1. ISA Transactions 40 (2001) 223±234
www.elsevier.com/locate/isatrans
Improving performance using cascade control and a
Smith predictor
Ibrahim Kaya *
Inonu University, Engineering Faculty, Electrical and Electronics Department, 44069, Malatya, Turkey
Abstract
Many investigations have been done on tuning proportional-integral-derivative (PID) controllers in single-input
single-output (SISO) systems. However, only a few investigations have been carried out on tuning PID controllers in
cascade control systems. In this paper, a new approach, namely the use of a Smith predictor in the outer loop of a cascade
control system, is investigated. The method can be used in temperature control problems where the secondary part of the
process (the inner loop) may have a negligible delay while the primary loop (the outer loop) has a time-delay. Two dif-
ferent approaches, including an autotuning method, to ®nd the controller parameters are proposed. It is shown by
some examples that the proposed structure as expected can provide better performance than conventional cascade
control, a Smith predictor scheme or single feedback control system. # 2001 Elsevier Science Ltd. All rights reserved.
Keywords: PID controller; Cascade control; Smith predictor; Process control
1. Introduction control, the corrective action for disturbances does
not begin until the controlled variable deviates
Many investigations have been done on tuning from the set point. A secondary measurement point
proportional-integral-derivative (PID) controllers and a secondary controller, Gc2, in cascade to the
in single-input single-output (SISO) systems, such main controller, Gc1, as shown in Fig. 1, can be
as Refs. [1±3]. However, the standard single feed- used to improve the response of the system to load
back control loop does not sometimes provide a changes.
good enough performance for processes with long A typical example is the natural draft furnace
time delays and strong disturbances. Cascade con- temperature control problem [10,11], shown in
trol loops can be used and are a common feature in Fig. 2. When there is a change in hot oil temperature,
the process control industries for the control of which may occur due to a change in oil ¯ow rate, the
temperature, ¯ow and pressure loops. conventional single feedback control system, Fig. 2,
Cascade control (CC), which was ®rst introduced will immediately take corrective action. However,
many years ago by Franks and Workey [4], is one if there is a disturbance in fuel gas pressure no
of the strategies that can be used to improve the correction will be made until its e€ect reaches the
system performance, particularly in the presence of temperature-measuring element. Thus, there is a
disturbances. In conventional single feedback considerable lag in correcting for a fuel gas pres-
sure change, which subsequently results in a slug-
* Corresponding author. Fax: +90-422-3401046. gish response. With the cascade control strategy
E-mail address: ikaya@inonu.edu.tr (I. Kaya). shown in Fig. 3, an improved performance can be
0019-0578/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved.
PII: S0019-0578(00)00054-9

2. 224 I. Kaya / ISA Transactions 40 (2001) 223±234
outer loop of a cascade control. In temperature
control, as given above, the process can often be
treated as having two transfer functions in cascade
with the one in the inner loop, generally, having,
no or a small time-delay, while the one in the outer
loop has a signi®cant time-delay. It is well known
Fig. 1. Block diagram of a cascade control system. [9] that a Smith predictor structure provides good
control for processes with a large time delay. Thus,
combining a Smith predictor and cascade control
strategy can provide a more advantageous struc-
ture in the aforementioned situations. The main
contribution of the paper is to provide compar-
isons, which are fully compatible since optimiza-
tion of an integral performance criterion has been
used, between the performance with this structure
and conventional CC, single feedback loop control
and a Smith predictor scheme. The comparisons
are carried out by simulations using SIMULINK.
The next section outlines some important aspects
of the cascade control strategy when compared to a
single feedback loop control. Section 3 introduces
Fig. 2. The natural draft furnace temperature control with the improved cascade control (ICC) strategy and
single feedback control.
also a tuning method to ®nd parameters of the ICC
method using an optimization approach which can
be used when a mathematical model of the plant is
available. Section 4 gives an autotuning method,
which makes use of results from relay autotuning
where the plant is ®rst approximated by a ®rst
order plus dead time (FOPDT) or second order
plus dead time (SOPDT), where it is relevant,
determined from exact analysis of the limit cycle.
The identi®cation procedure is not given here. The
interested readers may refer to [16] or [17]. Illus-
trative examples are given in Section 5 to show the
value of the proposed structure. The paper ends
with some conclusions in Section 6.
Fig. 3. The natural draft furnace temperature control with
cascade control.
achieved, since any change in the fuel gas pressure 2. Cascade control
is immediately detected by the pressure-measuring
element and the pressure controller takes corrective Cascade control can improve the response to a
action. set point change by using an intermediate mea-
Recent contributions on the tuning of PID con- surement point and two feedback controllers. This
trollers in cascade loops include [5±8] which have improvement can be shown by examination of the
only concentrated on the conventional cascade transfer functions between the system output and
control con®guration. In this paper, an improved the disturbance for the cascade and conventional
cascade control (ICC) structure is proposed. The single feedback loop control con®gurations,
idea is to use a Smith predictor structure in the respectively.

3. I. Kaya / ISA Transactions 40 (2001) 223±234 225
For a cascade control, from Fig. 1, the dis- disturbance. The time constant of designed inner
turbance transfer function with d1=0 is given by closed loop should at least be three times larger
than the outer loop time constant. The outer loop
Y1 Gp1 controller is chosen to be a PI or PID. The integral
ˆ …I†
D2 1 ‡ G™2 Gp2 ‡ G™1 G™2 Gp1 Gp2 action is now required to eliminate the low fre-
quency disturbance and o€set in the system.
Without the inner feedback loop, and of course
no controller Gc2, the transfer function between
the same variables is 3. Improved cascade control (ICC)
Y1 Gp1 In many process control systems the plant can
ˆ …P†
D2 1 ‡ G™1 Gp1 Gp2 often be regarded as a cascade system in which the
secondary part of the process (the transfer func-
which is quite di€erent from Eq. (1). Therefore, tion in the inner loop) has no or a negligible time-
because of the extra degrees of freedom, when delay while the primary part of the process (the
appropriate values of the parameters of the two transfer function in the outer loop) has a time-
controllers are chosen, the cascade control will delay. A Smith predictor strategy can give a satis-
generally result in a better response. factory performance for set point changes, but the
A cascade control structure has the following performance for disturbance rejection may not be
advantages over a single feedback loop control satisfactory. In this case, a cascade control can be
system [12]: used to improve the response of the system to dis-
turbances. However, a cascade control strategy
1. The secondary controller is used to correct
alone may not be enough if a long time delay
disturbances arising within the inner loop
exists in the outer loop, since it may result in a
before they can a€ect the controlled variable.
poor response for set point changes.
2. The e€ect of parameter variations in the
Thus, this paper reports on the use of a Smith
process Gp2 are corrected in the inner loop by
predictor strategy, shown in Fig. 4, in cascade
the secondary controller.
control systems. The general strategy to design
3. The e€ect of any phase lag existing in Gp2
two controllers in CC structure can be followed
may be reduced by the secondary loop, thus
for the ICC as well. In the next subsection, opti-
allowing the speed of response of the primary
mization is used to ®nd the tuning parameters for
loop to be improved.
the two controllers, Gc1 and Gc2, in the ICC
Thus, if there are additional measurable vari- structure. Then in section 4, an autotuning
ables, cascade control can provide better results. method, which makes use of results from the relay
However, it must be stressed that the inner loop autotuning, is given.
should include the major disturbances and be fas-
ter reacting than the outer loop in order to achieve 3.1. Tuning ICC using optimization
a signi®cant improved system performance.
To complete the design of a cascade control In this section integral performance criteria have
system, the parameters of two controllers have to been used to determine tuning parameters for the
be determined. The general strategy to ®nd two controllers. There are several integral performance
controller parameters can be outlined as follows: criteria which may be used to minimize the error
The inner controller is usually chosen to be a P or signals, e1 and e2, such as the integral of squared
PI controller. The derivative action is not required error, ISE, the integral of absolute error, IAE, and
as the inner process disturbance is noisy and its the integral of time-moment weighted squared
frequency range is higher than the outer process error, ISTE. In this paper, the ISTE criterion has
disturbance. The inner controller gain should be been chosen to ®nd the tuning parameters for the
chosen high to reduce the e€ects of inner loop two controllers, since it normally results in step

4. 226 I. Kaya / ISA Transactions 40 (2001) 223±234
Fig. 4. Improved cascade control system.
responses with a relatively small overshoot and a inner loop process transfer function is usually low
reasonable settling time and the integral can easily order and has no time delay. In this case, either
be evaluated in the s-domain [13]. one of the controller parameters, usually the con-
In cascade control systems, the auxiliary con- troller gain, can be constrained to a value and the
troller, Gc2, is usually chosen as a PI or P controller remaining controller parameters can be found or a
and the main controller, Gc1, as a PI or PID con- small time delay can be added in the optimization
troller. This is understandable, since the dynamics procedure to ®nd the controller parameters.
of the secondary part of the process, Gp2, are Once the parameters of the inner controller are
usually low order and derivative action is often obtained, the optimization procedure can be carried
not required. Thus it is assumed that the two con- out to obtain the parameters of the outer or main
trollers have the following ideal transfer functions controller, Gc1, for the resultant plant Gp where
1 G™2 Gp1 Gp2
G™1 ˆ Kp1 1 ‡ ‡ Td1 s …Q† Gp ˆ …T†
Ti1 s 1 ‡ G™2 Gp2
1 The transfer function of the error signal e1 is
G™2 ˆ Kp2 1 ‡ …R†
Ti2 s then given by
The easiest approach to applying optimization R1
E1 ˆ …U†
techniques in a cascade control system to ®nd the 1 ‡ G™1 Gp
parameters of the controllers is to optimize the
errors in the loops individually. For this, the inner which is used in the optimization procedure to ®nd
loop is considered ®rst. The parameters of the the parameters of the main controller. The main
inner loop controller, Gc2, can be obtained by di€erence between the proposed improved cascade
minimizing the performance criterion for e2 which control and conventional cascade control is given
is given in the s-domain by by Eq. (7). A similar equation for a conventional
cascade control will involve a time-delay in its
R2 characteristic equation while here, since ideal
E2 ˆ …S†
1 ‡ G™2 Gp2 matching has been assumed for the Smith pre-
dictor, it is free of the time-delay.
It should be noticed that the e€ect of a step dis- Finally, it should be said that, although a Smith
turbance d2 on the output y2 gives this same predictor con®guration is used in the outer loop of
equation. Another point which should be pointed the proposed ICC method and it is well known
out is that, the optimization procedure may not that a Smith predictor structure is sensitive to a
always lead to a solution for the inner loop as the mismatch in the plant and model dynamics, the

5. I. Kaya / ISA Transactions 40 (2001) 223±234 227
ICC design gives quite satisfactory results in the For the outer loop a ®rst order plus dead time
case of a mismatch. This is illustrated later by (FOPDT) model may be inadequate, as the overall
examples. transfer function for this loop may be of a higher
order. Here, a ®rst order or FOPDT model transfer
function is used for the inner loop and a second
4. Automatic tuning of ICC order plus dead time (SOPDT) model transfer func-
tion for the outer loop. The procedure to ®nd the
The controller design given in the previous section controller parameters can be carried out as follows.
is based on the assumption that the plant transfer
functions are known. Model based design methods . When the controllers need to be tuned, switch
su€er performance degradation when modelling from the two controllers to the two relays. Set
errors exist and in this respect Smith predictor the heights of the relay in the outer loop to
structures are known to be particularly sensitive. A zero and in the inner one to an appropriate
relay autotuning procedure, which is shown in the value. Measure the limit cycle frequency, wo2,
block diagram of Fig. 5, has the advantage that and amplitude, a2, for the inner loop.
retuning can be easily accomplished. The Before proceeding further, it is necessary
approach has been proposed by Hang et al. [14] to comment on tuning of the inner loop. The
and Zhuang and Atherton [5] for a conventional preferred approach is to use an ideal relay
cascade system. Hang et al. [14] used limit cycle but if the inner loop plant transfer function is
information to directly tune loops using Ziegler± of a low order, then one may not get a limit
Nichols rules [1] or re®ned Ziegler±Nichols rules cycle oscillation. To avoid the waste of time
[15]. Zhuang and Atherton [5] stated that the and guarantee a limit cycle oscillation, a
loops could be tuned by one of several methods relay with hysteresis and an extra known
which use the critical point information, such as circuit, such as an integrator, or a small time
Ê
the Ziegler±Nichols rules [1] or Astrom and Hag-
È È delay can be used in the inner loop. The
glund's phase and gain margin method [2]. In their amount of hysteresis or time delay may have
paper, however, they used a model based tuning to be chosen relatively large, as a small
where ®rst an approximate ®rst order plus dead time amount may not yield a limit cycle of low
(FOPDT) plant transfer function was found from enough frequency. In most practical cases
limit cycle measurements followed by describing some knowledge of the form of plant transfer
function analysis and then the controller para- function will be available to assist in selecting
meters were obtained using analytic equations the best approach.
obtained from optimal responses of a ®rst order . Use the limit cycle information to estimate
plus dead time (FOPDT) model [3]. the unknown parameters, based on exact
Fig. 5. , Autotuning of improved cascade control (ICC) strategy.

6. 228 I. Kaya / ISA Transactions 40 (2001) 223±234
analysis, for the inner plant transfer function design method gives a very satisfactory perfor-
[16]. mance. Both examples two and three are given to
. If the process is modelled by a FOPDT trans- show the use of the autotuning method. Example
fer function the tuning rules given by Zhuang two assumes a third order plant transfer function
and Atherton [3] can be used, if it is modelled in the inner loop so that the ideal relay feedback
by a ®rst order transfer function the tuning test can be carried out. The third example con-
rules given by Kaya [17] can be used to ®nd siders a ®rst order plant transfer function in the
suitable PI controller parameters. For con- inner loop a situation which was not considered
venience, the tables needed in this paper are by either Hang et al. or Zhuang and Atherton.
given in the next section, where it is relevent, With a ®rst order plant transfer function in the
and formulae are given in the appendix. inner loop, the ideal relay feedback test cannot be
. Switch the inner PI controller with the deter- performed as a phase lag of 180 is not achieved.
mined parameters into the loop to replace the In the example an ideal relay and time delay are
relay. Set the heights of the outer relay to an included in the loop to obtain a limit cycle. In all
appropriate value. At the same time open cases, simulations are carried out in SIMULINK
switch `S' so the loop transfer function is the for comparison of design method performances.
CC con®guration. Measure the limit cycle
parameters, !o1 and a1, for the outer loop. Example 1. In this example the plant transfer
. For the outer loop it is assumed that functions are given by
Gpm eÀLm s , where Gpm is the delay free part of
the model and Lm is the model time delay, eÀ10s
can properly be modelled by a SOPDT Gp1 ˆ
…20s ‡ 1†…3s ‡ 1†
transfer function. Thus once the limit cycle
parameters are measured for the outer loop,
3
the unknown parameters, namely the steady Gp2 ˆ
state gain, Km, the two time constants, T1m …10s ‡ 1†
and T2m, and the time delay, Lm, for
Gpm eÀLm s can easily be evaluated using an Calculations show that no solution exists for
exact parameter estimation method based on optimal PI (PID) controller settings for the ISTE
asymmetric limit cycle data given by Kaya criterion when the plant has a transfer function
and Atherton [16]. Note that one does not with no time delay and a relative order of less than
try to ®nd a model for the outer plant three (four). The design problem can be overcome
Gp1 eÀLs but rather for the overall plant by constraining one or more controller para-
Gpm eÀLm s . Then, the PID tuning parameters meters. Thus, to ®nd optimal PI settings for the
for the outer loop can be calculated using inner controller in the CC and ICC designs, the
simple tuning formulae given by Kaya [17]. controller gain Kp2 was limited to 3.0 and optimal
value of ``I'' found. Then the outer controller
parameters for the cascade control strategy were
5. Illustrative examples found. All the parameters are given in Table 1.
Again, the aforementioned problem still exists for
Three examples are given to illustrate the value calculating the outer controller parameters in the
of the proposed method. In the ®rst example improved cascade control strategy and the PID
optimization is used to ®nd the tuning parameters controller parameters in the Smith predictor
for the controllers. The performance of the ICC structure, since exact matching is assumed. Thus,
design is compared to cascade control, Smith pre- the outer controller gain in the improved cascade
dictor control and single feedback control systems. control and gain of the PID controller in the
In addition simulation results for mismatching Smith predictor structure were constrained to
cases are also given and it is seen that the ICC $2.0$ to provide a fair comparison between the

7. I. Kaya / ISA Transactions 40 (2001) 223±234 229
Table 1
PI and PID parameters for Example 1
PID
parameters
Method Kp Ti Td
CC Gc2 3.0 9.600 ±
Gc1 1.622 24.965 6.099
Gc2 3.0 9.600 ±
ICC 1.0 15.512 À4.469
Gc1 2.0 19.385 0.363
5.0 22.246 2.773
Single 0.545 31.267 10.430
Smith 2.0 29.959 8.575
Fig. 7. Step and disturbance responses for +30% variations in
two design methods. A summary of the optimal time delay for Example 1.
PID settings are listed in Table 1 for all the cases.
Fig. 6 shows the responses to a set point change designs when there is a +30% and À30% varia-
and a disturbance d2=-0.2 at t=200 for the dif- tion in the plant time delay. A positive variation in
ferent controller con®gurations. the plant time delay causes the overshoot to
The cascade control gives a good response for increase while a negative variation in the plant time
disturbance rejection, but a poor response for set delay results in less overshoot. In both cases, the
point change due to the large time delay. In contrast, proposed method gives superior performance to
the Smith predictor results in a good response for a the other design methods.
set point change but a poor response for a dis- As the controller gains, Kp1 and Kp2, in the ICC
turbance. The proposed method gives good respon- design were constrained, it is appropriate to inves-
ses for both a set point change and disturbance tigate the e€ect of their choice on the system per-
rejection, since it combines the best features of cas- formance. For this, the inner loop controller gain
cade control and the Smith predictor con®guration. Kp2 was kept constant and the outer loop controller
The disturbance rejection of the CC and ICC con- gain was varied. Control e€ort corresponding to
®gurations is so good that it can hardly be seen. the three cases are shown in Fig. 9. As is seen from
Figs. 7 and 8, respectively, give the step and the ®gure, the initial control e€ort increases with an
disturbance responses for the di€erent controller increase in the gain Kp2. Therefore, the controller
Fig. 8. Step and disturbance responses for À30% variations in
Fig. 6. Step and disturbance responses for Example 1. time delay for Example 1.

8. 230 I. Kaya / ISA Transactions 40 (2001) 223±234
Table 2
PI tuning formulae for set-point changes
L/T range 0.1±1.0 1.1±2.0
Criterion ISTE IST2E ISTE IST2E
a1 0.712 0.569 0.786 0.628
b1 À0.921 À0.951 À0.559 À0.583
a2 0.968 1.023 0.883 1.007
b2 À0.247 À0.179 À0.158 À0.167
T1=7.213, T2=8.178 and L=10.030. Using
Table 5 for a PID controller in a Smith predictor
scheme gives Ti1=12.641 and Td1=1.397 when
Fig. 9. Control signals for the inner loop with three di€erent Kp1 is limited to 1.5. To illustrate the accuracy of
values of Kp1 for the ICC design. modelling the overall system by a SOPDT model,
Nyquist plots of both the overall system transfer
gain should not be constrained to very high function and the model obtained are given in
values. In general, the outer loop controller gain Fig. 10 and show good agreement.
Kp1 should be chosen smaller than the inner loop To compare the performance of the ICC method
controller gain Kp2 so that the inner loop is faster with the single feedback and Smith predictor con-
and a better performance can be achieved. trol, the relay feedback test was also performed
for the overall plant G=Gp1Gp2 to obtain a
Example 2. Assume the outer and inner loop plant FOPDT model to be used in the calculation of the
transfer functions are given by PID controller in a single feedback control system
and SOPDT model in the Smith predictor scheme.
eÀ9s The FOPDT model was used for the single feed-
Gp1 ˆ
…8s ‡ 1†…7s ‡ 1† back control since in reference [3], PI and PID
tuning parameters were obtained based on a
4 FOPDT model. With the overall plant in the relay
Gp2 ˆ
…s ‡ 1†3 feedback system, limit cycle parameters were
obtained as !=0.133, Át1=20.788, amax=2.547
With the height of the outer relay set to zero and
inner ideal relay to h1=1 and h2=À0.6 the limit
cycle parameters for the inner loop were measured
as 1.685, 1.484, 0.642 and À0.384 for !, Dt1, amax
and amin respectively, see reference [16] for nota-
tions. This data was used to ®nd the FOPDT
model parameters as K=3.994, T=5.194 and
L=0.906 using the estimation method given in
[16]. These model parameters lead to Kp2=0.889
and Ti2=5.616 when Table 2 for the PI controller
is used. Then the inner loop was switched from the
inner relay to the inner PI controller with the
determined tuning parameters and the outer ideal
relay was set to h1=1 and h2=À0.6. The outer
loop limit cycle parameters were measured as
0.146, 18.583, 0.572 and -0.363$ for !, Át1, amax Fig. 10. Nyquist plots for the overall system and SOPDT
and amin, respectively, which results in K=0.989, model for Example 2.

9. I. Kaya / ISA Transactions 40 (2001) 223±234 231
and amin=À1.579 which result in K=3.999, T= Example 3. This example shows how the autotun-
14.617 and L=15.673 for a FOPDT model and ing method can be used when the plant in the
K=3.999, T1=6.015, T2=9.729 and L=11.773 inner loop does not have a phase lag of 180 . The
for a SOPDT model. Therefore, the tuning para- plant transfer functions are given by
meters for the PID controller in a single feedback
control system are Kp=0.270, Ti=20.028 and 2eÀ8s
Gp1 ˆ
Td=6.080 from Table 3 and in a Smith predictor …10s ‡ 1†…5s ‡ 1†
scheme are, from Table 5, Ti=15.682 and
Td=3.554 when Kp is limited to 1.5. The step and 5
Gp2 ˆ
disturbance responses are given in Fig. 11 for dif- …7:5s ‡ 1†
ferent controller design methods. The ICC method
gives better performance both for a set point Since Gp2 is a ®rst order transfer function, a
change and disturbance rejection. Fig. 12 illus- limit cycle cannot be obtained for the inner loop
trates the case when the plant time delay has using an ideal relay. In this case, a time delay can
changed to 11.7, which corresponds to +30% be added in the relay feedback control loop to
increase. The tuning parameters were kept the
same as before. Fig. 13 shows the responses
obtained after retuning for the changed time
delay. A better closed loop step response, which is
almost the same as for the matching case, is
obtained with the retuning.
Table 3
PID tuning formulae for set-point changes
L/T range 0.1±1.0 1.1±2.0
2
Criterion ISTE IST E ISTE IST2E
a1 1.042 0.968 1.142 1.061
b1 À0.897 À0.904 À0.579 À0.583
a2 0.987 0.977 0.919 0.892
b2 À0.238 À0.253 À0.172 À0.165 Fig. 12. Step and disturbance responses for +30% change in
a3 0.385 0.316 0.384 0.315 time delay for Example 2.
b3 0.906 0.892 0.839 0.832
Fig. 13. Step and disturbance responses for +30% change in
Fig. 11. Step and disturbance responses for Example 2. time delay for Example 2 after retuning.

10. 232 I. Kaya / ISA Transactions 40 (2001) 223±234
Table 4
PI tuning formulae based on a ®rst order model
KKp 1.50±2.50 2.60±5.00 5.10±15.00
2 2
Criterion ISTE IST E ISTE IST E ISTE IST2E
a 1.9443 1.3037 1.5556 1.1209 1.2149 1.0196
b À0.4722 À0.2426 À0.2173 À0.0666 À0.0679 À0.0075
Table 5
PID tuning formulae based on a second order model
KKp 1.50±2.50 2.60±5.00 5.10±15.00
Criteron ISTE IST2E ISTE IST2E ISTE IST2E
a1 0.6781 0.6338 0.5961 0.5554 0.5169 0.4950
b1 À0.2709 À0.2410 À0.1144 À0.0888 À0.0323 À0.0184
a2 0.1058 0.1057 0.2446 0.2453 0.4049 0.3978
b2 1.3371 1.3041 0.3877 0.3637 0.0767 0.0640
obtain a limit cycle and hence estimate the
unknown plant transfer function parameters. In
this example, a time delay of 1.0 was used in the
inner loop to obtain a limit cycle oscillation. With
the chosen time delay and relay parameters of
h1=1, h2=À0.6 and Á=0, a limit cycle oscillation
with !=1.587, Át1=1.542, amax=0.624 and
amin=À0.374 were obtained. These limit cycle
data result in K=5.003, T=7.506 and L=0.999.
Table 4 was used to obtain the inner controller
parameters. Kp2 was limited to 2.5 and then Ti2
was calculated as 6.296. The inner controller was
then switched in with the calculated parameters.
To obtain a SOPDT model for Gpm eÀLm s , the
outer relay was set to the same parameters used in Fig. 14. Nyquist plots for the overall system and the model
the inner relay. Limit cycle parameters were mea- obtained for Example 3.
sured as !=0.167, Á=16.975, amax=1.058 and
amin=À0.759. Then the SOPDT model para-
meters were found as K=1.999, T1=10.368, were obtained as !=0.121, Át1=22.364, amax=
T2=4.704 and L=8.640. The parameters of the 5.139 and amin=À3.221 which results in K=
outer PID controller were calculated using Table 5 10.000, T=20.874 and L=16.066 for a FOPDT
as Ti1=14.448, Td1=3.253 when Kp1 is limited to model and K=10.000, T1=10.306, T2=10.306 and
4.0. Again Nyquist plots for the overall system and L=10.986 for a SOPDT model. Therefore, the tun-
the model obtained are given in Fig. 14 to see the ing parameters for a PID controller using Table 3 in
accuracy of modelling. The ®gure clearly shows a single feedback control system are Kp=0.132,
that the model obtained has a good accuracy. Ti=25.968 and Td=6.339 and in a Smith pre-
The FOPDT model was used for the single feed- dictor scheme, using Table 5, are Ti=21.477 and
back loop as in Example 2. With the overall plant in Td=4.979 when Kp is limited to 1.0. The responses
the relay feedback system, limit cycle parameters for a set point change and a disturbance of d2=

11. I. Kaya / ISA Transactions 40 (2001) 223±234 233
Appendix. Tuning formulae and tables
This section gives some tuning formulae which
are used in the paper to ®nd controller parameters.
These formulae were obtained using repeated opti-
mizations, using the ISTE or IST2E criteria, on an
error signal and then least square ®tting method was
used to ®nd coecients in the formulae.
A1. Tuning based on a FOPDT
The tuning formulae are given [3] by
a1 L b1
Fig. 15. Step and disturbance responses for Example 3. Kp ˆ …V†
K T
À0.2 at t=150 are given in Fig. 15. The far superior T
Ti ˆ …W†
performance of the ICC method is obvious. a2 ‡ b2 …L=T†
b3
L
6. Conclusions T d ˆ a3 T …IH†
T
An improved cascade control strategy has been where the (a, b) coecients for a PI and PID con-
introduced in this paper. The structure brings troller are given in Tables 2 and 3 respectively.
together the best merits of a cascade control and
the Smith predictor structure. The ICC structure A2. Tuning based on a ®rst order model
can be used in process control problems, such as
temperature, ¯ow or pressure control. Two design The tuning formula [17] is
methods have been given to ®nd the tuning para-
meters of the PI controller in the inner loop and 1 aÀ Áb
ˆ KKp …II†
PID controller in the outer loop. Since the plant Ti T
transfer functions are assumed to be low order,
constraining some tuning parameters in the opti- The controller gain is ®rst speci®ed, which is chosen
mization may be necessary. This is usually done by so that the normalized gain KKp falls into one of
limiting the gain of the controller to some value the ranges given in Table 4, the controller integral
since this a€ects the maximum magnitude of the time constant Ti is then calculated from Eq. (11).
control signal and then performing the optimiza- The coecients in the formula over the various
tion. An autoning method, which is an extention ranges for KKp are listed in Table 4.
of the relay feedback tuning for SISO systems, has
also been given. The autotuning method may be A3. Tuning based on a second order model
more useful in practice since in the case of a mis-
match retuning can be done to obtain new model The tuning formulae are given [17] by
parameters and therefore new tuning parameters.
The performance of the ICC is compared to single 1 a1 À Áb T1 1=2
ˆ KKp 1 …IP†
feedback control, Smith predictor structure and Ti T1 T2
cascade control strategy and it is shown by examples
À Áb2 T2 1=2
that the ICC structure can give a better perfor- Td ˆ T1 a2 KKp …IQ†
mance. T1

12. 234 I. Kaya / ISA Transactions 40 (2001) 223±234
Again the controller gain is ®rst limited to a value Dynamics and Control, John Wiley Sons, New York,
and then the remaining controller parameters are 1989.
calculated from Eqs. (12) and (13). [12] F.G. Shinskey, Process-Control Systems, McGraw Hill,
New York, 1967.
The (a, b) coecients are given in Table 5 for [13] M. Zhuang. D.P. Atherton, Tuning PID controllers with
di€erent ranges of KKp. integral performance criteria, in: Matlab Toolboxes and
Applications, Peter Peregrinus, 1993, Chapter 8, pp. 131±
144.
References [14] C.C. Hang, A.P. Loh, V.U. Vasnani, Relay feedback
auto-tuning of cascade controllers, IEEE Trans. Control
Systems Technology 2 (1) (1994) 42±45.
[1] J.G. Ziegler, N.B. Nichols, Optimum settings for auto- [15] Ê
C.C. Hang, K.J. Astrom, W.K. Ho, Re®nements of the
È
matic controllers, Trans. of ASME 64 (1942) 759±768. Ziegler±Nichols tuning formula, IEE Proc. D, Control
Ê
[2] K.J. Astrom, T. Hagglund, Automatic tuning of simple
È È Theory and Application 138 (2) (1991) 111±118.
regulators with speci®cations on phase and gain margins, [16] I. Kaya, D.P. Atherton, An improved parameter estima-
Automatica 20 (5) (1984) 645±651. tion method using limit cycle data, UKACC Int. Con-
[3] M. Zhuang, D.P. Atherton, Automatic tuning of optimum ference on Control '98, Swansea, UK 1 (1998) 682±687.
PID controllers, IEE Proceedings-D 140 (3) (1993) 216±
[17] I. Kaya. Relay Feedback Identi®cation and Model Based
224. Controller Design, PhD thesis, University of Sussex,
[4] R.G. Franks, C.W. Worley, Quantitive analysis of cascade Brighton, England, 1999.
control, Industrial and Engineering Chemistry 48 (6)
(1956) 1074±1079.
[5] M. Zhuang, D.P. Atherton, Optimum cascade PID con-
troller design for SISO systems, UKACC Control '94 Ibrahim Kaya was born in 1971,
(1994) 606±611. Diyarbakõr, Turkey. He grad-
[6] F.S. Wang, W.S. Juang, C.T. Chan, Optimal tuning of uated from Gaziantep Uni-
PID controllers for single and cascade control loops, versity, Electrical Electronics
Chem. Eng. Comm. 132 (1995) 15±34. Department, in 1994. In the
[7] C.T. Huang, C.J. Chou, L.Z. Chen, An automatic PID same year, he started to work
controler tuning method by frequency response techni- as a research assistant at Inonu
ques, The Canadian Journal of Chemical Engineering 75 University. In 1996, he started
(1997) 596±604. his PhD studies at Sussex Uni-
[8] Y. Lee, S. Park, M. Lee, PID controller tuning to obtain versity, Brighton, England. In
desired loop responses for cascade control systems, Ind. 2000, he ®nished his PhD and
Eng. Chem. Res. 37 (1998) 1859±1865. returned back to Inonu Uni-
[9] O.J. Smith, A controller to overcome dead-time, ISA J. 6 versity. He is interested in
(2) (1959) 28±33. Relay Feedback Identi®cation,
[10] E.F. Johnson, Automatic Process Control, McGraw Hill, Relay Autotuning, PID Controllers, Time Delay Systems,
New York, 1967 (pp. 244). Computer-Aided Control System Design and User Interface
[11] D.E. Seborg, T.F. Edgar, D.A. Mellichamp, Process Toolkits.