International Journal of Computer Applications (0975 8887)
International Conference on Innovations In Intelligent Instrumentation, Optimization And Signal Processing ICIIIOSP-2013
Frequency Response based PID Controller Design with
Set point Filter
N.Chermakani
V.Suresh
A.Abudhahir
Prithee Madan
PG Scholar
Dept. of Electronics
and Instrumentation
Engineering
National Engineering
College, India
Assistant Professor
Dept. of Electronics
and Instrumentation
Engineering
National Engineering
College, India
Professor
Dept. of Electronics
and Instrumentation
Engineering
National Engineering
College, India
Assistant Professor
Dept. of Electronics
and Instrumentation
Engineering
National Engineering
College, India
undesirable and to be minimized to the extent possible to
ensure safety and economical norms. V.Vijayan etal.
proposed a setpoint filter design with PID to minimize the
peak overshoot[5]. The present work aims to achieve the
robustness with minimum overshoot by fusing the setpoint
filter with FR based tuning. .
ABSTRACT
In the present paper a simple procedure to design PID
controller with setpoint filter is proposed. Designing a PID
controller to meet gain and phase margin specification is a
well-known design technique. Several frequency response
based tuning methods are available to achieve the requirement
but higher value of overshoot is still a problem. Simple
frequency response method (FR) is modified by considering
the setpoint filter to minimize the peak overshoot. Even if the
FR Method PID parameter calculation is simpler, it gives high
peak overshoot. The set point filter coefficient is based on the
zeroes of the controller. The performance of the closed loop
system is analyzed by using the criterion IAE, ISE, peak
overshoot and settling time. Bench mark system has been
considered for analyzing the performance of the tuned
parameter. The performance of proposed method is compared
with simple frequency response method and Ziegler-Nichols
method. The proposed procedure is valid for PI,PD and PID
controller design. The method is applicable to any linear
model structure with dead time process.
The layout of the paper is as follows : first, the PI and PID
design problem is stated. Then, the proposed method
compared with other methods found in the literature. The
conclusion section summarizes the analysis and inferences
made.
2. DESIGN METHODS
The conventional PID control loop with setpoint filter is
considered in this paper is shown in figure 1. Where r is the
reference signal,
is the filter coefficient, u is the controller
output, y is the controlled output and d is the disturbance.
Keywords
Frequency response, PID control, Set point filter, Phase
margin , Gain margin, Peak overshoot
1. INTRODUCTION
One of the most research areas in automatic control is the
development of tuning methods for Proportional Integral
Derivative (PID) control. In most industries PI controllers are
commonly used, because derivative part in PID amplifies the
feedback measurement noise. On the other hand, addition of
the derivative mode with P/PI controller brings a stabilizing
effect and improves the speed of response without excessive
oscillation. In this work a filtered derivative type PID
controller structure that attenuates the measurement noise
while preserving the merits of derivative mode is used.
Tuning of PID controller was initiated by Ziegler and Nichols
in 1942 [6]. The criterion used for Ziegler-Nichols tuning rule
is one quarter decay ratio only, but it gives poor robustness in
many application[3]. Several tuning methods have been
proposed like direct synthesis method [9] and Astrom and
Hagguland method [3], Among these a simple procedure to
design PID controllers in the frequency domain proposed by
Roberto sanchis, uses only one tuning parameter which makes
it simple. It provides excellent robustness at the cost of peak
overshoot. But in many process industries peak overshoot is
Gf(s)
CPID(s)
u G(s)
+
+
Fig 1. PID controller with setpoint filter
The setpoint filter is the first order filter , that transfer
function is shown in equation (1).
G f (s)
f s 1
(1)
A tuning parameter a , that is defined as the ratio of final
gain cross over frequency of the process with controller to the
zero of the controller is determined using (2) . For the PI and
PID controller, the maximization of the controller gain is
equivalent to minimization of the integral error [1].
25
International Journal of Computer Applications (0975 8887)
International Conference on Innovations In Intelligent Instrumentation, Optimization And Signal Processing ICIIIOSP-2013
The resulting equation for the PI controller is
wcg
(2)
zc
While sweeping the setpoint filter coefficient from ten
percentage of1/zc to 1/zc, overshoot is reduced gradually . It is
observed that ninety percentage of 1/zc yields better
performance
0. 9
(3)
zc
The PID parameters are those that maximize the controller
gain kc , subject to the following constraints;
1) The phase margin ( m ) should be equal to the required
(specified) phase margin( r ). 2) The gain margin ( m )
should be larger than or equal to the required (specified) gain
margin ( r ) .
Kc
wcg
0.81a
G ( jwcg
1 a
(9)
6) The PI controller parameters Kp and Ti are calculated using
from (5).
2.2. Design Method for PID control
The transfer function of the PID controller is given in (10)
[1].
Td s
C ( s) K P 1
Td s
1 1
1
Kc
Ti s
s
s
1 1
z
z
i
d
s 1
z N
(10)
Where
2.1 Design Method for PI control
The transfer function of the PI controller is represented in (4)
[1].
1
C ( s) K P 1
T s K c
z d zi
z z
d i
K P Kc
s
zi
(4)
Td
z d zi
where
N
KP
Kc
zi
; Ti
(5)
zi
The following six steps are involved to tune the controller by
using single tuning parameter a.
1) Phase of the controller at gain crossover frequency of the
process with controller (arg(C(jwcg)) is calculated using (6).
arg( C ( jwcg ) arctan( a )
(6)
2) phase of the process at gain crossover frequency of the
process with controller (arg(G(jwcg)) is calculated using (7)
arg( G ( jwcg ) r arg( C ( jwcg ))
(7)
zi
zd N
zd N
; Ti
z d zi
z d zi
zd N
zd N
z d zi
zi
(11)
The following six steps are involved to tune the controller by
using single tuning parameter a.
1) Phase of the controller at gain crossover frequency of the
process with controller (arg(C(jwcg)) is calculated using (12).
a
N 2
arg( C ( jwcg )) 2 arctan( a ) arctan
(12)
2) Phase of the process at gain crossover frequency of the
process with controller (arg(G(jwcg)) is calculated using (13).
arg( G ( jwcg ) r arg( C ( jwcg ))
(13)
3) By using the equation (7) and process transfer function the
value of wcg is calculated.
3) By using (13) and process transfer function the value of wcg
is calculated.
4) The zero of the controller zc = zi is obtained from (2).
4) To simply the design method, two zeros are imposed to be
equal (zc = zi = zd). The zero of the controller is calculated
using (2).
5) By equating the magnitude expression to unity after
substituting wcg , the value of kc is calculated.
C( jwcg )G( jwcg )G f ( jwcg 1
(8)
5) The value of kc is calculated using (8). The resulting
equation for the PID controller is given in (14).
26
International Journal of Computer Applications (0975 8887)
International Conference on Innovations In Intelligent Instrumentation, Optimization And Signal Processing ICIIIOSP-2013
Kc
wcg
a
N
2
2
The finally obtained phase margin for the process G1(s),G2(s)
and G3(s) are 35.0089 , 35.0007 and 34.9873 respectively.
2 2
b a 1
G ( jwcg 1 a
(14)
Closed loop responses of considered model G1(s) , G2(s)
and G3(s) with PI controllers for a step change in
setpoint are shown in figure 2 , 3 and 4 respectively .
6) The PID controller parameters Kp ,Ti , and Td are calculated
using (11).
3. SIMULATION AND RESULTS
To evaluate the efficiency of the setpoint filter method, it has
been applied to three benchmark transfer function.
The
three
G1 ( s )
G2 ( s )
G3 ( s )
benchmark
transfer
functions
[2]
1 2s
( s 1)
e
are:
(15)
5 s
( s 1)
(16)
Fig 2 :Simulation results of PI controllers for G1(S)
1
( s 1)
(17)
servo response of the chosen models with proposed tuning is
compared with Z-N and FR methods for both PI and PID
controllers
3.1 Simulation results for PI controllers
The specification for the PI controller design is r = 35. The
closed loop responses of three models with PI controllers are
obtained and the performances indices are given in Table 1.
The lower proportional gain attained by the proposed method
has minimized the overshoot and the settling time (10%
mismatch) over Simple frequency response method (FR) and
Ziegler Nichols (ZN) tuning method. The proposed method
also reduces the integral errors such as IAE and ISE to a
reasonable extent when compared with ZN and FR .
Fig 3 :Simulation results of PI controllers for G2(s)
Table 1.Performance of proposed method with existing
method for PI controller
Over
Settling
Process
Method
IAE
ISE
shoot
time (sec)
%
G1(s)
G2(s)
G3(s)
FRS
5.781
3.879
5.1
14.5270
FR
9.144
6.609
54.9
33.2088
ZN
11.92
49.7952
FRS
24.49
6.646
7
14.52
17.36
102.088
FR
46.16
25.62
64.5
213.473
ZN
65.49
31.3
293.514
FRS
6.392
3.389
7.19
27.1818
FR
9.723
5.297
40.91
45.6985
ZN
8.536
4.736
2.18
45.9632
Fig 4 : Simulation results of PI controllers for G3(s)
27
International Journal of Computer Applications (0975 8887)
International Conference on Innovations In Intelligent Instrumentation, Optimization And Signal Processing ICIIIOSP-2013
3.2Simulation results of PID controllers
The specification r = 35 is considered for the above process
.Table II shows the performance for the closed loop response
of three chosen model with PID controller are obtained .
Table 2 .Performance of proposed method with existing
method for PID controller
Over
Settling
Process
Method
IAE
ISE
shoot
time (sec)
%
G1(s)
G2(s)
G3(s)
FRS
4.424
2.806
15.37
18.7686
FR
7.801
6.491
63.14
24.006
4. CONCLUSION
ZN
5.57
4.907
17.216
FRS
39.11
14.69
39.07
329.35
FR
46.67
25.77
82.06
223.835
ZN
28.47
19.49
11.87
125.22.
FRS
4.424
1.91
2.12
27.75
FR
7.841
4.207
43.48
40.6641
ZN
5.832
3.94
22.23
24.405.
In this paper, performance of the PID controller with setpoint
filter for servo and regulator problem has been analyzed. The
performance has been tested on a set of bench mark transfer
function. The proposed method yields better result in
obtaining closed loop performance IAE , ISE, overshoot and
settling time for servo problem .than the existing methods
namely simple frequency response method and Ziegler Nichols method. One main drawback of the method is setpoint
filter coefficient is not optimum. Optimum value of filter
coefficient will produce better result than the present method.
By varying the filter coefficient can help to achieve the
overshoot to the desired level.
It is observed that the proposed method gives better
overshoot, settling time (10% mismatches), IAE, and ISE than
the Simple frequency response method. For process G2(s), the
ZN method gives better performance than the proposed
method. Even though the proposed method gives poor
performance, it gives better robustness by specification of
phase margin than the ZN method. The final obtained phase
margin for the process G1(s),G2(s) and G3(s) are 35.021 ,
35.0349 and 35.2446 respectively.
Fig 7:Simulation results of PID controllers for G3(s)
5. REFERENCES
[1]
Robert Sanchis , Julio A.Romero and P.Balaguer , A
Simple Procedure to Design PID Controllers in the
frequency domain, IEEE, 2009
[2]
Roberto Sanchis , Julio A.Romero and P.Balaguer
,Tuning of PID Controllers Based on Simplified Single
Parameter Optimization, Journal of Process Control ,
2010
[3] K .J Astrom ,T.Hagglund ,Revisiting the Ziegler
Nichols Step Response Method for PID Control, journal
of process control,vol.14,pp. 635-650,2004
Fig 5 :Simulation results of PID controllers for G1(s)
[4]
WengKhuenHo,Chang Hang and LishengS.Cao , Tuning
of PID Controllers Based on Gain and Phase margin
Specifications, Automatica vol. 31, no.3,pp. 497-502
1995.
[5]
V.Vijayan ,RamesC.Panda ,Design of a Simple Setpoint
Filter for minimizing overshoot for low order processes,
ISA Transaction ,vol.51, pp. 271-276,2012
[6] J.G.Ziegler and N.B.Nichols, Optimum Settings for
Automatic Controllers , ASME Transaction ,
vol.64,pp.759-768 , 1942.
[7]
R.PadmaSree and M.Chidambaram ,Setpoint Weighted
Pid
Controllers
for
Unstable
Systems,
Chem.Eng.Comm., 192,1-13,2005
[8] K.Ogata , Modern Control Engineering , Prentice Hall ,
2003
[9] Dan Chen and Dale E. Seborg , PI/PID Controller Design
Based on Direct Synthesis and Disturbance Rejection,
Ind. Eng. Res, 2002, 41, 4807-4822 ,2002.
Fig 6 : Simulation results of PID controllers for G2(s)
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