be based on robustness considera- columns. Kem. Teollisuus 29(5) 1972.
tions. A relation has also been established between [7] R.K. Wood, M.W. Berry, Terminal composition control
the presence of RHP zeros of a process in series of a binary distillation column, Chemical Engineering
Science 28 (1973) 1707±1717.
with its simpli®ed decoupler and the nominal [8] F.G. Shinskey, Process Control Systems: Application,
instability of the ideal and inverted decouplers for Design and Adjustment. McGraw-Hill, New York, 1988.
the same process. It is also shown that inverted [9] D.E. Seborg, T.F. Edgar, D.A. Mellichamp, Process
decoupling performance can depend on the imple- Dynamics & Control. John Wiley & Sons, New York, 1989.
mentation method. Finally, a table summarizes the [10] H.L. Wade, Inverted decoupling: a neglected technique,
ISA Transactions 36 (1) (1997) 3±10.
main advantages and limitations of each method. [11] J.C. Doyle, G. Stein, Multivariable feedback design: con-
cepts for a classical/modern synthesis, IEEE Transactions
on Automatic Control AC-26 (1) (1981) 4±16.
[12] M. Morari, E. Za®riou, Robust Process Control. Prentice-
References Hall, Englewood Cli€s, NJ, 1989.
[13] S. Skogestad, I. Postlethwaite, Multivariable Feedback
[1] W.L. Luyben, Distillation decoupling, AIChE Journal Control. Analysis and Design. John Wiley & Sons, Eng-
16(2), 1970, 198±203. land, 1996.
[2] K. Weischedel, T.J. McAvoy, Feasibility of decoupling in [14] L.H. Keel, S.P. Bhattacharyya, Robust, fragile, or opti-
conventionally controlled distillation columns, I&EC mal?, IEEE Transactions on Automatic Control 42 (8)
Fundam. 19 (4) (1980) 379±384. (1997) 1098±1105.
 E. Gagnon et al./ISA Transactions 37 (1998) 265±276 275

Appendix A: According to Eq. (A.4)

 
Demonstration P11 s†P22 s† ÿ P12 s†P21 s†
c2 s† ˆ u2 s†
  P11 s†P22 s†
P11 s†P22 s†
u1 s† ˆ c1 s† A:6†
P11 s†P22 s† ÿ P12 s†P21 s†
P21 s†
A:1† ‡ c1 s†
P22 s†
 
P12 s†P22 s†
ÿ c2 s†
P11 s†P22 s† ÿ P12 s†P21 s† Substituting Eq. (A.6) to Eq. (A.3) leads to
 
P11 s†P22 s† ÿ P12 s†P21 s†
  u1 s† ˆ
P21 s†P11 s† P11 s†P22 s†
u2 s† ˆ ÿc1 s†
P11 s†P22 s† ÿ P12 s†P21 s†
A:2†  
P12 s† P11 s†P22 s† ÿ P12 s†P21 s†
  c1 s† ÿ u2 s†
P11 s†P22 s† P11 s† P11 s†P22 s†
‡ c2 s†
P11 s†P22 s† ÿ P12 s†P21 s†
P12 s†P21 s†
ÿ c1 s† A:7†
Therefore, P11 s†P22 s†
 
P11 s†P22 s† ÿ P12 s†P21 s† Substituting Eq. (A.5) to Eq. (A.4) leads to
u1 s† ˆ
P11 s†P22 s†  
P11 s†P22 s† ÿ P12 s†P21 s†
A:3† u2 s† ˆ
P11 s†P22 s†
P12 s†
c1 s† ÿ c2 s†
P11 s†  
P21 s† P11 s†P22 s† ÿ P12 s†P21 s†
c2 s† ÿ u1 s†
P22 s† P11 s†P22 s†
and
  P12 s†P21 s†
P11 s†P22 s† ÿ P12 s†P21 s† ÿ c2 s† A:8†
u2 s† ˆ P11 s†P22 s†
P11 s†P22 s†
A:4†
Therefore,
P21 s†
c2 s† ÿ c1 s†  
P22 s† P11 s†P22 s† ÿ P12 s†P21 s†
u1 s† ˆ
P11 s†P22 s†
According to Eq. (A.3)  
  P11 s†P22 s† ÿ P12 s†P21 s†
P11 s†P22 s† ÿ P12 s†P21 s† c1 s†
c1 s† ˆ u1 s† P11 s†P22 s†
P11 s†P22 s†
A:5†  
P12 s† P11 s†P22 s† ÿ P12 s†P21 s†
ÿ u2 s†
P12 s† P11 s† P11 s†P22 s†
‡ c2 s†
P11 s† A:9†
276 E. Gagnon et al./ISA Transactions 37 (1998) 265±276

and Finally,
 
P11 s†P22 s† ÿ P12 s†P21 s†
u2 s† ˆ P12 s†
P11 s†P22 s† u1 s† ˆ c1 s† ÿ u2 s† A:11†
P11 s†
 
P11 s†P22 s† ÿ P12 s†P21 s†
c2 s†
P11 s†P22 s† and
 
P21 s† P11 s†P22 s† ÿ P12 s†P21 s†
ÿ u1 s†
P22 s† P11 s†P22 s† P21 s†
u2 s† ˆ c2 s† ÿ u1 s† A:12†
A:10† P22 s†