 Open Access
 Total Downloads : 26
 Authors : Alok Prakash, Ashutosh Prasad Yadav, Raj Kumar
 Paper ID : IJERTCONV4IS15001
 Volume & Issue : ACMEE – 2016 (Volume 4 – Issue 15)
 Published (First Online): 24042018
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Design of IMC based PID Controller for Coupled Tank System
Alok Prakash Electrical and Instrumentation department SLIET, Longowal, Sangrur, Punjab India148106
Ashutosh Prasad Yadav Electrical and Instrumentation department SLIET, Longowal, Sangrur, Punjab India148106
Raj Kumar Electrical and Instrumentation department SLIET, Longowal, Sangrur, Punjab India148106
AbstractThe performance of the Proportional Integral Derivative (PID) controllers commonly used in process control industries depends upon the controller tuning parameters. In this paper an Internal Model Control (IMC) based PID controller is proposed to estimate controller tuning parameters in terms of single parameter, known as closedloop time constant which provides improved performance and robustness to control system. An IMC based controller is designed and presented here for a coupled tank level control system which is of noninteracting type. The transfer function of the system is obtained from the equipment specifications. The obtained transfer function is approximated into first order plus delay time (FOPDT) model for the estimation of the IMCPID controller tuning parameters in terms closedloop time constant. The process is simulated in MATLAB/Simulink to record the closedloop performance with IMCPID based tuning parameters. The result are compared with the ZieglerNichols, CohenCoon and TyreusLuyben tuning methods in terms of time response characteristics and various performance Indices like Integral of Absolute Error (IAE), Integral Squared Error (ISE) and Integral Time Absolute Error (ITAE). The robustness is checked by incorporating uncertainties in the process. The results indicate PID controller tuned with IMC has better performance and robustness as compared to other tuning techniques.
Keywords Proportional Integral Derivative, Internal Model Control (IMC), First Order plus Delay Time Model, Closedloop time constant, tuning, robustness.

INTRODUCTION
PID Controllers are extensively employed in process control industries because of their relatively simple structure and design. Tuning technique is adopted for determining the proportional, integral and derivative constants of these controllers which depend upon the dynamic response of the plants. ZieglerNichols and CohenCoon [13] tuning methods are the most popular methods used in process control to determine the parameters of a PID controller. Although these methods are very old, they are still widely used because of their capability to achieve desired optimal performance for specific inputs with less tolerance to plant variations. PID controller tuned with these methods shows less robust results. A controller is said to be robust if it is insensitive to small changes in process or to inaccuracies in process model. Robustness can be defined as amount of Uncertainty in process parameters or inaccuracy in Process model that can be tolerated by controller before the closedloop system becomes
unstable [45]. In reality a, model is never perfect, so controllers must be designed to be robust (to remain stable even when the true plant characteristics are different from the model). Internal model control (IMC) based PID controller has gained attention because of its robustness and single tuning parameter selection [67].
Maintaining the level at a desired state is an important and common task in all process industries. IMC based PID controller is developed in this paper to control the liquid level in the coupled tank system. Among the other tuning methods, IMC based PID controllers tuning methods has gained widespread acceptance in the process industries because of its easy in design and simple in understand, robustness and fast in real time applications.

INTERNAL MODEL CONTROL

Internal Model Control Strategy
Internal Model Control (IMC) has been presented by Garcia and Morari [6] which is developed upon Internal Model principle to combine the process model and external signal dynamics. The IMC controller is a model based procedure, where a process model is embedded in the controller, and is considered to be robust. Mathematically, robust means that the controller must perform to specification, not just for one model but also for a set of models [4]. The IMC controller design philosophy adheres to this robustness by considering all process model errors as bounded and stable. IMC Theory states that a perfect control can be achieved only if the control system encapsulates, either implicitly or explicitly, some representation of the process to be controlled.
The IMC basic structure is shown in Fig.3 is characterized by a controller Gc(s), actual process or plant Gp(s) and predictive model of the plant Gp(s). d(s) is an unknown disturbance affecting the system. The manipulated input U(s) is introduced to both the process and its model. The process output is Y(s). d*(s) is the difference between the output of the actual process Gp(s) and process model Gp(s)which is the result of model mismatch and the disturbances; this is used by the internal model controller.
Fig.3 IMC basic structure
The design procedure of IMC involves factorization of the plant model Gp(s) as invertible Gp (s) and non invertibleGp + (s) parts as shown in Eq. (1) by simple factorization or all pass factorization. Ideal IMC controller Gc(s) is the inverse of the invertible portion Gp (s) of the process model Gp(s).
() = + () () (1)
() = [ ()]1 (2)
Gc(s) will be stable, but may not be proper.A low pass filter f(s) of the form of Eq. (3) is added to Gc(s) for making it proper, which also attenuates the effects of process model mismatch, which usually occurs at high frequency and provides good set point tracking.
controller in Eq. (5) is compared to ideal PID Controller Eq.
(6) to find out PID parameters (Kc,i,d) in terms of closed loop time constant whose value is adjusted to give IMCPID tuning.
Fig.4 Feedback control structure


MATHEMATICAL MODELLING AND CONTROLLER DESIGN
A. Obtaining Transfer Function of the Process
The control objective in a coupled tank system is that a desired level of the liquid in tank is to be maintained when there is an inflow and outflow of water out of the tank respectively. The coupled tank system [89] is a multiinput multi output system (MIMO) consisting of two independent singleinput singleoutput systems (SISO) with control voltage as input and water level as the output.
Consider the process consisting of two noninteracting liquid tanks in the Fig.1 here Load Changes in first tank affects the second tank but not the viceversa. Qi is the volumetric flow rate into Tank1, Q is the volumetric flow rate
() = 1
( + 1)
(3)
from Tank 1 to Tank 2 and Qo is the volumetric flow rate out of Tank 2.Height of liquid level in Tank1 is H1 and in Tank 2 is H2.Both tanks are having same crosssectional area A.Two
() = ()() (4)
Value of n is chosen to make Gc(s) proper or semiproper. is filter time constant or closedloop time constant or filter tuning parameter whose value is adjusted to vary the speed of response of the closedloop system. Gc(s) in
ball valves V1 and V2 having Hydraulic resistances R1 and R2 are connected at the outlet of each tanks.Vi is the control input voltage to pump.
Assuming linear resistance to flow, transfer function of the coupled tank system through mathematical modeling is
Eq. (4) is the final form of the IMC Controller.
() = 2() = 2
(7)

IMC based PID Controller
() (1 + 1)(2 + 1)
Although the Internal Model Control (IMC) procedure is simple but it cannot be implemented practically since most industries still uses the PID controller. So the IMC structure can be modified and rearranged to the form of a standard feedback control diagram or Conventional PID structure shown in Fig.4.
GPID(s) is standard feedback controller which is a function
where 1 = AR1 and 2 = AR2 are the time constants of Tank 1 and Tank 2 related to operating levels in the tank
Flow rate of the pump is related as:
() = () ; is pump constant relating to control voltage
Hence, overall transfer function of the process becomes
of plant model Gp(s) and IMC Controller Gc(s) shown in Eq. (5) which can be obtained by rearrangement of IMC basic structure Fig.3 to Feedback control structure Fig.4
() =
2()
()
= 2
(1 + 1)(2 + 1)
(8)
() = ()
1 () ()
2 + + 1
(5)
(6)
Here H2 is controlled variable and Vi is manipulated variable
() = [
]
In IMC based PID design procedure Gc(s) is made semi proper or even improper to give the resulting PID controller derivative action. A first or second order pade approximation is used if a process model has a time delay. The standard PID
coupled tank system is approximated into a FOPDT transfer function using the same method as:
The method is based on times, t1 and t2, which can be estimated from a step response curve (Fig.2), corresponding to the 35.3% and 85.3% response times, respectively.
Fig.2 FOPDT approximation curve
The time delay and time constant are then estimated from the following equations:
= 1.3t1 .29t1 (11)
= .67(2 1) (12)
The FOPDT Transfer function is given by:
( + 1)
(13)
FOPDT model of Coupled Tank System is represented as:
Fig.1 Two tank noninteracting process
Therefore, Obtained Transfer function of coupled two tank
() 2.9646
16.22 + 1
7.1 (14)
noninteracting level process using coupled tank parameters from Table 1 is
C. IMC based PID Controller Design
FOPDT transfer function of Coupled Tank System obtained in Eq. (8) is:
() = 2.9646 (11.2509 + 1)2
(9)
() = 2.9646 7.1
16.22 + 1
(15)
() = 2.9646
126.58272 + 22.5018 + 1
(10)
Process Model Gp(s) after first order pade approximation
[6] for time delay is:Table 1 parameters of coupled tank system
Parameter 
Description 
Value 
Unit 
A 
Crosssectional area of tanks 
138.9 
cm2 
R1 
Hydraulic resistance of ball valve 1 
0.081 
sec/ cm2 
R2 
Hydraulic resistance of ball valve 2 
0.081 
sec/ cm2 
Pump constant related to flow rate into tank 
36.6 
cm3/v.sec 
B. FOPDT Approximation of process model
Industrial processes are of higher order so finding a real value of it is very difficult. The transfer functions of plants
() = 2.9646(1 1.35)
(16.22 + 1)(1 + 1.35)
Performing Simple factorization: Invertible part
() = 2.9646 (16.22 + 1)(1 + 3.5)
Noninvertible part
(16)
(17)
that can be approximately modelled by some definite transfer function. Sundaresan and Krishnaswamy [10] have proposed a simple method for fitting the dynamic response of higher order systems in terms of first order plus time delay transfer functions. The obtained second order transfer function of the
+ () = (1 3.5) (18)
Adding Filter with n=1 to make our Controller Improper in order to obtain an ideal PID Controller
(16.22 + 1)(1 + 3.5) 1
() = (
) (19)
B. Simulation Results for Performance
2.9646
Using Eq. (13) and (19)
1 +
The Controller performance is measured by calculating performance indices like ISE, IAE and ITAE and determining the time response characteristics like rise time(tr), settling time(ts) and peak overshoot(Mp) through closedloop
(16.22 + 1)(1 + 3.5) () = 2.9646( + 3.5)
(20)
simulation in MATLAB/Simulink. Performance results for IMCPID tuning were compared with zieglernichols, cohen coon and tyreusluyben tuning methods to see its
Again, by comparing Eq. (19) with ideal PID Controller Eq. (20), PID parameters are obtained in terms of closed loop time constant which has been easily adjusted to tune the controller.
effectiveness. The Simulations performed for step changes in setpoint and in the disturbance at t=100sec for different tuning methods. Simulation responses in Fig 9 and Fig. 10 shows setpoint tracking and disturbance rejection capability of IMCPID tuning in comparison with other tuning methods
= 6.6397
+ 3.5
(21)
used.
Table 3 Performance results for different tuning methods
Specifications 
IMCPID 
Ziegler Nichols 
Cohen Coon 
Tyreus Luyben 
Rise Time(sec) 
17.9511 
8.9575 
10.3185 
22.7988 
Settling Time(sec) 
38.8085 
73.6954 
51.5864 
115.8182 
Peak Overshoot (%) 
2.2617 
30.1450 
16.1750 
0 
IAE 
158.3 
182.1 
135.3 
233.1 
ISE 
1430 
1166 
1029 
1346 
ITAE 
1424 
3382 
1505 
7138 
Specifications 
IMCPID 
Ziegler Nichols 
Cohen Coon 
Tyreus Luyben 
Rise Time(sec) 
17.9511 
8.9575 
10.3185 
22.7988 
Settling Time(sec) 
38.8085 
73.6954 
51.5864 
115.8182 
Peak Overshoot (%) 
2.2617 
30.1450 
16.1750 
0 
IAE 
158.3 
182.1 
135.3 
233.1 
ISE 
1430 
1166 
1029 
1346 
ITAE 
1424 
3382 
1505 
7138 
= 19.72 (22)
= 2.9199 (23)
Thus we can easily do IMCPID tuning by adjusting .Rivera et al. [7] recommend that > 0.8 because of the model uncertainty due to the pade approximation.

SIMULATION RESULTS
Simulation results are presented to illustrate the effectiveness of IMC based PID Controller for coupled tank liquid level control system.
A. Simulation Results for different values of
Simulation was performed for IMCPID tuning for different values of .Simulation response in Fig.5 shows the behavior of response with increase in value of .Results in Table 2 indicates the variation of Gain and Phase Margin with as Gain and Phase Margins are related to Robustness of the Controller.
Table 2 PID parameters for different values of
= /
p> = Gain Margin
Phase Margin
0.9
0.6713
0.03404
1.901
2.7862
69.5402
0.6263
0.03175
1.8287
2.9864
70.9729
1.1
0.5870
0.02977
1.7139
3.1865
72.1926
1.2
0.5523
0.02801
1.6126
3.3866
73.2730
1.3
0.5215
0.02644
1.5227
3.5866
74.2363
Fig.5 simulation Response for Step input for various values
Fig.6 simulation response for step input for different tuning methods
Fig.7 simulation response of Integral of Absolute value of error (IAE) for different tuning methods
Fig.8 simulation response of Integral square error (ISE) for different tuning methods
Fig.9 Simulation response of different tuning methods for step change in disturbance
Fig.10 Simulation response of different tuning methods for step change in setpoint
C. Simulation Results for Robustness testing
The robustness testing of IMC tuned PID Controller was evaluated by incorporating uncertainty in the actual process by a factor of 20% and 25% in gain(), delay time() and time constant().Results in Table 49 and simulation responses Fig.1116 were presented to show the robustness of IMC tuned PID Controller in comparison with other tuning techniques.
Table 4 results with 20% change in gain()
Specifications
20% change in gain()
IMC PID
Ziegler Nichols
Cohen Coon
Tyreus Luyben
Rise time(sec)
10.4603
4.4962
4.4384
6.7058
Settling time(sec)
18.8504
20.555
18.6782
65.9047
Peak Overshoot (%)
0
40.6757
16.0683
0
IAE
124
150
109.1
194.9
ISE
1630
2220
1891
1730
ITAE
597.2
1021
541
5435
Fig.11 simulation response for step input for different tuning methods for 20% change in gain (K)
Table 5 Performance results with 25% change in gain()
Specifications
25% change in gain()
IMC PID
Ziegler Nichols
Cohen Coon
Tyreus Luyben
Rise time(sec)
9.7258
4.2
4.0947
5.9553
Settling time(sec)
17.5648
19.6499
17.4233
61.2076
Peak Overshoot (%)
0
41.5147
17.0456
0
IAE
119.1
147.3
108.1
187.1
ISE
1617
2303
1970
1745
ITAE
655.6
1080
621.5
5317
Table 6 Performance results with 20% change in delay time()
Specifications
20% change in delay time()
IMCPID
Ziegler Nichols
Cohen Coon
Tyreus Luyben
Rise time(sec)
12.6976
5.9046
6.0346
9.3253
Settling time(sec)
19.3810
50.9195
22.9978
84.0347
Peak Overshoot (%)
0.5638
63.8238
28.7914
0
IAE
150.2
249.5
152.7
233.3
ISE
2021
3130
2318
2069
ITAE
941.3
3376
1086
6924
Fig.12 simulation response for step input for different tuning methods for 25% change in gain ()
Fig.13 simulation response for step input for different tuning methods for 20% change in delay time ()
Table 7 Performance results with 25% change in delay time()
Specifications
25% change in delay time()
IMC PID
Ziegler Nichols
Cohen Coon
Tyreus Luyben
Rise time(sec)
12.5172
5.9184
6.0447
9.1274
Settling time(sec)
18.4789
52.1761
22.9321
83.5019
Peak Overshoot (%)
1.2969
71.1971
33.1745
0
IAE
153.2
280.3
164.4
233.4
ISE
2097
3514
2495
2186
ITAE
978.8
4211
1238
6840
Fig.14 simulation response for step input for different tuning methods for 25% change in delay time ()
Table 8 Performance results with 20% change in time constant()
Specifications
20% change in Time Constant()
IMC PID
Ziegler Nichols
Cohen Coon
Tyreus Luyben
Rise time(sec)
16.1031
6.9982
7.5528
15.9628
Settling time(sec)
24.1089
45.5994
32.5747
86.0580
Peak Overshoot (%)
2.3230
41.1355
15.7591
0
IAE
170.1
199.1
140.2
233.8
ISE
1853
2090
1713
1809
ITAE
1735
2677
1231
6604
Fig.15 simulation response for step input for different tuning methods for 20% change in time Constant ()
Table 9 Performance results with 25% change in time constant ()
Specifications
25% change in Time Constant()
IMC PID
Ziegler Nichols
Cohen Coon
Tyreus Luyben
Rise time(sec)
16.5442
7.2542
7.8865
16.7595
Settling time(sec)
44.8506
47.2821
34.1925
85.5761
Peak Overshoot (%)
3.00
41.9782
16.4586
0
IAE
176.7
207.8
145.6
233.9
ISE
1886
2118
1725
1825
ITAE
1943
2976
1339
6448
Fig.16 simulation response for step input for different tuning methods for 25% change in time Constant ()

REAL TIME RESULTS
Real time closedloop responses obtained for IMCPID tuning and ZieglerNichols tuning for CoupledTank System are shown in Fig.1719.Step changes in set point is made at t=350sec for IMCPID tuning to see its setpoint tracking capability.
Fig.17 Real time response for IMC based PID tuning
Fig.18 Real time response for ZieglerNichols tuning
Fig.19 Real time response for step changes in setpoint

CONCLUSION
Simulation Results were presented to illustrate the IMC based PID tuning and to demonstrate its effectiveness, we considered Coupled Tank System for Liquid level Control. The four tuning methods IMCPID, ZieglerNichols, Cohen
Coon and TyreusLuyben are considered for PID Controller and are comparatively analyzed based on performance and robustness. From Table 3 it is evident that IMC based PID tuning provides better time response characteristics i.e. optimum settling time and reduced overshoot as compared to other tuning methods. Table 3 and Fig.68 also shows that IMC based PID tuning exhibits minimum Integral error criterias i.e. ISE, IAE and ITAE compared to other tuning methods. Simulation responses in Fig.9 and Fig.10 shows that IMCPID tuning has better setpoint tracking and disturbance rejection capability than other tuning methods. The Robustness of IMC tuned PID Controller was tested by incorporating uncertainty in the actual process by a factor of 20% and 25% in gain(), delay time() and time constant().Results in Table 49 and simulation responses in Fig.1116 indicates that IMC based PID tuning shows robust performance in Comparison with other tuning techniques. It is evident from the robustness analysis that Gain Margin is related to the amount of gain uncertainty that can be tolerated, and the Phase Margin is related to the amount of delay time uncertainty that can be tolerated. Therefore we can say that Gain and Phase Margin indicates the Robustness of the Controller. The result can be found from Table 3 that as we increase the value of the Gain and Phase Margin values increases which indicates Robustness increases. Decreasing the value of makes the closedloop response fast whereas increasing its value makes the closedloop system more robust. Hence, the IMC based PID controller tuning has the advantage of using only a single tuning parameter () whose value is adjusted to achieve a clear tradeoff between the closed loop performance and robustness.
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