32

CHAPTER 3

IDENTIFICATION AND CONTROLLER DESIGN FOR

UNSTABLE SYSTEM

3.1 INTRODUCTION

In this chapter various methodologies used for system identification

and model based controller design are discussed. System identification

techniques such as classical / modified relay based closed loop analysis and

step response test based identification of lower order models with P/PI

controller are explained in detail. In particular, the relay based identification

procedure and P controller based step response test is discussed for unstable

FOPTD model. Further, model based classical and modified structured PID

controller design procedures for time delayed unstable systems are discussed.

A detailed study with classical PID, I-PD, Two Degree of Freedom (2DOF)

PID, setpoint weighted PID (SWPID), PID controller with prefilter, and

double feedback control loop are presented.

3.2 IDENTIFICATION

In model based controller design procedure, simple models

(FOPTD and SOPTD) are used to characterise the dynamics of a given

process. Based on simple dynamic model obtained, the controller parameters

are then computed using existing methods (Johnson and Moradi 2005).

Identification of lower order transfer function models from

experimental data is essential for model based controller design practice.

33

Often, derivation of a rigorous mathematical model is difficult due

to the complex nature of chemical processes (Vivek and Chidambaram 2005).

Hence, system identification procedure is a viable tool to develop

mathematical model of t process. A wide range of analytical techniques exist

for the identification of stable systems. However, for unstable systems, only a

limited progress has been made with analytical approaches such as relay

method and step response test. For a class of unstable systems, closed loop

step test with a P/PI/PID controller provides a satisfactory reduced order

model. This model is employed to design a controller.

3.2.1 Relay Based Methods

Astrom and Hagglund (1984) have proposed the relay based

autotuning procedure and it is widely adopted in industries because of its

simplicity. Process model identification using Auto Tuning Variation (ATV)

is one of the successful methods, which is used to identify the lower order

process models (Luyben 1987).

3.2.1.1 Basic relay

The basic block diagram of closed loop relay test is depicted in

Figure 3.1. The classical relay test generates sustained oscillation of the

controlled variable to obtain vital process information such as ultimate gain

( uk ) and ultimate frequency ( u ).

ah4ku (3.1)

uu P

2 (3.2)

where h is the relay height, a is the amplitude and uP is the period of

oscillations in the process output.

34

Figure 3.1 Relay based identification scheme

Luyben (1987) have proposed the relay based closed loop test for

the identification of unstable FOPTD process model from the following

equations;

11

Kk2

u2

u (3.3)

0)(tan u1

u (3.4)

where K is the process gain, is the process time constant, and is the

process delay. In this, it is assumed that the process gain or process delay is

known a priori. Li et al (1991) reported that, the model identified by the

symmetrical closed loop relay test offers an error of 27 to -18% in the value

of uk for stable first order systems.

Consider the open loop unstable FOPTD transfer function which is

given as

se1s

K)s(G (3.5)

In the process control literature, the process parameters for the

above model are considered as K =1 and ranging from 0.1 to 0.8. For the

model with K =1 and = 0.1, Marchetti et al (2001) developed unstable

UnstableSystem

+h

-h

Y (S)R (S)

+ _

e

35

FOPTD model with K = 1.001, = 0.101, = 0.87 using Basic Relay (BR)

and K = 1.004, = 0.101, = 0.816 with ATV.

Figure 3.2 Nyquist assessment of process with K=1 and = 0.1

Figure 3.2 depicts the Nyquist comparison of real and identified

process models. The model by BR shows the best fit with the real process

compared to ATV.

For )s(G with K =1 and = 0.4, Marchetti et al (2001)

developed a model with K = 1.001, = 0.426 and = 0.953 using BR based

identification. Further the ATV based identification provided K = 0.928,

= 0.396, = 0.757. With a symmetrical relay test, Vivek and

Chidambaram (2005) obtained K = 0.9841, = 0.4372, = 1.1332. The

relay based system identification proposed by Padhy and Majhi (2006)

offered, K = 1, = 0.3998, = 0.9932. The model attained by Liu and Gao

(2008) using the relay test has, K = 1.0001, = 0.4, = 0.9954.

-1 -0.8 -0.6 -0.4 -0.2 0

-0.4

-0.2

0

Real axis

Real processBR modelATV model

36

Figure 3.3 Nyquist assessment of process with K=1 and = 0.4

Figure 3.3 shows a Nyquist comparison of real and identified

models. The model by BR, PM (Padhy and Majhi 2006) and LG (Liu and Gao

2008) shows the best fit with the real process. The model obtained by VC

(Vivek and Chidambaram 2005) is better and the model by ATV method

shows larger deviation than other methods considered.

3.2.1.2 Modified relay

In order to improve the performances such as wide range of

applicability compared to conventional relay method and shorter time

duration to reach sustained oscillations, modified relay feedback methods are

proposed.

Tan et al (2006) discussed a preloaded relay to improve the

estimation of critical points. In this method, a gain parameter 'K is

considered to boost the fundamental frequency in the forced oscillations.

Ramakrishnan and Chidambaram (2003) presented an asymmetrical relay

feedback test to improve the estimation accuracy of SOPTD transfer function

-1 -0.8 -0.6 -0.4 -0.2 0

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Real axis

Real processBRATVVCPMLG

37

model. In this method, the relay amplitude is chosen as h and h , which

improves the estimation of critical point compared to a conventional relay

based procedure. Validation of this proposed method is carried out with a

class of stable process models and improved results are achieved. The work

by Hang et al (2002) summarises the developments of relay tuning methods.

The variation in the classical relay autotuning methods such as biased relay

and parasitic relay are elaborately discussed and examined using a class of

SISO and MIMO process models.

3.2.2 Step Response Test

Closed loop step response is one of the practical methods,

considered to identify the reduced order model of unstable processes. The

major merit of this method is, it does not require an additional component like

relay, and the existing controller (P / PI / PID) can be considered to obtain the

necessary closed loop data for model development. In this scheme, the

controller values are selected by trial and error until the system exhibit an

under damped like response (Padmasree and Chidambaram 2006). The

reduced order unstable model is then derived from the process response

details.

3.2.2.1 P controller based step test

The P controller based identification of unstable system is initially

proposed by Srinivas and Chidambaram (1996). This procedure identifies a

closed loop unstable SOPTD model. From the SOPTD model, open loop

unstable FOPTD model is derived. Figure 3.4 depicts the block diagram of P

controller based step response test.

38

Figure 3.4 P controller based step response test

It is assumed that, a real unstable process has an estimated transfer

function representation ranging from FOPTD to TOPTD. The unstable

FOPTD model with K =1 and = 0.1 widely discussed in the literature is

considered (Padmasree and Chidambaram 2006, Chen et al 2008). An

existing closed loop reaction curve method (Padmasree and Chidambaram

2006) is employed to identify the model of the said process. In the proposed

study, three process responses namely are considered; the nearly stable

response with approximately null oscillation (small pK value), response with

under damped like oscillation (modest pK value) and response with more

oscillation (high pK value).

Steps in P controller based system identification procedure:

Step 1 : Consider the closed loop system with pK only,

Step 2 : Excite the system with an unity step signal,

Step 3 : Adjust the value of pK until the closed loop system show an

under damped like response,

Step 4 : Calculate the values of pY , vY , t and Y ,

Step 5 : Find the reduced transfer function model of the system using

Equations 3.6 to 3.13,

Step 6 : Validate the model.

P Controller UnstableSystem

Y(s)R(s)

+ _

39

Figure 3.5 Parameters for P controller based system identification

where pY = First peak value, vY = First valley value, t = time difference

between pY and vY , Y = Final steady state value for process output.

The parameters of the process model are identified by considering

the following equations;

)1K)(1(P k2

1 (3.6)

1)(K(1)(K)1K(P k2

kk2 (3.7)

pk KKK (3.8)

22 )Vln(

)Vln( (3.9)

YYYYV

P

V (3.10)

where 1P , 2P , V = variables, = damping ratio, kK = closed loop gain.

40

The process model parameters such as K , and can be

evaluated using the following mathematical relations;

Process gain ( K ) =)1Y(K

Y

p (3.11)

Process time constant ( ) =)PPt( 21 (3.12)

Closed loop delay ( ) =P

)Pt(2

2

1 (3.13)

P controller based system identification procedure is attempted for

the above said process model and Figure 3.6 presents the process response for

various pK values. When the pK value is small, the damping ration of the

system is large. The increase in pK decreases the damping ratio, so the

oscillation in the system increases.

Figure 3.6 Closed loop reaction curve for various Kp values

0 1 2 3 4 5 6 70

0.5

1

1.5

2

Time (sec)

Kp =4 Kp =5 Kp =6 Kp =7

41

Table 3.1 lists the parameters obtained during the P controller

based identification scheme and the identified unstable FOPTD models for

various values of pK . In order to find the model with best fit with real

process, Nyquist plot comparison is attempted.

Table 3.1 Parameters of P controller based step response scheme

S.

NoKp Yp Yv Y t V P1 P2

Identified

Model

1. 4 1.45 1.32 1.33 0.6 1.12 1.73 2.17

K = 0.9994

= 0.3042

= 0.7205

2. 5 1.57 1.18 1.25 0.4 1.12 1.99 2.28

K = 0.9993

= 0.2221

= 0.5804

3. 6 1.70 0.99 1.20 0.4 0.42 1.91 3.84

K = 0.9994

= 0.3043

= 0.7229

4. 7 2.25 0.51 1.13 0.3 0.94 2.83 3.05

K = 1.0000

= 0.1070

= 0.8357

When pK is small (system with small oscillation), the model

mismatch is larger. From Figure 3.7, it is noted that, the model identified with

pK = 7 provides a best model compared to other pK values.

42

Figure 3.7 Nyquist comparisons for various Kp values

Figure 3.8 Nyquist comparison of the existing and identified model

-1 -0.8 -0.6 -0.4 -0.2 0 0.2-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

Real axis

Real processKp =4Kp =5Kp =6Kp =7

-1 -0.8 -0.6 -0.4 -0.2 0-0.5

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

Real axis

Real processBR modelATV modelStep test with Kp=7

43

Figure 3.8 shows the comparison of identified process models. The

model obtained with P controller based step response test is validated with a

relay based model developed by Marchetti et al (2001). The model obtained

with pK = 7 is very close to models by BR and ATV methods.

Consider the unstable FOPTD process model with K =1 and

= 0.4. The P based step response test is executed and FOPTD models are

identified with different pK values. When pK = 2, the identified model

parameters are K = 1.001, = 0.3994, = 1.0015; for pK = 2.25, the

model parameters are K = 1.0001, = 0.4121, = 0.9842 and for pK =

2.5, the process model with parameter K = 0.9992, = 0.4207, = 0.9805

are obtained. In this, the model identified with pK = 2 shows the best fit

with the real process compared to other process models.

For an unstable second order process with one unstable pole as

given in Equation 3.14,

1s5.012s1esG

s5.0 (3.14)

Vivek and Chidambaram (2005) identified unstable FOPTD model

with K = 0.8266, = 0.9951, = 1.8757 using conventional relay, the

identification process with improved relay offers K = 0.7534, = 1.0412,

= 2.1642. With a relay feedback test, Liu and Gao (2008) obtained

K = 1.0001, = 1.0486, = 2.1459. Attempted P controller based step

response test with a pK value of 1.5 provides K = 0.9974, = 0.9931,

= 2.2064.

44

Figure 3.9 Nyquist comparison for second order process with various

unstable FOPTD models

Figure 3.9 shows the Nyquist comparison of real and identified

models for the process. The relay and improved relay based model by VC

shows larger deviation with the real process. The relay based model by LG

shows better fit with the real process. The attempted relay test provides a

FOPTD model comparatively superior than LG. The above said second order

process is also considered with =1 (Chen et al 2008).

-1 -0.8 -0.6 -0.4 -0.2 0-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Real axis

Real processRelay (VC)Improved relay (VC)Relay (LG)Step test (Kp=1.5)

45

Table 3.2 Step response parameters and identified FOPTD models for

the second order process with = 1

S.

NoKp Yp Yv Y t V P1 P2

Identified

Model

1. 1.20 8.86 4.64 5.99 9.3 0.47 0.43 1.60

K= 1.0001

= 1.5934

= 2.0483

2. 1.275 7.56 2.80 4.63 7.8 0.62 0.52 1.55

K =1.0006

= 1.667

= 2.0143

3. 1.35 6.81 1.61 3.77 7.0 0.71 0.60 1.61

K = 1.008

= 1.664

= 2.1427

Table 3.2 provides the parameters and unstable FOPTD models for

various values of pK . Figure 3.10 depicts the Nyquist comparison of the real

and identified process model. The model for proportional gain 1.275 and 1.35

are approximately similar, and illustrates good fit with real system compared

to the model with pK = 1.2.

46

Figure 3.10 Nyquist comparison for second order process with = 1

From the above results, it is evident that the accuracy of model

parameter depends on controller setting. A heuristic algorithm can avoid this

dependence and with this motivation, a heuristic algorithm is proposed and

the proposed algorithm is also used for controller design.

3.2.2.2 PI / PID controller based step test

For the complex unstable systems, P controller based identification

procedure sometime fails to provide the expected model due to large offset.

Hence, a PI controller based system identification technique is considered for

such systems.

Figure 3.11 PI/PID controller based step response scheme

-1 -0.8 -0.6 -0.4 -0.2 0-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Real axis

Real processKp = 1.2Kp = 1.275Kp = 1.35

PI / PIDController

UnstableSystem

Y(s)R(s)

+ _

47

The identification procedure for PI based closed loop reaction

curve method is similar to the P controller method, except the initial

parameters to be adjusted are proportional gain pK and integral gain iK

(Padmasree and Chidambaram 2006).

3.3 CONTROLLER DESIGN

Controller tuning is an essential preliminary procedure in almost all

the industrial process control systems. Despite the significant developments in

advanced process control schemes such as predictive control, internal model

control, sliding mode control, PID controllers are still widely used in

industrial control application because of their structural simplicity, reputation

and easy implementation. The merits of PID controller are as follows: (i)

obtainable in variety of structures such as, academic PID, series PID, parallel

PID and IMC-PID (Vijayan and Panda 2011), (ii) provides optimal and robust

performance, (iii) supports online/offline tuning and retuning based on the

process performance requirement, (iv) advanced arrangement such as 2DOF

and 3DOF are possible. Many researchers proposed PID tuning rules to

control various stable and unstable systems by different schemes to enhance

closed loop performance ( ström and Hägglund 1995, 2006; O'Dwyer

2009). For stable systems, PID controller offers a viable result for both the

reference tracking and disturbance rejection. However, for unstable systems,

it can effectively work either for reference tracking or disturbance rejection.

The proportional and derivative kick in the controller also results in large

overshoot and large settling time.

In process control applications, PID and modified structured PID

are still widely used in industrial control system where reference tracking and

disturbance rejection are a major task.

48

3.3.1 PID Controller

Industrial PID controllers usually available as a packaged form and

to perform well with the industrial process problems, the PID controller

requires optimal tuning. Figure 3.12 shows the diagram of a simple closed

loop control system. In this structure, the controller Gc(s) has to provide

closed loop stability, smooth reference tracking, shape the dynamic and the

static qualities of the disturbance response (Johnson and Moradi 2005).

Figure 3.12 General closed loop system framework

In process industries, PID controller is used to improve both the

steady state as well as the transient response of the plant. In Figure 3.11, Gp(s)

represent the process under control and Gc(s) is the controller. The main

objective of this system is to make Y(s) = R(s). In this framework, the

controller continuously adjusts the value of Uc(s) until the error E(s) is zero

irrespective of the disturbance signal D1(s) and/or D2(s).

Closed loop response of the system with setpoint R(s), supply

disturbance D1(s) and load disturbance D2(s) can be expressed as,

)s(D)]s(G[)s(G)s(G1

1)s(R)s(G)s(G1

)s(G)s(G)s(Y 1p

CpCp

Cp

Gc(s)Y(s)R(s)

+ _

D2(s)

+

+Gp(s)

D1(s)

+

+E (s) Uc(s)

49

)s(D)s(G)s(G1

12

Cp (3.15)

where, the complementary sensitivity function and sensitivity function of the

above loop is represented in Equations (3.16) and (3.17) respectively.

)s(G)s(G1)s(G)s(G

)s(R)s(Y)s(T

Cp

Cp (3.16)

)s(G)s(G11)s(S

Cp(3.17)

The final steady state response of the system for the set point

tracking and the disturbance rejection is presented below,

AsA

(s)(s)GG1(s)(s)GG

xslim(s)Yslim)(yCp

Cp

tR

tR

(3.18)

0s

L(s)(s)GG1

(s)Gxslim)(y 1

Cp

p

tD1

(3.19)

0s

L(s)(s)GG1

1xslim)(y 2

CptD2

(3.20)

where, A = amplitude of the reference signal and L = disturbance amplitude.

To achieve a satisfactory )(yR , )(y 1D and )(y 2D , it is

necessary to have optimally tuned PID parameters.

In this research work, a non-interacting form of PID controller

structure is considered. For real control applications, the feedback signal is

50

the sum of the measured output and measurement noise component. A low

pass filter is used with the derivative term to reduce the effect of measurement

noise. The PID structure is defined below:

1N

sT sT

sT11K(s)G

d

d

ipc (3.21)

where pK / iT = iK , pK * dT = dK , N = filter constant.

Figure 3.13 Parallel form of PID structure

Figure 3.13 shows the PID controller. The output signal from the

controller is ;

T0

f

dipC dt

de(t)1sT

Kdte(t)Ke(t)K(s)U (3.22)

The chief limitation of the classical PID structure is a step change

in the reference input R(s) will cause an immediate spiky change in the

control signal )s(Uc . This abrupt change in the controller output is

Gp(s)Y(s)R(s)

+ _

Kp

Ki / s

Kd s / (Tf s+1)

Uc(s)

Gc(S)

E(s) ++

+

51

represented as the proportional and/or derivative kick. These kick effects

rapidly change the command signal to the actuator which controls the entire

operation of the plant. The kick effect can be nullified by using modified PID

structures.

3.3.2 Two Degree of Freedom PID

For unstable systems, the one degree of freedom controller fails to

provide a smooth reference tracking performance due to the occurrence of

proportional and derivative kick (Johnson and Moradi 2005). In order to

improve the overall closed loop performance, it is essential to consider a

2DOF PID structure. A detailed study on various 2DOF structures are clearly

presented by Araki and Taguchi (2003).

Figure 3.14 Feedback structure of 2DOF PID

Figure 3.15 Feed forward structure of 2DOF PID

In this research work, an attempt has been made with the Feedback

(FB) and the Feed Forward (FF) 2DOF PID structures. Figure 3.14 depicts the

FB 2DOF structure with a PD controller in the inner loop and a PID controller

C2(s)

Unstablesystem

Y(s)R(s)

+ _C1(s)

E (s) U(s)

_

+

D(s)

+

+

Unstablesystem

Y(s)R(s)

+ _C3(s)

E (s)

U(s)_

+

D(s)

+

+C4(s)

52

in the outer loop. In this structure, the PID controller responds on error signal

)t(e and the PD controller works on the process output )t(y . Hence, the

controller )s(C2 is free from proportional and derivative kick effect and the

response is very smooth compared to classical PID controller.

The outer and inner loop controller values are presented in

Equation (3.23) and (3.24) respectively.

)s(D)1(s

1)1(K)s(C fdi

p1

)s(DK)1(K)1(K fdip (3.23)

)s(DKK)s(DK)s(C fdpfdp2 (3.24)

where ipi /KK , dpd KK , and are controller weighting

parameters ranging from 0 to 1 and )s(D f is the derivative filter term given

by )sN1/(s f .

Figure 3.15 shows the FF type controller structure with a PD

controller in the feed forward loop and a PID controller in the closed loop.

The PID controller responds on error signal )t(e and the PD controller works

on the reference input )t(r .

The controller values for this structure are presented in Equations

(3.25) and (3.26).

)s(DKKK)s(Ds

11K)s(C fdipfdi

p3 (3.25)

53

)s(DKK)s(DK)s(C fdpfdp4 (3.26)

In this structure, the number of parameters to be tuned are pK , iK ,

dK , and . The comparative study by Araki and Taguchi (2003) reports

that, even though there exists a structural difference, the 2DOF FB and FF

controller offers similar reference tracking and disturbance rejection

performances.

3.3.3 Setpoint Weighted PID Controller

Figure 3.16 shows the structure of Setpoint Weighted PID

(SWPID) controller widely considered to offer a smooth reference

performance (Padmasree and Chidambaram 2005; Pillay and Govender

2011). In this the number of controller parameters to be tuned is pK , iK , dK ,

and .

Figure 3.16 Structure of setpoint weighted PID control system

Td D(s)

E(s)

UnstableSystem

R(s)Y(s)

_

_U(s)

Kp

D(s)

1-

1-

_

1/Tis

+

+

+

+

+ +

+

+

54

The above SWPID structure can be mathematically represented as

follows;

)s(U)s(U)s(U 21 (3.27)

)s(DT)1(sT

1)1(K)s(U di

p1 (3.28)

)s(DTK)s(U dp2 (3.29)

Recently, Chen et al (2008) have discussed a SWPID controller

design for a class of unstable system. The work reports that, based on the

values of setpoint weighting parameters such as and as represented in

Table 3.3, a simple PID-PD controller provides the classical and modified

PID structures.

Table 3.3 Various setpoint weighted controller structures

Weighting parameters Controller

structure

0 0 PID

0 1 PI-D

1 0 ID-P

1 1 I-PD

0 < <1 1 PI-PD

0 < <1 0 < <1 PID-PD

55

When = = 0, the structure offers a PID controller which

works based on the error signal )t(e . At t = 0, the )t(e is

maximum, since the PID structure results in large overshoot

because of proportional and derivative kick.

When = 0 and = 1, it represents a PI-D structure. In this,

PI part responds for )t(e and D works on )t(y . In this

structure proportional kick by the P is maximum and the kick

by D is minimum (since, when t=0, )t(e = max, and )t(y =

0). The reference tracking response of PI-D structure is similar

to PID.

When = 1 and = 0, SWPID forms an ID-P structure,

which is free from proportional kick. The effect of derivative

kick by this structure is considerably small and it can provide

a smooth reference tracking response compared to PID, PI-D.

When = 1 and =1, we can get an I-PD structure, which is

free from proportional and derivative kick and offers a very

smooth reference tracking performance.

When ranges from ‘0 to 1’ and = 1, the PI-PD structure is

constructed. Where the PI part works based on equation 3.28

and PD part works based on equation 3.29. From equation

3.28, the observation is that, the value of proportional gain is

)1(*K p . Since, in PI-PD controller, the effect of

proportional kick is 1 times lesser than the PID controller.

In this the PD part is available in feedback loop and it is free

from proportional and derivative kick effect. Since the

overshoot by PI-PD is lesser than PID and PI-D structures.

56

3.3.4 PID Controller with Prefilter

Figure 3.17 depicts the structure of prefilter / setpoint filter based

PID controller discussed by Araki and Taguchi (2003). Jung et al (1999)

reported that, when the filter time constant f is set equal to integral time

constant iT , the controller offers a smooth reference tracking performance.

Figure 3.17 Structure of PID controller with prefilter

Recently, Vijayan and Panda (2011) developed analytical

expression to assign the above controller parameters and validated using a

class of stable and unstable process models.

3.3.5 Double Feedback Control Loop

Figure 3.18 depicts the double feedback controller structure. The

inner loop with a P controller is used to stabilise the unstable system and the

outer loop with PID is used to obtain the required reference tracking and

disturbance rejection responses.

FilterPID

ControllerUnstablesystem

Y(s)R(s)

+_

D(s)

+

+

57

Figure 3.18 Double feedback controller structure

Recently, Vijayan and Panda (2012) proposed detailed analytical

expression for the above said controller and validated the performance using a

class of stable and unstable process models. This structure provides smooth

reference tracking performance for unstable FOPTD and SOPTD process

models

3.4 DISCUSSION

In this chapter various system identification procedures and various

controller structures are presented. The relay based system identification

procedure provides better result for unstable system with < 0.693 (Ali and

Majhi 2006; Yu 1999). The modified relay tuning scheme can be used for

more complex system where basic relay method fails (Hang et al 2002). The P

controller step response test provides better result for unstable systems with

< 0.5 (Padmasree and Chidambaram 2006). In conventional P/PI

controller based step response test, the identified model accuracy greatly

depends on the controller setting. Compared to the P controller based

identification procedure, PI controller based method offers better model

accuracy for the system with > 0.5 (Padmasree and Chidambaram 2006).

The simulation study performed in this chapter demonstrates that, the model

R(s)

_

UnstableSystem

Y(s)

+

e

_P1PID

+

58

accuracy can be improved by choosing appropriate value of Kp during the P

controller based step test.

This chapter also summarizes the classical and modified structured

PID controllers such as parallel form of PID, two degree of freedom PID

(feedback and feed forward structures), setpoint weighted PID, PID controller

with prefilter, and double feedback control loop existing in the literature. The

Setpoint Weighted PID (SWPID) controller has five parameters such as Kp,

Ki, Kd, and . In SWPID, by choosing proper values of and , it is

possible to realize the classical and modified structured PID controllers.