Guidelines for the tuning and the evaluation of decentralized and decoupling controllers for processes with recirculation

ISA Transactions 40 (2001) 341±351
www.elsevier.com/locate/isatrans

Guide lines for the tuning and the evaluation of decentralized
and decoupling controllers for processes with recirculation
Dominique Pomerleau a,*, Andre Pomerleau b
Â
a Âs Âbec, Canada, J3H 6C3
Breton, Banville et associe s.e.n.c., 375 Boul. Laurier, Mont-St-Hilaire, Que
b Á Ârale), Department of Electrical and Computer
GRAIIM (Groupe de recherche sur les applications de l'informatique a l'industrie mine
Â
Engineering, Universite Laval, Ste-Foy, Que Âbec, Canada, G1K 7P4

Abstract
This paper gives guidelines for the pairing, the time response speci®cation, and the tuning for processes with recir-
culation when decentralized controllers are used. This selection is based on the condition number, which is an indicator
of the process directionality, and on the generalized dynamic relative gain (GDRG), which is a measure of the inter-
action. Simple tuning rules are developed and results are compared to algebraic controllers with decouplers. Perfor-
mances are evaluated for set-point changes as well as disturbance rejection using the generalized step response (GSR).
The GSR gives a 3D graphic of the system response as a function of the input direction. # 2001 Published by Elsevier
Science Ltd. All rights reserved.
Keywords: Pairing; PID tuning; Decentralized control

1. Introduction main conclusion is a decrease in the understanding
of the circuit. This leads to important problems in
Mineralurgical and chemical industries have a operating the process in manual. A lack of proper
multitude of multivariable processes which, for choice of control structure and ecient tuning
reasons of e€ectiveness, have circulating loads. methods are also major reasons. The addition of
Blakey et al. [1] made a study of the advantage of circulating loads creates zeros in the process
recirculating loads on a ¯otation circuit. They transfer functions and requires tuning rules taking
demonstrated that a signi®cantly higher grade± into account these zeros [4]. It is thus important
recovery relationship is possible for rougher-sca- to be able to anticipate the e€ect of the open-
venger circuit designs that incorporate circulating loop system characteristics on the closed-loop
loads. A recent trend in Canadian mineral industry, system response and to develop simple rules for
though, has been the reduction of the number of the tuning of controller for such multivariable
recirculating loads in processing ¯ow sheet design. processes.
This philosophy results from diculties observed Good tuning of decentralized PI controllers for
in day-to-day plant operability. Stowe [2] and multivariable processes is relatively complex. In
Edwards and Flinto€ [3] discussed the operation particular, the design of single-input single-output
problems of ¯otation circuits with recycle. The (SISO) controllers for highly coupled multivariable
processes often leads to poor performance because
* Corresponding author. Fax: +1-418-656-3159 of a bad choice of manipulated variables, poor
E-mail address: dpomerle@hotmail.com (D. Pomerleau). speci®cations and poor tuning of the controllers.
0019-0578/01/$ - see front matter # 2001 Published by Elsevier Science Ltd. All rights reserved.
PII: S0019-0578(00)00040-9

342 D. Pomerleau, A. Pomerleau / ISA Transactions 40 (2001) 341±351

Despite considerable work on decoupling con- mathematical measure of directionality is given by
trollers, decentralized PI controllers remain the the singular value decomposition (SVD) [12]. The
standard for most industries. According to Skoges- singular values give, for each frequency, the max-
tad and Morari [5], they have fewer tuning para- imum [ …j!†] and the minimum [ …j!†] values of the
meters, are easier to understand, and are more easily gain of the process and the singular vectors give the
made failure tolerant. Furthermore, decoupling directions of theses maximum and minimum. The
controllers are complex, require excessive engi- gain of a multivariable process, at a given fre-
neering manpower, have a lack of integrity, have a quency, is not limited to a single value but a range
lack of robustness and often result in operator non of possible values between …j!† and …j!†. A pro-
acceptance, according to Luyben [6]. cess with a wide range of possible gains has a large
For decentralized controllers, Desbiens et al. [7] directionality and a process with a small range of
have proposed a method where the time speci®ca- possible gains has a low directionality. This char-
tions in closed loop have to be given and the con- acteristic is important because processes with large
trollers are evaluated by solving two quadratic directionality can show control problems [13±15].
equations. Some authors have proposed tuning The ratio …j!†= …j!† is called the condition num-
methods which take into account the process uncer- ber. It is an indicator of the directionality or how
tainty. Skogestad and Morari [8] have proposed an ill-conditioned the process is.
independent tuning method for decentralized con- Here, a more intuitive representation, based on
trollers based on individual loop conditions. They the step response, is given for measuring two
have derived their conditions from the global inputs±two outputs (TITO) processes direction-
robust performance condition of the m-synthesis ality and for the evaluation of the closed-loop
environment. Chiu and Arkun [9] and Ito et al. system characteristics. The process input u…t† or
[10] have proposed sequential design methods for the disturbance d…t† can be represented at a time t
decentralized controllers. Gagnon et al. [11] have in a condensed form by a vector d…t† with ampli-
also use the robust performance concept de®ned in tude given by its L2-norm. Similarly, the outputs
the m-synthesis environment. can also be represented by a vector, y…t†, with an
In this paper, the condition number, which is a amplitude given by its L2-norm. The method con-
measure of directionality, and the generalized sists in simulating the TITO process when the
dynamic relative gain (GDRG), which is a measure inputs di …t† are step functions. Keeping the ampli-
of interaction, are used to determine the most tude of d…t† constant (Fig. 1) and simulating y…t†
appropriate control structure (decentralized or
decoupling controllers). They also give the possibi-
lity to determine the pairing and the time response
speci®cations. From there, in decentralized con-
trol, the tuning of the SISO controllers based on
an approximation of the transfer functions seen by
each one is given. The controllers obtained are
compared to the corresponding controllers where
a decoupler is inserted between the process and the
controllers. Both control structures are compared
for set-point changes as well as in regulation using
the generalized step input (GSR).

2. Process characteristics

Multivariable processes are mainly character-
ized by their directionality and interaction. A Fig. 1. Input vector for TITO process.

D. Pomerleau, A. Pomerleau / ISA Transactions 40 (2001) 341±351 343

for all possible directions of d…t† generates the is preferred to the RGA, which only considers the
generalized step response (GSR). steady-state.
The GSR gives information about directionality,
but it is not a measure of the interaction. Indeed, a
process with a large directionality can have no 4. Tuning
interaction. A TITO process with zero gains in the
cross-coupled transfer functions and a large and a 4.1. Decentralized control
low gain in the direct branches is an example of a
multivariable process with high directionality. As for SISO processes, the tuning of decen-
Directionality can come from the intrinsic properties tralized controllers consists in opening the loop
of a process (ex. system with an even number of under study and evaluating the transfer function
positive gain for a TITO process) or a system seen by the controller as presented in Fig. 2. The
where the actuators are badly sized. transfer function seen by controller Gc1 …s† is G1 …s†
and the one seen by controller Gc2 …s† is G2 …s† where
G1 …s† and G2 …s† are given by:
3. Interaction and pairing
G12 …s†G21 …s†Gc2 …s†
G1 …s† ˆ G11 …s† À …2†
An interesting interaction measure is the gen- 1 ‡ Gc2 …s†G22 …s†
eralized relative dynamic gains (GRDG) of Huang
et al. [16]. The GRDG takes into account the G12 …s†G21 …s†Gc1 …s†
G2 …s† ˆ G22 …s† À …3†
dynamics of the closed-loops. The GRDG l11 …s† 1 ‡ Gc1 …s†G11 …s†
for a TITO process is de®ned as follows:
Eqs. (2) and (3) show that the tuning of one
R 2 … s† controller depends on the other one controller. The
Gp11 …s†Gp22 …s† system can then be separated into two SISO
Y 2 … s†
l11 …s† ˆ …1† systems as seen in Fig. 3. A set-point change on one
R 2 … s†
Gp11 …s†Gp22 …s† À Gp12 …s†Gp21 …s† loop is seen as a disturbance by the other loop.
Y 2 … s†
Di€erent approximations can be used to evalu-
ate G1 …s† and G2 …s†. Since the controllers include
where R2 …s†=Y2 …s† is the desired dynamics of the an integrator to prevent static errors, a possible
second loop. The variables R2 …s† and Y2 …s† are the approximation at frequencies lower than the
set point and the process output of the other loop cross-over frequency (!co ) is:
respectively. The transfer functions Gp11 …s†, Gp12 …s†,
Gp21 …s† and Gp22 …s† are the elements of the process G12 …s†G21 …s†
G1 …s† ˆ G11 …s† À for Gc2 …s†G22 …s† 1
transfer matrix Gp …s†. In this paper, a representa- G22 …s†
tion of the GRDG is given as a function of both …4†
closed-loop bandwidths [17]. For easier control
and tuning, the speci®cations on the closed-loop
set point responses have to be chosen in a frequency
band where interaction is reduced so the system
behaves more like SISO systems. In order to do so,
the closed-loop response speci®cations are chosen in
frequency band where the GRDG is close to one
since, as for relative gain array (RGA), it means
that the interaction is low.
Because the zeros in a transfer function a€ect
the process dynamic, the GRDG, which takes in
account the dynamic part of the transfer function, Fig. 2. Decentralized control.


with G21 …s†. The error made by the approximation
has then a reduced importance for the transfer
function seen by the controller. If the time con-
stants of G12 …s† and G21 …s† are smaller than the
crossover frequency !co , the ®ltering e€ect will be
reduced but, in this case, the gains K12 and K21
should be much smaller than K11 and the transfer
function seen by the controller will depend mostly
on G11 …s†. If this is not the case, the wrong pairing
has been used.

Fig. 3. Equivalent system (decentralized control). 4.2. Decoupling controllers

and For the tuning of the controllers when a decou-
pler is inserted between the process and the con-
G12 …s†G21 …s† troller, as seen in Fig. 4, one has:
G2 …s† ˆ G22 …s† À for Gc1 …s†G11 …s† 1
G11 …s†
…5† ÀG12 …s†D22 …s†
D12 …s† ˆ …8†
G11 …s†
This facilitates the tuning since the transfer
function seen by one controller is independent of ÀG21 …s†D11 …s†
D21 …s† ˆ …9†
the other controller. For the other output variable, G22 …s†
the system is in regulation. The process dynamics
on the regulated variable depends primary on the The transfer functions seen by each controller
dynamic of the manipulated variable where the are then given by:
set-point change occurred. From Eqs. (4) and (5),
one can expect a slow response if the transfer G12 …s†G21 …s†D11 …s†
G1 …s† ˆ G11 …s† À
functions in the direct branches contain a large G22 …s†
time constant in the numerator since it is trans- G12 …s†G21 …s†D22 …s†
G2 …s† ˆ G22 …s† À …10†
lated as a pole in the controller. G11 …s†
This relation cannot be applied if the transfer
functions of G11 …s† or G22 …s† contain an unstable
zero or a delay longer than 12 …s† ‡ 21 …s†, where It is observed that the transfer functions seen by
represents the process delay. In these cases, the each controller are the same as the ones seen by
approximation given by Eqs. (6) and (7) can be the decentralized controllers when Eqs. (4) and (5)
used. are used.

G12 …s†G21 …s†
G1 …s† ˆ G11 …s† À …6†
K22

and

G12 …s†G21 …s†
G2 …s† ˆ G22 …s† À …7†
K11

where K11 and K22 are, respectively, the gains of
G11 …s† and G22 …s†. Generally, the transfer function
Gc2 …s†
1‡Gc2 …s†G22 …s† is low pass ®ltered by G12 …s† in series Fig. 4. Control with decouplers.


5. Evaluation Gc …s†Gp …s†
U … s† ˆ L…s† …12†
1 ‡ Gc …s†Gp …s†
It is always dicult to make a valuable evalua-
tion of di€erent controllers. Here, since the speci-
®cations are given for set-point changes, similar Where L…s† is the disturbance and where Gc …s†
dynamics will be taken as a reference point for represents the controller. When a decoupler is
both control structures and the controllers will be used for the system, Gc …s† includes the decoupler.
evaluated for both set-point changes and in reg- Here, a limited number of cases will be studied
ulation for a GSR at the process inputs. For sym- and we will try to generalize the results. The dif-
metrical process, the GSR [Y…s†] to set point ferent processes under consideration are given in
changes is equivalent to the manipulated variables Table 1. System A and B are only di€erent in the
[U…s†] in regulation for a disturbance at the pro- signs of the gain of G12 …s†. Only two di€erent signs
cess input, as illustrated by Eqs. (11) and (12). of the gain are studied, since all the other cases can
be deduced from these two as seen in Table 2.
Gc …s†Gp …s† Cases 1, 6, 7, 8, 9, 10, 11 and 16 are similar to
Y…s† ˆ R…s† …11†
1 ‡ Gc …s†Gp …s† system A where the number of positive gain sign is

Table 1
Di€erent processes under considerations

System A System B

4 4
Initial process G11 ˆ G22 ˆ G11 ˆ G22 ˆ
1 ‡ 10s 1 ‡ 10s

3 À3 3
G12 ˆ G21 ˆ G12 ˆ G21 ˆ
1 ‡ 10s 1 ‡ 10s 1 ‡ 10s

4…1 À 10s† 4 4…1 À 10s† 4
Process 1 G11 ˆ G22 ˆ G11 ˆ G22 ˆ
…1 ‡ 10s†2 1 ‡ 10s …1 ‡ 10s†2 1 ‡ 10s

3 À3 3
G12 ˆ G21 ˆ G12 ˆ G21 ˆ
1 ‡ 10s 1 ‡ 10s 1 ‡ 10s

4 4
1 ‡ 10s 1 ‡ 10s

3…1 À 10s† 3 À3…1 À 10s† 3
G12 ˆ G21 ˆ G12 ˆ G21 ˆ
…1 ‡ 10s†2 1 ‡ 10s …1 ‡ 10s†2 1 ‡ 10s

4…1 ‡ 50s† 4 4…1 ‡ 50s† 4
…1 ‡ 10s†2 1 ‡ 10s …1 ‡ 10s†2 1 ‡ 10s

3 À3 3
G12 ˆ G21 ˆ G12 ˆ G21 ˆ
1 ‡ 10s 1 ‡ 10s 1 ‡ 10s

4 4
1 ‡ 10s 1 ‡ 10s

3…1 ‡ 50s† 3 À3…1 ‡ 50s† 3
G12 ˆ G21 ˆ G12 ˆ G21 ˆ
…1 ‡ 10s†2 1 ‡ 10s …1 ‡ 10s†2 1 ‡ 10s


Table 2
The di€erent processes under consideration

Case Gp11 …s† Gp12 …s† Gp21 …s† Gp22 …s†

1 + + + + System A
2 À + + + Identical to 3, with opposite gain sign for Gc
3 + À + + System B
4 + + À + Identical to 3
5 + + + À Identical to 3, with opposite gain sign for Gc2
6 À À + + Identical to 1, opposite gain sign for Gc1
7 À + À + Identical to 1, opposite gain sign for Gc1
8 À + + À Identical to 1, opposite gain sign for Gc1 and Gc2
9 + À À + Identical to 1
10 + À + À Identical to 1, opposite gain sign for Gc2
11 + + À À Identical to 1, opposite gain sign for Gc2
12 À À À + Identical to 3, opposite gain sign for Gc1
13 À À + À Identical to 3, opposite gain sign for Gc1 and Gc2
14 À + À À Identical to 3, opposite gain sign for Gc1 and Gc2
15 + À À À Identical to 3, with opposite gain sign for Gc2
16 À À À À Identical to 1, opposite gain sign for Gc 1 and Gc2

even. The other cases are similar to system B where functions seen by the controllers and the approx-
the number of positive gain sign is odd. The condi- imations used for controllers tuning are identical.
tion number and the GRDG are given for all pro-
cesses in Figs. 5 and 6 . On the basis of the condition 5.1. System A
number, which is a measure of directionality, system
A processes 1 and 2 should be accelerated while On the basis of the condition number, system
system B should not be. Fig. 5 shows that the ``A'', which has an even number of positive sign,
condition number, at high frequencies, is lower for presents a high directionality. For the initial process,
system A, and is higher for system B. On the basis which is symmetrical and has equal time con-
of the GRDG, system B process 3 should also be stants, this value is constant and equal to 16.9 dB.
accelerated in order to reduce interaction since it is The gain seen by the controllers is low (K11 À
K12 K21
near 1 at high frequencies as shown in Fig. 6. K22 ˆ 1:75) since the outputs are in¯uenced by
The transfer functions seen by each controller are components acting in opposite directions. For set-
given in Table 3 with the corresponding tuning. The point changes, decentralized and decoupling con-
tuning method proposed by Poulin et al. [18] has trollers give similar results on the output variable
been used. For the process under study where for which the set-point has occurred. The GSR for
G11 …s† ˆ G22 …s† for the initial system, the controllers a disturbance at the process inputs are given in
are symmetrical when a zero is incorporated in one Figs. 9 and 10, respectively. It is observed that the
of the cross-coupled transfer function. decentralized controller gives a response much less
Fig. 7 gives the approximation used for the directional that decoupling controller. This can be
transfer function seen by the controller in decen- explained by the fact that the disturbance is ®rst
tralized control and the real function seen for ampli®ed in the direction of the maximum sin-
initial system A while Fig. 8 gives these approx- gular vectors for both types of controllers but is
imation for initial system B. The full line of the corrected very slowly with decouplers since they
Bode plot refers to the approximation and the eliminate the directionality. The decentralized
dotted line refers to real function seen by the con- controllers although have a uniform directionality
trollers for the initial process. At frequencies lower for a symmetrical process. It is also observed, for
than the cross-over frequency (!co ), the transfer the case of a symmetrical process, that the


Fig. 6. Dynamic generalized relative gain for system A and B.

Fig. 5. Condition number for System A and System B. opposite directions. At the opposite, process 2
which has a non-minimal phase zero in the cross-
coupled transfer function has a stable zero in the
manipulated variable for a step output dis- transfer function seen by the controller. This could
turbance gives the process outputs for set point be deduced from Eqs. (4) and (5). A consequence
changes as given in Eqs. (11) and (12). As a result, of the latter is that process 2 will be easy to accel-
for highly directional and coupled processes, erate and process 1 will be impossible to accel-
decoupling controller will be much less robust to erate. For process 2, another advantage is that
modelling errors. accelerating will reduce directionality as given by
For process 1, which has a non-minimal phase the condition number. The GSR plots shown, for
zero in the direct branch, this non-desired char- process 2 in decentralized control, con®rm this as
acteristic is ampli®ed for the transfer function seen shown on Fig. 11 for Kc ˆ 0:57 (!co ˆ 0:1) and in
by the controller since the action coming from the Fig. 12 for Kc ˆ 4 (!co ˆ 0:7). This explains why a
cross-coupled transfer functions are acting in the PID has been used for the tuning of process 2.


Processes 3 and 4 have a stable zero. For a stable
zero in the direct branch, it means that the transfer
function seen by the controller will also have a
stable zero. The tuning will be easy and the process
can be easily accelerated. However, the controller
will contain a large pole to satisfy set-point chan-
ges speci®cations. As a result of this large pole,
one might expect slow time response for the out-
put in regulation in decentralized control. For
process 4, as one might expect, the stable zero in
the cross-coupled transfer function is seen as a
non-minimal phase system by the controller. This
will limit the system response in both types of
control structures.
Fig. 7. Bode plot for system A initial process.
5.2. System B

System B presents no directionality on the initial
system. The components coming from the direct
and the cross-coupled transfer functions are acting
on the same directions (K11 À K122221 ˆ 6:25). It
K
K

means that for the systems, which have a zero, the
e€ect of the zero will be reduced for the transfer
function seen by the controller. This is con®rmed
by the results shown in Table 3.
For process 3, according to the GRDG, it should
be accelerated in order to reduce the interaction.
This is shown in Figs. 13 and 14 where the devia-
tion of the regulated variable is reduced from 0.25
to 0.15 for a gain of 0.16 and 0.64 of the controller,
respectively. It also shows that the presence of an
Fig. 8. Bode plot for system B initial process. important zero in G11 …s† has a determinant e€ect

Fig. 9. (a) GSR output for system A, initial process (decentralized control). (b) GSR input for system A, initial process (decentralized
control).


Fig. 10. (a) GSR output for system A, initial process (control with decouplers). (b) GSR input for system A, initial process (control
with decouplers).

Fig. 11. GSR for system A process 2 (Kc ˆ 0:57). Fig. 13. Step response for process 3 of system B (Kc ˆ 0:16).

Fig. 12. GSR for system A process 2 (Kc ˆ 4). Fig. 14. Step response for process 3 of system B (Kc ˆ 0:64).


Table 3
Transfer functions seen by each controller and the corresponding tuning

Initial process Process 1 Process 2 Process 3 Process 4

System A
À0:75…1 À 10s† À0:75…1 ‡ 10s† À0:75…1 ‡ 50s†
D12 (decoupler) À0.75 0
…1 ‡ 10s† …1 ‡ 50s† …1 ‡ 10s†
D21 (Decoupler) À0.75 À0.75 À0.75 À0.75 À0.75
1:75 1:75…1 À 36s† 1:75…1 ‡ 36s† 1:75…1 ‡ 101s† 1:75…1 À 42s†
G1
…1 ‡ 10s† …1 ‡ 10s†2 …1 ‡ 10s†2 …1 ‡ 10s†2 …1 ‡ 10s†2

1:75 4 1:75…1 ‡ 36s† 1:75…1 ‡ 84s† 1:75…1 À 42s†
G2
…1 ‡ 10s† …1 ‡ 10s† …1 ‡ 10s†2 …1 ‡ 23s†2 …1 ‡ 10s†2

0:57…1 ‡ 10s† 0:127…1 ‡ 15s† 0:57…1 ‡ 10s†2 0:57…1 ‡ 15s† 0:11…1 ‡ 15s†
GC1
10s 15s 15s…1 ‡ 36s† 15s…1 ‡ 101s† 15s

0:57…1 ‡ 10s† 0:25…1 ‡ 10s† 0:57…1 ‡ 10s†2 0:57…1 ‡ 15s† 0:11…1 ‡ 15s†
GC2
10s 10s 15s…1 ‡ 36s† 15s…1 ‡ 84s† 15s

System B
0:75…1 À 10s† 0:75…1 ‡ 10s† 0:75…1 ‡ 50s†
D12 (decoupler) 0.75 0
…1 ‡ 10s† …1 ‡ 50s† …1 ‡ 10s†
D21 (decoupler) À0.75 À0.75 À0.75 À0.75 À0.75
6:25 6:25…1 À 2:8s† 6:25…1 ‡ 2:8s† 6:25…1 ‡ 35:6s† 6:25…1 ‡ 24:4s†
G1
…1 ‡ 10s† …1 ‡ 10s†2 …1 ‡ 10s†2 …1 ‡ 10s†2 …1 ‡ 10s†2

6:25 4 6:25…1 ‡ 2:8s† 6:25…1 ‡ 36s† 6:25…1 ‡ 24:4s†
G2
…1 ‡ 10s† …1 ‡ 10s† …1 ‡ 10s†2 …1 ‡ 10s†…1 ‡ 50s† …1 ‡ 10s†2

0:16…1 ‡ 10s† 0:125…1 ‡ 15s† 0:16…1 ‡ 15s† 0:16…1 ‡ 10s†2 0:16…1 ‡ 15s†
GC1
10s 15s 15s…1 ‡ 2:8s† 15s…1 ‡ 35:6s† 15s…1:24:4s†

0:16…1 ‡ 10s† 0:25…1 ‡ 10s† 0:16…1 ‡ 15s† 0:16…1 ‡ 52:6s† 0:16…1 ‡ 15s†
GC2
10s 10s 15s…1 ‡ 2:8s† 52:6s…1 ‡ 36s† 15s…1 ‡ 24:4s†

on the time response of the regulated variable y2 …t† observed the transfer functions seen by the decen-
for a set point change on y1 …t†. In both cases, a PID tralized controllers are the same as the ones seen
with a pole zero cancellation method has been used when a decoupler is used. For system that have a
in order to be able to accelerate the process. non-minimal zero in the direct transfer functions
another approximation has been used since the
formed cannot be inverted and fully decoupled
6. Conclusion systems are impossible.
For processes which have high directionality
Simple tuning rules have been developed for (even number of positive sign) and are highly
decentralized controllers. The approximations coupled, decentralized control should be used in
used remain valid for most systems since the order to reduce this directionality in regulation for
transfer functions seen by the controllers are low process input disturbances. For these processes, since
pass ®ltered by the cross-coupled functions. It is the components of the direct and cross-coupled


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