Colorado School of Mines CHEN403
Ziegler-Nichols Controller Tuning Example
The Ziegler-Nichols method uses a closed controller loop & requires the following steps:
• Bring system to steady state operation.
• Put on P control. Introduce a set point change and vary gain until system
oscillates continuously. This frequency is ωCO and M is the amplitude ratio.
• Compute the following:
1
Ultimate Gain = K u =
M
2π
Ultimate Period = Pu =
ωCO
The original Z-N tuning settings are given in the following table.
Controller
Kc
τI
τD
P
K u /2
—
PI
K u /2.2
—
Pu /1.2
PID
K u /1.7
Pu /2
—
Pu /8
These controller settings were developed to give a 1/4 decay ratio. However, other
settings have been recommended that are closer to critically damped control (so that
oscillations do not propagate downstream). These PID controller settings are shown in the
following table.
Controller
Kc
τI
τD
Original
0.6K u
Pu /2
Pu /8
Some Overshoot
0.33K u
Pu /2
Pu /3
No Overshoot
0.22K u
Pu /2
Pu /3
The question becomes one of how to find these parameters when we don’t have a real
process but rather the idealized transfer functions for the process.
Ziegler-Nichols Example
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December 21, 2008
Colorado School of Mines CHEN403
General 3rd Order Overdamped System Example
U′
R′
+
-
Gp =
+
+
Gc
Kp
( τ1s + 1)( τ2 s + 1)( τ3 s + 1)
Y′
As an example, let's assume we have a 3rd order overdamped system with negligible
dynamics in the final control & measurement elements. Then:
Kp
Gp ( s ) =
( τ1 s + 1)( τ2 s + 1)( τ3 s + 1)
G f ( s ) = Gm ( s ) = 1
The transfer function between the set point and the output under P control will be:
G(s) =
=
Kc
Gc Gp
1 + Gc Gp
=
Kp
( τ1 s + 1)( τ2 s + 1)( τ3s + 1)
1 + Kc
Kp
( τ1 s + 1)( τ2 s + 1)( τ3s + 1)
KcK p
τ1 τ2 τ3 s + ( τ1 τ2 + τ2 τ3 + τ3 τ1 ) s 2 + ( τ1 + τ2 + τ3 ) s + (1 + K p K c )
3
The characteristic equation of this transfer function:
τ1 τ2 τ3 s3 + ( τ1 τ2 + τ2 τ3 + τ3 τ1 ) s 2 + ( τ1 + τ2 + τ3 ) s + (1 + K p K c ) = 0
will control the stability & oscillatory nature of the system’s response. Undamped
oscillatory behavior occurs at the very edge of the system’s stability limit.
Routh Array. We can use a Routh array to determine this point.
Ziegler-Nichols Example
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December 21, 2008
Colorado School of Mines CHEN403
Row
1
τ1 τ2τ3
τ1 + τ2 + τ3
2
τ1 τ2 + τ2 τ3 + τ3 τ1
1 + K pK c
3
τ1 + τ2 + τ3 −
τ1 τ2τ3 (1 + K p K c )
τ1 τ2 + τ2 τ3 + τ3τ1
1 + K pK c
4
The stability limit from the 4th row is:
1 + K pK c > 0 ⇒ K c > −
1
Kp
which will be satisfied for any positive K c . The stability limit from the 3rd row is:
τ1 + τ2 + τ3 −
τ1 τ2 τ3 (1 + K p K c )
τ1 τ2 + τ2τ3 + τ3τ1
> 0 ⇒ Kc <
Kc <
( τ1 + τ2 + τ3 )( τ1τ2 + τ2τ3 + τ3τ1 ) −
K p τ1 τ2 τ3
1
Kp
1 1 1 1
1
( τ1 + τ2 + τ3 )  + +  −
Kp
 τ1 τ2 τ3  K p
So, the ultimate gain will correspond to the stability limit:
K cu =
1 1 1 1
1
( τ1 + τ2 + τ3 )  + +  −
Kp
 τ1 τ2 τ3  K p
The period of oscillation at this ultimate gain must be calculated from the inverted Laplace
expression.
Direct Substitution. We could also use direct substitution to determine the stability limit for
this process. Substituting s = ωj into the characteristic equation:
τ1τ2τ3 ( ωj ) + ( τ1 τ2 + τ2τ3 + τ3τ1 )( ωj ) + ( τ1 + τ2 + τ3 )( ωj ) + (1 + K p K c ) = 0
3
2
−τ1 τ2τ3 ω3 j − ( τ1 τ2 + τ2τ3 + τ3τ1 ) ω2 + ( τ1 + τ2 + τ3 )( ωj ) + (1 + K p K c ) = 0
( )
( 1 + K K ) − ( τ τ

p
c
1 2
( )
+ τ2 τ3 + τ3 τ1 ) ω2  + ( τ1 + τ2 + τ3 ) ω − τ1 τ2τ3ω3  j + = 0 .
Setting the imaginary part to zero gives us the frequency of oscillation:
Ziegler-Nichols Example
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December 21, 2008
Colorado School of Mines CHEN403
( τ1 + τ2 + τ3 ) ωu − τ1τ2τ3ω3u = 0
⇒ ωu =
τ1 + τ2 + τ3
τ1 τ2τ3
and setting the real part to zero gives us the controller gain at the stability limit:
(1 + K K ) − ( τ τ
p
cu
1 2
+ τ2 τ3 + τ3 τ1 ) ωu2 = 0 ⇒ K cu =
=
( τ1τ2 + τ2τ3 + τ3τ1 )
τ1 + τ2 + τ3
−1
τ1 τ2τ3
Kp
1 1 1 1
1
 + +  ( τ1 + τ2 + τ3 ) −
K p  τ1 τ2 τ3 
Kp
(which is exactly the same as that given by the Routh array analysis).
Specific 3rd Order Overdamped System Example
Let us take for example the process:
Gp ( s ) =
1
( s + 1)(5s + 1)( 0.2s + 1)
The ultimate gain will be:
K cu =
1
1 1 1
1
(1 + 5 + 0.2)  + +  − = 37.44
1
 1 5 0.2  1
and the frequency of oscillation will be:
ωCO =
1 + 5 + 0.2
= 2.48998
1 ⋅ 5 ⋅ 0.2
so the period of oscillation at the ultimate gain is:
Pu =
2π
= 2.52339
ωCO
The following will be the Ziegler-Nichols controller settings:
•
P control:
Ziegler-Nichols Example
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December 21, 2008
Colorado School of Mines CHEN403
Kc =
•
Ku
= 18.72
2
PI control:
Ku
= 17.02
2.2
P
τI = u = 2.10
1.2
Kc =
•
PID control:
Ku
= 22.02
1.7
P
τI = u = 1.26
2
P
τD = u = 0.32
8
Kc =
Let’s look at a step change to the load for PI control. The closed loop transfer function will
be:
G(s) =
Gp
1 + Gc Gp
1
( s + 1)(5s + 1)( 0.2s + 1)
=

1 
1
1 + Kc 1 +

τI s  ( s + 1 )(5s + 1 )( 0.2s + 1)

τI s
=
τI s ( s + 1)(5s + 1)( 0.2s + 1) + K c ( τI s + 1)
=
τI s
τI s + 6.2τI s + 6.2τI s 2 + τI (1 + K c ) s + K c
4
3
With a step change to the load, U = 1/ s and the Ziegler-Nichols settings:
1
8.1039 + 18.0182s + 6.2s2 + 6.2s3 + s 4
Y ( t ) = −0.00573006e −5.62165t + 0.0626769e −0.495759t
Y (s) =
−e −0.0412961t 0.00204812sin (1.70472t ) + 0.0569469cos (1.70472t ) 
Ziegler-Nichols Example
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December 21, 2008
Colorado School of Mines CHEN403
0.10
0.08
0.06
0.04
y(t) 0.02
0.00
-0.02
-0.04
-0.06
0
5
10
15
20
t
This figure shows the response to this disturbance. The controller settings are very
conservative – there is very little damping to the response. In actual practice, these
settings would only be used as initial settings and additional tuning would be done.
Ziegler-Nichols Example
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December 21, 2008