 Open Access
 Total Downloads : 67
 Authors : D. Gayathri , D. Rathikarani, L. Thillai Rani, D. Sivakumar
 Paper ID : IJERTV6IS040745
 Volume & Issue : Volume 06, Issue 04 (April 2017)
 DOI : http://dx.doi.org/10.17577/IJERTV6IS040745
 Published (First Online): 26042017
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Multiloop Adaptive Controllers for a Nonlinear Interacting Coupled Tank Process
D. Gayathri *,
* PG Student,
Dept. of Electronics and Instrumentation Engineering, Annamalai University, Annamalai Nagar

Rathikarani**, L. Thillai Rani***, D. Sivakumar**
** Faculty,
Dept. of Electronics and Instrumentation Engineering, Annamalai University, Annamalai Nagar Chidambaram, India
Abstract This paper presents the design and implementation of Direct Adaptive Controller (DAC) to control the level of liquid in a nonlinear two tank interacting process. Mean Square error is chosen as minimization criteria for the design of the controller. The objective of this work includes performance comparison of Adaptive MIT and Adaptive PI controller with two tank interacting process. The Constant PI, Adaptive MIT and Adaptive PI controllers are implemented for four different operating regions. White box model of the process is used in this work. The design and simulation studies are carried out in MATLAB/SIMULINK.
Keywords Nonlinear system, Interacting two tank system, MRAC, MIMO, Mathematical modeling, White box model

INTRODUCTION
In process industries one of the major problem is to control the liquid level in tanks. Vital industries such as Petro chemical industries, Paper industries, Water treatment industries have tanks used for chemical treatment and/or mixing the process fluids. In two tank interacting process, the level of liquid in two tanks must be controlled to improve the quality of the product. The difficulty in level control of Multiple Input Multiple Output (MIMO) process is due to its complex dynamics and the interacting nature. Control of the nonlinear process is a difficult task by itself. Deepa et al.[1] have compared the performances of MRAC with fuzzy control for a Single Input Single Output (SISO) system. Anna Joseph et al.[2] and Rathikarani et al.[3] have used Model Reference Adaptive Controller (MRAC) to control nonlinear process . For a coupled tank process, a Model Predictive Controller is designed by Gireesh Kumar et al.[4].
First principle based model of the MIMO process is used in this work. The most widely used PI controllers in the industrial applications have simple structures and good dynamic performances. These Constant PI (CPI) controller are popular in industrial applications, as they are easy to install and reasonably robust. It is necessary to develop advanced PI controllers for controlling nonlinear processes. Adaptive controllers parameters are adjusted automatically to compensate the variation in the process characteristics. These controllers performs better when compared to Constant Gain PI controllers for Nonlinear processes. Hence Adaptive MIT and Adaptive PI controller are designed and implemented in this work. The MRAC based Adaptive MIT and Adaptive PI controllers are designed to control the liquid level in interacting tanks. When production rate changes, the dynamics
of the process along with the amount of interaction varies. The adaptive controller has to decrease the error vector between the reference model and plant to zero. The proposed method can adjust the controller parameters in response to changes in plant and disturbances by referring to the reference model that satisfies properties of the desired closed loop control system

DESCRIPTION OF LEVEL PROCESS
In this work, two tank interacting process available in the laboratory is considered. This process is a MIMO process with two controlled variables. Hence two controllers are designed and implemented to control the level in two tank interacting process. The level in tank1 and tank2 are the controlled variables.Pump1 and pump2 are used to feed inflow to the tank1 and tank2. The Hand Valves (HV) are adjusted so that the levels in both the tanks are brought to nominal condition initially. Disturbances are applied to the tanks by varying the position of the hand valves HV21 and HV26. When the flow to the tank 1 is varied, the inflow to tank 2 also varies. When the level and/or flow of tank2 varies, tank1 level change, due to interaction between the tanks (Fig. 1).
. The volumetric inflow rate into the tank1 and tank2 are qin1 and qin2. The volumetric flow rate from the tank1 and tank2 are q01 and q02. Flow rate between tank1 and tank2 is q12. The height of he liquid level is p in tank1 and p in tank2.
Fig.1 Piping and Instrumentation Diagram for two tank interacting level process
The schematic diagram of the two tank interacting level process is shown in Fig. 2. The controlled variables in the
process are level in tank1 (p) and level in tank2 (p). The 20
Manipulated variables to the process are qin1 (l/hr) for tank1,
qin2 (l/hr) for tank2. 15
Level(cm)
10
qin1 qin2 p
TANK2
TANK1
qin1 5 p
q12
p
p
RESERVOIR TANK
q01 q02
00 200 400 600 800 1000 1200 1400 1600 1800 2000
Sampling instants
Fig. 3 Open loop responses of p and p for +10LPH change in qin1
20
Level(cm)
15
10
Fig. 2 Schematic diagram of Two tank interacting level process
The mathematical model of the process is obtained from mass balance equations, and are given below,
p
5 p
00 200 400 600 800 1000 1200 1400 1600 1800 2000
Sampling instants
Fig. 4. Open loop responses of p and p for +10LPH change in qin2
A1 dp
dt
A2 dp
dt
qin1 q01 q12
qin2 q02 q12
(1a)
(1b)
In the same manner, step change is given in qin2 maintaining qin1 in nominal condition [Fig. 4]. Using these responses the model of the process in continuous time domain for the change in qin1 and qin2 are computed using
where A1= A2=1130.4cm2 are the cross sectional area of the tank1 and tank2. p=p=25cm are the height of the tank1 and tank2. a1=5.3 cm2; a2=10.6 cm2 are the restriction areas in the outlet pipes of tank1 and tank2. g=9.81cm2/s acceleration due to gravity and cd= 0.8 the discharge coefficient.

Modelling of Level Process
The four models relating the two controlled outputs p and p with two manipulated inputs qin1 and qin2 are essential to design the multiloop controllers [5]. The model transfer functions with the flow rates as manipulated inputs and the levels as controlled outputs can be written as follows:
process reaction curve method and are tabulated in Table I.
TABLE I. IDENTIFIED MODELS
Operating Regions
qin (LPH)
Level (cm)
Models
1
5
12.6
to 9.8
0.5486e5.5S 0.338e4.5s
c c
11 57s 1 12 48s 1
0.449e10..5s 0.6084e3S
c c
21 58.5s 1 22 27s 1
2
+5
12.6
0.586e4S 0.502e8.5s
c c
11 54s 1 1 2 73.5s 1
0.387e6.5s 0.5806e2S
c c
21 64s 1 22 71s 1
to
16.02
3
10
12.6
0.5635e3.5S 0.3742e145s
c c
11 34.5s 1 1 2 43.5s 1
0.4607e8s 0.6345e4.5S
c c
21 52.5s 1 22 22.5s 1
to
7.45
4
+10
12.6
to 20
0.737e3S 0.508e3s
c c
11 69s 1 12 75s 1
0.4607e7.5s 0.587e2S
c c
21 52.5s 1 22 75s 1
p p
G11(s) ; G21(s)
qin1 qin1
qin2
h
qin2
h
G12(s) 1 ; G22(s) 2
q q
in2 qin1
in2 qin1

Validation of the Models
Interacting two tank process is modeled from the process reaction curve [Fig. 3 and Fig. 4]. The levels in the tanks are initially maintained at nominal operating condition (p=12.6cm, p=12.1cm). In 800th sampling instant 10LPH change is given in qin1 . which causes p to change from 12.6 to 20cm. Due to interaction between the tanks p has reached the steady state value of 16cm from its nominal value.
Time domain validation of the models are shown in Fig. 5 and Fig. 6. To evaluate the degree of closeness of the model with actual process the validation is done. The actual response (white box) of the process is compared with model response (black box) for the same input.
Fig. 5 Time domain validation for the Model (Tank1)
Fig. 6 Time domain validation for the Model(Tank2)


DESIGN OF CONTROLLERS
The process considered in this work is a Two Input Two Output (TITO) process. The constant PI, Adaptive MIT and Adaptive PI are designed and implemented to control level in the tanks. Hence two control loops are designed and implemented.

Constant PI (CPI) Controller
The CPI controller can be used to improve the dynamic response as well as reduce or eliminate the steady state error. The strategy used for controlling the interacting process with controllers are shown in Fig.7. The reference set points are hsp1 and hsp2. Manipulated inflow rates to tank1 and tank2 are qin1 and qin2. The process outputs from tank1 and tank2 are p and p. The variables e1 and e2 are the modeling errors to the controllers. Gc1 and Gc2 are the controllers in loop1 and loop2.
p
p
Table II
PI Controller Parameters
Operating Regions
PI Controller parameters
Gc1
Gc2
1
K c1 =14.084; Ti1 4.05
K c2 =12.344;
Ti2 =3.86
2
K c1 =7.771; Ti1 =5.65
K c2 =7.728;
Ti2 =3.90
3
K c1 =12.450; Ti1 =2.54
K c2 =15.567;
Ti2 =1.22
4
K c1 =9.056; Ti1 =2.89
K c2 =7.997;
Ti2 =1.08

Adaptive Controller
To control level in each tank, MRAC is used. The structure of the MRAC system with MIT rule used in this work is shown in Fig. 8. Each control loop consists of a reference model, adjustment mechanism and controller. The reference model describes the desired input/output character of the closed loop system. The controller drives the control signal so that the plants closed loop characteristics from the command signal; hspi to the plant output hi is equal to the dynamics of the reference model, hm. The suffix i in the variables represents the control loop, nos. 1 and 2.
Matching the plant and the reference model characteristics guarantees the convergence of the modeling error to zero for any given command signal (hspi). The controller drives the difference between the process response and desired model output to zero asymptotically at a rate constrained by the adaptation gain [6,7].
The designed controller has a conventional inner loop followed by a separate adaptive outer loop to adjust the controllers feedback gains ( 1i , 2i ) based on equating the coefficients of closed loop plant to coefficients of the desired model.
The advantage is that the proposed technique can deal with the nonlinear nature of the process and also retain the designers intuition and insight through the relatively simple design scheme that is proposed. This controller design is based on grey box model which combines both black and white box models.
The outer loop adjusts the controller parameter in such a way that the model error (ei), the difference between process output hi (i=1,2) and model output hm is small.
ei hi hm
(2)
Fig. 7 Block Schematic representation of closed loop system
The controller parameters for various operating ranges of the taken up process using the Ziegler Nichols tuning method are presented in Table II.
e bm u
1 s 2 s
2 2 c
n
n
n
n
e
2
bm s 2 2
n
s 2
( y)
(7)
n
The controller parameters are obtained
1
n
n
1 s s 2 2
s 2
uc e
1
n
n
2 s s 2 2
s 2
ye
Fig. 8 Block Schematic diagram of the system with adaptive control

Adaptive MIT Controller
The controller parameters () may be adjusted with the

Adaptive PI (API) Controller
The design of API controllers leads to large improvement in industries. API controllers are simple and easy to implement [8,9]. Hence an API based on MRAC is designed and implemented in this work. The API algorithm used in this work is given by equation (8).
following loss function,
u k (u y) ki (u

y)
(8)
J ( ) 1 e2
(3)
p c s c
ni 2 i
where kp and ki are the proportional and integral gains of the
where n=1,2 represents controller parameter number. In order to minimize the loss function J, the parameters can be changed in the direction of negative gradient of J.
controller[10,11]. Based on apriori knowledge the process considered for control is represented by equation (5). The closedloop transfer function is given by
The control law is
u(t) 1uc (t) 2 y(t)
(4)
y
uc
bk p ki
s2 (a bk p )s bki
(9)
The closed loop transfer function is given by equation
For perfect model matching
n
s 2 s(a bK p ) bKi s 2 2 n s 2
y b
1
c
2
u s2 a s a
(5)
The adapted PI Controller parameters based on MIT
where uc is the command signal (input). The controllers algorithm are shown in equations (10) and (11).
parameters are to be adapted such that the process output (equation 5) follows the model output (equation 6)
k e
p s
s
n
s2 2
s 2 n
[uc
y]
(10)
ym
uc
bm s 2 2
n
s 2
(6)
k e
i s
1
n
s 2 2
s 2 n
[uc 
y]
(11)
n
The modeling error is as follows
e y ym
Substituting u(t) in equation (5)


SIMULATION RESULTS
The servo and regulatory responses of interacting tank
y b1
1
2
s 2 a s (b
uc
a2 )
(white box model) are plotted in Fig. 9. The damping ratio ( ) of the reference model is 0.7.
ym
bm s 2 2
s 2 uc
The set point tracking for level p with Conventional PI , Adaptive MIT and Adaptive PI are presented in Fig. 9. The
n
n
For perfect model following the controller parameters are chosen as(when e=0)
bm
1 b
am a
2 b
The sensitivity derivatives are obtained by the partial derivatives of modeling error with respect to the controller parameters
psp is the set point for tank1 a nominal operating condition. The set point variation of p from 12.6 to 17.6cm is applied at 950th sampling instant. Due to interaction the level of tank 2 increases. The pAPI is the level of the tank1 using API controller. The pCPI is the level of tank1 with CPI controller. The pMIT is the level of tank1 when Adaptive MIT controller is used.
The set point tracking for level p with Conventional PI , Adaptive MIT and Adaptive PI are shown in Fig.10. Due to set point variation in tank1, the level in tank 2 varies at 950th
sampling instant. Due to controllers in loop2, the level variations are nullified and brought to nominal operating condition (12.1cm). A set point variation for level p from
12.1 to 17.1cm is applied at 1800th sampling instant, due to
1.4
1.2
Controller parameters
1
kc1CPI
interaction there is considerable rise in p.
kc2CPI
0.8 kc1API
kc2API
21 0.6
20
0.4
19
18
Level(cm)
17 psp
pMIT
0.2
0
800 1000 1200 1400 1600 1800 2000 2200 2400
16 pAPI
pCPI
Sampling instants
15
X: 802
Y: 12.58
14
13
12
800 1000 1200 1400 1600 1800 2000 2200 2400
Sampling instants
Fig. 9 Servo and Regulatory responses of the Interacting Coupled tank process for p( = 0.7)
19
18
17
1.4
Controller parameters
1.2
1
0.8
0.6
Fig. 12 Adaptation of Proportional Gains
ki1CPI ki2CPI ki1API ki2API
Level(cm)
16
15 psp
pMIT
14 pAPI
pCPI
13
12
0.4
800 1000 1200 1400 1600 1800 2000 2200 2400
Sampling instants
Fig. 13 Adaptation of Integral Gains
The vanishing nature of adapted controller
11
800
1000 1200 1400 1600 1800 2000 2200 2400
Sampling instants
parameters(1, Fig. 14.
2 ) of MIT1 and MIT2 can be visualized from
Fig. 10 Servo and Regulatory responses of the Interacting Coupled tank 60
50
process for p( =0.7)
Fig. 11 shows the response of the controllers for tank1
Controller Parameters
and tank2. At 950th sampling instant, the inflow rate to the 40
30
tank 1 increases as well as inflow rate to tank2 decreases. In tank 2 , the 1800th sampling instant the inflow rate of tank 2
increases due to this change the flow rate of the tank 20
1decreases in order to bring back the level p to set point. 10
TpMIT1 TpMIT1 TpMIT2 TpMIT2
55
50 qin1API
qin2API
45 qin1CPI
qin2CPI
40 qin1MIT
Flow(LPH)
qin2MIT
35
30
25
20
15
10
800 1000 1200 1400 1600 1800 2000 2200 2400
Sampling Instants
Fig. 11 Response of controllers for tank1 and tank 2
The adaptation of Proportional and Integral gains(Kc,Ki) for CPI and API can be visualized from Fig. 12 and Fig. 13.
0
800 1000 1200 1400 1600 1800 2000 2200 2400
Sampling instants
Fig. 14 Adaptation of the controller parameters(MIT)
The reference models damping ratio( ) is changed from
0.7 to 1 and 0.7 to 2. The set point tracking for level p with Conventional PI, Adaptive MIT and Adaptive PI for ( =1)
are presented in Fig. 15. The set point tracking for level p with Conventional PI , Adaptive MIT and Adaptive PI for ( =1) are shown in Fig.16. Fig.17 shows the corresponding response of the controllers for tank1 and tank2. The adaptation of Proportional and Integral gains(Kc,Ki) for CPI and API can be visualized from Fig. 18 and Fig. 19. The vanishing nature of adapted controller parameters (1, 2 ) of
MIT1 and MIT2 can be visualized from Fig. 20.
20
Level(cm)
18
16 psp
pMIT
14 pAPI
pCPI
1.4
Controller parameters
1.2
1
0.8
0.6
0.4
ki1CPI
ki2CPI ki1API ki2API
12
800 1000 1200 1400 1600 1800 2000 2200 2400
Sampling instants
Fig. 15 Servo and Regulatory responses of the Interacting Coupled tank
process for p ( = 1)
800 1000 1200 1400 1600 1800 2000 2200 2400
Sampling instants
Fig. 19 Adaptation of Integral Gains
60
19
50
18
Controller Parameters
17 40
Level(cm)
16
30
15 psp
pMIT
14 pAPI
pCPI 20
13
TpMIT1 TpMIT1 TpMIT2 TpMIT2
12 10
11
800 1000 1200 1400 1600 1800 2000 2200 2400
Sampling instants
Fig. 16 Servo and Regulatory responses of the Interacting Coupled tank process for p ( =1)
55
50 qin1API
qin2API
45 qin1CPI
qin2CPI
40 qin1MIT
Flow(LPH)
qin2MIT
35
30
25
20
15
0
800 1000 1200 1400 1600 1800 2000 2200 2400
Sampling instants
Fig. 20 Adaptation of the controller parameters(MIT)
Table IIIA and IIIB shows the Time Integral Criteria of the process for various controllers with various reference model parameters
Table IIIA
Performance comparison of Adaptive controllers
Parameters
IAE
ISE
CPI
API
MIT
CPI
API
MIT
0.7
Tank1
2025
2165
911.5
18550
19450
7673
Tank2
7829
2361
883.8
115700
2142000
6923
1.0
Tank1
2024
2167
908.7
18540
19310
7520
Tank2
7817
2367
867.4
117400
2134000
7626
2.0
Tank1
2022
2175
904
18070
18920
7091
Tank2
7795
2388
804
116900
2121000
7268
10
800 1000 1200 1400 1600 1800 2000 2200 2400
Sampling Instants
Fig. 17 Response of controllers for tank1 and tank
1.4
1.2
Controller parameters
1
0.8
0.6
0.4
0.2
0
kc1CPI
kc2CPI kc1API kc2API
Table IIIB
Performance comparison of Adaptive controllers
Parameters
ITAE
CPI
API
MIT
0.7
Tank1
5063000
5413000
2279000
Tank2
19650000
5902000
2038000
1.0
Tank1
5062000
5418000
2268000
Tank2
19540000
5814000
20136000
2.0
Tank1
506000
5438000
2254000
Tank2
19480000
5468000
1934000
800 1000 1200 1400 1600 1800 2000 2200 2400
Sampling instants
Fig. 18 Adaptation of Proportional Gains

CONCLUSION

This paper has presented aGradient approach based MRAC to control level in the interacting tanks. Identification of the first principle based process is done in simulation. The ISE and IAE values of the process with Adaptive MIT controller has lesser values compared to the process with conventional and Adaptive PI controller. From the implementation of MRAC based MIT, it is inferred that by increasing damping factor the time integral absolute error are minimized. Peak overshoot and undershoot are minimum in conventional PI controller. Hence Adaptive MIT controller is suitable for control of the Interacting coupled tank process when compared to Conventional and Adaptive PI controllers.
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