Chapter 3

Dead Time Compensation – Smith Predictor, A Model Based Approach

1. Effects of Dead-Time (Time Delay)

Process with large dead time (relative to the time constant of the process) are difficult to control by pure feedback alone:

Effect of disturbances is not seen by controller for a while

Effect of control action is not seen at the output for a while. This causes controller to take additional compensation unnecessary

This results in a loop that has inherently built in limitations to control

Example – A Bode Plot Example

31)(

s

esG

s

h=tf(1,[1 3 3 1],'inputdelay',1);

Example- continued, case without time delay

31

1)(

s

sG

PID Control Results- Case with delay (Kc=1,Ki=1,Kd=1)

PID Control Results- Case without delay (Kc=1,Ki=1,Kd=1)

2. Causes of Dead-Time

Transportation lag (long pipelines) Sampling downstream of the process Slow measuring device: GC Large number of first-order time

constants in series (e.g. distillation column)

Sampling delays introduced by computer control

3. The Smith Predictor

SGc

Controller

＋

－

Process

SG stde Sysp Sy

Sy

SGc

Controller

＋

－

Process

SG stde Sysp Sy

Sy*

SGc

Controller

＋

－

Process

SG stde Sysp Sy

Sy SGe std )1( ＋

＋ Sy '

Sy*

Dead-time compensator

Controller mechanism

Control with Smith Predictor: (Kc=1,Ki=1,Kd=1)

Alternate form

SG

SG SGc＋

－

d

SG

stdespy

y＋

－ ＋－stde

d

Example 2: Long time delay

-10

-8

-6

-4

-2

0

Mag

nitu

de (

dB)

10-2

10-1

100

-360

-270

-180

-90

0

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

1G(s)=

s^3 + 3 s^2 + 3 s + 1exp(-3*s) * ---------------------

Mag=-3.3dB=20log10(AR);AR=0.6839;Wu=0.5;Pu=2π/0.5=12.5664Ku=1/0.6839=1.4622Kc=0.8601Taui=Pu/2=6.2832Taud=Pu/8=1.5708

Load Disturbance Response

0 5 10 15 20 25 30-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Gc(s) G(s)

(1-e-td’s)G’(s)

e-tds

ProcessController mechanism

)(sy sp )(sy

)(sy

+

-

+

-

)(sya

)(' sy

The Effect of Modeling Error

spstst

c

spcstst

c

yeGGeGGsy

or

yGGeGeGsy

dd

dd

'

'

'')(*

'1)(*

Errors in td’,G’ both cause the control system to degrade

Smith Predictor with Modeling Error

Model=e-s/(s3+s2+s+1)

Plant=e-s/(s3+3s2+3s+1)

Model=e-0.5s/(s3+s2+s+1)

Analytical Predictor

GC(s)GPe-DS

Analytical Predictor

ySP

y(t+D)

y(s)

Analytical Predictor is a model that projects what y will beD units into the future (on-line model identification)

y(t)

time

Step response to manipulate variable

Processes with inverse responseThe output goes to the wrong way first

Example:

This can occurs in (1) reboiler level response to change in heat input to the reboiler. (2) Some tubular reactor exist temperatureTo inlet flow rate .Inverse response is caused by a RHP zero

11)(

2

2

1

1

s

K

s

Ksy

Example: Inverse response of concentration and temperature to a chanbe in process flow of 0.15 ft3/min

Gc(s)

Process with inverse response

)(sy sp )(sy

)(sy

+

-

11

1

s

k

Controller

12

2

sk

+

-

Gc(s)

Process with inverse response

Controller mechanism

)(sy sp )(sy

)(sy

+

-

+

-

)(sya

)(' sy)

1

1

1

1(

12

ssk

11

1

s

k

Controller

12

2

sk

+

-

Open Loop Response Analysis

sy

ss

KKsKKsGsy spc 11

)(21

211221

LHP in the is zero the if and

11)('* then

1

1

1

1)( response loopopen the toadd weIf

positive is then ,1:

at zero RHP

21

2112

21

21211221

12

2

1

2

1

1221

21

KKK

yss

KKsKKKsGsysysy

syss

KGsy'

sK

KifNote

KK

KKs

sc

spc