The proportional integral derivative (PID) controller is the most common form of feedback in the various control systems.PID control is also an important ingredient of a distributed control system and as such these controllers
and robust performance with a simple algorithm, the PID
been widely accepted in most of the industrial applications
published their classical methods and also a lot of research
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 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 controller with Internal Model Control (IMC) tuning method is used for series chemical reactor plant. The non process industries [14- 18]. The analytical method based on
IMC principle for the design of PID controller is developed linear equations so obtained are linearize and converted in to the transfer function. These model equations were solved
[19-20]. The resulting structure of the control system is capable of controlling a fast dynamic process by integral at steady and dynamic mode, and the concentration is obtained as a function of time. The transfer function control, which results in a striking improvement in performance. Its advantage is even being implemented in 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 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 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. 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.
Simulated results given in the paper show the feasibility and versatility of the IMC tuning technique in chemical
reactor
In process control industry, model-based control strategy is
Controller; Internal Model Control (IMC);
Series Chemical reactor; MIGO Frequency ; Stability;
Tuning; used to track set point and reject load disturbances.
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
At steady state, from
(2) 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,
F* = 0.085 mole/min
CA0* = 0.925 mol/min
V = 1.05 m 3 k= 0.04 min -1
We obtain the steady state values of the concentrations in two reactors:
CA1* = K CA0* = 0.6191 mol/m3
CA2* = K 2 CA0* = 0.4144 mol/m3 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) and the disturbance function is
Gd=CA0(S)/ CA2(S) (6)
Fig. 2(a) and 2(b) show the block diagrams of IMC control and equivalent classical feedback control structures, where G
P the process is, G
P
is the process model, q is the IMC controller, G c
is the equivalent feedback controller. In the IMC control structure, the controlled variable is related as
Fig. 2. (a) IMC Structure; (b) Classical Feedback Control
C

1


P

G
P
 R




1

1

 q G
P

G
P



 ( 7)
For the nominal case (i.e., G
P
= G
P
), the set-point and disturbance responses are simplified as
C
R

( 8)
C d
 
D
( 9)
According to the IMC parameterization the process model
G
P
is factored into two parts:
G
P

P P
A
(10)
Where P
M
is the portion of the model inverted by the controller; P
A
is the portion of the model not inverted by the controller and P
A
(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 q

P

M
1 f ( 11)

where the IMC filter f is usually set as f

1


1
 n
( 12)
The ideal feedback controller equivalent to the IMC controller can be expressed in terms of the internal model,
G
P
, and the IMC controller, q
G
C
 q
1
 qG
P
 K
C 


1

1
 


1


1 sT f n
 ( 13) where K , T
I
and T
D
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.
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.
Case a: Ziegler-Nichols closed loop design tuning
Case b: MIGO frequency based design tuning
Case c: IMC based 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
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].
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
G m
= 9.44dB and
 m
= 16.5deg.Since,
 gc
is less than
 pc
and hence in this case system is stable.
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 is 0.0732rad/sec and phase crossover frequency
 pc
 gc
is
0.135 rad/sec. The gain and phase margins are G m
=
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.
Fig. 5. Frequency response for MIGO frequency 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.
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 29.1
o 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
10.7dB Gain Margin 9.44dB
Gain crossover
Frequency
0.105r/s
Phase margin 16.5
o
Phase crossover
Frequency
0.175r/s
0.0732r/s
26.2
o
0.1358r/s
IMC
Based
Tuning
14.3dB
0.13r/s
29.1
o
0.28r/s
Fig.6 Frequency response for IMC based design tuning
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.
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 G m
= 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.
[1] George Stephanopoulos, ”Chemical Process Control”,
Eastern Economy Edition, pp. 3-20, 55-87.

[2] J.Ingham, I.J.Dunn, ”Chemical Engineering Dynamics -
An introduction to modeling and computer simulation”,
2nd Edition, Wiley VCH, pp.1-6, 19
[3] Brian Roffel & Ben Betlam, ”Process dynamics and control – Modeling for prediction and control”, John Wiley and Sons,P. 169-178. c udra Pratap, ”Getting started with
MATLAB 7 – A quick introduction for scientists and engineers”, Oxford University Press.
[4] Rudra Pratap, ”Getting started with MATLAB 7 - A quick introduction for scientists and engineers”, Oxford
University Press.
[5] R.P. Vyas, ”Process Instrumentation and Control”,
Dennett and Co., P.99-105.
[6] B.Wayne Bequette, ”Process Control - Modeling, design and Simulation”, Prentice Hall of India, pp.80-
83,195-199, 573-578
[7]Vrancic D, Kristiansson B, Strmcnik S, Oliveira PM.
Improving performance/activity ratio for PID controllers.
Int. Conf. Control and Automation 2005; p. 834-839.
[8] Astrom KJ, Panagopoulos H, Hagglund T. Design of PI controllers based on non-convex optimization.
Automatica 1998; 34: 585-601
[9] Seborg DE, Edgar TF, Mellichamp DA. Process dynamics and control. John Wiley & Sons, Second edition.
New York; 2004.
[10] Smith CL., Corripio AB, Martin JJ. Controller tuning from simple process model Instrumentation
Technological 1975; 22: 39-45.
[11] Ziegler JG, Nichols NB. Optimum settings for automatic controller. Transactions ASM E1942; 64: 759-
766.
[12] Chien IL, Fruehauf. Consider IMC tuning to improve controller performance.
Program 1990; 86: 33-38.
Chemical Engineering
[13] Aidan O, Dwyer. Handbook of PI and PID controller tuning rules . Imperial College Press: London; 2003.
[14] Horn IG, Arulandu JR, Christopher JG, VanAntwerp
JG, Braatz RD. Improved filter design in internal model control. Industrial Engineering Chemical Research 1996;
35:33-37.
[15] Lee Y, Park S, Lee M. Consider the generalized IMC-
PID method for PID controller tuning of time-delay processes. Hydrocarbon Processing 2006: p. 87-91.
[16] Lee Y, Park S, Lee M, Brosilow C. PID controller tuning for desired closed-loop responses for SISO systems.
AICHE Journal 1998; 44:106-15.
[17] Morari M, Zafiriou E. Robust Process Control .
Prentice Hall, Englewood Cliffs: NJ; 1989.
[18] Rivera DE, Morari M, Skogestad S. Internal model control, 4. PID controller design. Industrial Engineering
Proceeding Design Deu 1986; 25: 252-58,
[19] Shamsuzzohal M, Lee M. IMC Based Control System
Design of PID Cascaded Filter. SICE-ICASE International
Joint Conference 2006: p.2485-2490.
[20] MATLAB/SIMULINK ver. 7.6