# Details about ZN and it's applicability

```Article:
Ziegler-Nichols’ Closed-Loop Method
Finn Haugen
TechTeach
17. July 2010
1
Introduction
Ziegler and Nichols published in 1942 a paper [1] where they described two
methods for tuning the parameters of P-, PI- and PID controllers. These
two methods are the Ziegler-Nichols’ closed loop method 1 , and the
Ziegler-Nichols’ open loop method2 . The present article describes the
closed-loop method, while the open-loop method is described in another
article (available at http://techteach.no).
Ziegler and Nichols [1] used the following definition of acceptable stability
as a basis for their contoller tuning rules: The ratio of the amplitudes of
subsequent peaks in the same direction (due to a step change of the
disturbance or a step change of the setpoint in the control loop) is
approximately 1/4, see Figure 1:
A2
1
=
A1
4
(1)
However, there is no guaranty that the actual amplitude ratio of a given
control system becomes 1/4 after tuning with one of the Ziegler and
Nichols’ methods, but it should not be very different from 1/4.
Note that the Ziegler-Nichols’ closed loop method can be applied only to
processes having a time delay or having dynamics of order higher than 3.
Here are a few examples of process transfer function models for which the
method can not be used:
K
H(s) =
(integrator)
(2)
s
K
H(s) =
(first order system)
(3)
Ts + 1
K
H(s) = s 2
(second order system)
(4)
( ω0 ) + 2ζ ωs0 + 1
1
2
Also denoted the Ultimate gain method.
Or the Process reaction curve method
1
v
t
y
A1
A2
yr
t
Figure 1: If A2 /A1 ≈ 1/4 the stability of the system is ok, according to Ziegler
and Nichols
2
The Ziegler-Nichols’ PID tuning procedure
The Ziegler-Nichols’ closed loop method is based on experiments executed
on an established control loop (a real system or a simulated system), see
Figure 2.
The tuning procedure is as follows:
1. Bring the process to (or as close to as possible) the specified
operating point of the control system to ensure that the controller
during the tuning is “feeling” representative process dynamic3 and to
minimize the chance that variables during the tuning reach limits.
You can bring the process to the operating point by manually
adjusting the control variable, with the controller in manual mode,
until the process variable is approximately equal to the setpoint.
2. Turn the PID controller into a P controller by setting set Ti = ∞4
and Td = 0. Initially set gain Kp = 0. Close the control loop by
setting the controller in automatic mode.
3
This may be important for nonlinear processes.
In some commercial controllers Ti = 0 is a code that is used to deactivate the I-term,
corresponding to Ti = ∞.
4
2
Controller
v
u0
Manual
e
ySP
u
PID
y
Process
Auto
Tp
Measured y
Sensor
Figure 2: The Ziegler-Nichols’ closed loop method is executed on an established
control system.
3. Increase Kp until there are sustained oscillations in the signals in the
control system, e.g. in the process measurement, after an excitation
of the system. (The sustained oscillations corresponds to the system
being on the stability limit.) This Kp value is denoted the ultimate
(or critical) gain, Kpu .
The excitation can be a step in the setpoint. This step must be
small, for example 5% of the maximum setpoint range, so that the
process is not driven too far away from the operating point where the
dynamic properties of the process may be different. On the other
hand, the step must not be too small, or it may be difficult to
observe the oscillations due to the inevitable measurement noise.
It is important that Kpu is found without the control signal being
driven to any saturation limit (maximum or minimum value) during
the oscillations. If such limits are reached, you will find that there
will be sustained oscillations for any (large) value of Kp , e.g. 1000000,
and the resulting Kp -value (as calculated from the Ziegler-Nichols’
formulas, cf. Table 1) is useless (the control system will probably be
unstable). One way to say this is that Kpu must be the smallest Kp
value that drives the control loop into sustained oscillations.
4. Measure the ultimate (or critical) period Pu of the sustained
oscillations.
5. Calculate the controller parameter values according to Table 1, and
use these parameter values in the controller.
If the stability of the control loop is poor, try to improve the stability
by decreasing Kp , for example a 20% decrease.
3
P controller
PI controller
PID controller
Kp
0.5Kpu
0.45Kpu
0.6Kpu
Ti
∞
Pu
1.2
Pu
2
Td
0
0
Pu
8
=
Ti
4
Table 1: Formulas for the controller parameters in the Ziegler-Nichols’ closed
loop method.
Eksempel 1 Tuning a PI controller with the Ziegler-Nichols’
closed loop method
I have tried the Ziegler-Nichols’ closed loop method on a level control
system for a wood-chip tank with feed screw and conveyor belt which runs
with constant speed, see Figure 3.5 6 The purpose of the control system is
to keep the chip level of the tank equal to the actual, measured level.
The level control system works as follows: The controller tries to keep the
measured level equal to the level setpoint by adjusting the rotational speed
of the feed screw as a function of the control error (which is the difference
between the level setpoint and the measured level).
Figure 4 shows the signals after a step in the setpoint from 9 m to 9.5 m
with a ultimate gain of Kpu = 3.0. The ultimate period is approximately
Pu = 1100 s. From Table 1 we get the following PI parameters:
Kp = 0.45 · 3.0 = 1.35
Ti =
1100 s
= 917 s
1.2
Td = 0 s
(5)
(6)
(7)
Figure 5 shows signals of the control system with the above PID parameter
values. The control system has satisfactory stability. The amplitude ratio
in the damped oscillations is less than 1/4, that is, which means that the
stability is a little better than prescribed by Ziegler and Nichols’.
[End of Example 1]
5
This example is based on an existing system in the paper pulp factory Södra Cell Tofte
in Norway. The tank with conveyor belt is in the beginning of the paper pulp production
line.
6
A simulator of the system is available at http://techteach.no/simview.
4
Process & Instrumentation (P&I) Diagram:
Process (tank with belt and screw)
Conveyor
belt
Feed screw
u
Control
variable
Level
controller
Wood chip
ym
LC
ySP
Process
measurement
Reference
or
Setpoint
Sensor Process
(Level output
transmitter)variable
y [m]
LT
n
Measurement
noise
Process disturbance
(environmental variable)
Block diagram:
Reference
or
Setpoint
y SP
Level controller (LC)
Control
error
e
y m,f
Filtered
measurement
PID
controller
Measurement
filter
Wood
chip tank
Control
variable
u
d [kg/min]
Process disturbance
(environmental variable)
d
Process output
Process
variable
y
(tank with
belt and
screw)
Control
loop
Sensor
ym
(Level
Process Transmitter
- LT)
measureMeasurement
ment
n noise
Figure 3: P&I (Process and Instrumentation) diagram and block diagram of a
level control system for a wood-chip tank in a pulp factory
3
Some comments to the Ziegler-Nichols’ closed
loop method
1. You do not know in advance the amplitude of the sustained
oscillations. The amplitude depends on the size of the excitations of
the control system.
2. If the operating point varies and if the process dynamic properties
depends on the operating point, you should consider using some kind
of adaptive control or gain scheduling, where the PID parameter are
adjusted as functions of the operating point.
If the controller parameters shall have fixed value, they should be
5
A1
A2
Pu
Figure 4: Example 1: The tuning phase of the Ziegler-Nichols’ closed-loop
method.
tuned in the worst case as stability is regarded. This ensures proper
stability if the operation point varies. The worst operating point is
the operation point where the process gain has its greatest value
and/or the time delay has its greatest value.
3. The responses in the control system may become unsatisfactory with
the Ziegler-Nichols’ method. 1/4 decay ratio may be too much, that
is, the damping in the loop is too small. A simple re-tuning in this
case is to reduce the Kp somewhat, for example by 20%.
References
[1] J. G. Ziegler and N. B. Nichols: Optimum Settings for Automatic
Controllers, Trans. ASME, Vol. 64, 1942, s. 759-768
6
Figure 5: Example 1: Time responses with PI parameters tuned using the
Ziegler-Nichols’ closed loop method
7
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