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A Novel Method of PID Tuning for Integrating Processes

Jianghua Xu, Huihe Shao

Institute of Automation, Shanghai Jiao Tong University, Shanghai 200030, China

Abstract: PID control is widely used to control stable processes,

however, its application to integrating processes is less common.

In this paper, we proposed a new PID controller tuning method

for integrating processes with time delay to meet a new robust

specification. With the proposed PID tuning method, we can

obtain a loop transfer function with the real part close to –0.5.

This guarantees both robustness and performance. Simulation

examples are given to show the performance of the method.

1. Introduction

The proportional-integral-derivative (PID) controllers are still

widely used in the process industries even though more advanced

control techniques have been developed. The main reason is that

the PID controllers have simple structure and are robust to

modeling error and that many advanced control algorithms, such

as model predictive control, are based on the PID control. As

indicated in [1], more than 95% of the control loops are of PID

type in process control. Over the years, there are many formulas

derived to tune the PID controllers for stable processes, such as

Ziegler-Nichols, Cohen-Coon, internal model control, integral

absolute error optimum (ISE, IAE, and ITAE), and recently

proposed tuning methods[10][11]. However, it is difficult to control

integrating processes with time delay. Recently, many tuning

methods for integrating processes have been proposed[3]-[9],[12].

However, they usually either show poor closed-loop response

such as excessive overshoot and large settling time or have

complicated formulas.

For controller design purposes, many integrating processes

are often approximated by low-order plus time delay model,

which can be identified by P control method [12] or relay control

method[5]. Because the resulting models are usually imprecise

and the parameters of all physical systems vary with the working

condition and time, robustness is always a primary concern when

analyzing and designing the control system. In this paper, a new

tuning method for the PID controller with setpoint weighting[1] is

proposed for integrating processes to meet robustness

specification. The control scheme first presents the internal loop

design strategy. Then, simple and effective PID controller with

setpoint weighting is designed based on robustness specification.

Because of a good loop transfer with the real part closed to –0.5

in low frequency, the control system guarantees both robustness

and performance. Simulation examples show that the proposed

method achieves better control performance and robustness

compared with other methods.

2. PID Tuning method

For controller design purposes, many of the integrating

processes are adequately described by low-order plus dead-time

transfer function

) 1(

)(

+

=

−

Ts s

Ke

sG

sL

p

(1)

We first present an inner feedback loop for integrating

processes, the block diagram of the proposed method is shown in

Fig.1. Here, the PD controller (PD) in the inner feedback loop

plays an important role in changing the integrating process to the

stable process.

PID

G (s)

p

PD

r(s) e(s)u(s)

y(s)

Fig.1. Block diagram of a two-loop controller for integrating

process

Denote the PID controller transfer function by GPID(s) and is

given by

sK

s

K

KsG

d

i

pPID

++=

)(

(2)

Denote the PD controller transfer function by GPD(s) and is given

by

skksG

dp PD

+=

)(

(3)

Robustness is always of primary concern for process control

when the control systems are designed and analyzed because the

models used for the design of controllers are usually imprecise

and the parameters of all physical systems vary with the working

condition and time. We introduce a new robustness

Proceedings of the 42nd IEEE ?

Conference on Decision and Control ?

Maui, Hawaii USA, December 2003

TuA04-6

0-7803-7924-1/03/$17.00 ©2003 IEEE

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specification λ[11], which is defined as

)]()(Re[ max

0

ω

≤

1

λ

ω

j

ω

jGG

pc

∞

<

=

(4)

where the loop transfer function is Gl(s) = Gc(s)Gp(s), Gc(s) is

controller transfer function and the quantity λ is simply the

inverse of maximum of absolute real part of loop transfer

function Gl (jω), as shown in Fig.2. The new specification is

similar with the gain margin and phase margin specifications.

The following relations are obtained

Am > λ (5)

>

λ

φ

1

arccos

m

(6)

Fig.2. Nyquist curve of the loop transfer function.

It is obvious that the new specification satisfies both the gain

margin and phase margin requirement to a degree. Reasonable

values of λ are in the range from1.5 to 2.5.

The PD controller transfer function (2) is also written as

)()(

1

bsakskksG

dpPD

+=+=

(7)

We choose

a=1, b=T (9)

With the PD controller in the inner feedback loop, the internal

loop transfer function can be obtained as

Ls

pPDl

e

s

Kk

sG)sGsG

−

==

1

)(()(

(10)

where k is determined based on the new robustness

specification λ.

From equation (10), we can obtain

ω

ω

ω

j

L

KkGl

sin

)(Re

1

−=

(13)

To find the maximum, we note from (13) that

KLk

LL

Kk

L

Kk

11

0

1

0

1

cos

lim

→ω

sin

lim

→ω

=−=−

ω

ω

ω

(14)

Hence,

KLk1

1=

λ

(15)

Considering a internal loop transfer function with a good

shape, i.e. the real part close to –0.5 in low frequency, we choose

λ=2. From equation (15) and λ=2, we get

KL

k

2

1

1=

(16)

So the PD controller is

s

KL

T

KL

sGPD

22

1

)(

+=

(17)

The PD controller in equation (17) can guarantee both

robustness and performance of the inner feedback loop.

With the PD controller, the internal closed-loop transfer

function is given by

Ls

Ls

pPD

p

l

eTs(Kks Ts

Ke

sGG

sG

sG

−

−

+++

=

+

=

) 1

)(1

)(

)(

1

2

'

(18)

Considering Taylor series expansion, the time delay term in

the denominator of equation (18) can be approximated by

Lse

Ls

−≅

−

1

(19)

Then equation (18) is given by

KksKTkKLkTs)KLk

Ke

−

sGsG

sL

ml

111

2

1

'

)1 ( 1 (

)()(

+++−

=≅

−

(20)

Here, Gm(s) denotes the second-order plus time-delay model

obtained from the Taylor series expansion method.

Because the characteristic equation of Gm(s) should have

negative poles to be stable, the following condition must be

satisfied from the Routh-Hurwitz stability criterion:

KL

k

1

1<

(21)

The equation (16) satisfies the stability criterion (21).

For convenience of the outer loop controller design, the equation

(20) is rewritten as

1

11

2

1

'

11

)()(

ks

K

KTkKLk

Ts

K

KLk

e

−

sGsG

sL

ml

+

+

+

−

=≅

−

(22)

The PID controller transfer function (2) is also written as

++

=

s

CBsAs

ksGPID

2

)(

(23)

Where

k

K

A

d

=

,

k

K

B

p

=

and

k

K

C

i

=

. The controller

zeros are chosen to be equal to the poles of model Gm(s), that is

T

K

KLk

A

1

1−

=

,

,

1

11

K

KTkKLk

B

+−

=

and

1 kC =

. Hence

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s

ke

sG

)

sG

sL

mPID

−

=

)((

(24)

where k is determined based on new robustness specification λ.

From equation (24), we can obtain

ω

ω

ω

j

ω

j

)sin(

)()(Re

Lk

GG

mPID

−

=

(25)

To find the maximum, we note from (25) that

kL

L kLLk

=

−

=

−

→→

1

)cos(

lim

ω

)sin(

ω

lim

ω

00

ωω

(26)

Hence,

kL

=

λ

1

(27)

Considering a outer loop transfer function with a good shape,

i.e. the real part close to –0.5 in low frequency, we choose λ=2.

From equation (27) and λ=2, we get

L

k

5 . 0

=

(28)

Hence PID settings are given as

−

KL

+−

=

TKLk

L

k

KL

KTkKLk

K

K

K

d

i

p

)1 ( 5 . 0

5 . 0

)1 ( 5 . 0

1

1

11

(29)

From equation (16) and (29), we get

+

2

=

KL

T

KL

25. 0

KL

. 0

TL

K

K

K

d

i

p

25

2

)(25 . 0

(30)

Let’s reconsider the block diagram of Fig.1, where

)()()(sysrse

−=

. The process input u(s) can be written easily

as the following equation.

)()())()(()(syskksysrsK

s

K

Ksu

dpd

i

p

+−−

++=

(31)

Introducing

pp

p

kK

K

b

+

=

,

ppp

kKK

+=

'

,

ii

KK =

'

,

dd

d

kK

K

c

+

=

and

ddd

kKK

+=

'

, we have

()

))()(()()()()(

'

'

'

p

syscrsKse

s

K

sysbrKsu

d

i

−++−=

(32)

The expression (32) is the same as PID controller with setpoint

weighting[1], thus the block diagram of Fig.1 can be changed into

a PID control loop without an inner feedback loop, where K’

p, Ki,

Kd and setpoint weighting b, c are new PID settings. The new

PID setting for integrating processes are given by

2

'

)3 (

KL

25. 0TL

Kp

+

=

(33)

2

'

25 . 0

KL

Ki=

(34)

KL

T

Kd

75. 0

'=

(35)

LT

LT

b

3

+

+

=

(36)

3

1

=

c

(37)

3. Simulation examples

The following will give the comparisons between the proposed

PID tuning method and other method.

Example 1 Consider an integrating process with small dead time.

) 1

+

8 . 0 (s

)(

2 . 0

−

=

s

e

sG

s

p

The proposed method yields the PID control settings as

K’

p=8.75, K’

i=6.25, K’

d=3.0, b=0.714 and c=0.33. The control

performance of the proposed method is compared with

Astrom-Hagglund PID tuning method [2] for integrating

processes. The comparison of performance is shown in Fig.3

(proposed method, solid line; Astrom-Hagglund method, dashed

line), where a step load disturbance is added at t=20s. The

proposed methods shows better control performance for both

setpoint change and load disturbance.

Fig.3. Comparison of process responses

Example 2 Consider an integrating process with large dead time

) 1

+

(

)(

2

=

−

ss

e

sG

s

p

The proposed method yields the PID control settings as

K’

p=0.4375, K’

i=0.0625, K’

d=0.375, b=0.4286 and c=0.333. The

control performance of the proposed method is compared with

Tan’s PID tuning methods [8] for integrating processes. Fig.4

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shows the closed-loop process output response (proposed method,

solid line; Tan’s method, dashed line), where a step load

disturbance is added at t=100s. The proposed method shows the

better control performance for both the setpoint change and load

disturbance.

Fig.4. Comparison of process response

Because we introduced the Taylor series expansion method in

designing PID controllers, this produced model error especially

in high frequency. However, we consider desired robust

specification and a loop transfer function with the real part close

to –0.5 in low frequency[1], which guarantees both robustness and

performance.

4. Conclusions

In this paper, we proposed a novel PID tuning formulas for the

integrating processes with time delay. We adopted two-loop

design technique and obtained a tuning method for PID

controllers with setpoint weighting. The proposed tuning method

is very simple and shows better performance in controlling

integrating processes compared with other method. With the

proposed tuning method, we can obtain a loop transfer function

with the real part close to –0.5 in low frequency, which guarantee

both robustness and performance. Simulation results have been

given to show the performance of the method.

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Design, and Tuning, Instrument Society of America.

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regulators with specifications on phase and amplitude

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[3] Kwak, H.J., Sung, S.W. and Lee, I. On-line process

identification and autotuning for integrating processes,

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