Compensation
Using the process field GPF(s)
step response
G(j)  A()e j()
Analysis of step response
The following analysis technique are also used using the
original recorded values or the identified model created
from these recorded values.
 First it must be concluded if the process field has or
hasn't got integral effect.
It possible from the reaction curve of the process field. It takes a
new steady-state or uniformly changing the amplitude.

It should be noted the time constants of the
approximation model.
It is possible editing the reaction curve.
Self-adjusting process field
PI or PIDT1 compensation
European structure
Without integral effect
In this case the most commonly used controller
type the PI or if the reaction curve starts relatively
slowly PIDT1
The 1 can also be used if the process field has got large dead time.
The approximation models, whose parameters can be
determined without computers the next:
PTn
HPT1
GPFa (s)  K P
1
sTg  1
esTu
G PFa (s)  K P
1


sTg  1
n
The transfer functions
PI

1 
G C (s)  G PI (s)  K C 1 

 sTI 
PIDT1

sTD 
1
G C (s)  G PIDT (s)  K C 1 


 sTI sT  1
In case of PIDT1 the transfer function has got four variables.
You must be determine the AD differential gain to define the
T time constants!
The principle of compensation





Be plotted the step response of process field.
The ratio of the steady-state amplitude of the input
(energizing) and output (response) signals is the KP.
You should look for the inflection point of the
reaction curve.
You need to edit the crossover points of beginning
and final values of reaction curve with the line which
overlaid on the inflection point.
The intersection can be defined the apparent Tu dead
time and the apparent Tg first order time constant.
HPT1 model
from the reaction curve of process field
yM , u %
ym 1
Gp ( s) 
e sTu
u 1  sTg
55%
KP
yM
u
Tu
45%
Tg
t
Edited parameters
Determination of KP, Tu and Tg
The above figures are made MATLAB software. The amplitude
of step command of MATLAB is unit, and so the read final
value equals the process field gain KP = 0.72.
Determination with editing of Tg and Tu is quite inaccuracy.
Recommendation of Piwinger:
0
7.8
3.3
I
PID
50
PI
Tg
Tu
Recommendation of Chien-Hrones-Reswick
KC
P
1 Tg
0.3
K p Tu
PI
1 Tg
0.3
K p Tu
PID
0.6
1 Tg
K p Tu
TI
TD
1.2Tg
Tg
0.5Tu
The initial conditions for optimization parameters:
The process field is an ideal HPT1;
The objective function is the fastest aperiodic transient at setpoint tracking;
The optimization is based on the square-integral criterion.
Determination of KC and TI
Defined values: KP = 0.72, Tg = 10.6 sec., and Tu = 0.9 sec.
The ratio of the time constants 11.8, and so the
recommended compensation is PI.
Using the above table:
1 Tg
1 10.6
KC  0.3
 0.3
 4.9
K p Tu
0.72 0.9
TI  1.2Tg  1.2*10.6  12.7sec.
The PI compensation is:

1
GPI ( s)  KC 1 
 sTI
 62.2s  4.9

12.7 s

Step response of closed loop
Important: It is not an optimal parameter choice!
Chien-Hrones-Reswick recommendations
KC
P
1 Tg
0.7
K p Tu
PI
1 Tg
0.6
K p Tu
PID
0.95
1 Tg
K p Tu
TI
TD
Tg
1.35Tg
0.47Tu
The initial conditions for optimization parameters:
The process field is an ideal HPT1;
The objective function is the fastest periodically transient with maximum 20%
overshoot at setpoint tracking;
The optimization is based on the square-integral criterion.
Determination of KC and TI
Defined values: KP = 0.72, Tg = 10.6 sec., and Tu = 0.9 sec.
The ratio of the time constants 11.8, and so the
recommended compensation is PI.
Using the above table:
1 Tg
1 10.6
KC  0.6
 0.6
 9.8
K p Tu
0.72 0.9
TI  Tg  10.6sec.
A PI kompenzáló tag:

1
GPI ( s)  KC 1 
 sTI
 103.9s  9.8

10.6s

Step response of closed loop
It can be seen that the approximation of process field the
objective function is not satisfied.
PTn model
yM , u
G E (s)  K P
55%
1
 sT  1n
yM
70%
u
30%
10%
t10
45%
t 30
t
t 70
Determination of system parameters
The number of the first order time constant (n)
N
t10
t 30
t10
t 70
1
2
3
4
5
6
0.30
0.48
0.58
0.63
0.87
0.70
0.09
0.22
0.31
0.37
0.42
0.45
0.36
1.10
1.91
2.76
3.63
5.52
1.20
2.44
3.62
4.76
5.89
7.01
T  T2
Time constant T  1
2
t 30
T1
t 70
T2
Step response of process field
Determination of n and T
Defined values t10 = 1.95sec, t30 = 4 sec., és t70 = 10.1 sec.
The process gain KP = 0.72
t10 1.95

 0.49
t30
4
t10 1.95

 0.19
t70 10.1
Based on the table above the PT2 is the closest approximation:
n = 2.
T1 
t30
4

 3.64sec.
1.1 1.1
T
T2 
t70
10.1

 4.14sec.
2.44 2.44
T1  T2 3.64  4.14

 3.9sec.
2.
2
Proposed parameters for PTn model
The fastest periodically transient with maximum 20%
overshoot at setpoint tracking
KC
TI
P
n=1
20
KP
PI
n=1
3
KP
T
2
PI
n=2,3
1
KP
2n
T
n2
PID
n=4,5
3 n
KP n  2
2n
T
n 1
I
n=6
2nT
TD
T
5
Proposed parameters for PTn model
n = 2, and so you choose PI.
KC 
1
1

 1.4
K P 0.72
TI 
2n
4
T  3.9  3.9sec
n2
4
In the industrial area you never use a pure P compensation
to control a self-tuning process field!
Step response of the closed loop
Compare the two models the PTn is the better approximation, if the process
field has not got a real dead time.
Process field with integral
effect
P or PDT1 compensation
European structure
Process field with integral effect
In this case the most popular compensation is the P
or if the response signal without noise than PDT1,
but in the later case be applied the PIDT1 too.
The approximate models IT1 or HIT1
1
1
G P (s) 
sTI sTg  1
1
1
G P (s) 
esTu
sTI sTg  1
IT1 model from the reaction curve of process field
yM , u
G P (s) 
65%
1
1
sTI sTg  1
u
Tg
45%
TI
t
Recommendation of Friedlich for IT1
Típus
P
PDT1
PIDT1
KC
TI
TD
T
0.5 I
Tg
TI
0.5
Tg
TI
0.4
Tg
Tg
3.2Tg
0.8Tg
The initial conditions for optimization parameters:
The process field is an ideal IT1;
The objective function is the fastest periodically transient with maximum 20%
overshoot at setpoint tracking;
The optimization is based on the square-integral criterion.
Step response of the process field
The compensation type does not depend on the ratio of
the TI and Tg.
Parameters of the
P, PDT1, and PIDT1
P
PDT
TI
9.9
KC  0.5  0.5
 3.6
Tg
1.26
T
KC  0.5 I  3.6
Tg
It is possible other AD value too.
TD  Tg 1.26sec
1
T  Tg  0.14sec
9
TI
K

0.4
 3.15
PIDT C
Tg
TI  3.2Tg  4sec
TD  0.8Tg 1sec
1
T  Tg  0.11sec
9
Step response of closed loop with P
compensation
The steady-state error is 0; settling time is 11.4 sec.; overshoot is 6.1%
Step response of closed loop with PDT1
compensation
The steady-state error is 0; the settling time is 10.1 sec.; there is not overshoot.
Step response of closed loop with
PIDT1 compensation
Very bad! It is convenient the open-loop transfer function analysis.
The Bode plot of open-loop (G0(s))
with PIDT1 compensation
It can be seen that increasing the gain of the compensation up to 17.4 a better
phase margin value is obtained.
The result of the PIDT1 compensation
with the new parameters
Better, but it is not good!
Tuning the PDT1 compensation
Replace the phase margin value from 95° to 90° the KC increasing by 2.8-fold.
The result of the PIDT1 compensation
with the new parameters
It is good enough!