### Lecture four

```‫بسم هللا الرحمن الرحيم‬
Lecture four
Tuning of PID Controllers
1- Off-line methods (Ziegler-Nichols, Tyreus-Luyben, Ciancone-Marlin, IMC)
2- On-line methods (Ziegler-Nichols, Relay feedback)
Ref: C. C. Yu, Autotuning of PID controllers, 2nd ed., springer, 2006. Chapters 2, 3 & 4
Lecturer: M. A. Fanaei
Off-Line Tuning of PID Controllers
About 250 tuning rules are exist for PI and PID Controllers
What is the suitable tuning rule?
It really depends on your process (Type, Order, Parameters,
Nonlinearity, Uncertainty, etc)
1. Ziegler-Nichols(1942): Recommended for 0.1< D/t <0.5 (
K pe
 Ds
t s 1
)
2
Off-Line Tuning of PID Controllers
2. Tyreus-Luyben(1992): Recommended for time-constant
dominant processes ( D/t <0. 1 )
dominant processes ( D/t > 2.0 )
3
Off-Line Tuning of PID Controllers
4. PID tuning based on IMC (Rivera et al., 1986)
4
Recommended Tuning Formulas
The following formulas are recommended by Luyben and
Yu for tuning of PI controllers:
5
On-Line Tuning of PID Controllers
 Ziegler-Nichols Test (1942)
1. Set the controller gain Kc at a low value, perhaps 0.2.
2. Put the controller in the automatic mode.
3. Make a small change in the set point or load variable and observe the
response. If the gain is low, then the response will be sluggish.
4. Increase the gain by a factor of two and make another set point or load
change.
5. Repeat step 4 until the loop becomes oscillatory and continuous
cycling is observed. The gain at which this occurs is the ultimate gain
Ku , and the period of oscillation is the ultimate period Pu.
6
On-Line Tuning of PID Controllers
 Relay Feedback Test (Astrom & Hagglund, 1984)
Luyben popularized relay feedback method and called this method
“ATV” (autotune variation).
7
On-Line Tuning of PID Controllers
 Relay Feedback Test
1.
Bring the system to steady state.
2.
Make a small (e.g. 5%) increase in the
manipulated input. The magnitude of
change depends on the process
sensitivities and allowable deviations in
the controlled output. Typical values are
between 3 and 10%.
3.
As soon as the output crosses the SP, the manipulated input is switched
to the opposite position (e.g. –5% change from the original value).
4.
Repeat step 3 until sustained oscillation is observed .
5.
Read off ultimate period Pu from the cycling and compute Ku from
the following Equation:
Ku = 4h/(πa) , ωu = 2π/Pu
8
On-Line Tuning of PID Controllers
 Advantages of Relay Feedback Test
1.
It identifies process information around the important frequency, the
ultimate frequency (where the phase angle is -π).
2.
It is a closed-loop test; therefore, the process will not drift away
from the nominal operating point.
3.
The amplitude of oscillation is under control (by adjusting h ).
4.
The time required for a relay feedback test is roughly equal to two
to four times the ultimate period.
5.
If the normalized dead time D /t is less than 0.28, the ultimate
period is smaller than the process time constant. Therefore the relay
feedback test is more time efficient than the step test. Since the dead
time can not be too large, the temperature and composition loops in
process industries seem to fall into this category.
9
On-Line Tuning of PID Controllers
 Advantages of Relay Feedback Test
Kpe
 Ds
t s 1
10
On-Line Tuning of PID Controllers
 Process model identification from a Relay Feedback Test
In theory, the steady state gain can be obtained from plant data (Kp =
Δy/Δu). For highly nonlinear process, the change in input (Δu) must be small
as 10-3 to 10-6 % of the full range. Therefore for highly nonlinear processes,
trying to obtain reliable steady state gains from plant data is usually
impractical.
Luyben showed that the simple relay feedback test provides an effective
way to determine the linear model for such processes.
If necessary, the dead time (D) in the
transfer function can be easily read off
from the initial part of the relay
feedback test.
11
On-Line Tuning of PID Controllers
 Process model identification from a Relay Feedback Test
Kpe
Model 1 (FOPDT) :
   D   tan
1
 Ds
t s 1
(t ) ,
    D  u  tan
1
AR  K p /
(t u )
t 
,
KuK p 
(t )  1
2
(t u )  1
2
tan(   D  u )
u
(t u )  1
2
Kp 
Ku
12
On-Line Tuning of PID Controllers
 Process model identification from a Relay Feedback Test
Model 2 (integrator plus dead time) :
Kpe
 Ds
s

u

   D    / 2 , AR  K p / 
 D 
,Kp 

2 u
Ku

    D  u   / 2 , K u K p   u
Kpe
 Ds


1
   D  , AR  K p

D

,
K


p




D

,
K
K

1

Ku

u
u
p
u

13
On-Line Tuning of PID Controllers
 Relay feedback responses of FOPDT processes
Assume an integrator plus dead time
(Time constant dominant processes)
Assume a FOPDT (Most slow processes)