A NEW METHOD OF TUNING THREE TERM

CONTROLLERS FOR DEAD-TIME PROCESSES

WITH A NEGATIVE/POSITIVE ZERO

K. G. Arvanitis, G. D. Pasgianos

Department of Natural Resources Management and Agricultural Engineering

Agricultural University of Athens, 75 Iera Odos Str., 11855, Athens, Greece

A. K. Boglou

Technology Education Institute of Kavala, 65404, Agios Loukas, Kavala, Greece

N. K. Bekiaris-Liberis

Department of Mechanical and Aerospace Engineering, University of California

San Diego, La Jolla, CA 92093-0411, U.S.A.

Keywords: Process Control, PID Control, Dead-Time Processes, Process Zeros.

Abstract: The use of the Pseudo-Derivative Feedback (PDF) structure in the control of stable or unstable dead-time

processes with a negative or a positive zero is investigated. A simple direct synthesis method for tuning the

PDF controller is presented. Moreover, a modification of the proposed method, which ensures its

applicability in the case of large overshoot response processes with dead time, is also presented. The PDF

control structure and the proposed tuning method ensure smooth closed-loop response to set-point changes,

fast regulatory control and sufficient robustness against parametric uncertainty. Simulation results show

that, in most cases, the proposed method is as efficient as the best of the most recent PID controller tuning

methods for dead-time processes with negative/positive zeros, while its simplicity in deriving the controller

settings is a plus point over existing PID controller tuning formulae.

1 INTRODUCTION

In the process industry, stable second order dead-

time models as well as second order dead-time

models with one right-half-plane pole are frequently

used to adequately describe numerous processes for

the purpose of designing controllers. However, these

types of process models are inadequate in the case

where a process controlled variable encounters two

(or more) competing dynamic effects with different

time constants from the same manipulated variable

(Waller and Nygardas, 1975). This composite

dynamics results to a process behaviour that exhibits

an inverse response or a large overshoot response.

Inverse response or large overshoot response is port-

rayed by the presence of one (or an odd number of)

positive or negative zeros, respectively, in the

process models, and they can cause, together with

the process dead-times, serious problems in

designing and tuning simple controllers for the pro-

cess under consideration.

Inverse response second order dead-time process

models (SODT-IR) are used to represent the

dynamics of several chemical processes (like level

control loops in distillation columns and temperature

control loops in chemical reactors), as well as the

dynamics of PWM based DC-DC boost converters

in industrial electronics. In the extant literature, there

is a number of studies regarding the design and

tuning of three-term controllers for SOPD-IR

processes (Waller and Nygardas, 1975; Scali and

Rachid, 1998; Zhang et al, 2000; Luyben, 2000;

Chien et al, 2003; Padma Sree and Chidambaram,

2004; Chen et al, 2005; Chen et al, 2006). In

particular, Waller and Nygardas (1975) presented an

74

G. Arvanitis K., D. Pasgianos G., K. Boglou A. and K. Bekiaris-Liberis N.

A NEW METHOD OF TUNING THREE TERM CONTROLLERS FOR DEAD-TIME PROCESSES WITH A NEGATIVE/POSITIVE ZERO.

DOI: 10.5220/0002170500740083

In Proceedings of the 6th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2009), page

ISBN: 978-989-674-001-6

Copyright

c

2009 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

empirical tuning of PID controllers based on the

Ziegler-Nichols method for SOPDT-IR processes. In

Scali and Rachid (1998) and Zhang et al (2000),

analytical design methods based on the Internal

Model Control framework and the H

∞

control

theory, have been proposed for inverse response

processes without time delay. In Luyben (2000), an

empirical method that gives large overshoot and

oscillatory response has been proposed to design PI

controllers for SODT-IR processes. In Chien et al

(2003), a direct synthesis tuning method is presented

to tune PID controllers for both under-damped and

over-damped SODT-IR processes. In Chen et al

(2005), an analytical PID controller design for SOD-

IR processes is derived based on conventional unity

feedback control. In Chen et al (2006), an analytical

design scheme based on IMC theory has been

proposed to control SODT-IR processes. Finally, in

Padma Sree and Chidambaram (2004), a method of

tuning set-point weighted PID controllers for

unstable SODT processes with a positive or a

negative zero is presented. This method is based on

appropriately equating coefficients of like powers of

s in the numerator and the denominator of the

closed-loop transfer function.

In contrast, controller tuning for large overshoot

response dead-time processes have received less

attention in the past, although they used to model

several physical phenomena, like blending

processes, mixing processes in distillation columns

and temperature of heat exchangers (see Chien et al

(2003), for details). In Chang et al (1997) a tuning

method of controllers in first order lead-lag form has

been proposed for such processes. Furthermore in

Chien et al (2003), a direct synthesis tuning method

is presented in order to tune PID controllers for both

under-damped and over-damped large overshoot

response processes.

The present paper investigates some aspects of

the controller configuration proposed by Phelan

(1978), and called the “pseudo-derivative feedback

controller” (PDF), which is put forward here as an

alternative means of tuning three-term controllers

for stable or unstable dead time processes with a

negative or positive zero. The aim of the paper is to

propose a set of tuning rules for the PDF controller

when it is applied to such processes. The proposed

method is a direct synthesis tuning method and it is

based on the manipulation of the closed loop transfer

function through appropriate approximations of the

dead-time term in the denominator of the closed

loop transfer function as well as appropriate

selection of the derivative gain, in order to obtain a

second order dead-time closed-loop system. On the

basis of this method the settings of the PDF

controller are obtained in terms of two adjustable

parameters, one of which can further be

appropriately selected in order to achieve a desired

damping ratio for the closed-loop system, while the

other is free to designer and can be selected in order

to enhance the obtained regulatory control

performance. Moreover, an appropriate modification

of the proposed method, that makes it applicable in

the case of large overshoot response processes with

dead time, is also presented. For assessment of the

effectiveness of the proposed tuning method and in

order to provide a comparison with existing tuning

methods, a series of simulation examples are

presented. Simulation results verify that the PDF

control structure and the proposed direct synthesis

tuning method ensure smooth closed-loop response

to set-point changes, fast regulatory control and

sufficient robustness in case of model mismatch.

2 THE PSEUDO-DERIVATIVE

FEEDBACK CONTROLLER

The Pseudo-Derivative Feedback (PDF) controller

has first been proposed by Phelan (1978), and its

general feedback configuration is shown in Figure 1.

The transfer function

CL

G(s)of the closed loop

system is given by

()

IP

CL

n2

D,n 1 D,1 D,0 I P

KG (s)

G(s)

s K s ... K s K s K G (s)

−

=

+++++

(1)

The PDF controller is essentially a variation of

the conventional PID controller. In contrast to the

PID controller, the PDF controller does not

contribute to closed-loop zeros, and hence it is

expected that it will not render worst the overshoot

of the closed-loop response. The two configurations

differ in the way they react to set-point changes (as

it can be easily checked, they are equivalent for load

or disturbance changes). The PID controller often

has an abrupt response to a step change because the

step is amplified and transmitted directly to the

feedback control element and downstream blocks.

This can induce a significant overshoot in the

response that is unrelated to the closed loop system

damping. For this reason, it is a common practice to

ramp or filter the set-point. The PDF structure

avoids this because naturally ramps the controller

effort, since it internalizes the pre-filter that one

would apply to cancel any closed-loop zeros

introduced in the PI/PID control configuration.

A NEW METHOD OF TUNING THREE TERM CONTROLLERS FOR DEAD-TIME PROCESSES WITH A

NEGATIVE/POSITIVE ZERO

75

R(s)

E(s)

+

_

+

+

Y(s)

L(s)

PDF control structure

_

G

P

(s)

U(s)

n1 1

D,n 1 D,1 D,0

Ks ...KsK

−

−

++ +

I

K

s

Figure 1: The general PDF control structure.

In the present paper, we focus our attention on

the specific form of the general PDF control

structure which contains proportional as well as a

single derivative action in the feedback path (i.e.

K

D,i

=0 for i=2,…,n-1 and K

P

≠0, K

D,1

≠0). We call

this feedback scheme, the PD-1F control structure,

in contradistinction with the PDF controller without

derivative action (i.e. the controller with

D,1

K0

=

),

which is designated as the PD-0F controller. We

shall next analyze its performance, in the case where

the system under control is a second order process

with both dead-time and a minimum or a non-

minimum phase zero, which can be described by the

following general transfer function model

() ( )( )

P12

G (s) K ps 1 exp( ds) / s 1 s q=+ −τ+τ+

⎡⎤

⎣⎦

(2)

where

z

z

in the case of a positive zero

p

in the case of a negative zero

−τ

⎧

=

⎨

τ

⎩

, τ

z

>0

and q=1, in the case of a stable process or q=-1, in

the case of an unstable process, and where, K, d, τ

z

,

τ

1

and τ

2

, are the process gain, the dead-time, the

zero’s time constant and the process time constants,

respectively.

To this end, observe that, equation (1), in the

case of a PD-1F controller and for process models of

the form (2), takes the form

()( )

()

()

()

()

CL

I1

2

2dPI

1

G (s)

KK ps 1 / s 1 exp( ds)

ps 1

ssq KKsKKsKK exp(ds)

s1

=

+τ+ −

⎡⎤

⎣⎦

+

τ+ + + + −

τ+

(3)

Relation (3) will be the starting point for the

development of the tuning method that will be

presented in the sequel.

3 A SIMPLE TUNING METHOD

In order to systematically present the proposed

tuning method, observe that by making use of the

approximation

(

)

(

)

11

p

s1/ s1 1( p)s+τ+≈−τ− (4)

in the numerator of (3), and observing that

()

exp 1 ds

exp( ds)

exp( ds)

−−α

⎡

⎤

⎣

⎦

−=

α

for some α

∈ℜ, we obtain

()

() ()

()

CL

1

2

d

2P

IIII 1

G (s)

1psexp(ds)

p

s1exp 1 ds

K

K

q

ss s s1

KK KK K K s 1 exp( ds)

=

−τ− −

⎡⎤

⎣⎦

+−−α⎡⎤

⎛⎞⎛ ⎞

τ

⎣⎦

++ ++

⎜⎟⎜ ⎟

τ+ α

⎝⎠⎝ ⎠

(5)

Next, using the approximations,

(

)

[

]

[

]

p

s1exp (1 )ds p(1 )ds1

+

−−α = −−α +

(

)

(

)

11

s 1 exp( ds) d s 1

τ

+α=τ+α+

in (5), we obtain

()

()

()

CL

1

2

d2P

IIII 1

G (s)

1 p s exp( ds)

p1 ds1

K

K

q

ss s s1

KK KK K K d s 1

=

−τ− −

⎡⎤

⎣⎦

⎡⎤

−−α +⎡⎤

⎛⎞⎛ ⎞

τ

⎣⎦

++ ++

⎢⎥

⎜⎟⎜ ⎟

τ+α +

⎢⎥⎝⎠⎝ ⎠

⎣⎦

(6)

Relation (6) may further be written as

(

)

1

CL

2

II

1psexp(ds)

G(s)

q

ss P(s)

KK KK

−τ− −

⎡⎤

⎣⎦

=

⎛⎞

τ

++

⎜⎟

⎝⎠

(7)

where

() ()

()

()

dd

P

2

1I1I

1I

KK

K

P(s) s Q(s)

dK dK

dK

p 1 d s 1

⎡

⎤

⎛⎞

⎢

⎥

⎜⎟

=+−+

⎜⎟

τ+α τ+α

⎢

⎥

τ+α

⎝⎠

⎣

⎦

⎡⎤

×−−α +

⎡⎤

⎣⎦

⎣⎦

()

()

()

d

P

2

1I

1I

1

K

K

1

dK

dK

Q(s)

ds 1

−+

τ+α

τ+α

=

τ+α +

Observe now that by selecting

ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics

76

()()

2

d1 P1 I

KdKdK=τ+α −τ+α

(8)

we obtain Q(s)=0 and

()

p

1

I

K

P(s) d s 1 p 1 d s 1

K

⎡⎤

⎛⎞

⎡⎤

=−τ−α+−−α+⎡⎤

⎢⎥

⎜⎟

⎣⎦

⎣⎦

⎢⎥

⎝⎠

⎣⎦

Therefore, relation (7) yields

()

()

CL

1

p

2

1

II I

G (s)

1psexp(ds)

K

q

ss ds1p1ds1

KK KK K

=

−τ− −⎡⎤

⎣⎦

⎡⎤

⎛⎞⎛⎞

τ

⎡⎤

+ + −τ −α + − −α +

⎡⎤

⎢⎥

⎜⎟⎜⎟

⎣⎦

⎣⎦

⎝⎠⎝⎠

⎣⎦

(9)

which can further be written in the form

()

1

CL

22

1psexp(ds)

G(s)

s2s1

−τ− −⎡⎤

⎣⎦

=

λ+ζλ+

(10)

[]

p

2

1

II

K

d(1 )dp

KK K

⎛⎞

τ

λ= − −τ −α −α −

⎜⎟

⎝⎠

(11)

[]

p

1

II

p

2

1

II

K

q

dp

KKK

K

2d(1)dp

KK K

−−τ++

ζ=

⎛⎞

τ

−−τ−α−α−

⎜⎟

⎝⎠

(12)

Τhe Routh stability criterion about (10) yields

()

P1I

q

Kd pK

K

>+τ− −

(13)

and

()

[]

2

P1 I

KdK

K(1 )d p

τ

<τ+α +

−α −

(14)

Therefore, as for K

P

one can choose the middle

value of the range given by inequalities (13) and

(14). That is

()

[

]

[]

2

1I

P

q(1 )d p

21dpK

K(1 )d p

K

2

τ− −α −

τ+ +α − +⎡⎤

⎣⎦

−α −

=

(15)

Then, from (15), we obtain

()

[

]

[]

2

1

I

P

I

q(1 )d p

21dp

KK (1 )d p

K

K2

τ− −α −

τ+ +α − +

⎡⎤

⎣⎦

−α −

β= =

(16)

which yields,

[

]

[]

()

2

I

1

q(1 )d p

K

K(1 )d p 2 2 1 d p

τ− −α −

=

−α − β− τ − +α +

⎡

⎤

⎣

⎦

(17)

Therefore,

[

]

[]

()

2

P

1

q(1 )d p

K

K(1 )d p 2 2 1 d p

⎡⎤βτ− −α −

⎣⎦

=

−

α− β−τ−+α+

⎡

⎤

⎣

⎦

(18)

[

]

[]

()

2

11 2

d

1

(d)(d) q(1)dp

K

K(1 )d p 2 2 1 d p

⎡⎤

⎡⎤

β τ +α − τ +α τ − −α −

⎣⎦

⎣⎦

=

−α − β− τ − +α +

⎡

⎤

⎣

⎦

(19)

Clearly, relations (17)-(19) provide the settings

of the desired PD-1F controller as functions of two

adjustable parameters α and β, which must be

selected in order to guarantee positive controller

settings (in the case where the process parameters

take positive values), as well as to fulfil inequalities

(13) and (14). For a pre-specified value of α∈ℜ,

parameter β can be selected in order to assign a

specific damping ratio ζ

des

of the closed-loop system.

Indeed, using relations (12), (17) and the definition

of β, and after some trivial algebra, one can resort

the following quadratic equation with regard to β,

2

210

A0

β

+Α

β

+Α =

(20)

()

()

()

()

()

[]

()

2

1des

2

2

1

1

2

2

A4 11 dp

Tq1 dp

2q 1 d p

2 1

Tq1 dp

q1 dp2 (1 )dp

d q

Tq1 dp

⎡⎤

=ζ − −α −⎡⎤

⎢⎥

⎣⎦

−−α−⎡⎤

⎢⎥

⎣⎦

⎣⎦

⎡⎤

−α −⎡⎤

⎣⎦

−+

⎢⎥

−−α−⎡⎤

⎢⎥

⎣⎦

⎣⎦

⎡

⎤

−α − τ + +α −⎡⎤

⎣⎦

×+τ−+

⎢

⎥

−−α−⎡⎤

⎢

⎥

⎣⎦

⎣

⎦

(21)

()

()

2

2

2

2q 1 d p

A1

Tq1 dp

⎡

⎤

−α −⎡⎤

⎣⎦

=+

⎢

⎥

−−α−

⎡

⎤

⎢

⎥

⎣

⎦

⎣

⎦

(22)

()

[]

()

[]

()

()

2

1

01

2

21

2

des 1

2

q1 dp2 (1 )dp

Ad q

q1 dp

2(1)dp

4 d 1 d p

q1 dp

⎡⎤

−α − τ + +α −

⎡⎤

⎣⎦

=+τ−+

⎢⎥

τ− −α −

⎡⎤

⎢⎥

⎣⎦

⎣⎦

⎡⎤

ττ++α−

−ζ

−τ −α −α −

⎡⎤

⎢⎥

⎣⎦

τ− −α −

⎡⎤

⎢⎥

⎣⎦

⎣⎦

(23)

Then, β is chosen as the maximum real root of (20)

Clearly, the method presented above is

applicable when p=τ

Ζ

or – τ

Ζ

and q=1 or -1.

However, extensive simulations show that, in the

case where τ

Ζ

>>0 (i.e. in the case of large overshoot

A NEW METHOD OF TUNING THREE TERM CONTROLLERS FOR DEAD-TIME PROCESSES WITH A

NEGATIVE/POSITIVE ZERO

77

processes), the method provides controller settings

that renders the closed-loop unstable or marginally

stable. This is due to the swings of the controller

R(s)

E(s)

+

_

+

+

Y(s)

L(s)

PD-1F control structure

_

G

P

(s)

U(s)

dP

Ks K+

I

K

s

G

F

(s)

Figure 2: PD-1F control structure in the case of large

overshoot response processes.

output induced by the excessive derivative action.

One way to avoid this problem is to filter the

controller output using a first order filter of the form

(see Figure 2).

(

)

FF

G(s) 1/ s 1=τ+ (24)

while calculating the PD-1F controller settings as

suggested by relations (17)-(19). The time constant

of the filter can be selected as τ

F

= τ

Z

.

Alternatively, one can select the controller

settings according to the following method, which is

a modification of the method resulting in the settings

given by relations (17)-(19): In the case where the

filter of the form (24) is introduced in the control

loop, relation (3) is modified as

()()

()

()

()()

CL

I

F1

2

2dPI

F1

G (s)

ps 1

KK exp( ds)

s1 s1

ps 1

ssq KKsKKsKK exp(ds)

s1 s1

=

⎡⎤

+

−

⎢⎥

τ+ τ+

⎣⎦

⎡⎤

+

τ+ + + + −

⎢⎥

τ+ τ+

⎣⎦

Then, making use of the approximations

(

)

(

)

(

)

F1 1

p

s1/ s1 s1 1( p)s+τ+τ+≈−τ−

()

exp 1 ds

exp( ds)

exp( ds)

−−α⎡⎤

⎣⎦

−=

α

[][]

F

ps 1

exp(1)ds p(1)ds1

s1

⎛⎞

+

−−α = − −α +

⎜⎟

τ+

⎝⎠

(

)

(

)

11

s 1 exp( ds) d s 1τ+ α =τ+α +

where,

F

pp=−τ

, we finally obtain

()

()

CL

1

p

2

1

II I

G (s)

1 p s exp( ds)

K

q

ss ds1p1ds1

KK KK K

=

−τ− −⎡⎤

⎣⎦

⎡⎤

⎛⎞⎛⎞

τ

⎡⎤

+

+−τ−α+−−α+⎡⎤

⎢⎥

⎜⎟

⎜⎟

⎣⎦

⎣⎦

⎝⎠⎝⎠

⎣⎦

(25)

It is now obvious that relation (25) is similar to

relation (9) when p is replaced by

F

pp=−τ.

Therefore, following an argument similar to that

used above to produce relations (17)-(19), we may

easily conclude that, in the present case

[

]

[]

()

2

I

1

q(1 )d p

K

K(1 )d p 2 2 1 d p

τ− −α −

=

−α − β− τ − +α +

⎡

⎤

⎣

⎦

[

]

[]

()

2

P

1

q(1 )d p

K

K(1 )d p 2 2 1 d p

⎡⎤

βτ− −α −

⎣⎦

=

−α − β− τ − +α +

⎡

⎤

⎣

⎦

[

]

[]

()

2

11 2

d

1

(d)(d) q(1)dp

K

K(1 )d p 2 2 1 d p

⎡⎤

⎡⎤βτ+α − τ+α τ − −α −

⎣⎦

⎣⎦

=

−α − β− τ − +α +

⎡

⎤

⎣

⎦

Now, it only remains to select the filter time

constant. A suitable choice of τ

F

, is τ

F

=τ

Z

. With, this

selection, the PD-1F controller settings in the case of

large overshoot processes are obtained as suggested

by the relations

[

]

[]

()

2

I

1

q(1 )d

K

K(1 )d 2 2 1 d

τ− −α

=

−α β− τ − +α

⎡

⎤

⎣

⎦

(26)

[

]

[]

()

2

P

1

q(1 )d

K

K(1 )d 2 2 1 d

⎡⎤

βτ− −α

⎣⎦

=

−α β− τ − +α

⎡

⎤

⎣

⎦

(27)

[

]

[]

()

2

11 2

d

1

(d)(d) q(1)d

K

K(1 )d 2 2 1 d

⎡⎤

⎡⎤β τ +α − τ +α τ − −α

⎣⎦

⎣⎦

=

−α β− τ − +α⎡⎤

⎣⎦

(28)

4 SIMULATION RESULTS

For assessment of the effectiveness of the proposed

tuning methods and in order to provide a comparison

with existing tuning methods, a series of simulation

examples are carried out for different dead-time

processes.

4.1 Inverse Response Processes with

Two Stable Poles

Consider the typical inverse response process with

K=1, τ

1

=1, τ

2

=1, d=0.8, p=-0.5, q=1. Applying the

proposed method with α=0.6 and ξ

des

= 0.8225, yields

β=2.15. The PD-1F controller settings are then

obtained as K

I

=0.4221, K

P

=0.9076 and K

d

=0.4186.

The settings of the series form PID controller with

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78

filtered derivative, tuned according the method

proposed by Chien

et al (2003), are K

C

=0.3367,

τ

Ι

=1, τ

D

=1, while the low-pass filter parameter takes

the value a=0.1 and the inverse of the cyclic

frequency of the desired critically damped closed-

loop system takes the value τ

cl

= 0.8348. The settings

of the conventional PID controller that is tuned

according to the method reported in Chen

et al

(2006), are K

C

=0.71, τ

Ι

=2, τ

D

=0.5. Figure 3

illustrates the comparison of the servo-responses as

well as of the regulatory control responses obtained

by the proposed method and by the methods

reported in Chien

et al (2003) and Chen et al (2006),

in the case of nominal process parameters. In case of

set-point tracking, the proposed method provides a

0 20 40 60 80

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time (second)

Closed-Loop Response

Figure 3: Servo-responses and regulatory control

responses for the system G

P

(s)=(-0.5s+1)exp(-0.8s)/(s+1),

in case of nominal process parameters. Black line:

Proposed Method. Orange line: Method in Chien et al

(2003). Blue line: Method in Chen et al (2006).

0 20 40 60 80

0

0.5

1

1.5

2

Time (second)

Closed-Loop Response

Figure 4: Servo-responses and regulatory control

responses for the system G

P

(s)=(-0.5s+1)exp(-0.8s)/(s+1),

in case of a +20% mismatch in all process parameters.

Other legend as in Figure 3.

slightly more sluggish response as compared to the

abovementioned PID tuning methods, while the

initial jump obtained by our method is smaller. In

the case of regulatory control, our method gives a

better response in terms of maximum error, while

the settling time is comparable to that obtained by

the methods in Chien

et al (2003) and Chen et al

(2006).

A comparison in terms of the ISE criterion, in the

case of regulatory control, gives the values 1.2002

for the proposed method, while for the methods in

Chien

et al (2003) and Chen et al (2006), we obtain

ISE=1.5782 and ISE=1.4425, respectively. The

respective IAE values for the methods under

comparison are obtained as 2.443, 3.058 and 2.8941.

Figure 4 shows the comparisons of the servo-

responses and of the regulatory control responses in

the case where a simultaneous +20% uncertainty in

all process parameters is assumed. The responses

obtained by the proposed method are better in terms

of overshoot, maximum error and initial jump, while

the settling time is similar to that of the responses

obtained by the PID controllers tuned according to

the methods by Chien

et al (2003) and Chen et al

(2006). The ISE values, in case of regulatory

control, are 1.9514, for the proposed method, 2.3765

for the method of Chien

et al (2003) and 2.1673, for

the method of Chen

et al (2006). The respective IAE

values are 3.8221, 4.2492 and 3.9183.

As already mentioned, for a pre-specified value

of adjustable parameter α, parameter β is directly

related to the damping ration ζ of the second order

approximation (10) of the closed-loop system. In

0 20 40 60 80

0

0.5

1

1.5

2

Time (second)

Closed-Loop Response

Figure 5: Servo-responses and regulatory control

responses for the system G

P

(s)=(-0.5s+1)exp(-0.8s)/(s+1),

in case of nominal process parameters, for α=0.6 and for

three values of β. Orange line: β=2.05; Black line:

β=2.15;. Blue line: β=2.25.

A NEW METHOD OF TUNING THREE TERM CONTROLLERS FOR DEAD-TIME PROCESSES WITH A

NEGATIVE/POSITIVE ZERO

79

0 20 40 60 80 100

0

0.5

1

1.5

2

Time (second)

Closed-Loop Response

Figure 6: Servo-responses and regulatory control

responses for the system G

P

(s)=(-0.5s+1)exp(-0.8s)/(s+1),

in case of nominal process parameters, for β=2 and for

three values of α. Orange line: α=0.45; Black line: α=0.5;.

Blue line: α=0.55.

particular, as shown in Figure 5, β increases when ζ

is increased. This of course results to a more

conservative PD-1F controller. Therefore, a greater

value of β, renders the closed-loop system more

robust. Parameter α has an inverse effect on the

closed-loop system robustness: For a pre-specified

value of the parameter β, an increase of the

parameter α, leads to a less robust but faster closed-

loop system, as illustrated in Figure 6.

4.2 Control of a Continuous Stirred

Tank Reactor

Let us consider the transfer function model of a

CSTR reported in Padma Sree and Chidambaram

(2004), and having the form

P

2

2.07(0.1507s 1)

G(s) exp(0.3s)

2.85s 2.31s 1

2.07(0.1507s 1)

exp( 0.3s)

(0.8905s 1)(3.2005s 1)

−+

=−

+−

−+

=−

+−

The process has one dominant unstable pole and

one stable pole, at s=0.3125 and s= -1.123,

respectively, as well as a stable zero -6.6357. Here,

K=-2.07, τ

1

=0.8905, τ

2

= 3.2005, d=0.3, p=0.1507,

q=-1. Application of the proposed method with α=-

0.5, β=2.5, yields the PD-1F controller settings K

P

=-

4.3862, K

I

=-1.7545, K

d

=-2.2859. The settings of the

set-point weighted PID controller tuned according to

the method reported in Padma Sree and

Chidambaram (2004) are, K

C

=-0.7205, τ

I

=39.7228,

τ

D

=0.1494, while the tuning parameter used in the

above mentioned paper, as well as the set-point

weight b, take the values 0.15 and 0.3275,

respectively. Figure 7 illustrates the servo-responses

obtained by the two controllers. Figure 8 shows the

comparison of the regulatory control responses for a

negative unit step load change. Obviously, the PD-

1F controller tuned according to the proposed

method provides a considerably better performance,

particularly in the case of regulatory control, where

the response obtained by the controller tuned

according to the method in Padma Sree and

Chidambaram (2004) is practically unacceptable.

0 10 20 30 40 50

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (second)

Closed-Loop Response

Figure 7: Closed-loop servo-responses of the CSTR

model. Black line: Proposed method. Blue line: Set-point

weighted PID controller tuned according to the method

proposed by Padma Sree and Chidambaram (2004).

0 10 20 30 40 50

0

0.5

1

1.5

2

2.5

3

3.5

Time (second)

Closed-Loop Response

Figure 8: Regulatory control responses of the CSTR.

Other legend as in Figure 7.

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80

4.3 Second Order Unstable Process

with a Positive Zero

Consider the process with K=1, τ

1

=2.07, τ

2

=5,

d=0.939, p=-1, q=-1. The process has a stable pole,

an unstable pole and a strong non-minimum phase

zero. To the authors’ best knowledge, controller

design for second order processes with one or two

righ-half-plane poles and a right-half-plane zero has

not yet been addressed in the literature. Application

of the proposed method, with α=0.3 and β=25, yields

the PD-1F controller settings K

I

=0.0920, K

P

=2.3012,

K

d

=4.9027. The process model is next approximated

as

(

)

(

)

P

G (s) exp( 1.939s) / 2.07s 1 5s 1=− + −

⎡

⎤

⎣

⎦

, i.e.

the negative numerator time constant has been

approximated as a time delay term of the form exp(-

s). This is reasonable since an inverse response has a

deteriorating effect on control similar to that of a

time delay. We next apply the method reported in

Lee

et al (2000), in order to design a PID controller

with first order set-point filter for the given process,

on the basis of the approximated model. Application

of the method reported in Lee

et al (2000), with the

IMC parameter λ=6.25, yields the PID controller

settings K

C

=1.9570, τ

Ι

=34.9614 and τ

D

=2.4889.

Figure 9 illustrates the comparison of the servo-

responses and the regulatory control responses for a

unit step set-point change at t=0 sec and an inverse

unit step load change at t=75 sec. It is seen that the

proposed method results in an improved load

disturbance response as compared to the method in

Lee

et al (2000), while the set-point responses are

similar, with comparable settling times.

0 50 100 150

0

0.2

0.4

0.6

0.8

1

1.2

Time (second)

Closed-Loop Response

Figure 9: Servo-responses and regulatory control

responses for the system G(s)=(-s+1)exp(-0.939s) /

[(2.07s+1)(5s-1)]. Black line: Proposed method; Blue line:

Method in Lee et al (2000).

4.4 Stable Second Order Unstable

Process with a Positive Zero

Consider the process model of the form (2), with

K=1, τ

1

=2, τ

2

=1, d=1, p=0.3, q=1. Application of the

proposed method with α=0.4, β=3, yields Κ

Ι

=1.0358,

K

P

=3.1073, K

d

=1.6340. The settings of the series

form PID controller with filtered derivative, tuned

according the method proposed by Chien

et al

(2003) are K

C

=1.0355, τ

Ι

=2, τ

D

=1, while the low-

pass filter parameter takes the value τ

F

=0.3 and the

inverse of the cyclic frequency of the desired

critically damped closed-loop system takes the value

τ

cl

= d/ 2 = 0.5457. Figure 10 illustrates the

comparison of the servo-responses as well as of the

regulatory control responses obtained by the

proposed method and by the method reported in

Chien

et al (2003). In the regulatory control case our

method gives a considerably better response,

whereas, although our method provides a smooth

response, the method in Chien

et al (2003) is better

in the case of set-point tracking.

Let us now consider the case of a large overshoot

process with K=1, τ

1

=2, τ

2

=1, d=1.2, p=5, q=1.

Evaluating relations (17)-(19), while assuming

α=0.2, β=3, yields the PD-1F controller settings

K

I

=0.2208, K

P

=0.6624, K

d

= 0.3759. Application of

the above controller yields an unacceptable

oscillatory response, as shown in Figure 11. Let us

try, another design by evaluating relations (17)-(19)

in the case where we select a=0.6, β=3. This yields

K

I

=0.1951, K

P

=0.5853, K

d

= 0.1486, i.e. a more

conservative controller. The obtained servo-response

0 20 40 60 80

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time (second)

Closed-Loop Response

Figure 10: Servo-responses and regulatory control

responses for the system G

P

(s)=(0.3s+1)exp(-0.8s)

/(2s+1)(s+1). Black line: Proposed method. Blue line:

Method in Chien et al (2003).

A NEW METHOD OF TUNING THREE TERM CONTROLLERS FOR DEAD-TIME PROCESSES WITH A

NEGATIVE/POSITIVE ZERO

81

0 20 40 60 80 100

0

0.5

1

1.5

2

2.5

Time (second)

Closed-Loop Response

Figure 11: Closed-loop servo-response and regulatory

control response of the system G(s)=(5s+1)exp(-1.2s)

/(2s+1)(s+1), in the case of he PD-1F controller with

parameters K

I

=0.2208, K

P

=0.6624, K

d

= 0.3759.

0 20 40 60 80 100

0

0.5

1

1.5

2

2.5

Time (second)

Closed-Loop Response

Figure 12: Closed-loop servo-response and regulatory

control response of the system G(s)=(5s+1)exp(-1.2s)

/(2s+1)(s+1), in the case of he PD-1F controller with

parameters K

I

=0.1951, K

P

=0.5853, K

d

= 0.1486.

and regulatory control responses are given in Figure

12. In the later case, the servo-response is quite

smooth while the regulatory control response is less

oscillatory. However, the robustness of the closed-

loop system is marginal, and a small parameter

mismatch can readily lead to instability.

Let us now consider filtering the output of the PD-

1F controller that is designed for the case where

α=0.2, β=3, with settings K

I

=0.2208, K

P

=0.6624,

K

d

= 0.3759, by a filter of the form (24), where τ

F

=5.

Moreover, let us design a PD-1F controller with

filtered output as suggested by relations (26)-(28),

with α=-0.2, β=2, τ

F

=5. In this case the controller

0 20 40 60 80 100

0

0.5

1

1.5

2

2.5

Time (second)

Closed-Loop Response

Figure 13: Closed-loop servo-response and regulatory

control responses of the system G(s)=(5s+1)exp(-1.2s)

/(2s+1)(s+1). Black line: PD-1F controller with filtered

output tuned according to relations (17)-(19); Orange line:

PD-1F controller with filtered output tuned according to

relations (26)-(28); Blue line: Series form PID controller

with filtered derivative tuned according to the method in

Chien et al (2003).

settings are K

I

=0.3183, K

P

=0.6366, K

d

= 0.1344.

Figure 13 shows the obtained servo-responses and

regulatory control responses for both designs,

together with the respective responses obtained by a

series PID controller with filtered derivative,

designed according the method reported in Chien

et

al (2003). It is seen that, in the regulatory control

case our method gives a considerably better

response, whereas, although our methods provide

smooth responses, the method in Chien

et al (2003)

is better in the case of set-point tracking. A

comparison in terms of ISE in the case of regulatory

control gives the ISE values 1.8326 and 1.5892, for

the proposed methods and 4.5754 for the method in

Chien

et al (2003). The respective IAE values are

4.5287, 3.7058 and 4.9453.

5 CONCLUSIONS

A new direct synthesis method of tuning the PDF

controller for stable or unstable dead-time processes

with a negative or a positive zero has been

presented. The proposed tuning method ensures

smooth closed-loop response to set-point changes,

fast regulatory control and sufficient robustness

against parametric uncertainty. Numerical

simulation examples verify the advantages of the

proposed method over known PID controller tuning

methods for the classes of dead-time processes under

ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics

82

study. Extension of the proposed tuning method in

the case of frequency domain specifications of the

closed-loop system in terms of gain and phase

margins is currently under investigation.

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IEEE Conference on Industrial Technology (ICIT

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Chen, P.-Y., Zhang, W.-D., Zhu, L.-Y., 2006. Design and

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A NEW METHOD OF TUNING THREE TERM CONTROLLERS FOR DEAD-TIME PROCESSES WITH A

NEGATIVE/POSITIVE ZERO

83