Control Theory and Applications 10.3 SISO System Controls 255
Part A | 10.3
Ziegler–Nichols’ Second Method
Similar to the first method, this method can be used to
design a P, PI, or PID controller. Unlike the first method,
the PID parameters for a system are obtained by observing its closed-loop response via experimentation.
During the PID parameter identification phase, the system is driven with only a proportional controller, with
its controller gain increased until the closed-loop response becomes marginally stable. The amplitude (K cr )
and period (P cr ) of oscillation are recorded and used to
design a P, PI, or PID controller using the expressions
in Table 10.3. As expected, this tuning method is not
applicable to systems that cannot be driven to marginal
stability.
Integrator Anti-Windup
In almost all physical systems, actuators saturate because they have finite range of operation, typically
restricting the actuator output (control action) in terms
of absolute value u.t/ D u c .t/, ju c .t/j Ä u lim ; u lim
sgn.u c .t//, ju c .t/j > u lim .
If the integral control action is used, its error can
be accumulated (or windup) excessively and drastically
deteriorate the system performance. The integral control law thus needs to be modified
P
u I D
k P
T I
e.t/ C ˛.u.t/ u c .t// ;
(10.127)
where ˛ is a user-defined coefficient for tracking the actuator saturation u, u c are the control actions and control
signals, respectively, (refer to Fig. 10.1). When there is
actuator saturation, or u ¤ u c , the two terms will have
opposite signs, and thus the effect of windup can be kept
to a minimum. As expected, ˛ must be chosen carefully
so that it does not overcompensate the effect of windup; [10.10] suggests that ˛ D 1=
p
T I T D .
Example 10.4 AUV Forward Speed P-I-D Control
Consider the AUV forward speed model in Example 10.2 with the first tuning method. From the simulation results, we found K D 1; L D 1:3; T D 18:7. These
are used to set up P, PI, and PID controllers for the
system. Figure 10.29 shows the closed-loop responses
using the P (blue), PI (red), and the PID (green) controllers. One can see that the PI controller has the worst
transient performance whereas the PID controller has
Table 10.3 PID controller parameters based on the second
tuning method
Type of controller
K p
T i
T d
P
0:5K cr
1
0
PI
0:45K cr
P cr =1:2
0
PID
0:6K cr
0:5P cr
0:125P cr
P
PI
PID
0
2
4
6
8
10
12
First tuning method
14
16
18
20
Closed loop response
Time (s)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Fig. 10.29 Closed loop system responses using a P (in blue), PI
(red), and a PID (in green) controller
Desired
Actual
0
5
10
15
20
25
α = 0, u lim = 9999
30
35
40
45
50
Output
Time (s)
2
1.5
1
0.5
0
–0.5
–1
–1.5
–2
Fig. 10.30 A closed-loop AUV forward speed response using a PI
controller with varying setpoints, no actuator saturation or antiwindup
the best compromised performance in terms of overshoot and rise time. It should be noted that this tuning
method only provides a starting point for the controller
parameters, and manual fine tuning is almost always
needed to further improve the system performance.
To study the effect of anti-windup algorithm, the
same forward speed model is controlled with a PI controller. Figure 10.30 shows the closed-loop response
subject to varying setpoints over a period of 50 s. In this
case, no actuator saturation is considered and ˛ D 0.
Again, the sluggish transient response can be noticed.
Part A | 10.3
Ziegler–Nichols’ Second Method
Similar to the first method, this method can be used to
design a P, PI, or PID controller. Unlike the first method,
the PID parameters for a system are obtained by observing its closed-loop response via experimentation.
During the PID parameter identification phase, the system is driven with only a proportional controller, with
its controller gain increased until the closed-loop response becomes marginally stable. The amplitude (K cr )
and period (P cr ) of oscillation are recorded and used to
design a P, PI, or PID controller using the expressions
in Table 10.3. As expected, this tuning method is not
applicable to systems that cannot be driven to marginal
stability.
Integrator Anti-Windup
In almost all physical systems, actuators saturate because they have finite range of operation, typically
restricting the actuator output (control action) in terms
of absolute value u.t/ D u c .t/, ju c .t/j Ä u lim ; u lim
sgn.u c .t//, ju c .t/j > u lim .
If the integral control action is used, its error can
be accumulated (or windup) excessively and drastically
deteriorate the system performance. The integral control law thus needs to be modified
P
u I D
k P
T I
e.t/ C ˛.u.t/ u c .t// ;
(10.127)
where ˛ is a user-defined coefficient for tracking the actuator saturation u, u c are the control actions and control
signals, respectively, (refer to Fig. 10.1). When there is
actuator saturation, or u ¤ u c , the two terms will have
opposite signs, and thus the effect of windup can be kept
to a minimum. As expected, ˛ must be chosen carefully
so that it does not overcompensate the effect of windup; [10.10] suggests that ˛ D 1=
p
T I T D .
Example 10.4 AUV Forward Speed P-I-D Control
Consider the AUV forward speed model in Example 10.2 with the first tuning method. From the simulation results, we found K D 1; L D 1:3; T D 18:7. These
are used to set up P, PI, and PID controllers for the
system. Figure 10.29 shows the closed-loop responses
using the P (blue), PI (red), and the PID (green) controllers. One can see that the PI controller has the worst
transient performance whereas the PID controller has
Table 10.3 PID controller parameters based on the second
tuning method
Type of controller
K p
T i
T d
P
0:5K cr
1
0
PI
0:45K cr
P cr =1:2
0
PID
0:6K cr
0:5P cr
0:125P cr
P
PI
PID
0
2
4
6
8
10
12
First tuning method
14
16
18
20
Closed loop response
Time (s)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Fig. 10.29 Closed loop system responses using a P (in blue), PI
(red), and a PID (in green) controller
Desired
Actual
0
5
10
15
20
25
α = 0, u lim = 9999
30
35
40
45
50
Output
Time (s)
2
1.5
1
0.5
0
–0.5
–1
–1.5
–2
Fig. 10.30 A closed-loop AUV forward speed response using a PI
controller with varying setpoints, no actuator saturation or antiwindup
the best compromised performance in terms of overshoot and rise time. It should be noted that this tuning
method only provides a starting point for the controller
parameters, and manual fine tuning is almost always
needed to further improve the system performance.
To study the effect of anti-windup algorithm, the
same forward speed model is controlled with a PI controller. Figure 10.30 shows the closed-loop response
subject to varying setpoints over a period of 50 s. In this
case, no actuator saturation is considered and ˛ D 0.
Again, the sluggish transient response can be noticed.