Ind. Eng. Chem. Res.

2010, 49, 761–771 761

Multivariable Process Control: Decentralized, Decoupling, or Sparse?

Yuling Shen,† Wen-Jian Cai,*,‡ and Shaoyuan Li†
Department of Automation, Shanghai Jiao Tong UniVersity, 800 Dongchuan Road, Minhang,
Shanghai 200240, China, and School of Electrical and Electronic Engineering, Nanyang Technological
UniVersity, Singapore 639798, Singapore

In this article, a systematic approach is proposed to design PI-/PID-based multivariable control systems in
which the designs and analyses of decentralized, decoupling, and sparse control schemes are all treated under
a unified framework. First, based on the relative normalized gain array (RNGA), a best loop pairing is obtained
using the RGA (relative gain array)-Nederlinski index (NI)-RNGA rules. The index matrix is then calculated,
and the control structure is determined according to a selection criterion. Finally, the selected loop controllers
are independently designed based on equivalent transfer functions. The effectiveness of the proposed design
approach is verified by analysis of several multivariable industrial processes, demonstrating that the selected
control structure results in better overall system performance.

1. Introduction eliminating the effects of the undesirable cross-couplings, the

process can then be treated as multiple single loops, and less-
Energy and material recycling design and tight control
conservative single-loop PID control design methods can be
requirements for modern industrial processes justify the need
directly applied. Several decoupling control schemes are cur-
to implement multivariable control schemes. However, multi-
rently available, such as ideal decoupling,8 simplified decou-
variable controllers are much more difficult to design and
pling,9 inverted decoupling,10 decoupling via linear matrix
implement than their single-input-single-output (SISO) coun-
inequality (LMI),11 decoupling-based step/relay test,12,13 and
terparts, because of the existence of interactions among the
normalized decoupling.14,15 Even though decoupling control can
loops. Even though several model predictive control (MPC)
provide better performance than decentralized control schemes
schemes have emerged in the process control industry to handle
for closely coupled processes, it can result in very complex
constraints and interactions, their computational complexity
structures, especially when a multivariable process is of high
makes them more suitable for a higher level in the control
dimension, making it impossible to implement in practice.
structure, whereas control schemes based on proportional-integral
(PI)/proportional-integral-derivative (PID) controllers are still To compromise between the two control schemes, sparse
widely used at the lower levels for regulating controls. As such, control structures were proposed, including “block diagonal
research on PI-/PID-based multivariable control system analysis control” and “triangular control” structures.16-19 By adding
and design has received great interest from academia and some extra off-diagonal controllers, sparse control is simpler
industry alike. than full multi-input-multi-output (MIMO) control, but it is
PI-/PID-based control can generally be classified into two able to achieve superior performance compared to decentralized
main groups: decentralized control and decoupling control. In control if it is properly designed. However, the design of sparse
decentralized control, a multivariable process is decomposed control is not as simple as it appears to be because two important
into multiple single-input-single-output (SISO) processes, and but difficult issues need to be resolved: (1) Which loops should
the controllers are designed for each individual loop by taking be controlled other than the diagonal ones? The consideration
the loop interactions into account. So far, decentralized control should be based on a balance of control structure simplicity,
is still the dominant control scheme in multivariable control system performance, and robustness. (2) How should the
because it has many advantages, such as flexibility in operation, controllers be designed? Design of a controller for one loop is
failure tolerance, and simplified design and tuning. Over the dependent on all other loop controllers because of the existence
years, several methods for decentralized control system design of interactions. Ideally, a good design should be simple for
have been proposed, including detuning factor methods,1-3 practical engineers while taking the loop interactions into
sequential loop closing methods,4,5 and equivalent transfer consideration. Unfortunately, these two issues remain unsolved,
function methods.6,7 Decentralized control can generally work which has limited algorithm development for and real-world
well when loop interactions are modest. However, when the applications of sparse control.
processes are closely coupled, it inevitably leads to performance In this article, a systematic approach is proposed to design
degradation compared to full-dimensional control schemes, as PI-/PID-based multivariable control systems in which decentral-
the tuning of the loop controller is aimed at a compromise ized, decoupling, and sparse control schemes are all placed under
between improving system performance and conquering loop a unified framework. The main features of the approach include
interactions. the following: 1) By employing the concept of relative normal-
On the other hand, decoupling control schemes are often ized gain array (RNGA),20 an interaction index is defined to
employed for those processes that are closely coupled and/or evaluate the seriousness of the interactions of the off-diagonal
require tight control. In this type of control scheme, by first elements. (2) A control structure selection criterion is proposed
through an analysis of relative gain and relative average time
* To whom correspondence should be addressed. Tel.: +65 6790
6862. Fax: +65 6793 3318. E-mail: ewjcai@ntu.edu.sg. constant, as well as engineering experience. (3) The process
†
Shanghai Jiao Tong University. equivalent transfer functions (ETFs)21 when other loops are
‡
Nanyang Technological University. closed are derived based on the relative gain and relative average
10.1021/ie901453z  2010 American Chemical Society
Published on Web 12/01/2009
762 Ind. Eng. Chem. Res., Vol. 49, No. 2, 2010

Hence, it is desired that the control structure can be configured

based on an effective evaluation of control-loop interactions in
terms of both steady-state and transient information. The steady-
state information can be easily extracted from the process steady-
state gain matrix, whereas the dynamic information can be
obtained from the process average residence time, τar,ij, defined
as the time constant plus dead time of the transfer function.
For FOPTD processes, the average residence time τar,ij can be
obtained as

Figure 1. Closed-loop multivariable control system. τar,ij ) τij + θij (3)

A smaller τar,ij value indicates that the transfer function has a

time constant of each loop for the control system design. (4)
faster response to the inputs and disturbances, whereas a larger
An independent design method is proposed for control system
τar,ij value indicates slower process dynamics.
design such that the controllers can be designed independently
To use both steady-state gain and average residence time for
of each other. By this design approach, the conversion between
interaction measuring and loop pairing, we defined the normal-
decentralized control, decoupling control, and sparse control can
ized gain kN,ij for a particular transfer function gij(s) as16
be simply achieved by adding or dropping corresponding
controllers. The method is simple, straightforward, and easy to gij(j0)
understand and implement. Several multivariable industrial kN,ij ) (4)
processes with different interaction characteristics are employed τar,ij
to demonstrate the design procedure, as well as the simplicity and for all elements, the normalized gain matrix KN is
and effectiveness of the design method.
KN ) [kN,ij]n×n ) G(j0) . Tar (5)
2. Preliminaries
where Tar ) [τar,ij]n×n and . indicates element-by-element
Consider an open-loop stable n × n multivariable system as division.
shown in Figure 1, where ri, i ) 1, 2, ..., n, represents the When the relative normalized gains are calculated for all
reference inputs; ui, i ) 1, 2, ..., n, represents the manipulated input/output combinations of a multivariable process, the result
variables; and yi, i ) 1, 2, ..., n, represents the system outputs. is an array with a form similar to that of the relative gain array
G(s) is the process transfer function matrix with compatible (RGA), called the RNGA, ΛN ) [λN,ij]n×n, which can be
dimensions, described by

[ ]
calculated as
g11(s) g12(s) ... g1n(s)
g21(s) g22(s) ... g2n(s) ΛN ) KN X KN-T (6)
G(s) ) · (1)
l l ·. l 2.2. Loop Pairing Rules Based on RGA-NI-RNGA. For
gn1(s) gn2(s) ... gnn(s) the Nederlinski index (NI), given as

Without loss of generality, the elements in the transfer |G(0)|

NI ) (7)
function matrix can be considered first-order plus dead time n
(FOPDT) ∏ g (0) ii
i)1
kij
gij(s) ) e-θijs (2) where |G(0)| denotes the determinant of matrix G(0), RGA and
τijs + 1
NI tools can provide sufficient conditions to filter out the
for i, j ) 1, 2, ..., n, where kij, τij, and θij are the process open- structurally unstable control configurations. That is, the RGA
loop gain, time constant, and dead time, respectively. Because and NI tools can be used to eliminate those structures with
high-order transfer function elements can be simplified by either unstable pairing options, whereas the RNGA can find the pairing
analytical or empirical methods,22,23 FOPDT models will be with the best overall performances. Thus, the RGA-NI-RNGA-
used for interaction analysis and control system design for based control configuration criterion is given as follows:20
simplicity. Manipulated and controlled variables in a decentralized
Before introducing the main results of this work, we first control system should be paired in such a way that (i) all paired
present some concepts that are important to our later development. RGA elements are positive, (ii) the NI is positive, (iii) the paired
2.1. Normalized Gain and Relative Normalized Gain RNGA elements are closest to 1.0, and (iv) large RNGA
Array (RNGA). In control system configuration, two factors elements are avoided. Because this pairing criterion takes the
in the open-loop transfer functions affect the loop pairing RNGA, RGA, and NI into consideration, it offers important
decision and should be focused on when considering the effect insights into the issue of control structure selection. The RNGA
of interactions: steady-state information, because the steady- is used to measure the loop interactions over the whole
state gain of the transfer function gij(s) reflects the effect of frequency range, whereas the RGA and NI are used as sufficient
manipulated variable uj on controlled variable yi when the system conditions to rule out closed-loop unstable pairings.
is stable, and transient information, as the transient information 2.3. Equivalent Transfer Function (ETF). To reveal the
of the transfer function gij(s) is accountable for the sensitivity model relations between the conditions under which all loops
of the controlled variable yi to manipulated variable uj and, are open and all loops are closed, the relative average resident
consequently, the promptness of a particular output response time (RART), γij, was introduced. It is defined as the ratio of
to an input and the ability to reject the interactions from other loop yi-uj average resident times between when other loops
loops. are closed and when other loops are open14
 Ind. Eng. Chem. Res., Vol. 49, No. 2, 2010 763

γij }
τ̂ar,ij
τar,ij
i, j ) 1, 2, ..., n (8) κij ) { 1, gc,ij(s)
0, no controller
for i * j; i, j ) 1, 2, ..., n
(15)
where τ̂ar,ij is the average resident time when other loops are
closed. The advantages of sparse control lie in the fact that the control
When RARTs are calculated for all average resident times performance can be improved with limited structure complexity
of a transfer function matrix, the result is an array with a form while still preserving control system integrity.
similar to that of RNGA, called the relative average residence To unify the control structure selection procedure, we first
time array (RARTA), which is calculated as put the selected pairs on the diagonal positions by column
Γ } ΛN . Λ transformation of both the RNGA and the transfer function

][ ]
matrix, and we then rationalize the matrix by making all

[
diagonal elements equal to 1, which can be calculated by
λN,11 λN,12 ... λN,1n λ11-1 λ12-1 ... λ1n-1 defining an “interaction index”, B ) [ij]n×n, where
λN,21 λN,22 ... λN,2n λ21-1 λ22-1 ... λ2n-1

| | | |
) · X · (9) λN,ij λij γij
l l ·. l l l ·. l ij ) ) (16)
-1 -1 λN,ii λii γii
λN,n1 λN,n2 ... λN,nn λn1 λn2 ... λnn-1
To analyze the effects of ij on the selection of κij, consider
From the definition of the RARTA, the closed-loop average
two extreme cases:
residence time can then be written as
(1) If ij, j * i, is very small, then this implies that either
τ̂ar,ij ) γijτar,ij ) γijτij + γijθij (10) λij/λii is very small or γij/γii is very large. λij/λii being very small
would mean that kij is very small compared with kii. In this case,
} τ̂ij + θ̂ij
the input uj has very little influence on yi. On the other hand,
By assigning the equivalent transfer functions (ETFs) when other γij/γii being very small means that τar,ij is very small compared
loops are closed to have the same structure as the open-loop with τar,ii, the i-j loop reacts very rapidly, and the effect of the
transfer functions, the ETFs can be approximated in terms of fast loop appears as a high-frequency disturbance that can be
relative gains and relative average resident times when the effectively filter out by the relatively slow paired control loop.
control system is closed. The ETFs for FOPDT processes are (2) If ij, j * i, is very large, then this implies that either
given as λij/λii is very large or γij/γii is very small. If λij/λii is very large,
then, with this loop included, the steady-state gain matrix is
1 kij 1
ĝij(s) ) k̂ij e-θ̂ijs ) e-γijθijs (11) nearly singular. The system is very sensitive to modeling errors,
τ̂ijs + 1 λij γijτijs + 1 so that small modeling errors will be magnified into very large
errors in yi and a small change in controller output uj will also
3. Control Structure Selection result in large errors in yi. Control will be difficult to achieve
for such a loop, and it will also be very sensitive to modeling
Generally, a multivariable process can be controlled by
errors. On the other hand, the fact that γij/γii is very large means
decentralized, decoupling, or sparse control schemes. Each of
that τar,ij is very large compared with τar,ii, such that the loop
these types of control is briefly discussed here.
reacts very slowly. The effect of the slow loop appears as a
Decentralized control has a diagonal controller structure

[ ]
constant disturbance that can be effectively rejected by the paired
gc1(s) 0 ... 0 loop controllers.
0 gc2(s) ... 0 Based on the preceding analysis of the process characteristics,
Gc(s) ) · (12) the economic value of improved control, and the reasonable
l l ·. l
computing and process control resources, we propose the
0 0 ... gcn(s) following control selection criterion for κij as general guideline
and is the simplest control structure for multivariable processes. for engineering applications
Decoupling control is a full control structure

[ ]
gc,11(s) gc,12(s) ... gc,1n(s)
gc,21(s) gc,22(s) ... gc,2n(s)
κij ) { 1, 0.15 e ij e 8
0, otherwise
for i * j; i, j ) 1, 2, ..., n
(17)
Gc(s) ) · (13)
l l ·. l
gc,n1(s) gc,n2(s) ... gc,nn(s) Because the RNGA for a 2 × 2 system is symmetric, it should
be controlled by either a decentralized or decoupling control
that first eliminates the effects of the undesirable cross-couplings scheme, whereas for n > 2, decentralized, decoupling, and sparse
using off-diagonal controllers, such that the process can be control schemes all can be applied, at least in theory. However,
treated as multiple single loops, and less-conservative single- because a process suitable for full decoupling control can seldom
loop PID control design methods can be directly applied by be found, a decentralized or sparse control scheme is generally
diagonal controllers. more appropriate.
In sparse control, a sparse controller structure

[ ]
4. Independent Design
gc,11(s) κ12 ... κ1n
κ21 gc,22(s) ... κ2n For a closed-loop-controlled MIMO system, it is desirable
Gc(s) ) · (14)
l l ·. l that the forward transfer function be of the form
κn1 κn2 ... gc,nn(s) I
G(s) Gc(s) ≈ (18)
where κij is the off-diagonal controller index s
764 Ind. Eng. Chem. Res., Vol. 49, No. 2, 2010

Case 1: |λ̂ij| e 1, γ̂ij g 1. |λ̂ij| e 1 means that the retaliatory

effect when all the other loops are closed magnifies the effect
of uj on yi, so the controller gain should be reduced accordingly
to ensure system stability. In this case
kij
Figure 2. Block diagram of the independent design method. k̂ij ) (24)
λij
such that the sensitivity and complementary sensitivity functions λγ̂ij g 1 implies that the average residence time when the
approach 0 and 1, respectively. other loops are closed is not less than that when the other loops
From the relationship between G-1(s) and the ETF matrix, are open. The enlarged residence time will make the critical
we have14 frequency shift to the left and reduce the phase margin. In this

[ ]
case
ĝ11-1(s) ĝ21-1(s) · · · ĝn1-1(s)
τ̂ar,ij ) γijτar,ij (25)
-1 ĝ12-1(s) ĝ22-1(s) · · · ĝn2-1(s)
G (s) ) Ĝ (s) )
T
· Case 2: |λ̂ij| > 1, γ̂ij < 1. |λ̂ij| > 1 indicates that the retaliatory
l l ·. l effect when all other loops are closed reduces the effect of uj
ĝ1n-1(s) ĝ2n-1(s) · · · ĝnn-1(s) on yi. Because the controller gain cannot be enlarged for better
(19) performance because of system integrity considerations, no

[ ]
magnitude detuning for gain is needed. If λ̂ij < 0, then the sign
Substituting 19 into eq 18 gives of the steady-state gain is still supposed to be changed. In this
case
ĝ11-1(s) ĝ21-1(s) ĝ -1(s)
· · · n1
s s s k̂ij ) kij (26)
ĝ12-1(s) ĝ22-1(s) ĝn2-1(s) γ̂ij < 1 means that the average residence time when the other
I ···
Gc(s) ≈ G-1(s) ) s s s loops are closed is less than that when the other loops are open.
s ·
l l ·. l The reduced residence time will make the critical frequency
ĝ1n-1(s) ĝ2n-1(s) ĝ -1(s) shift to the right and enlarge the phase margin. In this case, by
· · · nn considering system integrity, one obtains
s s s
Define an error function as τ̂ar,ij ) τar,ij (27)
Case 3: |λ̂ij|e1, γ̂ij < 1. |λ̂ij|e1 is the same as in case 1, eq
E(s) ) Gc(s) - s-1ĜT(s) (20) 23.
To minimize the error function, we define the objective function γ̂ij < 1 is the same as in case 2, eq 27.
Case 4: |λ̂ij| > 1, γ̂ij g 1. |λ̂ij| > 1 is the same as in case 2, eq
n n 26.
J ) min |E(s)| ) min ∑ ∑ |G (s) - sc
-1
ĜT(s)| ij γ̂ij g 1 is the same as in case 1, eq 25.
i)1 j)1 In summary, the integrity of sparse control can be preserved
(21) if the ETFs for controller design are updated as
where the subscript ij indicates the element in the ith row and
jth column of [ · ].
Because
k̂ij ) { kij /λij, |λij |<1
kij, |λij | g 1
(28)

{
γijτar,ij ) γijτij + γijθij, γij > 1

{
|gc,ij - s-1ĝji |, gc,ij(s) * 0 τ̂ar,ij ) (29)
-1 τar,ij, 0 < γij e 1
|Gc(s) - s Ĝ (s)| ij ≈
T
|-s-1ĝji |, gc,ij(s) ) 0
Because perfect control is impossible, a PI/PID controller can
(22) be used to design gc,ij(s) such that the closed loop of gc,ij(s) ĝji(s)
has good dynamic properties. Under the gain and phase margin
the minimization of J requires that gc,ij(s) be determined by
(GPM) design method, the loop forward transfer function is
1 usually expressed as
gc,ij(s) - s-1ĝij-1(s) ) 0 ⇒ gc,ij(s) ĝji(s) ) (23)
s kP,ij -Lijs
Imagine that gc,ij(s) ĝji(s) is the forward transfer function of an gc,ij(s) ĝji(s) ) e (30)
s
artificial closed control system as shown in Figure 2. Then,
gc,ij(s) ĝji(s) ) 1/s is the ideal control performance target for Let the PID controller be of the form
the loop, and the controller design is totally independent of the kI,ij
other loops. gc,ij(s) ) kP,ij + + kD,ijs (31)
In designing controllers for multivariable control systems, the s
closed-loop integrity is very important to guarantee that the The ETFs together with the PID controller parameters for
overall system remains stable regardless of the removal or FOPDT are summarized for different combinations of λij and
insertion of control loops. The integrity requires that each γij in Table 1.
individual loop controller be no more aggressive than the Remark 1. If the loop transfer function has a high D/τ ratio,
original single-loop controller without interactions. For different the GPM method will result in a very aggressive PI/PID
combinations of λ̂ij and γ̂ij, general rules for the design of each controller and deteriorate the whole stability. In such a case,
loop controller are discussed as follows: internal model control (IMC)-Maclaurin is recommended for
 Ind. Eng. Chem. Res., Vol. 49, No. 2, 2010 765
Table 1. ETFs and PI Parameters for FOPDT Process
mode ĝij(s) kP,ij kI,ij

kij 1 πλijTij πλij

|λij| < 1, 0 < γij e 1 e-θijs
λij τijs + 1 2Amθijkij 2Amθijkij

kij 1 πλijTij πλij

|λij| < 1, γij > 1 e-γijθijs
λij γijTijs + 1 2Amθijkij 2Amγijθijkij

sign(λij) · kij -γijθijs πTij π

|λij| g 1, γij > 1 γijTijs + 1
e
2Amθijkij 2Amγijθijkij

sign(λij) · kij -θijs πTij π

|λij| g 1,0 < γij e 1 Tijs + 1
e
2Amθijkij 2Amθijkij

processes with high D/τ ratios (D/τ g 8),24 where the PID The index matrix B is then calculated as
controller takes the form

gc,ij(s) ) kc,ij (1+

1
+
τD,ijs
) (32)
B) [ 1.0000 0.3601
0.3601 1.0000 ]
τI,ijs RτD,ijs + 1 For comparison, the gain and phase margins for all loops are
Remark 2. If λij is negative, λij < 0, but 0.15 < ij, this implies specified as Am,i ) 4 db and Φm,i ) 3π/8 rad, respectively.
that uj has a significant effect on yi in the opposite direction Decentralized Control. Because λii > 1 and γii < 1 for i )
under closed-loop control. In such a case, a controller for the 1, 2, the original transfer functions are used for controller design,
loop can still be added to improve the overall system perfor- that is
mance; however, the system integrity will be lost because of
12.8
the addition of the control loop. ĝ11(s) ) g11(s) ) e-s
Remark 3. The significance of the proposed independent 16.7s + 1
-19.4e-3s
design technique is that decentralized, decoupled, and sparse ĝ22(s) ) g22(s) )
control schemes can all be treated under a unified framework. 14.4s + 1
The transformation between schemes is simply realized by The decentralized controller is obtained as
adding or reducing the independent control loops.
Gc_decentralized(s) )

[ ]
5. Case Study 0.03068
0.5123 + 0
s
In this section, the proposed control design technique is -0.006747
applied to three typical industrial processes. Example 1 is 0 -0.09716 +
s
employed to show that decoupling control provides better
Decoupling Control. Because 12 ) 21 > 0.15, the interac-
performance than decentralized control for a 2 × 2 process,
tion between loops cannot be ignored according to the proposed
whereas examples 2 and 3 show that sparse control has obvious
criterion. gc,12 and gc,21 should be added to the above decentral-
advantages over the decentralized control structure.
ized control to make it a full decoupling control structure to
Example 1. Consider the Wood and Berry process (1973)25

[ ]
obtain better performance.
12.8e-s -18.9e-3s The ETFs of g12 and g21 are calculated as
16.7s + 1 21s + 1
G(s) ) ĝ12(s) )
18.9 -3s
6.6e-7s -19.4e-3s 21s + 1
e
10.9s + 1 14.4s + 1 -6.6
ĝ21(s) ) e-7s
The RGA (Λ), normalized gain matrix (ΚN), RNGA (ΛN), and 10.9s + 1
RARTA (Γ) can be calculated, respectively, as The resulting controllers are

Λ)
.
[ 2.0094 -1.0094
-1.0094 2.0094 ] gc,12(s) ) -0.09265 +
-0.0085
s

[ ]
0.006926
0.7232 -0.7875 gc,21(s) ) 0.1468 +
ΚN ) s
0.3687 -1.1149
. Upon addition of these controllers to the decentralized controller
ΛN ) [ 1.5628 -0.5628
-0.5628 1.5628 ] matrix, the full decoupling controller is obtained as
Gc_decoupling(s) )

[ ]
.
Γ) [
0.7778 0.5576
0.5576 0.7778 ] 0.5123 +
0.03068
s
-0.09265 +
-0.0085
s
The best pairing solution is 1-1/2-2, according to the 0.006926 -0.006747
0.1468 + -0.09716 +
RGA-NI-RNGA rules. s s
766 Ind. Eng. Chem. Res., Vol. 49, No. 2, 2010

Figure 3. Step responses of WB process.

The step responses under three different control structures are The RGA (Λ), normalized gain matrix (ΚN), RNGA (ΛN),
given in Figure 3. For comparison, the simulation results of RARTA (Γ), and index matrix B are calculated, respectively,
Zhang et al.26 are also presented. This figure shows that full as:

[ ]
decoupling control results in better overall performance than
can be obtained with decentralized and partial decoupling 2.0445 -0.7149 -0.3296
control, due to the fact that 0.15 < 12 ) 21 < 8. Because λ12 Λ ) -0.6275 1.8329 -0.2055
) λ21 ) -1.0094 < 0, however, the system integrity will be -0.4170 -0.1180 1.5351

[ ]
lost. Indeed, if the u1-y1 loop is placed in manual control mode .
while the other control loops are unchanged, the system becomes 0.0710 -0.0502 -0.0005
unstable. ΚN ) 0.1138 -0.2950 -0.0012
Example 2. Consider a 3 × 3 process given by Ogunnaike -1.9988 2.2759 0.0943

[ ]
.
and Ray (1979)27 1.4827 -0.3485 -0.1342

[ ]
G(s) ) ΛN ) -0.3443 1.407 -0.0614
0.66 -2.6s -0.61 -3.5s -0.0049 -s -0.1384 -0.0572 1.1956

[ ]
e e e .
6.7s + 1 8.64s + 1 9.06s + 1
1.11 -2.36 -3s -0.01 -1.2s 0.7252 0.4875 0.4071
e-6.5s e e Γ ) 0.5488 0.7669 0.2988
3.25s + 1 5s + 1 7.09s + 1
-34.68 -9.2s 46.2 0.87(11.61s + 1) 0.3318 0.4845 0.7788
e-9.4s e-s

[ ]
e .
8.15s + 1 10.9s + 1 (3.89s + 1)(18.8s + 1)
1 0.2395 0.0952
Using the least-squares method,21 the second-order plus dead B ) 0.2434 1 0.0460
time (SOPDT) element g33(s) can be simplified to
0.1211 0.0501 1
0.87(11.61s + 1) 0.7922
g33(s) ) [ (3.89s + 1)(18.8s + 1) ] 7.936s + 1
e ≈ -s
e -0.465s
For comparison, three different control structures are adopted,
and the gain and phase margins for all loops are specified as
For control system design, the original transfer function matrix Am,i ) 4 db and Φm,i ) 3π/8 rad, respectively.
becomes Decentralized Control. According to the adjustment rules,

[ ]
the original transfer functions are selected to be the ETFs
Ḡ(s) )
0.66 -2.6s -0.61 -3.5s -0.0049 -s 0.66 -2.6s
e e e ĝ11(s) ) g11(s) ) e
6.7s + 1 8.64s + 1 9.06s + 1 6.7s + 1
1.11 -2.36 -3s -0.01 -1.2s -2.36 -3s
e-6.5s e e ĝ22(s) ) g22(s) ) e
3.25s + 1 5s + 1 7.09s + 1 5s + 1
-34.68 -9.2s 46.2 0.7922 -0.465s 0.7922 -0.465s
e e-9.4s e ĝ33(s) ) g33(s) ) e
8.15s + 1 10.9s + 1 7.936s + 1 7.936s + 1
 Ind. Eng. Chem. Res., Vol. 49, No. 2, 2010 767

Figure 4. Step response of OR process.

[ ]
The corresponding decentralized controller is obtained as gc,11 gc,12 0
Gc_sparse1(s) ) gc,21 gc,22 0

[ ]
Gc_decentralized(s) )
0 0 gc,33
0.2288
1.533 + 0 0 The ETFs for the off-diagonal elements are
s
-0.05547
0 -0.2773 + 0 0.8533 -3.5s
s ĝ12(s) ) e
1.066 8.64s + 1
0 0 8.4601 + -1.7691 -6.5s
s ĝ21(s) ) e
Sparse Control 1. According to the structure selection 3.25s + 1
criterion in eq 17, sparse control should have the following Adding the two off-diagonal controllers, the sparse controllers
structure are obtained as
 [ ]
768 Ind. Eng. Chem. Res., Vol. 49, No. 2, 2010

0.2288 -0.03415
1.533 + -0.111 + 0
s S
0.1315 -0.05547
Gc_sparse1(s) ) 1.136 + -0.2773 + 0
s s
1.066
0 0 8.4601 +
s

Sparse Control 2.
Adding two more controllers that have relatively large  values, we obtain the control structure of

[ ]
gc,11 gc,12 gc,13
Gc_sparse2(s) ) c,21 gc,22 0
g
gc,31 0 gc,33
and for |λ13| < 1, γ13 < 1, and |λ31| < 1, γ31 < 1, the ETFs of g13 andg31 are
0.0149 -s
ĝ13(s) ) e
9.06s + 1
83.1630 -9.2s
ĝ31(s) ) e
8.15s + 1

[ ]
The final controller has the form
0.2288 -0.03415 0.0005133
1.533 + -0.111 + 0.004183 +
s S S
0.1315 -0.05547
Gc_sparse2(s) ) 1.136 + -0.2773 + 0
S s
0.2896 1.066
2.624 + 0 8.4601 +
S s
The results of the closed-loop step responses under the three control structures are given in Figure 4.
The simulation results reveal the following: (1) Sparse control indeed provides a great improvement in performance compared
with decentralized control. (2) The closed-loop performance shows no improvement upon addition of additional loops that do not
meet the structure selection criterion.

[ ]
Example 3.
Consider a 4 × 4 process given by Alatiqi (1985)28

4.09e-1.3s -6.36e-0.2s -0.25e-0.4s -0.49e-5s

(33s + 1)(8.3s + 1) (31.6s + 1)(20s + 1)
21s + 1 (22s + 1)2
-4s -1.01s -5s
-4.17e 6.93e -0.05e 1.53e-2.8s
45s + 1 44.6s + 1 (34.5s + 1) 2
48s + 1
G(s) )
-1.73e-17s 5.11e-11s 4.61e -1.02s
-5.48e-0.5s
(13s + 1)2 (13.3s + 1)2 18.5s + 1 15s + 1
-2.6s -0.02s -0.05s
-11.18e 14.04e -0.1e 4.49e-0.6s
(43s + 1)(6.5s + 1) (45s + 1)(10s + 1) (31.6s + 1)(5s + 1) (48s + 1)(6.3s + 1)
The RGA (Λ), normalized gain matrix (ΚN), RNGA (ΛN), and RARTA (Γ) can be calculated, respectively, as

[ ]
3.1058 -0.9007 -0.4749 -0.7302
-5.0308 4.6742 -0.0395 1.3961
Λ)
-0.0838 0.0543 1.5492 -0.5197
3.0088 -2.8278 -0.0348 0.8538

[ ]
0.2808 -0.2575 -0.0117 -0.0181
-0.0851 0.1519 -0.0013 0.0301
ΚN )
-0.0577 0.2103 0.2362 -0.3535
-0.8047 0.8570 -0.0115 0.3825

[ ]
1.8033 -0.4423 -0.1774 -0.1836
-1.0320 2.2582 0.0291 -0.2553
ΛN )
-0.0212 0.0246 1.2330 -0.2364
0.2498 -0.8405 -0.0847 1.6754

[ ]
0.5806 0.4910 0.3736 0.2515
0.2051 0.4831 -0.7361 -0.1829
Γ)
0.2530 0.4523 0.7959 0.4549
0.0830 0.2972 2.4314 1.9622
 Ind. Eng. Chem. Res., Vol. 49, No. 2, 2010 769

Figure 5. Response and IAE values of A2 process.

770 Ind. Eng. Chem. Res., Vol. 49, No. 2, 2010

According to the pairing rules, this is a diagonal pairing (i.e., the loops should be paired as 1-1, 2-2, 3-3, and 4-4), and the index
matrix B ) [ij]n×n is calculated as

[ ]
1.0000 0.2452 0.0984 0.1018
0.4570 1.0000 0.0129 0.1131
B)
0.0172 0.0199 1.0000 0.1917
0.1491 0.5017 0.0505 1.0000
Decentralized Control.
According to the adjustment rules, the ETFs of the diagonal transfer functions are calculated as

4.09e-1.3s
ĝ11(s) )
273.9s2 + 41.3s + 1
6.93e-1.01s
ĝ22(s) )
44.6s + 1
4.61e-1.02s
ĝ33(s) )
18.5s + 1
5.2588
ĝ44(s) ) e-1.1773s
302.4s2 + 27.6730s + 1

[ ]
The decentralized controller is designed using the gain and phase method (Am,i ) 4 db) as
0.0739
3.0503 + + 20.2295s 0 0 0
s
0.05611
0 2.502 + 0 0
s
Gc_decentralized(s) )
0.08351
0 0 1.545 + 0
s
0.0634
0 0 0 1.7530 + + 19.1809
s

Sparse Control.
Because 12 > 0.15, 21 > 0.15, 42 > 0.15, and 34 > 0.15, the sparse control structure should includegc,21, gc,12, gc,24, and gc,43 loops.
The ETFs for these transfer functions are obtained, respectively, as

(-6.36/-0.9007)e-0.2s 7.0612e-0.2s
ĝ12 ) )
(31.6s + 1)(20s + 1) (31.6s + 1)(20s + 1)
(-4.17/-1)e-4s 4.17e-4s
ĝ21 ) )
45s + 1 45s + 1
(14.04/-1)e-0.02s -14.04e-0.02s
ĝ42 ) )
(45s + 1)(10s + 1) (45s + 1)(10s + 1)
(-5.48/-0.5197)e-0.5s 10.5445e-0.5s
ĝ34 ) )
15s + 1 15s + 1
Considering the high D/τ ratios of the above four ETFs, the off-diagonal controllers are designed by the IMC-Maclaurin method.
The sparse controllers are then obtained as

[ ]
Gc_sparse(s) )
0.0739 0.02176 0.8628s
3.0503 +
s
+ 20.2295s 1.2227 1 +( s
+
8.631s + 1 ) 0 0
0.02037 10.4103s 0.05611 0.0190 8.1913s
(
0.6824 1 +
s
+
10.14s + 1 ) 2.502 +
s
0 (
-0.3732 1 +
s
+
64.95s + 1 )
0.08351
0 0 1.545 + 0
s
0.0665 0.0353s 0.0634
0 0 (
0.4074 1 +
s
+
3.532s + 1 ) 1.7530 +
s
+ 19.1809

The results of the closed-loop step response and integral absolute error (IAE) values are given in Figure 5.
It can be seen that the overall performance of sparse control is superior.

6. Conclusion
In this work, by introducing the concepts of interaction index, equivalent transfer function, and independent design, we have
presented a systematic design approach for multivariable processes. The structure selection criterion provides a compromise between
system performance and structural complexity, whereas the equivalent transfer function and independent design make the conversion
between decentralized, decoupling, and sparse control schemes possible simply through the addition or removal of loop controllers.
This method is very easy to understand and implement. The simulation results for several industrial processes show that the selected
 Ind. Eng. Chem. Res., Vol. 49, No. 2, 2010 771
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