CONTROLLER TUNING FOR NONLINEAR HOPPER PROCESS...

Journal of Engineering Science and Technology EURECA 2013 Special Issue August (2014) 59 - 67 © School of Engineering, Taylor’s University

59

CONTROLLER TUNING FOR NONLINEAR HOPPER PROCESS TANK – A REAL TIME ANALYSIS

SURESH MANIC KESAVAN*, TVN PADMESH, CHAN WOEI SHYAN

CIARG, School of Engineering, Taylor’s University,

47500 Subang Jaya, Selangor DE, Malaysia

*Corresponding author: [email protected]

Abstract

In this work, Proportional + Integral + Derivative (PID) controller design and

real time implementation was proposed for a nonlinear hopper tank system in order to solve servo and regulatory problem. Firstly, identification of process

transfer function is done through the real time implementation experiment and it

is applied to a conventional PID controller. Then tuning methods has been

proposed including Cohen-coon (CC), Ziegler & Nichols (ZN), Internal Model

Control (IMC) and Tyreus & Luyben (TL). The optimization of tuning

parameter is taken up with Integral Square Error (ISE), Integral Absolute Error (IAE) and Integral Time Absolute Error (ITAE). The experimental results show

that TL and ZN tuning method are suitable in controlling nonlinear hopper level

tank system.

Keywords: Nonlinear process, PID controller, Hopper liquid level tank, Tuning method.

1. Introduction

In many process industries (petrochemical industries, paper making industries,

water treatment industries, etc.) the main issue is to control the level and flow

between the tanks. Since many industries involves batch process, it is necessary to

control the first process parameter like liquid level or flow, failing to do so may

lead to serious shutdown process. So it is necessary to maintain the level of tank

at particular set point.

Most of the industries deal with nonlinear process tanks such as conical,

spherical, hopper type tanks. But the majority of the findings in control theory

deal only with the linear system designs. So the control of nonlinear shape of

process tank presents a challenging task mainly due to its non-linearity and

constantly changing cross section. Hence, the hopper type tank process is taken

up for present study.

60 Suresh Manic et.al.

Journal of Engineering Science and Technology Special Issue 8/2014

Nomenclatures

A Cross sectional area of hopper tank, cm2

Fin Liquid inflow to hopper tank, cm3/s

Fout Liquid outflow from hopper tank , cm3/s

h Instantaneous height of conical portion, cm

H Total height of conical portion, cm

k Process gain

r Radius of conical tank, m

R Radius of cylindrical tank, cm

Greek Symbols

θ Delay time, s

τ Time constant

Abbreviations

IAE Integral absolute error

ISE Integral square error

ITAE Integral time absolute error

PID Proportional integral derivative

Conventional PID controllers are been proved to best controllers for linear

type of processes. But for non-linear control systems, the controller parameters

have to monitor continuously for the changes in the process.

Many research works has been done for controlling nonlinear systems like

conical tank [1] and spherical tank [2]. Anandanatarajan et al. 2006 [3] studied

about two different controllers at two different operating points using fuzzy logic

controller for a conical tank. PID controller limitations of controlling conical tank

have been studied by Lee and Sung [4]. Design of fuzzy controller for conical

process tank was studied by Madhubala et al. [5] on tuning the membership

functions of the input variables and optimized the peak of the fuzzy sets using

genetic algorithm. The result shows that the proposed method is a trial and error

method and time consuming. On the other hand, the soft computing based

controller like Fuzzy logic Controller (FLC) and Genetic Algorithm (GA) has

been proven out performs well than the conventional controller [6].

Anandanatarajan et al. [7] studied on the problem of Zhao’s Fuzzy PI which

worked properly for servo but failed to regulatory problems. In addition, modified

Zhao’s Fuzzy PI in regulatory changes is proposed and is compared to conventional

control method as well Zhao’s Fuzzy PI. Araki [8] presented PID controller including

process models, performance evaluation of PID control system, action modes of PID

controllers, and design of PID control system on conical tank.

Only limited number of research work has been carried out for nonlinear

hopper type process tanks. Therefore our research work focused on hopper tank

process designing a suitable controller (PID) and comparing results with Internal

Model Control (IMC), and Tryeus and Luyben (TL) tuning rules. The error

analyses are compared with IAE, ISE and ITAE. The paper is organized as

follows. In Section 2, the process and the lab scale experimental hardware set-up

are described. In Section 3, simulation studies and real-time conventional control

Real Time Study on Controller Tuning for Nonlinear Hopper Process Tank 61


are discussed. In Section 4, optimization using error analysis tools are discussed.

In Section 5, Real time results are compared with simulation results and

discussed. Finally, results and conclusions are discussed.

2. Process Description

The design for the research work is constructed as a prototype of the non-linear

hopper process tank with the conical and cylindrical portion situated at School of

Engineering; Taylor’s University, Malaysia. The experimental model is to be used

to maintain the level of liquid at a desired constant value. This is achieved by

controlling the inflow of liquid into tank. A disturbance in the form of set point

change is introduced into the system during normal operating conditions. The

geometrical cross section of process tank is shown in Fig. 1. The inflow and

outflow rates are measured using suitable level sensors.

Fig. 1. Geometrical Cross-Section of the Process Tank.

2.1. Lab scale experimental setup

Figure 2 shows the fabricated set up of process tank. The experimental set-up, Fig.

3, consists of conical section at the bottom and cylindrical section at the top open

to the atmosphere. The setup also consists of centrifugal pump, level sensor

arrangement, inlet and outlet valve, air pump, air regulator, IP converter, Pressure

gauge, interfacing card, level indicator and PID controller. The height of the

conical and cylindrical portion of the tank is 50 cm each. The centrifugal pump is

capable of discharging liquid at the rate of 3500 L/h is used and a PVC pipe is

used to connect the pump and the control valve. The minimum voltage applied to

the pump for discharge is 104 V.

2.2. Process modeling

The process transfer function is obtained in terms of the process characteristics

namely the process gain and the process time constant.



Fig. 2. Fabricated Set-up of Fig. 3. Experimental Set-up.

the Process Tank.

The mass balance equation governing the system dynamics is given by Eqs.

(1) and (2) [9]

�� (1) � ��

�� (2)

where A = R2, Fin is inflow rate of the tank, 125 cm

3/s, Fout is the outflow rate

of the tank cm3/s, R is the top radius of the conical tank, 20 cm, H is the total

height of the tank, 100 cm, and r is the radius at any height hi= 20 cm.

The transfer function relating the height h and the inflow rate Fin with

parameters (k, τ) can be obtained as Eq. (3)

�(�) � �(�)��(�) � �� !

("#$�) (3)

where % � &�' ( ) � &�*

' ( + � ��

The nominal transfer function is shown in Eq. (4)

��(�) � �,�� !("#$,�) (4)

where k0 and τ0 are evaluated at a nominal height h0.

From the system identification, process parameters of different operating

regions are obtained (Table 1). The process tank is divided into four (I-IV). It is

evident from Table 1, when the level of the tank raises the process gain and delay

time decreases because of accumulation of integral error.

Table 1. Process Parameters at Different Valve Opening.

Inflow

%

Range

(cm3/s)

Level range

(cm) k0 τ0 (s) θ

40 (I Region) 0-15 2.70 0.75 0.15

60 (II Region) 15-30 0.68 1.50 0.70

80 (III Region) 30-40 0.18 0.78 0.22

100 (IV Region) 40-50 0.09 0.30 0.30



2.3. Controller identification

Identified four PID controller tunings methods are as follows:

� Cohen-Coon (C-C) � Internal Model Control (IMC)

� Ziegler and Nichols (Z-N) � Tryeus and Luyben (1997) (TL)

Tuning technique, C-C can be fit to a first order process with dead time model

with the advantages that the procedure does not involve trial and error with only a

single experiment is necessary [9]. As for ZN method is one of the popular

methods among others in tuning PID controllers. The controller settings are easily

calculated and a process model is not necessary [10]. For IMC tuning method, it

involves “lamda tuning (λ)” which is obtained by trial and error. At last TL

method is an alternative method for ZN, with modified formulas for the controller

parameter that provide better performance.

3. Simulation

Block diagram shown in Fig. 4 for the hopper tank process system was created by

using MATLAB Simulink software. The system simulation response is analysed

for 40% to 100% valve opening.

Fig. 4. Simulink Block Diagram for Nonlinear Hopper Process Tank.

4. Criteria of Error Integral Analysis

The best performance of the controller was selected by using three different

tuning criteria IAE, ISE and ITAE. Each of this error integral is a form of penalty

function representing the size and duration of error [11]. Table 2 gives the details

of the tuning methods that aim to minimize the penalty

Table 2. Integral Criteria for Load Changes of PID Tunings.

Error Integral ISE IAE ITAE

-. � /0- 12

3450

a1 1.495 1.435 1.357

b1 -0.945 -0.921 0.947

36 � 3/7

1234

57

a2 1.101 0.878 0.842

b2 0.771 0.749 0.738

38 � /9 3 1234

59

a3 0.560 0.482 0.381

b3 1.006 1.137 0.995

5. Results and Discussion

The hopper tank process includes controller tuning settings using C-C, Z-N, IMC

and LT methods. These tuning method results were compared by means of

MATLAB Simulink software. The performance of the controller is compared on

the time domain specification like rise time, settling time and overshoot.



5.1. Performance analysis based on simulation results

Results are simulated for CC, IMC, ZN and TL methods and discussed in this

section. Four regions of step input have been analysed and observed from 40%

valve opening to 100% valve opening from 20% of initial liquid level.

Figure 5 shows the simulated results of C-C tuning method. From the simulated

results step response for 40% step input is better than 60%, 80% and 100% in terms of

time domain specifications. In the simulation the process system behaved with a faster

response with 2.25 seconds rise time and minimum offset of 0.005 cm as compared to

others step changes. In addition, it had a minimum time delay, 2.13 seconds.

Figure 6 shows the simulated results of IMC tuning method. From the

simulated results step response for 60% step input is better than 40%, 80% and

100% in terms of time domain specifications. However, process responded with

longest time delay among the others step changes, 2.7 seconds. For 40% step

input, it gives an unsteady response with continually aggressive oscillation while

the remaining step inputs have given a large offset value about 0.75 cm.

Fig. 5. Simulated Response Fig. 6. Simulated Response

for C-C Tuning Method. for IMC Tuning Method.

Figure 7 shows the simulated results of ZN tuning method. From the

simulated results step response for 40% step input is better than 60%, 80% and

100% in terms of time domain specifications. 60% step response also shows

better results closer to 40% step change, but 40% step change has a fast response

than 60% with a longer time in achieving steady state or the set point value.

Figure 8 shows the simulated results of LT tuning method. From the simulated

results step response for 40% step input is better than 60%, 80% and 100% in terms

of time domain specifications with fast response and less delay time. 80% and

100% have shown an offset value more than 0.25 cm as well as slow response. As

for 60% step input, it has an aggressive oscillation that persist for a long time.

Fig. 7. Simulated Response for Fig. 8. Simulated Response for

Z-N Tuning Method. L-T Tuning method.



5.2. Performance analysis based on real time implementation

The tuning rules of CC, IMC, ZN and TL from the simulation performance are

implemented and analysed in real time hopper tank control system, Figs. 9 (a)-(d).

Different set point had been set on each controller tuning settings which clearly

showed that both CC and IMC tuning method had given an aggressive response in

the real time implementation. The response gives an oscillation that persists over

a long time and it does not reach steady state.

(a) Cohen-Coon Method. (b) Ziegler-Nichols Method.

(c) IMC Method (d) Tyreus-Luyben Method.

Fig. 9. Real Time Responses with Different Methods.

The performance of the ZN and TL are much better as compared to others

tuning methods. By comparing ZN and TL tuning, it is observed that TL has lesser

overshoot and faster time that track the set point changes than ZN tuning method.

As over all, TL performs much significantly stable than ZN and others controller

(CC, IMC). Indeed, it has shown oscillatory behaviour and exhibits lesser peak time

as compared to Cohen Coon and IMC controller tuning, but it is completely

different than simulated results as mentioned in previous section. This has been

proven that Cohen Coon and IMC method does not fit well in controlling a closed-

loop control system of a real time hopper tank.

Therefore both ZN and LY tuning method for 80% and 100% has been selected

and tested for the further studies on comparison between simulated and real time

experimental results. Comparison of performance response between ZN and TL

tuning method based on ISE, IAE and ITAE is shown in Table 3.

Table 3. Comparison of Performance Response between ZN and TL Methods.

Tuning rules Zeigler Nichols Tyreus & Luyben

Rise time (min) 10.4 10.1

Settling time (min) 10.2 9.5

Overshoot (cm) 18.8 17



5.3. Comparison of experimental and simulated results

The tuning methods of ZN and TL have been chosen as a comparison between real

time and experimental results of the hopper tank system for 80% step input by

minimum error criterion of ISE, IAE and ITAE. The performances of ZN and TL

tuning method based on error analysis are shown in Table 4 and it is evident that TL

is better than ZN for 80% step input response.

Table 4. Error Analysis for Different Tuning Methods at 80% Step Input.

Method ISE IAE ITAE

ZN (Ziegler & Nichols) 27.86 26.01 25.44

TL (Tyreus & Luyben) 21.38 20.55 19.67

For 80% step input, TL tuning has shown a similarity response with ZN

tuning. From Figs. 10(a) and (b), it shows that both simulated and real time

experimental responses eventually achieved the steady state or set-point of 10 cm.

On the other hand, ZN does not reach the steady state although real time result

has achieved it, at set-point of 7.5 cm. From this observation, both controllers

have proven that at 80% step input for TL will be the suitable tuning method than

ZN tuning method with set-point change and disturbance responses.

(a) Tyreus-Luyben Method.

(b) Ziegler-Nichols controller Method.

Fig. 10. Comparison of 80% Step Input between

Simulated and Real Time Response for Different Methods.



6. Conclusions

The control systems of a nonlinear hopper tank response vary with different types

of tuning rules. In general from the simulated results as the step input increases,

settling time increase with a larger offset value. However, in decreasing of step

input there is more aggressive response behaviour that exhibits a high overshoot.

The result between simulated and real time experimental will never be exactly

similar due to human factor may occurs when running the process.

Tuning method of TL in PID controller has given good and acceptable

performance with less aggressive oscillation as well as fast settling response at 80%

step input. Besides, it also shows good result for disturbance rejection with fast

settling response. In addition, it has the minimum error integral value and followed

by ZN method. This concludes TL method is most suitable for the chosen nonlinear

hopper process tank based on error analysis.

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