journal 2007 jcej jp
TRANSCRIPT
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Journal of Chemical Engineering of Japan, Vol. 40, No. 2, pp. 1451 63, 2007 Research Paper
Copyright 2007 The Society of Chemical Engineers, Japan 145
PID Control of Unstable Processes with Time Delay: A ComparativeStudy
Han-Qin ZHOU1,2
, Qing-Guo WANG1
and Leang-San SHIEH2
1Department of Electrical and Computer Engineering,
National University of Singapore, Singapore 1192602Department of Electrical and Computer Engineering,
University of Houston, Houston, TX 77204, U.S.A.
Keywords: Unstable Processes, Time Delay, PID Control, IMC, Nonlinear Control
In this paper, several linear PID control methods for unstable processes with time delays, which have
recently been reported in the literature, are studied in terms of a comprehensive set of control perform-
ance specifications. Their applicability and achievable performance are indicated with regard to differ-
ent cases of normalized time delay for ease of users choice of the methods. By analyzing these methods
pros and cons, a modified method which combines their respective strengths using simple linear time-
variant and nonlinear control strategies is obtained and demonstrated with performance enhancement.
Introduction
It is well-known that unstable dynamic systems
are inherently more difficult to control than their sta-
ble counterparts. For an unstable process, open-loop
control is impossible, and controller gain cannot be
tuned gradually from zero value. A sufficiently large
feedback gain must be used to stabilize the process
before addressing performance and robustness, and yet
the feedback loop may become unstable again if the
gain is too large. This stability range for the feedback
gain is limited. Moreover, as the time delay increases
relatively with the time constant, the range gets nar-
row and thus the system performance could be further
deteriorated. These and other difficulties have moti-
vated active research in control system design for un-
stable processes in recent years.
Design of PID controllers, the most commonly
used controllers in industrial process control, for un-
stable time-delay processes have been reported (DePaor
and OMalley, 1989; Venkatashankar and
Chindambaram, 1994; Shafiei and Shenton, 1994;Huang and Lin, 1995; Poulin and Pomerleau, 1996)
and surveyed by Chidambaram (1997). More recently,
Ho and Xu (1998) derived PID tuning formulas based
on gain and phase margin specifications. Visioli (2001)
proposed optimal PID parameters tuning in terms of
IAE, ISTEand ITSE specifications via genetic algo-
Received on November 29, 2004; accepted on May 28, 2005.
Correspondence concerning this article should be addressed to
Q.-G. Wang (E-mail address: [email protected]).
Partly presented at 5th Asian Control Conference, at Melbourne,
Australia, July 2023, 2004.
rithm. However, these two PID design methods exhibit
excessive overshoot and large setting time. To reduce
them, Park et al. (1998) and Majhi and Atherton
(2000a) developed PID-P and PI-PD strategies respec-
tively, both using inner feedback loops. Besides, Wang
and Cai (2002) employed the PID-P structure to de-
sign an equivalent two degree-of-freedom (DOF) sin-
gle loop PID control scheme in terms of gain and phase
margin specifications.
Owing to power and popularity of internal model
control (IMC) in process industry (Morari and Zafiriou,
1989), many efforts have been made to exploit the IMC
principle to design the equivalent feedback controllers
for unstable processes. Satisfactory results have been
obtained for SISO applications (Chen, 1988; Wang et
al., 2001). Rotstein and Lewin (1991) proposed explicit
PI and PID settings for first-order plus dead time
(FOPDT) and second-order plus dead time (SOPDT)
unstable processes. However, performance limitation
of implementing the IMC controller in an equivalent
feedback structure for unstable processes was not ad-
dressed. Lee et al. (2000) derived a set of PID tuningrules for FOPDT and SOPDT unstable processes, us-
ing Maclaurin series expansion to approximate the ideal
IMC controller with PID. Yang et al. (2002) developed
a unified framework for an IMC-based single loop con-
troller design method using frequency response fitting
for both high-order controllers as well as PID ones for
general processes, where stability is ensured. These two
methods give very good control performance with rela-
tively wide applicability.
The Smith predictor (Smith, 1959) has been
known as a very effective dead-time compensator,
which can eliminate the delay term from the closed-loop
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characteristic equation. Thus standard PI/PID control-
lers can be applied to the delay free systems. How-
ever, the Smith predictor will become internally un-
stable when applied to unstable processes (Wang et al.,
1999). Therefore, many modified Smith schemes have
been proposed to overcome this obstacle. DePaor
(1985) suggested a modified Smith predictor for time-
delayed unstable processes, by changing the denomi-nator of the process model from the original unstable
one to a stable one, but no time response is presented
in this work. Astrom et al. (1994) presented a modi-
fied Smith predictor for integrator plus dead-time
processes, which achieved fast setpoint response and
good disturbance rejection, by decoupling the setpoint
and load responses. Matausek and Micic (1996) con-
sidered the same problem and provided similar but
easier tuning rules. Majhi and Atherton (1999) pro-
posed another modified Smith predictor with good per-
formance particularly for integral unstable processes.
Another paper of Majhi and Atherton (2000b) extended
it to unstable FOPDT processes. Kaya (2003) proposed
a more systematic tuning formula for this modified
Smith predictor structure (Majhi and Atherton, 2000b).
With so much work done, we are motivated to
conduct a comparative study to show a comprehensive
review and assessment of time-delayed unstable
processes control. Seven recent and representative
PID control methods are chosen in our investigation:
(A) Optimal PID Tuning Method (Visioli, 2001);
(B) PID-P Control Method (Parket al., 1998); (C) PI-
PD Control Method (Majhi and Atherton, 2000a);
(D) Gain and Phase Margin Method (Wang and Cai,
2002); (E) IMC-Maclaurin PID Tuning Method (Leeet al., 2000); (F) IMC-based Approximate PID Tuning
Method (Yang et al., 2002); (G) Modified Smith Pre-
dictor (Majhi and Atherton, 2000b). A brief summary
of each of them is given first, their performance is
evaluated against a set of most popular and important
specifications and is commented on, and their applica-
bility is concluded. It is observed from the compari-
son that the best achievable control performance among
the investigated methods has been very good already,
and further enhancement within these frameworks
could be quite difficult though it is not impossible.
However, these methods employ linear time invariant(LTI) controllers only. By using linear time variant
(LTV) and nonlinear components in the controller, the
best performance from LTI controllers can be further
improved by such modifications.
The rest of this paper is organized as follows. The
existing control schemes are reviewed in Section 1,
and their performance is evaluated in Section 2. In
Section 3 some new results on performance improve-
ment are presented.
1. Review of Control Methods
In this section, we will briefly review seven meth-
ods for control of time-delayed unstable processes. The
PID controller to be discussed below is represented in
the form of
G s KT s
T sc pi
d( ) = + +
( )1 1 1
1.1 Optimal PID tuning method
For the FOPDT unstable process
G sK
Tse
Lsp ( ) =
( )1
2
Visioli (2001) proposed three sets of PID auto-
tuning rules to minimize one of the following three
specifications, respectively:
ISE e t t = ( ) ( )
20 3d
ITSE te t t = ( ) ( )
20 4d
ISTE t e t t = ( ) ( )
2 20 5d
The optimal controller parameters are obtained by
means of genetic algorithms, which is well-known toprovide a global optimum in a stochastic framework,
in order to avoid local minima in the optimization pro-
cedure. The value ofKin the process transfer function
results in a simple scaling of the PID proportional gain
Kp, and thus is not required to address in the genetic
algorithm. Based on the optimal PID coefficients for
various values of process normalized dead time n
=
L/Tand time constant T, the tuning rules are obtained
by analytical interpolation. Each interpolation function
was selected manually and its parameters were deter-
mined again by genetic algorithms to minimize the sum
of the absolute values of the estimation errors. Two
sets of tuning formulas are displayed in Table 1, onefor setpoint response, while the other for disturbance
rejection.
It is noted that the configuration Figure 1 is only
of one degree-of-freedom (DOF), small overshoot and
fast settling-time can hardly be obtained at the same
time. The methods below are however all of two-DOF
structure (Figure 2), and have potential to perform
better than 1DOF counterparts. Therefore, for a fair
comparison of all the methods, a pre-filter is used and
tuned to best performance by trial and error when
Method A is implemented in the next section.
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1.2 PID-P control method
For an FOPDT unstable process, Parket al. (1998)
proposed the configuration in Figure 3, where a pro-
portional controller Giis used in the inner loop to sta-
bilize the unstable process, and the main PID control-
ler Go
is designed by viewing the inner closed-loop
system as a stable process G.With relay auto-tuning, the unstable process is
modelled by an FOPDT unstable process:
G s G s
K e
T s
L s
p mm
m
m
( ) ( ) = ( )
1 6
The P controller gain that can stabilize this unstable
FOPDT process is within the range:
kK
kK
T kmin max= < < + ( ) = ( )1 1
1 72
mci
mm u
where u
is the ultimate frequency. To have the opti-
mal gain margin, the controller gain derived by DePaor
and OMalley (1989) is used on the inner loop:
PID parameter ISE ITSE ISTE
Setpoint t racking
Kp 1 320 92
..
K
L
T
1 38
0 90.
.
K
L
T
1 35
0 95.
.
K
L
T
Ti
4 00
0 47
.
.L
T T
4 12
0 90
.
.L
T T
4 52
1 13
.
.L
T T
Td3 87 1 0 84
0 02
0 95
. .
.
.
TL
T
L
T
3 62 1 0 85
0 02
0 93
. .
.
.
TL
T
L
T
3 70 1 0 86
0 02
0 97
. .
.
.
TL
T
L
T
Disturbance rejection
Kp 1 371
.
K
L
T
1 37
1.
K
L
T
1 37
1.
K
L
T
Ti2 42
1 18
.
.L
TT
3 76
1 39
.
.L
TT
4 68
1 52
.
.L
TT
Td 0 60.L
TT
0 55.
L
TT
0 50.
L
TT
Table 1 Optimal PID tuning formulas of Method A
Fig. 1 Unity feedback system with PID controller
Fig. 2 Two-degree-of-freedom unity feedback control sys-
tem
Fig. 3 Double-loop control scheme
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k k kci = ( )min max 8
As a result, the closed-loop transfer function of the
inner feedback loop becomes
H sk e
T s k k e
L s
L s( ) = +( )
m
m m ci
m
m19
which can be approximated by a stable SOPDT sys-
tem
G ske
s s
s
( ) =+ +
( )
2 2 110
by either (i) a model reduction technique, or (ii) Taylor
series expansion. Their study of two methods shows
that the model reduction technique is superior to the
Taylor series expansion regarding the system perform-
ance. However, the Taylor series expansion is much
easier to carry out.
Since the unstable process has been stabilized by
the proportional controller on the inner loop, the pri-
mary PID controller then acts for a stable plant, G(s).With k
1,
1,
1, and
1in Eq. (10) available, the tuning
rules proposed by Sung et al. (1996) in terms ofITAE
criteria are used to obtain PID controller settings, as
shown in Table 2.
Essentially, this PID-P scheme is equivalent to a
two-degree-of-freedom controller. Therefore,
stabilization and performance problems can be treated
separately and better performance than 1DOF control
can be expected. However, the normalized dead-time
of the process should be no larger than 0.693, which is
the limitation imposed by the normal relay feedback
identification. Robustness is not analyzed in this work.1.3 PI-PD control method
For an FOPTD unstable process, Majhi and
Atherton (2000a) presented another double-loop PID
control. The structure is similar to that of Method B as
shown in Figure 3. But now Go
is a PI controller, while
Gion the inner loop is a PD controller.
The unstable FOPDT process is described by
G s G sK e
T s
Ke
s
L s s
p mm
m
m n
( ) ( ) =
( )
1 111
Setpoint response
kKp = + +
0 04 0 333 0 949 0 9
0 983
. . . , .
.
kKp = + + >
0 544 0 308 1 408 0 9
0 832
. . . , .
.
Ti
= +
2 055 0 072 1. . ,
Ti
= + >
1 768 0 329 1. . ,
Td
= +( )
( )[ ]1
0 870 55 1 683
1 061 09
exp.
. .
..
Disturbance rejection
kKp = + +
0 67 0 0297 2 189 0 9
2 001 0 766
. . . , .
. .
kKp = + + >
0 365 0 26 1 4 2 189 0 9
2 0 766
. . . . , .
.
Ti
=
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where n =Lm/Tm is the normalized dead-time. A directrelay feedback identification is appplied to the plant
to obtain the parametersLm, T
mand K
mof Eq. (11). For
processes with n
< 0.693, the normal relay feedback
can be used. However, ifn
is large, i.e., n
> 0.693,
the limit cycle does not exist in the normal relay feed-
back, which explains why the Method B is only appli-
cable for processes with normalized dead-time less than
0.693. Thus an additional inner loop P controller has
to be added to overcome this problem, by which the
range of normalized dead time for the existing limit
cycle is extended to n
< 1. As a result, the method will
be applicable to control the FOPDT unstable processwith 0 <
n< 1.
The inner controller, Gi(s), is to stabilize the un-
stable process and takes a PD form:
G s KT
Ts K T si f
d
if f( ) = +
= +( ) ( )1 1 12
To approximate the resultant closed-loop transfer func-
tion by a stable FOPDT process after Pade approxima-
tion of esn , T
iis set toL/2. K
fis given by 1/K 2 n
as in Eq. (8) with the optimal gain margin. The main
controller is of the PI form:
G s KT s
o pi
( ) = +
( )11
13
and is tuned for satisfactory setpoint tracking. The in-tegral square time error optimization criterion, ISTE,
is used to design the PI controller and tuning formulas
for this PI-PD controller are shown in Table 3. Ro-
bustness of the control method is examined for
perturbations in process time delay.
1.4 Gain and phase margin PID tuning method
For the unstable FOPDT process, Wang and Cai
(2002) employed the same control configuration of
Method B as in Figure 3, where Go(s) is the primary
PID controller, and Klis the proportional controller on
the inner loop. But they designed the primary PID con-
troller based on gain and phase margin specifications.Such a double-loop configuration can be implemented
in an equivalent single-loop PID feedback system with
a setpoint weighting in Figure 4, where Kp, T
i, and T
d
are PID settings and b is the setpoint weighting.
With the P controller in the inner loop, the inner
closed-loop transfer function Gl(s) is
G ll
sKe
Ts KK e
Ls
Ls( ) = +( )
114
By its Taylor series expansion and truncation of the
time delay term in the denominator:
e Ls L sLs + ( )1 0 5 152 2.
Equation (14) becomes
G l pl l l
s G sKe
KK L s T KK L s KK
Ls
( ) ( ) =+ ( ) +
( )
0 5 1
16
2 2.
Table 3 Tuning rule for PI-PD controller of Method
C
Apeak
, h and =Apeak
/(kmh) are peak output amplitude,
relay amplitude and normalized peak output respec-
tively.
Fig. 4 PID controller with setpoint weighting
0 < 0.693
K Km p
= +
+
( )[ ]( )
0 8011 1 0 9358 1
1
. . ln
T
T
m
i= + + +
( ) ( )[ ]2
0 1227 1 4550 1 1 2711 1
1
2
tanh
. . ln . ln
T
T
d
m
=+( )
ln
tanh
1
41
Kf
=+( )
2
1ln
0.693 < < 1
K Km p =+
+
+
+
( )0 8011 0 99460 0682
0 7497 0 9946
0 0682
. .
.
. .
.
T
T
m
i =
+ + +
+ + +
3 2
3 2
5 2158 4 481 0 2817
0 0145 0 5773 2 6554 0 3488
. . .
. . . .
T
T
d
m
=+
+
( )0 0237 34 53384 1530
. .
.
Kf =
+
+
( )2 0 99460 0682
.
.
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To make Gp(s) stable, it follows from the Routh
Hurwits criterion that
KK
KT
LKKmin max= < < = ( )
117l
Again, to have the optimal gain margin, the P control-
ler gain is chosen as
K K KK
T
Ll = = ( )mi n ma x
118
Substituting it into Eq. (16) yields
( ) =+ +
( )
G se
as bs c
Ls
p 219
where a = 0.5L/K TL , b = 1/K(T TL ) and c =
1/K( T L 1 ).The transfer function of the PID controller is re-
written as
G s kAs Bs C
sc ( ) =
+ +
( )
2
20
whereA = Kd/k,B = K
p/kand C= K
i/k. The controller
zeros are chosen to cancel the poles of model Gp(s),
i.e.,A = a,B = b and C= c. This leads to
( ) ( ) = ( )
G s G s k e
s
Ls
p c 21
kis to be determined based on gain and phase margin
specifications. By assigning a gain margin ofAm
= 3
and a phase margin ofm
= 60, one has
kA L
= = ( )
2 622
mL
Hence, the tuning formulas are given as follows:
KK
T
L K
T
L
T
Lp = +
( )
1
623
TK
KL
T
L
i
p=
( )
61
24
TK K
TLdp
= ( )1
1225
b
L
T
L
T
=
+
( )
1
16
126
For implementation, the 2DOF control system in
Figure 4 is applied with all parameters set according
to Eqs. (23)(26), which is simpler and more straight-
forward than Figure 3. However, this method is lim-
ited into FOPDT unstable processes with the normal-
ized dead-timeL/Tless than 1, as indicated by the in-
equality of Eq. (17).
1.5 IMC-Maclaurin PID tuning method
For both FOPDT and SOPDT unstable processes,Lee et al. (2000) proposed explicit PID tuning rules
based on the internal model control (IMC).
The IMC scheme is shown in Figure 5 and the
closed-loop transfer functions are
HGq
q G GH
Gq G
q G Gyr yd
D=
+ ( )=
( )+ ( )11
1,
where G is the plant, G is the model, and q is the IMC
controller. They reduce to
H Gq H Gq Gyr yd D= = ( ), 1
in the case of perfect model-plant match, i.e., G = G .
The IMC system can be put into the equivalent con-
ventional single loop control scheme in Figure 2 with
HG G
G GH
G
G Gyr
c
cyd
D
c
=+
=+1 1
,
where
Fig. 5 IMC control system
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Gq
Gqc = 1
Write an unstable process model as G(s) = PM
(s)PA(s),
where PM
(s) is the invertible portion, while PA(s) is
the non-invertible part, i.e., RHP zeros and time delay.
Ifq has zeros to cancel the unstable poles ofG, and if
(1 G q) has zeros to cancel the unstable poles ofGD,
the closed-loop responses under both a setpoint change
and a load disturbance will be stable. To satisfy the
above conditions, the IMC controller is set as q =
PM
1(s)f. Here,f=fsf
dis comprised of two parts:f
s=
1/(s + 1)n is to make the controller proper by choos-ing a suitable n; andf
d= (
im=1 is
i + 1)/(s + 1)m is tocancel the unstable/stable poles near the zeros ofG
D,
in which iis to be determined to cancel the m unsta-
ble poles. Thus,fis the IMC filter with an adjustable
time constant , and the IMC controller is
q P s f P s
s
s
sn
ii
i
m
m= ( ) =
( )
+( )
+
+( )( )
=
MM1
11
1
1
127
It follows that
H GqP s
s
s
sn
ii
i
m
myrA= =
( )
+( )
+
+( )( )=
1
1
1281
H Gq GP s
s
s
sG
n
ii
i
m
myd DA
D= ( ) = ( )
+( )
+
+( )
( )
=
1 11
1
1
29
1
The term (im=1 is
i + 1) in Hyr
causes an overshoot in
the closed-loop response for setpoint changes. This
problem could be solved by adding a pre-filter FR
(s) =
1/( im=1 is
i + 1). The equivalent conventional feedback
controller from this IMC controller is
Gq
Gq
P s
s
s
s
P s
s
s
s
n
ii
i
m
m
n
ii
i
m
m
c
M
A
=
=
( )
+( )
+
+( )
( )
+( )
+
+( )
( )
=
=
11
1
1
11
1
1
30
11
1
Gc(s) is further approximated by a PID controller with
its settings taken as the first three terms of its Maclaurin
series expansion in s:
G ss
f f sf
sc ( ) = ( ) + ( ) +( )
+
( )1
0 00
2312
!K
The tuning formulas for first-order and second-order
unstable time-delayed processes are given in Table 4.
The limitation is due to the stability constraint: for the
FOPDT unstable process, no stabilizing PID param-
eters can be found unless its normalized dead-time
L/T 2. Robustness is fully analyzed for parametricuncertainty.
1.6 IMC-based approximate PID tuning method
For quite general unstable processes, Yang et al.
(2002) developed a frequency response fitting approach
to derive feedback controllers from an IMC one with
stability. It has options to choose between PID and
high-order forms. The ideal controller is first formu-
lated according to the standard IMC design. Stability
is ensured when converting to the single-loop control-
ler. It is shown that high-order controllers become nec-
essary for high performance for processes of essential
high orders, where PID controllers become inappro-
priate. For simple processes, model reduction is ex-
ploited to approximate the ideal IMC equivalent feed-
back controller in Eq. (30) by a standard PID control-
ler.
G s K T s T s K K s K sC,PID p
id p
id( ) = + +
= + + ( )11 1
32
Given the desired closed-loop bandwidth b, the
standard non-negative least squares method is used to
find the optimal PID parameters {Kp, K
i, K
d} to mini-
mize the criterion
EG j G j
G j=
( ) ( )( )
( )( )max ,
033
b
C,PID C
C
where is the desired accuracy of the PID approxima-tion to the IMC controller. The default is set as 5%.Once this criterion is satisfied, the controller design
procedure is completed. Similar to Lee et al. (2000)s
method, a pre-filter is added to eliminate the overshoot.
1.7 Modified Smith predictor control method
For an FOPDT unstable process Gp(s) in Eq. (2),
Majhi and Atherton (2000b) presented a modified
Smith predictor control scheme depicted in Figure 6,
where three controllers serve different objectives. Gc1
in the inner loop is to stabilize the integrating and un-
stable process. The other two controllers, Gc
and Gc2
are used for set-point tracking and disturbance rejec-
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tion, respectively, by regarding the inner loop as an
open-loop stable process. This structure is similar to
that of Matausek and Micic (1996) when Gc1
= 0 and is
equivalent to the standard Smith predictor when Gc1
=
Gc2
= 0.
Suppose that the method perfectly matches the
process dynamics, i.e., Gm(s) e
L s m= Gp(s) with Gm(s)= K
m/T
ms 1. The closed-loop setpoint response and
disturbance response are given by
Y sG G e
G G GY s e
L sL s
rc m
m c c1r
m
m
1+( ) =
+( )= ( ) ( )
34
Y sG e
G G G
G G G G G e
G G e
Y s e
L s L s
L s
L s
lm
m c c1
m c c1 c m
m c2
l
m m
m
m
1 +
1 +( ) =
+( )
+( )
+
= ( ) ( )
1
35
Since the time delay term is eliminated from the
denominator of the setpoint response transfer function
Yr(s), the G
ccan be taken as a PI controller G
c(s) =
Kp(1 + 1/T
is) for pole placement on the delay free sys-
tem. The controller Gc1
is chosen in the PD form: Gc1
(s)
= Kf(1 + T
fs). The proportional controller G
c2= K
d,
whose another job is to reject unwanted load distur-
bances, is designed to stabilize the second part of the
characteristic equation of Eq. (35), and a possible
choice is Kd
= 1/Km
T Lm m under the constraintLm/Tm< 1, as suggested by DePaor and OMalley (1989).
Hence, it follows that Yr(s) and Yl
(s) in Eqs. (34)and (35) become
( ) =+( )[ ] +
=+
( )Y sT T K K s s
r
m f m p
1
2 1
1
136
( ) =+ ( )
+( ) +( )( )
Y s
K s e
s s K K e
L s
L sl
m
m d
m
m
1
1 137
where is the tuning parameter. Taking = Tm
+ 2Tf
results in the following controller parameters:
Table4
IMC-PIDtuningrulesofMethodE
Fig. 6 Modified Smith predictor control system
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KK
pm
= ( )1
38
T T Ti m f= + ( )2 39
KK
fm
= ( )2 40
TT
fm=
( )
241
KK
T
Ld
m
m
m
= ( )1
42
Kaya (2003) proposed an alternative formula for tun-
ing , from the user-specified settling time and assumesthat the settling time t
sis proportional to the time con-
stant , i.e.,
t ks = ( ) 43
In the coefficient diagram method (CDM)
(Hamamci et al., 2001), k is chosen between 2.5 and
3.0 to perform well for processes with large time con-
stants, an integrator or unstable pole. They suggested
k= 2.5, or
= ( )ts2 544
.
This modified tuning rule looks systematic, compared
with Majhi and Atherton (2000b), where is chosenarbitrarily or equal to the estimated dead time.
2. Performance Comparison
To assess the performance of seven methods men-
tioned in the preceding section, the FOPDT unstable
process is used for simulation study as it is most popu-
lar. Three cases of different ranges of normalized dead-
time are considered: (i) 0 < L/T< 0.693, (ii) 0.693
L/T < 1, and (iii) 1 L/T < 2. Controller settingsused are obtained strictly following respective meth-
ods, and when there are some design parameters which
are not specified by certain methods, the optimum val-
ues are found by trial and error and applied. There-
fore, the true or best performance of seven methods
are compared. Performance is evaluated in terms of its
applicability, time domain specifications on set-point
response and disturbance rejection, control action size
and system robustness. The time domain specifications
under consideration are listed as follows:
(1) Rise t ime trof the set-point response: the time for
the step response to rise from 10% to 90% of its
steady state value.
(2) Settling-time tsof the set-point response: the time
for the step response to stay within 2% of its steady
state value.
(3) OvershootMp
of the set-point response: the ratio
of the difference between the first peak and thesteady state value to the steady state value of the
step response.
(4) Integral absolute error of the set-point response:
IAE = 0 |e(t)|dt.
(5) Integral square error of the set-point response: ISE
= 0 [e(t)]2dt.
(6) Recovery time tR
of disturbance response: the time
for the response to stay within 2% of its steady
state.
(7) Maximum error emax
of disturbance response: the
maximum absolute error during the response.
Control signals are observed in simulation and
commented. Maximum control signals should be of
similar magnitudes for judgement of controller designs.
Robustness is evaluated by varying some process pa-
rameters.
2.1 Small normalized dead-time: 0 < L/T< 0.693
Consider the following FOPDT unstable process
G se
s
s
p ( ) = ( )
4
4 145
2
which has been studied extensively in the literature.One seesL = 2, T= 4, K= 4, and the normalized dead-
time L/T = 0.5. A step change is introduced in the
setpoint at t= 0 with a unity magnitude and in the dis-
turbance at t= 75 with a magnitude of0.1. To have a
fair comparison, a pre-filter is added in Method A to
reduce the excessive overshoots.
The output time responses of Methods A to G are
shown in Figures 79. The performance specifications
are listed in Table 5.
Let us first look at the control signal u(t). Figure
10 exhibits input time responses of all seven methods.
It is seen that in this scenario the sizes of control ac-
tions are similar. In the case of stable processes, a bigcontrol action, when properly desiged, could poten-
tially give better output response. It seems that the case
becomes much complicated when an unstable process
is considered. As it is known to all, feedback system is
only conditionally stable for unstable processes. It is
guessed that control action is already limited due to
stabilization requirement and leaves less room to ma-
nipulate for performance than the stable process case.
This argument is more appealing for other two cases
of median and large normalized dead-time as they are
more difficult to stabilize than the small case. Therefore,
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we will drop this factor of control action from now on
and focus on the system performance and robustness.
In the sense of the setpoint response, we can eas-
ily see that Method G, the modified Smith predictor
control, gives the best performance: fastest settling,
no overshoot and lowest IAE/ISE. The merit of Smith
predictor is thus obvious: the time delay term is elimi-
nated from the characteristic equation of the setpoint
transfer function, and thus closed-loop time constant
can be taken as the design parameter. Consequently,
settling time is under control. Moreover, there is no
closed-loop zero introduced by the PI controller on the
forward transfer function, and thus no overshoot is to
be taken care of. However, for the disturbance rejec-
tion, the performance is not satisfactory compared with
other methods. Note also that the control system of
Method G is the most complicated, too, in which there
are three controllers to be designed.
Methods E and F, which are both IMC-based, give
the second best setpoint responses: short settling time,
Fig. 7 System response of Method A for small normalized dead-time
Fig. 8 System response of Methods B, C and D for small normalized dead-time
Fig. 9 System response of Methods E, F and G for small normalized dead-time
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very small or even no overshoots and small IAE/ISE
as well. The recovery times of Methods E and F from
the disturbance are the smallest. IMC based design is
essentially more complicated than the traditional PID
design, and the PID settings obtained by approximat-
ing the ideal IMC controller are superior to those from
traditional PID tuning rules. More desirable closed-
loop responses can be expected. Since Method E gives
an explicit tuning rule, it is more convenient for prac-tical implementation as a PID auto-tuning strategy.
Among the remaining four traditional PID con-
trol schemes, Method C provides the highest perform-
ance, although not as good as the former three. Except
for some oscillatory behaviors, it generates very low
overshoots but a fast rise-time. Method A has very sim-
ple tuning rules in line with different optimization
specifications. However, given a properly tuned pre-
filter, these simple tuning formulas gives PID settings
with relatively good system responses, especially, in
the optimized specifications they aim to achieve, for
example, ISE specification of Method A (ISTE setting)
is the lowest among all the methods. Also derived from
the double loop PID-P structure, Method D gives a
reasonable output response, which is similar but bet-
ter than Method B. In fact, Method B doesnt have a
very good performance in the comparison, but it pio-
neered in the double-loop PID controller design for
unstable plants.
The robustness performances of Methods A to G
are addressed by assuming a mismatch of 10% onthree parameters, K, T and L, of the process model,
respectively. The results are presented in Figures
1113 , from which it can be concluded that all the
Table5
Controlpe
rformanceforsmallnormalizeddeadtime
Fig. 10 Control signals of investigated methods for smallnormalized dead-time
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investigated methods are able to maintain stability and
satisfactory performance in small modelling mismatch.
However, in the case of +10% perturbation, Methods
A and C present evident oscillating behaviors. Over-
all, the IMC-based Methods E and F and the modified
Smith predictor Method G still outperform the others
under parametric uncertainty, as shown by specifica-
tions in Table 6.
2.2 Medium normalized dead-time: 0.693 L/T< 1
Consider now the time-delayed unstable process
G se
s
s
p ( ) = ( )
1 2
1 5 146
.
.
with the normalized dead-time ofL/T= 0.8. Method B
is excluded since it has been stated in Parket al. (1998)
that is only designed for 0 < L/T < 0.693. The same
Fig. 11 Robustness analysis of Method A
Fig. 12 Robustness analysis of Methods B, C and D
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Mp [%] tr ts IAE ISE
Method A (ISE) nominal 17.27 5.32 29.26 5.75 1.35
10% mismatch 20.83 5.06 26.45 6.04 1.63
+10% mismatch 18.82 3.25 33.56 5.46 1.24
Method A (ISTE) nominal 6.92 3.28 14.58 4.34 0.95
10% mismatch 10.60 4.13 16.72 4.49 1.05+10% mismatch 8.65 2.91 26.81 4.21 0.99
Method A (ITSE) nominal 11.05 3.56 19.16 4.58 1.00
10% mismatch 13.02 4.66 18.48 4.75 1.14
+10% mismatch 11.52 2.99 28.28 4.41 0.99
Method B nominal 42.47 4.10 50.13 10.16 5.49
10% mismatch 52.78 4.14 67.50 12.85 6.32
+10% mismatch 36.65 4.01 37.82 8.92 5.25
Method C nominal 10.81 2.68 15.62 4.45 3.88
10% mismatch 6.67 2.92 15.02 4.30 3.24
+10% mismatch 20.60 2.56 33.74 6.14 3.86
Method D nominal 25.28 4.33 27.20 7.28 4.6110% mismatch 35.14 4.34 41.25 8.76 4.88
+10% mismatch 21.19 4.26 30.34 7.09 4.69
Method E nominal 0 7.44 13.48 5.79 1.09
10% mismatch 3.03 6.42 18.91 5.93 1.32
+10% mismatch 0.45 9.74 21.20 5.23 0.98
Method F nominal 1.30 6.83 12.01 5.02 0.95
10% mismatch 3.52 5.79 17.66 5.18 1.10
+10% mismatch 3.84 3.92 23.65 4.69 0.91
Method G nominal 0 4.39 9.83 2.01 1.01
10% mismatch 0 3.58 20.73 2.43 1.03
+10% mismatch 8.23 5.17 19.75 2.43 1.02
Table 6 Performance robustness
Fig. 13 Robustness analysis of Methods E, F and G
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external signals as before are introduced to the system
to generate time responses for Methods A, C, D, E, F
and G, as shown in Figures 1416. The performance
specifications are listed in Table 7.
Note that Method G gives almost unchanged
setpoint specifications. It is because in this example,
the design parameter , closed-loop time constant, re-mains the same as in the first case. Although it still
performs the best in setpoint performance, Method G
becomes less effective for disturbance rejection for
larger normalized dead-time. It can be observed from
Eq. (35) that the disturbance transfer function of the
modified Smith predictor is not delay-free, which is
the reason why its disturbance rejection is much worse
than its setpoint response.
The IMC-based Methods E and F have no over-
shoot, very short settling time and recovery time for
the process Eq. (46). They are superior to all the other
Fig. 14 System responses of Method A for median normalized dead-time
Fig. 15 System responses of Methods C, D and E for median normalized dead-time
Fig. 16 System responses of Methods F and G for median normalized dead-time
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methods in overall behaviors.
Methods C and A are also working pretty well in
this range of normalized dead-time, especially with
regard to the disturbance rejection. And it is worth
mentioning that Method A outperforms other methods
at IAE and ISE indices. Method D gives the poorest
performance in this scenario. This may be due to the
choice of the setpoint weight value b provided byEq. (26) and the Taylor series approximation ofeLs.
However, if the process to be controlled is only of an
insignificant time delay, Method D will become a con-
venient design approach.
2.3 Large normalized dead-time: 1 L/T< 2Finally consider this process:
G se
s
s
p ( ) = ( )
1 5
147
.
with the normalized dead-time ofL/T= 1.5. Now, only
Methods E and F remain applicable to the process ofthe ratio L/T 1. The system responses are shown inFigure 17. The performance indices are listed in Ta-
ble 8. Note that in this case, Method E yields more
oscillatory behaviors than Method F does.
It can be concluded from the above three case stud-
ies that the following overall ranking in terms of gen-
eral applicabilities and achievable control perform-
ance: (1) Method F, (2) Method E, (3) Method G, (4)
Method C, (5) Method A, (6) Method D, (7) Method
B.
It has been reported in the literature that properly
tuned P/PI controllers can stabilize the FOPDT unsta-ble process if and only if its normalized dead timeL/T
1, and that the PD/PID controller can relax the con-straint to L/T 2, as the D portion contributes phaselead in the control system. Therefore, for those feed-
back configurations with PD/PID controllers, like
Method C, they have potential to be extended to the
case of 0 L/T 2, but need some more future re-search. And similar extension can also be made for the
modified Smith predictor structure of Method G, by
changing the controller Gc2
into the PD form to increase
its applicability. Improvements on disturbance attenu-
ation are also expected in Method G, while retaining
the merit of pole placement in its closed-loop setpointtransfer function, Eq. (34).
3. Nonlinear PID Control
It can be seen from comparative studies in the pre-
ceding section that the best achievable performance
among seven studied methods has been impressively
good. It would be difficult to devise a new and better
control scheme within their frameworks. However, we
noticed that all these methods use linear time invari-
ant (LTI) controllers only. As there are many nonlinear
Table7
Controlper
formanceformediumnormalizeddead
time
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Thus kmax
= k0
+ k1
when e(t) = , and kmin
= k0
when
e(t) = 0. k2
defines the change rate in the range from
kmin
to kmax
together with e(t). k(t) can also be of the
sigmoidal function:
k t k k k e t
( ) = ++ ( )( )
( )0 12
2
11 51
exp
where kmin
= k0k
1when e(t) = , k
max= k
0+ k
1when
e(t) = + and k(t) = k0
when e(t) = 0. Let us implement
the above three NPID control strategies in Method B
and repeat simulation example Eq. (45) to validate them
in controlling unstable processes. The NPID settings
are listed in Table 9, which are tuned by trial and er-
ror. From Table 9 and Figure 18, it is obvious that the
system performance has been improved greatly by
NPID controllers, especially the setpoint response,
which indicates the usefulness of NPID control.
It is however noted that the above modifications
of Method B yet could not achieve performance as good
as the best available ones in Table 5, say those of
Method F. Hence, it is logical to incorporate nonlinear
PID control into the best linear controller. As Method
F gives an overall most satisfactory control, we use its
controller settings and modify the proportional gain
into a nonlinear component, while the linear integral
and derivative gains are retained. Therefore,
u t k k t e t k e t t k e t t
( ) = ( ) ( ) + ( ) + ( ) ( )p i dd 0 52
where kp, k
i, k
dare the fixed gains provided by Method
F, and k(t) is in the form of our modified sigmoidal
function:
k t k k k e t k e t
( ) = +( )( ) + ( )( )
( )0 12 3
21 53
exp exp
The simulation examples, Eqs. (45), (46) and (47), have
been repeated to test such a nonlinear control strategy.
The parameters in the nonlinear equation, Eq. (53), are
determined by trial and error. The system responses
are shown in plots A, B and C of Figure 19, respec-
tively. The performance specifications and controller
Table 9 Control performance of nonlinear modification of Method B
Fig. 18 Nonlinear modification of Method B
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settings are listed in Table 10. From simulation results,we can see that performance improvement is achieved
from the nonlinear modification on the original con-
troller settings of Method F, although not very signifi-
cant. It is also worth noting that disturbance rejection
is not as good as that provided by linear controllers.
We also noticed that the setpoint response of
Method G is the best among all the investigated linear
methods. Due to the merit of Smith predictor control
structure, there is no RHP zero introduced in the
setpoint transfer function. Therefore, we try to further
accelerate the step response by using LTV setpoint
weighting. The time varying gain of the pre-filter is
chosen as follows:
f t M t
T M T t r ( ) = +
( ) ( )
1 54sgn
where
sgn,
,t
t
t( ) =