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CHAPTER 2

TWO CONICAL TANK INTERACTING

LEVEL SYSTEM

2.1 INTRODUCTION

Two Conical Tank Interacting Level System (TCTILS) is a class of

two tank benchmark setup used by researchers. TCTILS is a nonlinear TITO

system whose parameters vary with operating point. TCTILS has nonlinear

dynamic interaction. The process description, mathematical model and

input-output characteristics are discussed in detail in this chapter.

2.2 SYSTEM DESCRIPTION

The TCTILS as shown in Figure 2.1 is based on the two tank

benchmark setup that has been used by a number of researchers. The TCTILS

is regarded as a setup for investigating, theoretically and experimentally,

nonlinear multivariable feedback control schemes. TCTILS consists of two

identical conical tanks (TANK1 and TANK2), two independent pumps

(PUMP1 and PUMP2) that deliver the liquid flows FIN1 and FIN2 to TANK1

and TANK2 through the two control valves CV1 and CV2 respectively. These

two tanks are interconnected at the bottom through a manually controlled

valve, MV12 with a valve co-efficient 12. FOUT1 and FOUT2 are the two output

flows from TANK1 and TANK2 through manual control valves MV1 and

MV2 with valve coefficients 1 and 2 respectively. The 12, 1 and 2 are

adjustable coefficients representing the resistance of the respective valves

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opening orifice. The valve coefficient i may be evaluated using

Equation (2.1),

g2aV iii (2.1)

where Vi (i = 1, 2,12) are the cross sectional area of valves and ai is the

discharge coefficient, which is assumed as unity. The operating parameters of

TCTILS are presented in Table 2.1. In TCTILS, level h1 in TANK1 and level

h2 in TANK2 are considered as output variables and FIN1 and FIN2 are

considered as respective manipulated variables.

Figure 2.1 Schematic of TCTILS

Table 2.1 Operating parameters of TCTILS

Parameter Description Value R Top radius of conical tank 19.25 cm H Maximum height of TANK1 and TANK2 73 cm

FIN1(max) Maximum Inflow to TANK1 138.89 cm3/sFIN2(max) Maximum Inflow to TANK2 138.89 cm3/s

1 Valve co-efficient of MV1 50 cm2 / s 12 Valve co-efficient of MV12 35 cm2 / s 2 Valve co-efficient of MV2 50 cm2 / s

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2.3 MATHEMATICAL MODEL OF TCTILS

Using the mass balance equation, the mathematical model for

TCTILS is derived below.

Mass balance around TANK1 is given by Equation (2.2)

F F sign(h h ) |h h | = A(h ) + h )

(2.2)

Mass balance around TANK2 is given by Equation (2.3)

F F + sign(h h ) |h h | = A(h ) + h )

(2.3)

where

A(h1) - Cross sectional area of TANK1 (cm2) at h1 cm

A(h2) - Cross sectional area of TANK2 (cm2) at h2 cm

The cross sectional area of the conical tank is given by Equation (2.4)

A = r (2.4)

From the Figure 2.2,

tan = = (2.5)

Figure 2.2 Schematic of single conical tank

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= (2.6)

where R - Top radius of the tank (cm)

H - Total height of the tank (cm)

r - Radius of the tank at h (cm)

Substituting (2.6) in (2.4),

A = (2.7)

For TANK1, F = h (2.8)

For TANK2, F = h (2.9)

Substituting (2.7) and (2.8) in Equation (2.2) becomes

13 A(h )

dhdt + h

dA(h )dt = F sign(h h ) |h h | h

dh1

dt =FIN1- 1 h - sign(h1-h2) 12 |h1-h2| – 1

3 h1ddt (A(h1))

R2 h12

H2

(2.10)

Substituting (2.7) and (2.9) in equation (2.3) becomes

13 A(h )

dhdt + h

dA(h )dt = F + sign(h h ) |h h | h

dhdt =

F h + sign(h h ) |h h | 13 h d

dt A(h ))R h3 H

(2.11)

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Hence, the first principal model of TCTILS is given by following nonlinear Equations (2.12) and (2.13)

dhdt

=F h sign(h h ) |h h | – 1

3 h ddt A(h )

R h3 H

(2.12)

dhdt =

F h + sign(h h ) |h h | 13 h d

dt A(h ))R h3 H

(2.13)

2.4 INPUT-OUTPUT CHARACTERISTIC OF TCTILS

Using the nonlinear Equations (2.12) and (2.13), the mathematical model of TCTILS is developed with the help of Simulink/MATLAB. The Simulink model of TCTILS is shown in Figure 2.3. In all the simulation runs, the TCTILS is simulated using the nonlinear first principle model (represented by Equations (2.12) and (2.13)). The output variables are computed by solving the nonlinear differential equations using differential equation solver in MATLAB R2009a.

Figure 2.3 Simulink model of TCTILS

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The input-output steady state data required to plot input-output

characteristics of TCTILS are collected using Simulink model shown in

Figure 2.3. For this, the steady state data of tank levels (h1, h2) are generated

from TCTILS Simulink model by introducing different positive steps in FIN1

and keeping FIN2 at constant input flow of 138.6 LPH. Similarly, the steady

state data of tank levels (h1, h2) are generated from TCTILS Simulink model

by introducing different positive steps in FIN2 and keeping FIN1 at constant

input flow of 138.6 LPH. The various input-output steady state data collected

from Simulink model of TCTILS are presented in Table 2.2.

Table 2.2 Input-Output steady state data collected from TCTILS

FIN2=138.6 LPH FIN1=138.6 LPHFIN1 (LPH) h1 (cm) h2 (cm) FIN2 (LPH) h1 (cm) h2 (cm)

0 0.751 2.828 0 2.828 0.75120 1.732 3.445 20 3.445 1.732 40 2.567 3.880 40 3.880 2.567 60 3.499 4.416 60 4.416 3.49980 4.504 5.061 80 5.061 4.504

100 5.564 5.824 100 5.824 5.564120 6.649 6.713 120 6.713 6.649 140 7.756 7.756 140 7.756 7.756160 8.947 8.865 160 8.865 8.947 180 10.300 9.997 180 9.997 10.300 200 11.791 11.140 200 11.140 11.791220 13.420 12.310 220 12.310 13.420 240 15.180 13.510 240 13.510 15.180260 17.670 14.750 260 14.750 17.670 280 19.100 16.010 280 16.010 19.100 300 21.260 17.320 300 17.320 21.260320 23.540 18.660 320 18.660 23.540 340 25.960 20.040 340 20.040 25.960360 28.510 21.460 360 21.460 28.510 380 31.190 22.920 380 22.920 31.190400 34.010 24.410 400 24.410 34.010 420 36.950 25.950 420 25.950 36.950 440 42.020 27.520 440 27.520 42.020460 43.020 29.140 460 29.140 43.020 480 46.540 30.800 480 30.800 46.540500 50.000 32.500 500 32.500 50.000

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Using the input-output steady state data from Table 2.2, the input-

output characteristics of FIN1 vs (h1 and h2) and FIN2 vs (h1 and h2) for TCTILS

are obtained and are shown in Figures 2.4 and 2.5 respectively.

Figure 2.4 Input - Output characteristic of TCTILS for FIN1 vs (h1 and h2)

Figure 2.5 Input - Output characteristic of TCTILS for FIN2 vs (h1 and h2)

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From the Figures 2.4 and 2.5 the following observations are made:

The TCTILS is a nonlinear TITO process which has varying

parameters as the operating point changes.

TCTILS has an interaction between inputs and outputs. The

input – output characteristic shows that, the change in FIN1 not

only affect h1 but also affect h2. Similarly change in FIN2 not

only affect h2 and but also h1. Hence, TCTILS has dynamic

nonlinear coupling between input and output variables.

Hence, implementation of control algorithms for the TCTILS

is often complicated due to variations in process dynamics and

the characteristics of nonlinear dynamic coupling. Due to

change in process parameters and nonlinear dynamic

interaction, adaptive control scheme will be more appropriate

to control TCTILS effectively.

2.5 SUMMARY

The process description, mathematical model and input - output

characteristic of TCTILS are discussed in detail in this chapter. The Simulink

model for TCTILS is developed from the mathematical model and using that

input-output characteristic curve is plotted. From input-output characteristic

plot, it is concluded that implementation of control algorithms for the TCTILS

is often complicated due to variations in process dynamics and the

characteristics of nonlinear dynamic coupling. Due to change in process

parameters and nonlinear dynamic interaction, adaptive control scheme will

be more appropriate to control TCTILS effectively. The conventional control

schemes such as adaptive and non adaptive type decoupling based

decentralised PI control schemes using regime based MMAC strategy are

discussed in next chapter.