Frequency domain analysis of Optimal Tuned IMC- PID
The proportional integral derivative (PID) controller is the
Controller for Continuous Stirred
Tank
(CSTR)
most common
formReactor
of feedback in the
various control
systems.PID control is also an important ingredient of a
distributed control system and as such these controllers
Pankaj Negi
come in different forms [5-7]. And also due to its efficient
and robust performance with a simple algorithm, the PID
IMS Engineering College,
Ghazibad integral, and derivative) controllers have
(proportional,
been widely accepted in most of the industrial applications
[email protected]
[8-12]. Ziegler and Nichols have implemented and
published their classical methods and also a lot of research
ABSTRACT
Since the last three decades, structure of the chemical
processes has become increasingly complex, due to better
management of energy and raw materials. As a
consequence, the design of control systems has become the
focal point in industries today. Any chemical process needs
to be controlled for various purposes, such as
environmental
regulations,
safety,
economic
considerations; product quality etc. In the present work PID
controller with Internal Model Control (IMC) tuning
method is used for series chemical reactor plant. The non
linear equations so obtained are linearize and converted in
to the transfer function. These model equations were solved
at steady and dynamic mode, and the concentration is
obtained as a function of time. The transfer function
developed is used to tune the system. The tuning
parameters are then determined by a IMC tuning method.
The results of the proposed IMC tuning method have been
compared in the midst of controller with MIGO frequency
based tuning and Ziegler-Nichols (Z-N) closed loop tuning.
A remarkable improvement in stability of the system has
been observed with IMC tuning justifying its applicability.
Simulated results given in the paper show the feasibility
and versatility of the IMC tuning technique in chemical
reactor
is done along the conventional PID controller design
[13].However, the classic tuning methods involved in PID
controller suffers with a few systematic design problems.
Hence, in order to compensate these internal design
problems, internal model control (IMC) based tuning
approach has been developed. Due to its simplicity,
robustness, and successful practical applications it gained a
widespread acceptance in designing the PID controller in
process industries [14- 18]. The analytical method based on
IMC principle for the design of PID controller is developed
[19-20]. The resulting structure of the control system is
capable of controlling a fast dynamic process by integral
control, which results in a striking improvement in
performance. Its advantage is even being implemented in
many of the industries. However, it has been found from
the literature that the IMC-PID controller has not yet been
implemented in the chemical reactor. Consequently, the
present work is a step towards implementing an IMC
tuning based PID controller in chemical reactor plant. The
results with IMC tuned controller have been found to
outperform the MIGO frequency and Z-N tuned controllers.
Keywords: Controller; Internal Model Control (IMC);
In process control industry, model-based control strategy is
used to track set point and reject load disturbances.
Series Chemical reactor; MIGO Frequency ; Stability;
Tuning;
2 MODEL OF CHEMICAL REACTOR
PLANT
1. INTRODUCTION
Structure of chemical process has seen a major change in
the last decade. The change has been due to environmental
legislation, safety considerations, energy and raw material
minimization, product quality to name a few. During its
operation, a chemical plant must satisfy several
requirements imposed by the designers and the general
technical, social and economic conditions in the presence
of ever-changing external disturbances. As a consequence,
the design of control systems has become the focal point in
industries today [1]. Tuning of a controller is done by
various methods like a trial and error method(ZN tuning),
MIGO tuning and IMC tuning. This can be very tedious if
done manually as the optimum values of the parameters of
the same controller might be different for different
processes. Hence, we can use some computer aided
techniques to speed-up the controller tuning. SIMULIK is
one such widely used software provided by Mathworks
Inc., which is an add on tool of MATLAB.
Fig:1 Chemical reactor plant
The chemical reactor system shown in the above diagram
comprises two well mixed tanks. Both the reactors are
isothermal and the reactions are first order on component
A: rA = -kCA
Component balance is applied to both the tanks to generate
the dynamic mathematical model for the system. The tank
levels remain constant because the nozzle is at the same
point for both tanks.
We have the following differential equations to describe
component balances:
V*{dCA1/dt} = F(CA0 -CA1) - Vk CA1
(1)
V*{dCA2/dt} = F(CA1 -CA2) - Vk CA2
(2)
the IMC controller, Gc
is the equivalent feedback
controller. In the IMC control structure, the controlled
variable is related as
At steady state, from
dCA1/dt = 0
dCA2/dt = 0
We have the following material balances:
F*( CA0* -CA1*) - Vk CA1* = 0
(3)
F*( CA1* -CA2*) - Vk CA2* = 0
(4)
where variables with * denote steady state values.
By substituting the following design specifications and
reactor parameters,
Fig. 2. (a) IMC Structure; (b) Classical Feedback Control
F* = 0.085 mole/min
CA0* = 0.925 mol/min
V = 1.05
C
m3
 1  GPQ 
R
 GD d
1  q  GP  GP 
1  q GP  GP  
GP q
( 7)
k= 0.04 min-1
We obtain the steady state values of the concentrations in
two reactors:
For the nominal case (i.e., GP = GP ), the set-point and
disturbance responses are simplified as
CA1* = K CA0* = 0.6191 mol/m3
C
CA2* = K2 CA0* = 0.4144 mol/m3
R
where K = F*/{F*+Vk} = 0.6688
The outlet concentration of reactant from the second reactor
CA2 should be maintained by the molar flowrate of the
reactant F entering the first reactor in the presence of
disturbance in feed concentration CA0.
In this control design problem, the plant model is
Gp*=CA2(S)/ F(s)
(5)
C
d
 GP q
 1  GP q  GD
( 8)
( 9)
According to the IMC parameterization the process model
GP is factored into two parts:
GP  PM PA
(10)
and the disturbance function is
Where PM is the portion of the model inverted by the
Gd=CA0(S)/ CA2(S)
(6)
3. IMC-PID CONTROLLER DESIGN
Fig. 2(a) and 2(b) show the block diagrams of IMC
control and equivalent classical feedback control structures,
where GP the process is, GP is the process model, q is
controller; PA is the portion of the model not inverted by
the controller and PA (0) = 1. The noninvertible part
usually includes dead time and/or right half plane zeros and
is chosen to be all-pass.The IMC controller is designed by
1
q  PM f
( 11)
where the IMC filter f is usually set as
f 
Case a: Ziegler-Nichols closed loop design tuning
Case b: MIGO frequency based design tuning
1
T s  1
( 12)
n
Case c: IMC based design tuning
f
The ideal feedback controller equivalent to the IMC
controller can be expressed in terms of the internal model,
GP , and the IMC controller, q
GC 
q
1qGP

 K C 1 
1
 TI s

 TD s 
1
 1  sT f 
n
( 13)
4.2 Case 1: Ziegler-Nichols closed loop design
tuning
To achieve such a system of two well mixed tank, the
Fig. 3 is simulated in SISO tool. For this system also the
frequency response is computed using the linear
approximation (Bode plot
where K , TI and TD are the proportional gain, integral
time constant, derivative time constant of the PID
controller, respectively, and
T f is the filter tuning
parameters/filter time constant.
4.Results and Discussion
A standard test model as considered in [6] is taken for
stability study of chemical reactor plant with IMC tuning
controller. The test model below shown is completely
designed in SISO tool.
Fig. 4Frequency response for ZN based design tuning
The magnitude and phase as a function of frequency are
plotted and is as shown in Fig. 5(a). From Fig. 5(a), it is
determined that gain crossover frequency  gc is
0.105rad/sec and phase crossover frequency  pc
is
0.175rad/sec for this case. The gain and phase margins are
Gm = 9.44dB and m = 16.5deg.Since, gc is less than
Fig:3 Block Diagram of PID Controller combines with
plant
Considering the above block diagram, the plant and
disturbance transfer functions are:
CA2/ F(s) = {13.3259s+3.2239}/{(8.2677s+1)2}
(14)
CA2/ CA2= G{A1}G{A2} = {0.4480}/{(8.2677s+1)2} (15)
To show the robustness of the speed governing system with
IMC tuning controller, various cases as given below have
been considered. The cases considered have been simulated
and verified in SISO tool MATLAB/SIMULINK ver 2012
[21].
 pc and hence in this case system is stable.
4.2 Case b: MIGO frequency based design
tuning
To get the MIGO frequency based design tuning the
Fig. 3 is simulated in SISO tool. The frequency response
for such a system is computed using the linear
approximation (Bode plot). The magnitude and phase as a
function of frequency of such a system are plotted in Fig. 4.
From the plotted graph the gain crossover frequency  gc
is 0.0732rad/sec and phase crossover frequency  pc is
0.135 rad/sec. The gain and phase margins are Gm =
10.7dB and
m
= 26.2deg,  gc is less than  pc since  gc
should not be greater than  pc for stability of the system.
The system with MIGO frequency based design tuning is
stable.
Frequency domain responses have been determined to
investigate the effectiveness of the proposed controller in
IMC design tuning. It has been determined that the IMC
tuning provides the required stability and performance
specifications. Frequency-response characteristics allow
good insight into the tuning of the control systems
compared to time domain responses. The results show that
the gain and phase margins are significantly improved with
14.3dB gain margin and 29.1o phase margin. These are
obtained from the frequency response of the open-loop
system and are as given in Fig.6. It is found from Fig.6 that
the phase margin is significantly improved at the critical
frequency of inter-area modes between 0.13rad/sec and
0.28rad/sec. On the other hand, 9.44dB and 10.7dB gain
margins for the Ziegler-Nichols tuning and MIGO
Frequency controllers are obtained which are low
compared with the IMC tuning controller. Detailed results
are as summarized in Table 1.
Table 1. Frequency Domain Results
Specification
Z-N Closed
loop
Tuning
MIGO
Frequency
Tuning
IMC
Based
Tuning
Gain Margin
9.44dB
10.7dB
14.3dB
Gain
crossover
Frequency
0.105r/s
0.0732r/s
0.13r/s
Phase margin
16.5o
26.2o
29.1 o
Phase
crossover
Frequency
0.175r/s
0.1358r/s
0.28r/s
Fig. 5. Frequency response for MIGO frequency based
design tuning
4.3 Case c: Internal Model Control (IMC)
based design tuning
This tuning design can be obtained when the Fig. 3 is
simulated in SISO tool. The magnitude and phase as a
function of frequency for this case are plotted in Fig. 6.
5. CONCLUSIONS
Fig.6 Frequency response for IMC based design tuning
It is seen from the figure that gain crossover frequency  gc
is 0.13rad/sec and phase crossover frequency  pc is
0.28rad/sec. The gain and phase margins are Gm = 14.3dB
and
m
= 29.1deg. Since  gc is less than  pc (phase
crossover frequency) then the magnitude and phase values
of the bode plot are more and positive. The reactor plant
system
with
IMC
design
tuning
is
stable.
A robust IMC tuning based PID controller is proposed
for continuous stir tank reactor. The proposed tuning
method has been found to enhance the stability. Different
cases have been considered and compared to justify the
suitability of the IMC tuning controller. From Table 1 it is
found that the gain margins IMC tuning controller is
4.86dB higher compared with Z-N tuning controller and
3.6dB higher when compared with MIGO frequency tuning
controller.
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