In industrial practice, controller designs are performed based on an approximate model of the actual process. It is essential to design a control system which will exhibit a robust performance because the physical systems can vary with operating conditions and time. Gain and phase margins are well known parameters for evaluating the robustness of a control system. This paper presents a tuning algorithm to design and tune PI controllers for stable processes with a small dead time while meeting specified gain and phase margins. Simulation examples are given to demonstrate that the proposed design method can result, in a closed-loop system, in better performances than existing design methods which are also based on user-specified gain and phase margins.

1. ISA
TRANSACTIONS®
ISA Transactions 43 ͑2004͒ 297–304
Tuning PI controllers for stable processes with specifications
on gain and phase margins
Ibrahim Kaya*
Inonu University, Engineering Fac., Dept. of Electrical & Electronics Eng., 44069 Malatya, Turkey
͑Received 28 February 2003; accepted 17 August 2003͒
Abstract
In industrial practice, controller designs are performed based on an approximate model of the actual process. It is
essential to design a control system which will exhibit a robust performance because the physical systems can vary with
operating conditions and time. Gain and phase margins are well known parameters for evaluating the robustness of a
control system. This paper presents a tuning algorithm to design and tune PI controllers for stable processes with a
small dead time while meeting specified gain and phase margins. Simulation examples are given to demonstrate that the
proposed design method can result, in a closed-loop system, in better performances than existing design methods which
are also based on user-specified gain and phase margins. © 2004 ISA—The Instrumentation, Systems, and Automa-
tion Society.
Keywords: IMC design; PI controller; Gain margin; Phase margin; Time delay
1. Introduction ally, vary with operating conditions and time.
Hence robustness of a control system has always
Proportional-integral-derivative ͑PID͒ control- been an important issue. Gain and phase margins
lers are widely used in industrial systems despite are two well-known measures for maintaining the
the significant developments of recent years in robustness of the control system. Recently, there
control theory and technology. This is because has been a renewed interest in designing a control
they perform well for a wide class of processes. system to satisfy the specified gain and phase mar-
PID controllers give robust performance over a gins ͓1– 4͔.
wide range of operating conditions. Furthermore, This paper presents a controller design, with
they are easy to implement using analog or digital specifications on gain and phase margins, for con-
hardware and familiar to engineers. trolling stable processes with small time delays.
In the practice, the model used to analyze or To accomplish this, the classical single input
design control systems is only an approximation single output ͑SISO͒ feedback control system is
of the actual plant transfer function. The most represented as its equivalent internal model con-
common models used for stable plant transfer trol ͑IMC͒ ͓5,6͔. This representation provides the
functions are a first-order plus dead time ͑FOPDT͒ parameters of PID-type controllers used in the
or second-order plus dead time ͑SOPDT͒ model. SISO system to be defined in terms of the desired
Also, the parameters of the physical systems, usu- closed-loop time constant, which can be adjusted
by the operator, and the parameters of the process
*Tel: ϩ90 422 3410010, ext. 4499; fax: ϩ90 422 model. This means that only one parameter,
3410046. E-mail address: ikaya@inonu.edu.tr namely the desired closed-loop time constant, is
0019-0578/2004/$ - see front matter © 2004 ISA—The Instrumentation, Systems, and Automation Society.

2. 298 Ibrahim Kaya / ISA Transactions 43 (2004) 297–304
left for tuning, assuming that the model param-
eters have been obtained from a relay autotuning
͓7,8͔. The details of the identification method are
not given here and interested readers can refer to
the cited references. However, for the sake of easi-
ness, equations used for parameter estimation of
the FOPDT plant transfer function are given in the
Appendix. The value of the closed-loop time con-
stant is obtained from the specified gain and phase
margins. The proposed design method is compared Fig. 1. IMC control strategy.
with existing design methods based on the speci-
fied gain and phase margins and it is shown by This control structure is referred to as internal
examples that the proposed design method gives ˆ
model control ͑IMC͒ since the plant model G ( s )
better closed-loop performance. appears in the control structure. Here, G ( s ) and
The next section gives a brief review of the IMC ˆ
G ( s ) are the actual process and process model
design, since the tuning rules to tune/design PI transfer functions, respectively. When G ( s )
controllers are derived using IMC principles. ˆ
ϭG ( s ) , that is perfect modeling, and dϭ0, the
Simple tuning rules for obtaining parameters of a system is basically open loop. This provides the
PI controller based on the user-specified gain and ˆ
open-loop advantages. When G ( s ) G ( s ) or d
phase margins are derived in Section 3. Section 4
0 the system is a closed-loop system. Thus the
gives simulation examples which show that the
IMC control strategy has the advantages of both
design method given in the paper results in a bet-
the open-loop and closed-loop structures.
ter closed-loop system performance than existing
The first step in the IMC controller design is to
design methods which is designed with specifica-
factor the process model,
tions on gain and phase margins as well. Conclu-
sions are given in Section 5. ˆ ˆ ˆ
G ͑ s ͒ ϭG ϩ ͑ s ͒ G Ϫ ͑ s ͒ , ͑1͒
ˆ
where G ϩ ( s ) contains all the time delays and
right-half plane zeros.
2. Internal model control „IMC… The second step is to define the IMC controller
as
A control system design is expected to provide a
fast and accurate set-point tracking, that is, the G imc ͑ s ͒ ϭG Ϫ1 ͑ s ͒ F ͑ s ͒ ,
ˆ
Ϫ ͑2͒
output of the system should follow the input signal
where F ( s ) is a low pass filter with a steady-state
as close as possible. In addition, any external dis-
gain of 1. The filter is introduced for physical re-
turbances must be corrected by the control system
alizability of the IMC controller G imc ( s ) . The
as efficiently as possible. The first requirement is
simplest filter has the following form ͓5,6͔:
achieved by an open-loop control system. With an
open-loop control scheme, the stability of the sys- 1
tem is guaranteed provided that both the plant and F͑ s ͒ϭ . ͑3͒
controller transfer functions are stable. Also, the ͑ ␭sϩ1 ͒ n
design of the controller in an open-loop control From the block diagram of the IMC structure
scheme may simply be chosen as G c ( s ) shown in Fig. 1, the closed-loop transfer is given
ϭG Ϫ1 ( s ) , where G c ( s ) and G ( s ) are, respec- by
tively, the controller and plant transfer functions.
The drawback of an open-loop control system is G ͑ s ͒ G imc ͑ s ͒
T r͑ s ͒ ϭ ˆ . ͑4͒
the sensitivity to modeling errors and inability to 1ϩ ͓ G ͑ s ͒ ϪG ͑ s ͔͒ G imc ͑ s ͒
deal with external disturbances entering the sys-
tem. In this case, a closed-loop system can be used The disturbance transfer function of the IMC
to deal with disturbances and modeling errors. structure is
Rivera et al. ͓5͔ proposed the control structure ˆ
1ϪG ͑ s ͒ G imc ͑ s ͒
given in Fig. 1. T d͑ s ͒ ϭ ˆ . ͑5͒
1ϩ ͓ G ͑ s ͒ ϪG ͑ s ͔͒ G imc ͑ s ͒

3. Ibrahim Kaya / ISA Transactions 43 (2004) 297–304 299
Substituting Eq. ͑2͒ into Eqs. ͑4͒ and ͑5͒ gives
G͑ s ͒F͑ s ͒
T r͑ s ͒ ϭ ˆ ˆ ͑6͒
G Ϫ ͑ s ͒ ϩ ͓ G ͑ s ͒ ϪG ͑ s ͔͒ F ͑ s ͒
and
ˆ
G Ϫ ͑ s ͓͒ 1ϪF ͑ s ͔͒
T d͑ s ͒ ϭ ˆ ˆ . ͑7͒
G Ϫ ͑ s ͒ ϩ ͓ G ͑ s ͒ ϪG ͑ s ͔͒ F ͑ s ͒ Fig. 2. IMC representation of a SISO control system.
ˆ
For perfect modeling, G ( s ) ϭG ( s ) , and nonmini-
mum phase systems, G Ϫˆ ( s ) ϭG ( s ) , Eqs. ͑6͒ and
ˆ To find the tuning parameters for the controller
͑7͒ can be further simplified to give G c ( s ) in Fig. 2, stable first-order plus dead time
͑FOPDT͒ plant transfer function is considered. It
T r ͑ s ͒ ϭF ͑ s ͒ ͑8͒ should be pointed out that the FOPDT model is
only used for simplifying calculations and that the
and
actual process may be a higher-order process or a
T d ͑ s ͒ ϭ1ϪF ͑ s ͒ . ͑9͒ process with complex poles, etc. In order to obtain
the IMC controller, the process FOPDT model,
Eqs. ͑8͒ and ͑9͒ show that the performance of a
G ( s ) ϭKe Ϫ ␪ s / ( Tsϩ1 ) , must be factored as in
ˆ
closed-loop system designed based on the IMC
Eq. ͑1͒:
design method is determined solely by the filter
dynamics. For a filter with the form given by Eq. G ϩ ͑ s ͒ ϭe Ϫ ␪ s ,
ˆ ͑14͒
͑3͒ and for t→ϱ, Eqs. ͑8͒ and ͑9͒ give c r ( t ) →1
and c d ( t ) →0. ˆ K
G Ϫ͑ s ͒ ϭ . ͑15͒
Tsϩ1
3. Controller design
The IMC controller can be obtained from Eq. ͑2͒,
The closed-loop transfer function of a classical assuming a filter with nϭ1, as
SISO feedback system and IMC design, for per-
fect matching, are, respectively, given by Tsϩ1
G imc ͑ s ͒ ϭ . ͑16͒
K ͑ ␭sϩ1 ͒
G c͑ s ͒ G ͑ s ͒
T siso ͑ s ͒ ϭ ͑10͒ Using a first-order Taylor series expansion for the
1ϩG c ͑ s ͒ G ͑ s ͒
time-delay approximation, the classic controller
and G c ( s ) can be obtained from Eq. ͑13͒,
T imc ͑ s ͒ ϭG imc ͑ s ͒ G ͑ s ͒ . ͑11͒ Tsϩ1
G c͑ s ͒ ϭ . ͑17͒
In order to have the same output for both configu- K ͑ ␭ϩ ␪ ͒ s
rations, it is easy to illustrate, by comparing Eqs.
Eq. ͑17͒ is rearranged as an ideal PI controller,
͑10͒ and ͑11͒, that the IMC controller G imc ( s ) is
which has the following controller parameters:
related to the classic controller G c ( s ) through the
transformation T
K pϭ , ͑18͒
G c͑ s ͒ K ͑ ␭ϩ ␪ ͒
G imc ͑ s ͒ ϭ ͑12͒
1ϩG c ͑ s ͒ G ͑ s ͒ T i ϭT. ͑19͒
or The only unknown in the last two equations is the
G imc ͑ s ͒ filter time constant ␭, since it is assumed that the
G c͑ s ͒ ϭ . ͑13͒ plant transfer function model is obtained from the
1ϪG imc ͑ s ͒ G ͑ s ͒
exact relay feedback identification method given
Therefore from Eq. ͑12͒ a classical SISO feedback in Refs. ͓7,8͔. Thus if a proper value of ␭ is
system can be put into the IMC structure as shown achieved, then the design procedure will be com-
in Fig. 2. pleted. Here, the value of ␭ is obtained from gain

4. 300 Ibrahim Kaya / ISA Transactions 43 (2004) 297–304
and phase margin specifications, which are well-
known robustness parameters.
The characteristic equation of a SISO control
system is given by 1ϩG c ( s ) G ( s ) . Hence the
open-loop transfer function of the SISO control
system, with G c ( s ) given by Eq. ͑17͒, is
e Ϫ␪s
G c͑ s ͒ G ͑ s ͒ ϭ . ͑20͒
͑ ␭ϩ ␪ ͒ s Fig. 3. The block diagram for computing the controller,
G c (s), parameters.
Therefore from the basic definitions of the gain
and phase margins the following equations can be The recommended ranges for the gain and phase
obtained: margins are between 2 and 5 and 30° and 60°,
arg͕ G c ͑ j ␻ p ͒ G ͑ j ␻ p ͒ ͖ ϭϪ ␲ , ͑21͒ respectively ͓9͔. Choosing A m ϭ3, then ␾ m
ϭ60°. Therefore the closed-loop time constant ␭
A m ͉ G c ͑ j ␻ p ͒ G ͑ j ␻ p ͒ ͉ ϭ1, ͑22͒ is obtained, by rearranging Eqs. ͑25͒ and ͑26͒, as
͉ G c ͑ j ␻ g ͒ G ͑ j ␻ g ͒ ͉ ϭ1, ͑23͒ ␭ϭ ␪ ͩ 2A m
␲ ͪ
Ϫ1 ϭ0.91␪ . ͑31͒
␾ m ϭ ␲ ϩarg͕ G c ͑ j ␻ p ͒ G ͑ j ␻ p ͒ ͖ , ͑24͒
Hence the PI controller parameters are given by
where the gain margin is given by Eqs. ͑21͒ and Eqs. ͑18͒ and ͑19͒ with ␭ given by Eq. ͑31͒.
͑22͒, and the phase margin by Eqs. ͑23͒ and ͑24͒. Remark: If a second-order plus dead time
The frequency ␻ p is known as phase crossover ͑SOPDT͒ plant transfer function model, G ( s )ˆ
Ϫ␪s
frequency, where the Nyquist curve has a phase ϭKe / ( T 1 sϩ1 )( T 2 sϩ1 ) , instead of the
lag of Ϫ␲, and the frequency ␻ g is known as the FOPDT model, is assumed, then the controller
gain crossover frequency, where the Nyquist curve G c ( s ) will be a PID controller. However, it has
has an amplitude of 1. been observed during extensive simulations that
Substituting Eq. ͑20͒ into Eqs. ͑21͒–͑24͒, results using a PID controller provides a little improve-
in the following set of equations: ment in the closed-loop performance of the sys-
tem. Hence simulation results only for the PI con-
␲
␻ p␪ ϭ , ͑25͒ troller are given.
2
3.1. Tuning procedure
A m ϭ ␻ p ͑ ␭ϩ ␪ ͒ , ͑26͒
1 The block diagram for obtaining the PI control-
␻ gϭ , ͑27͒ ler, G c ( s ) , parameters, based on the specified gain
␭ϩ ␪ and phase margins, is shown in Fig. 3. The tuning
procedure can be carried out as follows:
␲
␾ mϭ Ϫ ␻ g␪ . ͑28͒ 1. When the controller needs to be tuned,
2 switch from the controller mode to relay mode.
2. Measure the limit cycle parameters and esti-
From Eqs. ͑26͒ and ͑27͒ one can obtain
mate the parameters of the FOPDT plant transfer
A m␻ gϭ ␻ p . ͑29͒ function using the relay feedback method. Equa-
tions to compute the parameters of the FOPDT
Multiplying both sides of Eq. ͑29͒ with ␪ and then plant transfer function are provided in the Appen-
substituting values of ␻ g ␪ and ␻ p ␪ from Eqs. ͑25͒ dix.
and ͑28͒, the relation between gain and phase mar- 3. Find the PI controller parameters using Eqs.
gin can be obtained as ͑18͒ and ͑19͒, in conjunction with Eq. ͑31͒.
ͩ ͪ
4. Switch from the relay mode to the controller
␲ 1 mode with calculated tuning parameters for the
␾ mϭ 1Ϫ . ͑30͒
2 Am control of the process.

5. Ibrahim Kaya / ISA Transactions 43 (2004) 297–304 301
Fig. 5. Control signals for example 1.
Fig. 4. Step responses for example 1.
that with the proposed design method less effort is
4. Simulation examples required for the control action, for both design
methods.
Several examples are presented to illustrate the Example 2: In this example, a real industrial
use of the proposed method. Since the presented HVAC system used in Wang et al. ͓3͔ with trans-
design method is model based, the identification fer function of G ( s ) ϭe Ϫ2s / ( 0.12s 2 ϩ1.33s
method given by Kaya ͓7͔ or Kaya and Atherton ϩ1.24) is considered. The FOPDT model was ob-
͓8͔ has been used to find the FOPDT model. The tained as G ( s ) ϭ0.81e Ϫ2.78s / ( 1.011sϩ1 ) , using
identification method has been used for all transfer the estimation method given in Refs. ͓7,8͔. The PI
functions in the examples but since it gives essen- controller parameters are K p ϭ0.235 and T i
tially exact results on simulation data the esti- ϭ1.011, when Eqs. ͑18͒ and ͑19͒ are used in con-
mated plant transfer functions are only given for junction with Eq. ͑31͒. The PID controller param-
original plants of higher order. In all the examples, eters used by Wang et al. ͓3͔ are K p ϭ0.611, T i
controllers, for both the proposed design method ϭ1.441, and T d ϭ0.564. The response of the
and design methods that are used for comparison, closed-loop system with designed controllers for
are designed for a gain and phase margin of 3 and both design methods to a unity step set-point
60°, respectively. change together with load disturbance introduced
Example 1: Consider a second-order plus dead at time 30 s are given in Fig. 6. The assumed load
time plant transfer function of G ( s ) ϭe Ϫ1.0s / ( s disturbance magnitude was Ϫ0.5. In terms of
ϩ1 )( 0.5sϩ1 ) , which was given in Ref. ͓1͔. The
identification method given in Refs. ͓7,8͔ was
used to obtain the FOPDT model as G ( s )
ϭe Ϫ1.34s / ( 1.44sϩ1 ) . Therefore the PI controller
parameters are obtained from Eqs. ͑18͒ and ͑19͒,
in conjunction with Eq. ͑31͒, to be K p ϭ0.563 and
T i ϭ1.44. The controller parameters for the design
method proposed by Ho et al. ͓1͔ are K p ϭ0.52,
T i ϭ1.00, and T d ϭ0.50. The closed-loop re-
sponses for both design methods are given in Fig.
4 for a unity step set-point change and a distur-
bance with magnitude of Ϫ0.5 introduced at time
30 s. As is seen from the figure, the proposed de-
sign method results in a better performance for
both the set-point response and disturbance rejec-
tion. Fig. 5 illustrates control signals, which show Fig. 6. Step responses for example 2.

6. 302 Ibrahim Kaya / ISA Transactions 43 (2004) 297–304
Fig. 8. Step responses for example 3.
Fig. 7. Control signals for example 2.
illustrates that with the proposed method less ef-
fort is necessary for the control action.
maximum overshoot, the proposed design method Example 4: A high-order oscillating plant trans-
gives better performance than the design method fer function of G ( s ) ϭe Ϫs / ( s 2 ϩsϩ1 )( sϩ3 ) ,
proposed by Wang et al. ͓3͔ for the set-point re- which was used by Wang and Shao ͓4͔, is consid-
sponse. In terms of settling time, both design ered. Again, the parameter estimation method
methods give similar responses. The load distur- given in Refs. ͓7,8͔ was employed to generate the
bance rejection of the design method suggested by FOPDT model as G ( s ) ϭ0.333e Ϫ3.1s / ( 0.075s
Wang et al. ͓3͔ is slightly faster than the proposed ϩ1 ) . Once a proper model is found, the PI con-
one. This is expected, since in the proposed design troller tuning parameters were calculated to be
method pole zero cancellations are used and this in K p ϭ0.038 and T i ϭ0.075. Wang and Shao sug-
some cases may lead to a sluggish load distur- gested a PID controller with settings of K p
bance rejection. However, when comparing the ϭ1.298, T i ϭ1.034, and T d ϭ1.017. With these
control signals, shown in Fig. 7, it is seen that the calculated controller settings, the step response of
proposed design method requires less attempt for the closed-loop system to a unity step set-point
the control action. change and a disturbance of magnitude of Ϫ0.5
Example 3: A high-order plant transfer function introduced at time 50 s is shown in Fig. 10. Again,
of G ( s ) ϭ1/( sϩ1 ) 8 , which was given in Wang the proposed design method results in a better per-
and Shao ͓4͔, is considered in this example. The formance, especially for set point response. In Fig.
identification method given in Refs. ͓7,8͔ was 11, control signals for both design methods are
used to find the FOPDT model, G ( s )
ϭe Ϫ5.10s / ( 4.35sϩ1 ) . Using Eqs. ͑18͒ and ͑19͒
together with Eq. ͑31͒, the PI controller settings
were found to be K p ϭ0.447 and T i ϭ4.340. The
PID controller parameters suggested by Wang and
Shao ͓4͔ are K p ϭ0.677, T i ϭ4.340, and T d
ϭ1.649. Fig. 8 illustrates responses for both de-
sign methods to a unity set-point change together
with load disturbance introduced at time 80 s. The
load disturbance magnitude was again assumed to
be Ϫ0.5. The proposed design method, as is seen
from Fig. 8, results in a better closed-loop system
performance in terms of maximum overshoot,
while in terms of the settling time both designs
give quite similar performances. Control signals
for this example are given in Fig. 9, which again Fig. 9. Control signals for example 3.

7. Ibrahim Kaya / ISA Transactions 43 (2004) 297–304 303
controller may lead to sluggish load disturbance
rejection, again due to pole zero cancellation used
in the design procedure. However, in the examples
given in this paper, it is seen that the load distur-
bance rejection of the proposed design method is
also satisfactory.
Appendix: Model identification
In this section, equations used to identify the
unknown parameters of the FOPDT plant transfer
function are given. The identification method
Fig. 10. Step responses for example 4. makes use of the relay autotuning. The given
equations will result in exact parameter estima-
tions, assuming no measurement errors. The de-
given. Again, it is seen that the process can be tails can be found in Refs. ͓7,8͔.
controlled with less effort by the proposed design Two equations for the limit cycle frequency ␻
method. and the pulse duration ⌬t 1 can be obtained and are
given by
ͩ ͪ
5. Conclusions
Ϫ ␻ ⌬t 1 ␲ ͑ e ⌬t 1 /T Ϫ1 ͒ e ␪ /T
Simple tuning rules for a PI controller for con- K ϩ
2 ͑ e 2 ␲ /␭ Ϫ1 ͒
trolling stable process with small time delays have
been derived using specified gain and phase mar-
gin specifications. The design method presented in
this paper is model based. Therefore first a
ϭ ͩ ␲
h 1 Ϫh 2 ͪͩ RϪ⌬Ϫ
G ͑ 0 ͓͒ h 1 ⌬t 1 ϩh 2 ⌬t 2 ͔
P ͪ
FOPDT plant transfer function model was ob- ͑A1͒
tained from a single relay feedback test with exact and
ͩ ͪ
limit cycle analysis. Once the model was found,
simple tuning rules provided in the paper were ␻ ⌬t 1 Ϫ2 ␲ ␲ ͑ e ͑ Ϫ ␻ ⌬t 1 ϩ2 ␲ ͒ /␭ Ϫ1 ͒ e ␪ /T
used to control the process. Since the proposed K ϩ
design method incorporates IMC design prin-
2 ͑ e 2 ␲ /␭ Ϫ1 ͒
ͩ ͪͩ ͪ
ciples, where pole zero cancellation is used, the Ϫ␲ G ͑ 0 ͓͒ h 1 ⌬t 1 ϩh 2 ⌬t 2 ͔
closed-loop system with designed PI controller re- ϭ Rϩ⌬Ϫ ,
sults in good set point responses. The designed h 1 Ϫh 2 P
͑A2͒
where h 1 and h 2 are the relay heights and ⌬ is the
hysteresis. ⌬t 1 and ⌬t 2 are the pulse durations
and Pϭ⌬t 1 ϩ⌬t 2 is the period of the oscillation.
R is a constant valued signal entering the system
and ␭ϭ ␻ T.
Two more equations can be obtained for the
maximum and minimum of the plant output wave
form which are given by the following equations:
a minϭ
G ͑ 0 ͓͒ h 1 ⌬t 1 ϩh 2 ⌬t 2 ͔
P
ϩ ͩ
h 1 Ϫh 2
␲ ͪ
Fig. 11. Control signals for example 4.
ϫ ͩ Ϫ ␻ ⌬t 1 ␲ ͑ e ⌬t 1 /T Ϫ1 ͒
2
ϩ
͑ e 2 ␲ /␭ Ϫ1 ͒
ͪ ͑A3͒

8. 304 Ibrahim Kaya / ISA Transactions 43 (2004) 297–304
and either Eq. ͑A3͒ if a min is measured or Eq. ͑A4͒ if
ͩ ͪ
a max is measured. Finally, with K and T known,
G ͑ 0 ͓͒ h 1 ⌬t 1 ϩh 2 ⌬t 2 ͔ h 1 Ϫh 2
a maxϭ ϩ the dead time ␪ can be calculated from either Eq.
P ␲ ͑A1͒ or Eq. ͑A2͒.
ϫ ͩ Ϫ ␻ ⌬t 1 ␲ e 2 ␲ /␭ ͑ 1Ϫe Ϫ⌬t 1 /T ͒
2
ϩ
͑ e 2 ␲ /␭ Ϫ1 ͒
. ͪ References
͓1͔ Ho, W. K., Hang, C. C., and Cao, L., Tuning of PID
controllers based on gain and phase margin specifica-
͑A4͒ tions. Automatica 31, 497–502 ͑1995͒.
͓2͔ Fung, H. W., Wang, Q. G., and Lee, T. H., PI Tuning
Although Eqs. ͑A1͒–͑A4͒ obtained for a stable in terms of gain and phase margins. Automatica 34,
FOPDT transfer function are sufficient to identify 1145–1149 ͑1998͒.
͓3͔ Wang, Q. G., Fung, H. W., and Zhang, Y., PID tuning
the unknown parameters, K, T, and ␪, initial with exact gain and phase margins. ISA Trans. 38,
guesses are required for the unknowns to solve 243–249 ͑1999͒.
these nonlinear equations. To reduce the number ͓4͔ Wang, Y. G. and Shao, H. H., PID autotuner based on
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͵ c͑ t ͒dt
0
P
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KϭG ͑ 0 ͒ ϭ ͑A5͒ ͓7͔ Kaya, I., Relay feedback identification and model
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͓9͔ Astrom, K. J. and Hagglund, T., PID Controllers:
¨
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