57

CHAPTER 3

TUNING METHODS OF CONTROLLER

3.1 INTRODUCTION

This chapter deals with a simple method of designing PI and PID

controllers for first order plus time delay with integrator systems (FOPTDI).

Controllers have been designed using various tuning methods like equating

coefficient method (EQ), direct synthesis method (DS), model reference

control (MRC) method and dual loop control (DLC) method. In order to

demonstrate the effectiveness of the developed methods to tune PI and PID

controller for setpoint tracking and disturbance rejection, the IMC and ZN

tuning methodologies are chosen for comparison. The performance of the

controller under uncertainty in model parameters has been analyzed and its

robustness is verified.

3.2 EQUATING COEFFICIENT METHOD

System whose dynamics are slow with large time constant can be

approximated as integrating systems. Integrating processes are frequently

encountered in the process industries. The process would be non-minimum-

phase if it contains time delays and/or right-half-plane zeros. The design of

controllers for such integrating processes is challenging and an interesting

problem. Conventional PI and PID methods with unity feedback control

structure have been developed by several authors (Lee et al 2003) have

58

proposed unity feedback control structure for the control of integrating

processes with time delay.

In this equating coefficient method, the closed loop transfer

function of the given process along with the given controller is being found

for the servo problem and the coefficients of the corresponding s, s2, s

3 in the

numerator and that in the denominator are equated to find the tuning

parameters. Tuning parameters can be of single or more than one. In this

work, tuning of both PI and PID controllers has been done. The equations for

the controller settings are simple. More the tuning parameter, the complexity

of the tuning is increased.

Closed loop identification method is preferred over that of the open

loop method since the former is insensitive to disturbances. Transfer function

models are used for designing PI/ PID controllers. The methods for designing

PID controllers for unstable FOPTD systems have been developed by DePaor

and O’Malley (1989), Ho and Xu (1998), optimization method by Cheng and

Hwang (1998), Manoj and Chidambaram (2001) and synthesis method by and

Jung et al (1999). An excellent review of the work reported on the design of

PID controllers is given by Astrom and Hagglund (1995).

Transfer function models are used for designing PI/PID controllers.

In many of these methods, one or two adjustable parameters are used to

calculate the PID settings but the design procedure is complicated. In this

work, a simple method proposed by Sakthe Vivek and Chidambaram (2005)

to design PI and PID controllers for both stable and unstable First Order Plus

Time Delay(FOPTD) system have been developed for FOPTDI systems.

59

3.2.1 EQ- PI Controller

The Equating Coefficient (EC) method developed to design PI

controllers for FOPTD system is extended for FOPTDI systems. The equating

coefficient method gives simple equations for the controller settings. This

method is based on matching the coefficient of corresponding powers of ‘s’ in

the numerator and that in the denominator of the closed loop transfer function,

since the objective of the controller is to make y/yr=1. Controllers are

designed using single tuning parameter, and to improve the performance of

the controller under uncertainty in model parameters, concept of two tuning

parameters is used. The performance of the closed loop system is evaluated

for both the original and the approximated model. The controllers are also

tuned using IMC and ZN, and its performance has been compared by

simulation.

The following sets of linear algebraic equations are obtained for PI

controller for FOPTDI system,

1 2(1 )k 0.5(1 )k 0-a + +a = (3.1)

1 20.5(1 )k (1 )k+a + -a= a (3.2)

By solving the equations (3.1) and (3.2) the following equations are

derived:

k1=kckptd (3.3)

12

1

d

kk = æ ötç ÷tè ø

(3.4)

q =std (3.5)

60

The value of a is greater than one and this parameter is considered

as tuning parameter. It has been found by simulation that a = 1.01 gives best

result for Rmodel1, Rmodel2 and Rmodel3. The value of a less than 1.01

doesn’t yield better Integral Square Error (ISE) and Integral Absolute Error

(IAE) values. Solving the equations (3.3) to (3.5), the values of k1and k2 are

obtained. Using the definitions of k1 and k2, the PI controller settings are

obtained as,

k kc p dt = 1.005 (3.6)

I

d

tt = 100.5 (3.7)

From the equations (3.6) and (3.7), the controller settings has been

obtained by substituting the kp and td of the given model. For kp=1.667,

td = 0.3526 and t = 0.558, the value of kc = 1.7 and ti = 0.354 are obtained.

By using these controller parameters, the transfer function model is simulated

for both servo and regulatory responses and its performance has been

compared with IMC and ZN methods.

3.2.2 EQ - PID Controller

A stable FOPTD system with an Integrator is represented by

( )ds

pk e

s s 1

-t

t + , wherepk is the process gain, d

t is the time delay and t is the time

constant of the process. For the purpose of designing controllers, the

dynamics of many processes can be described adequately by a FOPTD model.

The method gives simple set of equations for the controller settings. The

performance of the control system is compared with that of the IMC method

61

and ZN methods. The closed loop transfer function relating the output(y) to

the set point (yr) is given by,

2 q

1 2 3

2 2 q

r d 1 2 3

(k q k k q )ey(q)

y (q) [q [( )q 1] (k q k k q )e ]

--

+ += t t + + + + (3.8)

where q = t s, and s - is the laplace operator. Using pade’s approximation for

qe- as [(1-0.5q)/ (1+0.5q)] in the denominator of the equation (3.8), the

numerator and the denominator terms are expanded using the taylor series for

0.5qe and0.5qe- .

The coefficient of q in the numerator is equated to a1 times that of

the denominator of the closed loop transfer function. The coefficients of q2

and q3

of the numerator are equated to a1 times that of the denominator. The

following sets of linear algebraic equations are obtained for PID controller.

Since the objective of the control system is to make y to follow yr, the

corresponding coefficients of q, q2 and q

3 of the numerator with that of the

denominator are equated. Since the presence of integral model makes the

offset zero, the constant term in the numerator and that in the denominator is

the same.

(1 )k 0.5(1 )k 01 1 1 2

-a + +a = (3.9)

0.5(1 )k (1 )k (1 )k2 1 2 2 2 3 2

+a + -a + -a= a (3.10)

0.125(1 )k 0.0208(1 )k [0.5 ( )]2 1 2 3 2d

-a + +a= + t t a (3.11)

By solving these linear algebraic equations 3.9, 3.10 and 3.11, PID

controller settings are obtained. It has been found by simulation that1

a =1.1,

2a =0.8

1a gives best result for Rmodel1, Rmodel2 and Rmodel3. The ISE

62

and IAE values are also obtained to evaluate the performance criteria. The

performance of the EQ-PID controller is compared with the controller

designed by the EQ-PI method.

The Figure 3.1 and Figure 3.2 show the servo and regulatory

response of EQ-PI controller respectively and it has been compared with the

IMC and ZN methods of controller. The simulation shows that the IMC and

ZN method produces a response having large undershoots on the other hand

the EQ-PI method shows marginally quicker settling time than IMC and ZN

method.

Figure 3.1 Servo response of Rmodel 3

EQ-PI

IMC

ZN

63

Figure 3.2 Regulatory response of Rmodel 3

The PI controller settings obtained by using the equations 3.6 and

3.7 for the FOPTDI system with kp= 3.1818,t =1.0659, td =0.673 are

kc =0.296, ti = 0.002. By using these controller settings, the process has been

simulated and it shows a response having large overshoot. With the same

controller settings, the model3 has been simulated and response is shown in

Figure 3.3 where IMC and ZN controller gives sluggish response with offset.

EQ-PI

IMC

ZN

64

Figure 3.3 Regulatory response of model 3

The tuning parameters obtained using EQ-PID method are

kc = 0.3212, tI = 0.044 and td =0.4 for the average FOPTDI transfer function

(Rmodel3) model. The same controller settings give better response for the

Rmodel1 and Rmodel2 also.

EQ-PID

EQ-PI

65

Figure 3.4 Servo response under parameter uncertainty +12% in

process gain kp

The robustness of the EQ method is studied by using +12%

perturbation in process gain kp from the nominal value is shown in Figure 3.4,

whereas the controller settings are those calculated for the process with

nominal parameters. The servo response of the system under +12%

uncertainties in time constant t and +16% uncertainties in time delay td are

shown in Figure 3.5 and Figure 3.6 respectively. From these results, it is

observed that the EQ-PI method produce an oscillatory response with

overshoot and longer settling time following a setpoint.

EQ-PID

EQ-PI

66

Figure 3.5 Servo response under parameter uncertainty of +12% in

time constant t

Figure 3.6 Servo response under parameter uncertainty of +16% in

time delay td

EQ-PI

EQ-PID

EQ-PID

EQ-PI

67

Analyses of the results indicate that the tuning method IMC

referring to the response curves of Figures 3.4 to 3.6 under uncertainty in

process gain, time constant and time delay respectively, the EQ-PID method

settles with less oscillation and offset. The ISE and IAE value comparison for

EQ-PI and EQ-PID tuning method are shown in Table 3.1.

Table 3.1 ISE, IAE value comparison for EQ-PI and EQ-PID controller

(kc= 0.3212, tI =1.0659,td = 0.673); PI(kc=0.296, ti =0.002)

Models for

simulationMethod

Servo

Response

Regulatory

Response

ISE IAE ISE IAE

RModel 1PID 2.25 409.2 43.35 4320

PI 2.05 542.5 71.37 4500

RModel 2PID 2.108 133 65.26 5170

PI 6.41 333.9 70.75 7760

RModel 3PID 2.05 542.5 43.35 4510

PI 2.99 651.7 82.01 8710

The controller parameters obtained are kc= 0.3212,ti=0.0454,

td = 0.4 for the FOPTDI system with kp=3.1818,t =1.0659,td = 0.673. The

performance evaluation criteria ISE and IAE values for EQ-PID method gives

less value compare to EQ-PI controller. The performance under model

parameter uncertainty is better for EQ-PID method compared to EQ-PI

method. Also the Table 3.2 shows the ISE and IAE values comparison for the

transfer function of the system under uncertainty in process gain kp, time

constant t and time delay td.

68

Table 3.2 ISE, IAE value comparison for system under parameter

uncertainty (Parameters for controller design Kp= 2.1818,

t =1.0659, td= 0.673)

S.No Method ISE values for uncertaintyIAE values for

uncertainty

+12% in

kp

+12% in

t+16% in

td

+12% in

kp

+12% in

t+16%

in td

1. EQ-PI 2.53 2.51 2.58 651.8 652.0 652.2

2. EQ-PID 2.40 2.42 2.45 542.7 542.8 542.1

For uncertainty in process parameters, the performance evaluation

criteria such as ISE and IAE are comparatively less for EQ-PID method.

3.3 DIRECT SYNTHESIS METHOD

Time delay is common in all process industries due to

transportation delay, recycle loops, composition analysis loops and the like.

These types of processes can be approximated as integrating processes with

time delay for the purpose of designing controllers instead of controlling in

the original form.

Using direct synthesis method, a PID controller in series with a

lead/lag compensator is designed for control of closed loop FOPTD processes

with time delay. Guidelines are provided for selection of the desired closed

loop tuning parameter. The method gives significant load disturbance

rejection performance. With the reference to the literature studied, several PI

and PID controllers design methods have been proposed by Bhattacharya et al

(1993 and 1995) for control of integrating processes with time delay.

However, when there is large time delay, control of integrating processes is

69

difficult because of the limitations imposed by the time delay on system

performance and stability.

The controller for the processes such as Integrating Plus Unstable

First-Order Plus Time Delay (IUFOPTD), Double Integrating Plus Time

Delay(DIPTD) and Double-Integrating Plus First-Order Plus Time

Delay(DIFOPTD) have been addressed by Skogested (2003) and Hang et al

(2003). But most of the method uses modified form of Smith Predictor whose

structure is complicated.

The number of tuning parameters required is also more. Practically

a simple control structure with a simple controller is desirable as it is very

easy for the operator to tune. In general Gp is the process transfer function and

Gc is the controller transfer function. For designing the controller Gc, the

method developed by Seshagiri Rao et al (2008) has been considered. In this

DS method for set point tracking, a simple controller design method

with only one controller in a single feedback loop for all classes of integrating

processes has been considered. The desired output behavior of the closed loop

can be specified as a trajectory model based on the process to design the

required form of the controller. With the conventional controllers, there may

be problems like overshoot and settling time.

In this work, based on the nature of the integrating process, the

desired closed loop transfer function is chosen and correspondingly the

controller structure is derived. In this case PID controller in series with lead /

lag compensator is obtained. The controllers are also tuned using ZN method

(Ziegler 1942) and Rivera et al (1986), and the performance has been

compared by simulation. The direct synthesis method gives simple equations

for the controller settings. The performance of the closed loop system has

70

been evaluated for both original and approximated model. The transfer

function model is of the type,

s

p

keG

s( s 1)

-q= t + (3.12)

The desired closed loop transfer function is considered as,

2 s

2 1

r

y ( s s 1)e

y ( s 1)

-qh + h +l + (3.13)

This design method is chosen here because the desired output

behavior of the closed loop can be specified as a trajectory model based on

the process to design the required form of the controller. The closed loop

relation for setpoint changes is given by,

c p

r c p

G Gy

y 1 G G= + (3.14)

from equation (3.14) the controller is given by,

rc

p r

1 (y / y )G

G 1 (y / y )= - (3.15)

According to the direct synthesis method, the closed loop trajectory

model should be specified for designing the controller. The controller can be

written as

r dc

p r d

1 (y / y )G

G 1 (y / y )= - (3.16)

71

where (y/yr)d is the desired closed loop trajectory for set-point changes. The

PID controller is designed in series with the lead/ lag compensator. If the

process iss

p

keG

s( s 1)

-q= t + , the desired closed loop transfer function is

considered as,2 s

2 1

3

r

y ( s s 1)e

y ( s 1)

-qæ öh + h +=ç ÷l +è ø . Using first-order pade

approximation for the time delay, after simplification the controller is

obtained as,

( )c c d

i

s 11G k (1 s)

s ( s 1)

a += + + tt b + (3.17)

where

1c 2

1 2

kk(3 1.5 0.5 )

h= l + ql + qh - h (3.18)

1 1

2d

1

3

2

1 2

0.50.5 ,

(3 1.5 0.5 )

t = hht = h

qla = q b = t l + ql + qh - h

(3.19)

in which h1 =3l+ q and

( )( )3 2 2 2 2 2

2

0.5 ) 3 1.5 3 0.5

(0.5 )

q - t l + t - qt l + qt l + q th = t q + t (3.20)

The tuning parameter l should be selected in such a way that the

resulting controller gains should be positive for positive values of k. Hence to

get positive values of controller gain (kc), the constraint to be followed is,

72

2

2 13 1.5 0.5h < l + ql - h + qh (3.21)

In addition, lshould be selected in such a way that the resulting

controller gives good robust control performance. The initial value of the

tuning parameter can be taken as equal to half of the time delay of the process

to get good control performance. If not, then, the tuning parameter can be

increased from this value till good nominal and robust control performance

are achieved.

For suitable value of l and b, the controller designed on DS method

gives good control performances. However for high value of b, the phase lag

imposed by the term (bs+1) in the controller is more, thus the designed

controller with this value of b is not able to give robust control performances

which results in low gain and phase margins of the open loop system than the

required values (gain margin should be >1.7 and phase margin should be >350

for robust control of a process).

Based on many simulation studies, it is observed that taking ‘0.1 b’

instead of b gives good compromise between nominal performance and robust

control performance. Thus, in the present work, the value of b obtained is

modified as ‘0.1b’ for simulation studies. The desired closed loop transfer

function is chosen based on the nature of the integrating process and

correspondingly the controller structure is derived in this method.

73

.

Figure 3.7 Closed loop performance comparison of Rmodel 1, 2 and 3.

Controller designed on Rmodel 2

In Figure 3.7, performance of the closed loop system are evaluated

by giving a unit step input in the set point and a negative step input of 0.1 in

the load at t= 25s. In this work, the controller is designed using DS method

for Rmodel2 and simulated for other models such as Rmodel1 and Rmodel3.

From the response curves, it is clearly observed that the controller gives better

performance for the other models such as Rmodel1 and Rmodel2 too.

Referring to Figure 3.8, the IMC method settles at a faster rate than that of DS

method for model3 and the ZN gives a very sluggish response confirming that

it may not be suited for process of this nature. Whereas the same IMC

controller does not give good performance for model1 and model 2.

Rmodel1

Rmodel2

Rmodel3

74

Figure 3.8 Closed loop performance comparison of model 3

Table 3.3 gives the ISE and IAE values for the closed loop response

of the system. It is clearly seen that the DS method gives less ISE and IAE

values comparatively

Table 3.3 DS, IMC and NZ comparison of ISE and IAE values for

servo problem

Process Controller ISE IAE

RModel 1

DS

IMC

ZN

0.177

190.3

0.83

32.2

480.0

142.9

RModel 2

DS

IMC

ZN

0.73

245.0

57.61

142.9

629.0

173.0

RModel 2

DS

IMC

ZN

0.396

176.8

20.51

87.29

873.8

456.0

DS

IMC

ZN

75

For any closed loop system it is necessary to analyze the stability

and robustness for uncertainties in the process. The controller parameters

obtained for the FOPTDI system of Rmodel3 using the equations (3.17-3.19)

are Kc =1.25;it =0.125;

Dt = 0.463 and b = 0.089 with the tuning

parametera = 0.871 and q =0.67. Using these controller settings the closed

loop performance of the system is evaluated. Figure 3.9 shows the response of

the system for +34% perturbations in time delay and the DS method gives the

best performance for upto ± 28 % uncertainty in Kp, ± 26% uncertainty in tand ± 34% uncertainty in time delay

dt whereas other two methods IMC and

ZN gives unstable performance for such uncertainty values.

Figure 3.9 Responses for a perturbation of +34 % in process time delay

t for the Rmodel 2

Table 3.4 gives the corresponding ISE and IAE values for the

closed loop response of the system under +34% uncertainty in time delay.

IMC, ZN gives more higher error value.

DS

IMC

ZN

76

Table 3.4 DS method comparison of ISE and IAE values for

uncertainty in time delay

Process Controller ISE IAE

RModel 1

DS

IMC

Z-N

19.94

20.3

120.34

199

204

347

RModel 2

DS

IMC

Z-N

52.54

52.74

297.5

539

526

890

RModel 2

DS

IMC

Z-N

17.9

27.71

112.41

187

211

329

By giving positive and negative step input at certain point of time,

among the comparison of DS and EQ-PID, DS method gives faster settling

time for both servo and regulatory response. Figure 3.10 and Figure 3.11

shows the closed loop response comparison for the methods DS and EQ-PID

under uncertainty of +23% in process gain kp and +24% of time constantt .

Whereas the controller has been designed for the nominal values. With such

percentage of uncertainty the IMC method takes more settling time and gives

undershoot also.

77

Figure 3.10 Closed loop response of RModel2 for uncertainty of 23% of

process gain kp

Figure 3.11 Closed loop response of RModel 2 for uncertainty of

+24% times constant

DS

EQ-PID

DS

EQ-PID

78

Referring to the Table 3.5, the performance evaluation criteria ISE and

IAE are low for DS method of controller when compared to the other method.

The EQ-PID method produce large undershoot for the negative input and

takes more time to settle than DS method.

The response has been obtained after applying negative step input,

The DS controller settles faster comparatively. PID controller tuning

parameters have been obtained using different methods like EQ-PI, EQ-PID,

optimization method, and direct synthesis method. In the equating coefficient

method single and two tuning parameters have been used to design the

controller.

Table 3.5 Performance comparison of DS and EQ-PID method

Models for

SimulationMethod

Evaluation

Criteria

ISE IAE

Model 1

DS 0.18 2.52

EQ-PID 2.13 39.5

Model 2

DS 0.28 2.81

EQ-PID 0.59 56.51

Model 3

DS 0.32 2.95

EQ-PID 0.28 70.1

This method gives simple equations and the performance of the

system is evaluated by simulation comparing with IMC and ZN method.

Among EQ-PI and EQ-PID, EQ-PID performance is better in terms of

evaluation criteria like ISE and IAE. The controllers have also been tuned

using and direct synthesis method. Comparing the DS with EQ-PID, DS gives

less error value and overshoot.

79

3.4 MODEL REFERENCE CONTROL

For controller design purposes, the dynamics of the processes are

described by first order time delay model. In this model reference control

Jacob and Chidambaram (1996) have designed PI controller to force the error

to follow the given error dynamics for unstable first order plus time delay

system. In this work, for stable first order plus time delay system with

integrator, the model reference control method has been extended to design

PID controller. By specifying the settling time, ts, the value of the controller

gain is obtained. Thus the controller is robust. For nonlinear systems, the

values of time constant and gain will be varying. Hence, the controller

settings based on fixed values of time constant and gain have to be detuned.

However, the controller is shown to be robust for perturbation in time

constant and gain.

Normally due to certain non linearity of the systems, many real

systems exhibit multiple steady states. Sometimes for safety reasons, it may

also be necessary to operate the system at unstable steady state. In this work

PID controller for stable first order plus time delay systems has been

designed. The performance of the closed loop system is evaluated for both the

actual and the approximated model. The controllers are also tuned using

Internal Model Control (IMC) and the performance is compared by

simulation. The general transfer function of the process to be controlled is

given by,

p Lsk

es( s 1)

-t + (3.21)

where kp is the process gain,t is the time constant , and L is the process lag.

The corresponding time domain description is given by,

80

py (( y k u(t L))= - + - t&& & (3.22)

with t=0, y& =0

Let the error can be defined as,

re y y= - (3.23)

where ry is the setpoint value. Hence, the double derivative of the error is

given by,

re y y= -&& && &&

Figure 3.12 Block diagram representation of the process with the

control law equation

r pe y [ y k u(t L)]= - - + - t&& && & (3.24)

To get the expression for u(t), let the right side of equation 3.24 be

equated to

c

.t

0 DIk [e (1 ) edt e]- + t ò + t (3.25)

81

Then, the control law equation u (t) is given by,

c

t

r 0 D t LI

p

u(t) {y (y ) k [e (1 ) edt edt]}k

+t= + t + + t ò + t&& & (3.26)

t L

p

u(t) pk

+t= , where t Lp p Lp+ = + &

by solving the equation 3.26, the tuning parameters for PID controller are

obtained as,

c 2

s

25k

t= z (3.27)

2

I s D

I

10.4t (1 s)

st = z + + tt (3.28)

D s0.4tt = (3.29)

Thus specifying the settling time and damping coefficient ( z ) for

the error system, controller gain and integral time can be calculated. To use

the control law equation (3.26), it required the prediction of y at (t+ dt ). A

simple prediction formula for y(t+ dt ) can be obtained from the truncated

Taylor’s series expansion of y(t+ dt ) as y(t+ dt )=y(t)+ dt .'y(t) . The block

diagram representation of the feedback system of the control law is shown in

Figure 3.12. The closed loop system is thus stable and for a step input in yr,

we get y equals yr as tà ¥ . For nonlinear systems, the value of time constant

and gain will be varying. However, the present controller is shown to be

robust for perturbations in the values of time constant and gain. Hence, it is

expected that the present linear controller will give good performance on the

nonlinear system also.

82

The performance of the control system is evaluated for step change

in yr at the input and the load and the simulated results are shown in Figures 3.13

and 3.14 respectively. Referring to the servo response of Rmodel2 in Figure

3.13, MRC gives less overshoot and faster settling time. In the regulatory

response of the same model, MRC gives less undershoot comparatively.

Figure 3.13 Servo response of Rmodel 2

Figure 3.14 Regulatory response of Rmodel2

MRC

IMC

MRC

IMC

83

With the same controller settings, the other first order plus time

delay with integrator models like Rmodel1 and Rmodel2 and the original

models Model1, Model2 and Model3 are simulated for both servo and

regulatory responses. The values of the error in both cases give very less

value. The ISE and the IAE performance indices have been calculated and is

shown in Table 3.6. The parameters of the open loop system are kp=2.1818,

t = 1.0659; td = 0.672. The parameters obtained for PID controller are

kc= 0.452; tI = 0.96 and tD =0.366 for MRC method and kc= 0.12;tI = 0.012

and tD = 0.131 for IMC method. The controller settings are obtained for the

average model using MRC and IMC method.

Table 3.6 Performance comparison of MRC with IMC(Kc=0.452,

ti 0.96dt =0.36); IMC (Kc=0.12, ti =0.012; td =0.131)

Models for

simulationMethod

Servo

Response

Regulatory

Response

ISE IAE ISE IAE

RModel 1MRC 0.116 9.62 3.92 498.6

IMC 0.416 13.9 2.79 509.3

RModel 2MRC 0.166 0.07 3.99 487.6

IMC 0.33 120.92 6.78 500.9

RModel 3MRC 0.14 0.17 3.98 501.2

IMC 0.34 19.81 5.03 503.18

Figure 3.15 shows the step response of average original model. The

IMC controller shows offset whereas the model reference controller gives

faster settling time. The regulatory response of the same model is shown in

Figure 3.16 demonstrate the improvement obtained with the MRC controller.

The response of the MRC controller was comparatively better in terms of

reducing the overshoot and settling time.

84

Figure 3.15 Servo response of model 3

Figure 3.16 Regulator response of model 3

MRC

IMC

MRC

IMC

85

The behavior of manipulated variable versus time behavior is

shown in Figure 3.17. The performance in terms of ISE and IAE is listed in

Table 3.7. The performance of the model reference control is significantly

better than that of the IMC controller.

Figure 3.17 Manipulated variables versus time behavior

Table 3.7 Performance comparison for actual model 2

Method Servo Response

ISE IAE

MRC 0.017 1.88

IMC 0.2 5.96

Regulatory Response

ISE IAE

MRC 0.016 1.85

IMC 17.6 5.66

MRC

IMC

86

The simulation results demonstrate the capability of the MRC

controller in accommodating uncertainty in the process model. Uncertainty of

about -19% in time constantt, -29% in process lag L and -21% in process gain

kp are separately introduced and the response is shown in the Figures 3.18 to

3.20 respectively. When the IMC controller almost fails to cope with the

model-process mismatch, the MRC controller provides acceptable results. It

can also be noticed that the response of the MRC controller settles at the

desired setpoint with minimum oscillations compared to IMC controller

where oscillations persist for quite long time.

Figure 3.18 Comparison of MRC and IMC for uncertainty of -19% in

time constant t for Rmodel 3

MRC

IMC

87

Figure 3.19 Comparison of MRC and IMC for uncertainty of -29% in

process lag L for Rmodel 3

Figure 3.20 Comparison of MRC and IMC for uncertainty of -21% in

process gain kp for Rmodel 3

MRC

IMC

MRC

IMC

88

The results of the above simulation examples indicate, in general,

that the MRC controller reduced the sensitivity to modeling errors

comparatively. In addition, both the overshoot and settling time have been

considerably reduced.

3.5 DUAL LOOP PID CONTROL

In the Dual loop control, the process to be controlled is controlled

using dual loops. The general transfer function of the process to be controlled

is given by equation (3.21), where kp is the process gain,t is the time constant

and L is the process lag. The general block diagram of the dual loop PID is

shown in the Figure 3.21.

Figure 3.21 Block diagram of the dual loop control PID

The inner loop consists of PI controller and the outer loop consists

of PID controller designed using direct synthesis method developed by

Sheshagiri Roa et al (2009) for unstable system. In this work, it has been

extended for stable system. The closed loop transfer function of the inner

loop is given by

Ls

c p

Ls

in c p

k k ey

y s k k e

--= +

The delay is approximated asL s 1 0 .5 L s

e1 0 .5 L s

- -= + and the

resultant equation is given by,

89

( )( ) ( )c p

in c p

k k 1 0.5Lsy

y s 1 0.5Ls k k 1 0.5Ls

-= + + - (3.30)

( )( )( ) ( )( )Ls

c p

in c p

k k 1 0.5Ls ey

y s 1 0.5L s k k 1 0.5L s

-+= + + -

( )( )Ls

c p

2in c p c p

k k 1 0.5Ls ey

y 0.5Ls 1 0.5k k L s k k

-+= + - +

The closed loop transfer function of the inner loop is given by,

( )( )Ls

c p

2in c p c p

k k 1 0.5Ls ey

y 0.5Ls 1 0.5k k L s k k

-+= + - + (3.31)

Figure 3.22 Block diagram of the simplified dual PID

Figure 3.22 shows the simple block diagram of the simplified dual

PID controller. After simplification, the closed loop transfer function of the

controller is obtained as,

c c d

i

1 ( s 1)G k (1 s)

s ( s 1)

a += + + tt b + (3.32)

90

where

3

2

1 2

0.5

(3 1.5 0.5 )

qlb = t l + ql + h -hi2 1t = h

2d2

1

ht = h0.5a = q

h1=3l+qand

3 2 2 2 2 2

2

(0.5 ) (3 1.5 ) 3 0.5

(0.5 )

q - t l + t - qt l + qt l + q th = t q + tThe performance of the closed loop control system is evaluated for

a step input yr. The response of the dual loop controller is compared with

model reference controller. Simulation of the plant models have been

conducted to study the performance of the dual loop controller comparing

with MRC type controller. Referring to the figures 3.23 and 3.24, it is

observed clearly that the dual loop PID controller provides improved

performance and settles at a faster rate both in servo and regulatory responses

whereas the MRC controller shows sluggish response.

91

Figure 3.23 Servo response of Rmodel2

Figure 3.24 Regulatory response of Rmodel 2

Dual

MRC

Dual

MRC

92

The sensitivity of the controller to model inaccuracies is also

considered and is shown in the Figures 3.25 to 3.27. The robustness of the

controller is evaluated using ± 23% perturbations in the process gain and the

controller used was the one designed for the actual value of the process

parameters. Similarly robust response for perturbation in time delay and time

constant is obtained.

Figure 3.25 The regulatory performance of controller under -23%

uncertainty in process gain for Rmodel2

It is clear from the figures 3.26 and 3.27 that the MRC controller

has reduced the effects of mismatch between the process and the model on the

system performance and it is less sensitive to model errors. It provides stable

response up to ± 20% variation in time delay and ± 26% variation in time

constant.

Dual

IMRC

93

Figure 3.26 Comparison of MRC and dual loop for uncertainty of -20%

in time delay for Rmodel2

Figure 3.27 The servo performance of controller under -26%

uncertainty in time constant for Rmodel 2

Dual

MRC

Dual

MRC

94

The simulation results demonstrate the capability of the dual loop

controller in accommodating uncertainty in the process model. When the

MRC controller almost fails to cope with model-process mismatch, the dual

controller provides acceptable results. The response of the dual loop

controller settles at the desired setpoint with minimum oscillations compared

to the MRC where oscillations persist for quite long time.

The ISE and the IAE performance indices have been calculated,

and presented in Table 3.8. The content of the table shows the improvement

in performance in terms of the ISE and IAE.

Table 3.8 ISE, IAE value comparison of MRC and dual loop PID

controller

S.No Models Servo Response Regulator Response

ISE IAE ISE IAE

Model 1 MRC 0.94 201.3 42.2 498.6

Dual Loop 0.116 9.62 3.92 450.4

Model 2 MRC 2.34 410.5 71.09 487.6

Dual Loop 0.166 0.07 3.99 522.5

Model 3 MRC 1.75 375.65 37.8 501.2

Dual Loop 0.14 0.17 3.98 396.4

95

Figure 3.28 The regulatory performance of controller under -34%

uncertainty in time delay in the Rmodel2

Figure 3.29 Servo response of dual loop and MRC for RModel 2

Dual

MRC

Dual

MRC

96

The comparison of dual loop and MRC controller for both servo

and regulatory response are shown in figures 3.28 and 3.29 respectively. Dual

loop PID shows good response compared to MRC controller. Table 3.8 gives

the calculated performance evaluation criteria. For all types of FOPTDI

systems and the actual models like model1, model2 and model3 responses,

the Dual loop PID method gives comparatively less error value.

Figure 3.30 Servo Response of MRC, DS and Dual loop

Figure 3.30 shows the overall comparison of servo response of

MRC, DS and dual loop controller. The DS and Dual loop controller settles at

the faster rate without any overshoot whereas the MRC controller settles with

bit oscillation. Perturbation in process gain, time delay and time constant has

also been simulated separately with the same controller parameters.

Kharitonov’s theorem has been used to determine the range at which the

controller attains instability.

Dual

MRC

97

3.6 SUMMARY

The transfer function model of the web guide used to control the

position of the web in cold rolling mill is controlled using different tuning

methods and its simulation results have been compared. The performance

indices such as ISE and IAE values have been obtained and compared. The

process parameters are perturbed for certain percentage of process gain, time

constant and time delays separately and simulated with nominal controller

parameters in order to check the range at which the controller attains

instability.