3 1

3 2

Let

q -Inlet flow rate to the tank(m /min)

q -Outlet flow rate to the tank(m /min)

h-Height of the spherical tank(m)

H-Height of the liquid level in the tank at any time't'(m)

R-Top radius of the spherical tank(m)

r-Radius of the spherical tank at a particular level of height h(m)

**Abstract****— This paper describes a LabVIEW based data **
**acquisition and measurement for modeling of nonlinear system. **
**LabVIEW is gaining its popularity as a graphical programming **
**language especially for data acquisition and measurement. This is **
**due to the vast array of data acquisition cards and measurement **
**systems, which can be supported by the LabVIEW as well as the **
**relatively easy by which advanced software can be programmed. **
**One area of application of LabVIEW is the measurement of **
**process data for modeling of nonlinear system. From the process **
**data, system modeling is obtained for the nonlinear system by the **
**principle of Sundaresan and Krishnaswamy method **

**Index Term****— Dynamic system, differential pressure **
**transducer , signal conditioning , current to pressure converter, **
**data acquisition card , virtual instrumentation **

I. INTRODUCTION

Industrial processes are naturally multivariable nature and exhibit nonlinear behavior and complex dynamics. It is well known that virtually all processes of practical importance exhibit certain degree of nonlinear behavior. Spherical tanks find wide application gas plants. Measurement of steady state level in a spherical tank for modeling is important, because the change in shape gives rise to the nonlinearity. Level measurement represents an essential task in industrial automation systems for various processes. Sensing changes in physical parameters is a vital task for industrial control applications. Advances in the field of sensor devices and interfacing circuits can pave the way for better performing control systems.

Laboratory Virtual Instrumentation Engineering Workbench (LabVIEW) is a graphical programming environment based on the concept of data flow programming. This programming paradigm has been widely used for the data acquisition and measurement. There are three important components involved in test and measurement applications, namely data acquisition, data analysis and data visualization. LabVIEW features an easy-to-use graphical programming environment, which covers these vital components. Many exciting experiments can be designed and demonstrated by integrating these powerful Virtual Instrument technology products in a flexible laboratory environment with enormous possibilities of expansion and experiments.

The LabVIEW full development system features the analysis library. The function in this library is called Virtual Instruments (VIs). These VIs allow us to use classical digital

processing algorithms without writing a single line of code. The LabVIEW block diagram approach and the extensive set of analysis VIs simply the development of analysis applications. In addition to the analysis Library, there are number of specialized toolkits, such as Digital Filter Design, Fuzzy Logic Control Toolkit Control System Design Toolkit.

II. MATHEMATICAL MODELING

It is quite often the case that we have to design the control system for a process before the process has been constructed. In such a case we need a representation of the process in order to study its dynamic behavior. This representation is usually given in terms of a set of mathematical equations whose solution gives the dynamic or static behavior of the process. The process considered is the spherical tank in which the level of the liquid is desired to be maintained at a constant value. This can be achieved by controlling the input flow into the tank. The spherical tank is shown in Fig.1. Using the law of mass, Rate of accumulation of mass in the tank = Rate of mass flow in – Rate of mass flow out.

Fig. 1. Spherical tank

The

## LabVIEW Based Modeling of Dynamic System

### S.Rajendran

**1**

### , Dr.S.Palani

**2**

**1**

### Mookambigai College of Engineering, Keeranur, Pudukkottai, Tamil Nadu, India-622502

**2**

level in spherical tank at any instant is obtained by making mass balance equation as indicated below:

1 2 (1)
*dv*

*q* *q*

*dt*

where V is the Volume of the tank

3

4

(2) 3

*V*

###

*h*

Applying the steady state value,

2

4

3 ( ) (3) 3

*s* *s* *s*

*V**V* *h h h*

2

( ) 4 * _{s}* ( ) (4)

*V s*

###

*h H s*

2 *s* (5)

*q* *c h*

where „c‟ is the valve coefficient

1 2

2 2

1

( ) ( ) (6)

2 *s* *s*

*q* *q s* *ch* *h**h*

Linearizing the nonlinearity in the spherical tank

2

( )

( ) , ( ) (7)

2 *s* *t*

*c* *H s*

*Q s* *H s*

*R*
*h*

Where

2

2 ( )

*s*
*t*

*h*
*R*

*Q s*

1( ) 2( ) ( ) (8)
*Q s* *Q s* *sV s*

2 1

( )

( ) (4 *s*) ( )

*t*

*H s*

*Q s* *s* *h H s*

*R*

2

1( ) R1 (4 1 *s* 1) ( )

*Q s*

###

*R h s*

*H s*

1

( )

(9)

( ) 1

*t*
*R*
*H s*
*Q s* *s*

where

###

4###

*R h*2

_{t s}Thus the equation.9 gives the model of the system.

III. PRINCIPLES OF SYSTEM IDENTIFICATION

Fig. 2. Block diagram of open loop system

To design a feedback controller, we require a combined transfer function of the final control element (control valve), process and the measuring element to obtain this transfer function, we give a step input of known magnitude at the input signal of the control valve and record the response of the measuring element. A typical response is shown in the Fig.3. This response is called a process reaction curve

Fig. 3. Process reactive curve

** The response is similar to that of a higher order transfer **
function. There is a delay and the slope starts with zero and the
slope increases first and then decreases. There is a point in the
response with a maximum slope. This point is called an
inflection point. Let us identify the response as that of a delay
plus first order transfer function since; the controller design
methods are easily available for such transfer function models.

From the final value of the response and the given magnitude of the input signal at the control valve, we can calculate the steady-state gain of the combined system. Then, the slope at the inflection point is drawn. Since for first order system the maximum slope line starts at its origin, the delay is identified as the time at which the maximum slope line meets the time axis. The time constant of the first order system is calculated from this slope. Once the gain (K), the time constant (

###

) and the time delay (*L*) are known, the step response of the identified model is calculated and compared with the actual response of the system. Thus, the delay plus first order system approximates the given higher order system since both of them have the same maximum slope line. Locating the inflection point and drawing the slope at this point may be time consuming and may not be accurate. Let us discuss a more accurate method (S-K method) of identifying delay plus first order transfer function models.

A typical process reaction curve

*y t*

### ( )

and that of a first order plus time delay model are shown in the figure. The two curvesare allowed to meet at two points (

*t*

_{1},

*y*

_{1}) and the (

*t*

_{2},

*y*

_{2}). While intersection at more than two points would mean greater accuracy, is sufficient to describe the two parameters

###

and*L*of the model. From analytical expression for the step response of the first order plus time delay system:

### ( )

*A.* *Principles of S-K Method *

It was developed in the year 1978. It was the advanced method of the single point curve method

Fig. 4. Response of actual system

### ( )

*y t*

*= [1-*

### exp

*{-(t - L )}] for t*

_{ > L (11)}We get

*L* = [

*t*

_{2}ln (

*f*

_{1}) -

*t*

_{1}ln (

*f*

_{2})] / ln (

*f*

_{1}/

*f*

_{2})

_{ (12) }

and

###

= (*t*

_{2}-

*t*

_{1}) / ln (

*f*

_{1}/

*f*

_{2}) (13)

Where

*f*

_{1}= (1 -

*y*

_{1}) and

*f*

_{2}= (1-

*y*

_{2}) (14)

Denoting the complement of

*y t*

### ( )

and*y t*

### ( )

with respect to unity as*f t*( ) and ( )

*f t*respectively we would like to estimate the parameters

###

and*L*in order to minimize the shaded area between the actual system and that of the model. The expression for the shaded area is written

2

1

( ) 2 [ ( ) ( )] (15) 0

*t*

*I* *L* *f t dt* *f t* *f t dt*

*t*

_{}

_{}

To minimize

*I*

we have to differentiate equation(15) with
respect to ###

and*L*and set the derivative to zero. Noting that

and = , it can be shown that

1 1 2 2

*f* *f* *f* *f*

*f*1*f*20.5 (16)

*f*1ln(*f*1) *f*2ln(*f*2) (17)

These two equations are solved to get

*f*

_{1 0.647 and 2 0.147}

_{}

*f*

_{}

. That is, *y*_{1}0.353 and *y*_{1}0.853. With these values of

1

*f*

and *f*

_{2}equations (12)and (13)can be written as

*L*1.3*t*10.29 (18)*t*2

0.67(*t*2*t*1) (19)

It is interesting to note that the optimal values of the parameters of the delay plus first order model correspond to these intersections given by the two values of y.

From the experimental process reaction curve, we have to note down the time required to reach 0.353 and 0.853 of the fractional response. Then using the above equations (18)and (19), we calculate the model parameters of the first order plus time delay model. The gain of the model is always calculated as the ratio of the steady-state value of the actual response to the step magnitude given in the input of the final control element.

Therefore the model for the above system is given by in output

in input

*p*

*Change*
*K*

*Change*

,

(20) 1

*Ls*
*K p*
*TF*

*s*

###

###

Where

1 2

2 1

1.3 0.29 delay time

0.67( ) time constant

*L* *t* *t*

*t* *t*

IV. REAL TIME SYSTEM

*A.* *Block diagram *

Fig. 5. Block diagram of real time system

compressor, Differential Pressure Transmitter (DPT), VMAT01 DAQ CARD, I/V converter, V/I converter and a Personal Computer which acts as a controller, forms a closed loop system. The block diagram of this system is shown in Fig.5

*B.* *Real time system *

Fig.6 shows the real time implementation of the system. The flow rate to the spherical tank is regulated by changing the stem position of the pneumatic valve by passing control signal from computer to the I/P converter through VMAT01 DAQ CARD and V/I converter. The operation current for regulating the valve position is 4-20mA, which is converted to 3-15psi of compressed air pressure. The water level inside the tank is measured with the differential pressure transmitter which is calibrated for 0-40cm and is converted to an output current of 4-20mA.This output current is converted into 0-5V using I/V converter ,which is given to the controller through VMAT01 DAQ CARD. The VMAT01 USB based DAQ CARD is used for interfacing the personal computer with the spherical tank.

Fig. 6. Real time implementation

Fig. 7. Photo of NI6009 DAQ CARD

*C.* *Specifications of real time system *

The specifications of the real time can be represented as follows

**Parts Name ** **Details **

Spherical tank

Body Material :SS 316 Diameter :500mm Capacity :200 litres

Storage Tank Capacity :200 litres Body Material :SS 316

Differential Pressure Transmitter

Make : AB

Type : Capacitance Input : 2.5 -250)mbar Output : 4-20mA

Rotameter

Make :Tellien/Equivalent Flow Rate :(100 – 1000)LPH Type :Variable Area Float Material :SS 316

Control Valve Type : Pneumatic air to close Input : 3-15psi

Pump

Make :Tullu/Kirlosker Flow Rate :1500 LPH Supply Voltage :230VAC/50Hz

Air Regulator Size : ¼” BSP Range : 0-2.2bar

I/P Converter

Make : ABB Input : 4-20mA Output : 0.2 -1bar

V/I Converter

Model : Electronic Input : 1-5V Output :4-20mA

I/V Converter

Model : Electronic Input : 4-20mA Output :1-5V

Pressure Gauge Range : (0-30)psi Range :(0-100)psi

NI6009 DAQ CARD Input : 8 Nos. Output : 2 Nos.

V. SYSTEM IDENTIFICATION

*A Block box model *

time delay estimate; the time constant is estimated by calculating the tangent intersection with the steady state output value divided by the model gain. Cheng and Hung have also proposed tangent and point of inflection methods for estimating FOPTD model parameters. The major disadvantage of all these methods is the difficulty in locating the point of inflection in practice and may not be accurate. Prabhu and Chidambaram have obtained the parameters of the first order plus time delay model from the reaction curve obtained by solving the nonlinear differential equations model of a distillation column. Sundaresan and Krishnaswamy have obtained the parameters of FOPTD transfer function model by letting the response of the actual system and that of the model to meet at two points which describe the two parameters

###

and . The proposed times

*t*

_{1}and

*t*

_{2}, are estimated from a step response curve. This time corresponds to the 35.3% and 85.3% response times. Conventional PID controllers are

Fig. 8. Process reactive curve

widely used in industry since they are simple, robust provided the system is linear. But the process considered here has non-linear characteristics, is represented as piecewise non-linearalized models around 6 operating points as shown in Fig.8 Using process reactive curve method, the transfer function model parameters is obtained by Sundaresan and Krishnaswamy method.

*B. Model of the real time system *

TABLE I

Input, output and time for two inter section point of process curve

Fig. 9. Block diagram of LabVIEW for real time system

Fig. 10. Front panel of LabVIEW for real time system TABLE II

TABLE III

Transfer function of real time system

**Height ** **Transfer Function **

01-04 cm 2.01 60.6

270.68 1

*s*
*e*
*TF*

*s*

04-11cm 4.94 835.02

387.93 1

*s*
*e*

*TF*

*s*

11-18cm 6.273 822.5

575.53 1

*s*
*e*
*TF*

*s*

18-26cm 5.78 502.66

894.45 1

*s*
*e*

*TF*

*s*

26-33cm 6.32 506.47

866.98 1

*s*
*e*

*TF*

*s*

33-42cm 6.35 456.57

805.56 1

*s*
*e*

*TF*

*s*

VI. CONCLUSION

This paper presents system identification of a dynamic system using Lab VIEW. Lab VIEW provides a convenient environment for software development due to its block diagram based (graphical) programming environment and the large number of proper functions. It also provides accurate acquisition and measurement. The NI6009 Hardware really interfaces the real time physical system with computer. Now the real time is available in mathematical model. If any dynamic system in mathematical model, can easily develop controller to obtain desired output

REFERENCES

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[2] Grigore Stamatescu, Valentin Sgarciu, “PC –Based System for Level Transducer Interfacing” –First International Conference on Sensor Devices Technologies and Applications 2010

[3] Finn Haugen, Evinind Fjelddalen. “Demonstrating PID Control Principles using an Air Heater and Lab VIEW” –Dec 6 2007,

Telemark University College Norway [4] National Instruments, “Virtualization Technology under the Hood”

White Paper, 2008,

http://zone.ni.com/devzone/cda/tut/p/id/8709,14.04.2010

[5] G Nikolov and B. Nikolova, “Virtual Techniques for Liquid Level Monitoring using Differential Pressure Sensors” RECENT Journal for Industrial Engineering ISSN 2065-5529,vol9,no2, July2008 [6] R.Bitter, T.Mohiuddin and M.Nawrocki, “LabVIEW Advanced

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McGraw-Hill, Inc., New York, U.S.A

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**Dr. Palani Sankaran is Director and Professor, Department of Electronics **

**& Communication Engineering.** He has obtained Bachelor Degree in

Electrical Engineering from University of Madras1966, M. Tech in Control Systems Engineering from IIT, Kharagpur in 1968 and Ph.D. in Control Systems Engineering from University of Madras in 1982. He has wide teaching experience over four decades in National institute of Technology, Government of INDIA

He has published more than 100 research papers in reputed national and
international journals and conferences. He is the author of the books titled
**Automatic Control Systems, Signals and Systems, and Digital Signal **

**Processing**. Under his guidance six candidates were awarded Ph.D degree and

10 candidates are doing Ph.D in Anna University, Chennai

Currently he is working as Director and Professor in the department of Electronics and Communication Engineering, Sudarsan Engineering College, Anna University Chennai, India

**Rajendran Subbiah **received B.E. degree in Electrical and Electronics