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PID TuningPID Parameters
The Ziegler-Nichols RulesZiegler-Nichols: Method 1Ziegler-Nichols: Method 2
Computational Search
Unit 8: Part 3: PID Tuning
Engineering 5821:Control Systems I
Faculty of Engineering & Applied Science
Memorial University of Newfoundland
March 31, 2010
ENGI 5821 Unit 8: Design via Root Locus
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PID TuningPID Parameters
The Ziegler-Nichols RulesZiegler-Nichols: Method 1Ziegler-Nichols: Method 2
Computational Search
1 PID Tuning
1 PID Parameters
1 The Ziegler-Nichols Rules
1 Ziegler-Nichols: Method 1
1 Ziegler-Nichols: Method 2
1 Computational Search
ENGI 5821 Unit 8: Design via Root Locus
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PID Tuning
Designing a PID compensator using the analytical methodsdiscussed so far requires a mathematical model of the system.
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PID Tuning
Designing a PID compensator using the analytical methodsdiscussed so far requires a mathematical model of the system.However, PID controllers (compensator and controller aresynonymous in control systems) are often used even when no suchmodel exists.
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PID Tuning
Designing a PID compensator using the analytical methodsdiscussed so far requires a mathematical model of the system.However, PID controllers (compensator and controller aresynonymous in control systems) are often used even when no suchmodel exists. The process of adjusting the parameters of the PIDcontroller in this situation is known as tuning.
We will consider two options for tuning a PID (or PI or PD)controller in the absence of a system model:
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PID Tuning
Designing a PID compensator using the analytical methodsdiscussed so far requires a mathematical model of the system.However, PID controllers (compensator and controller aresynonymous in control systems) are often used even when no suchmodel exists. The process of adjusting the parameters of the PID
controller in this situation is known as tuning.
We will consider two options for tuning a PID (or PI or PD)controller in the absence of a system model:
Ziegler-Nichols rules (rules of thumb)
PID T i
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PID Tuning
Designing a PID compensator using the analytical methodsdiscussed so far requires a mathematical model of the system.However, PID controllers (compensator and controller aresynonymous in control systems) are often used even when no suchmodel exists. The process of adjusting the parameters of the PID
controller in this situation is known as tuning.
We will consider two options for tuning a PID (or PI or PD)controller in the absence of a system model:
Ziegler-Nichols rules (rules of thumb)Computational search
PID P
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PID Parameters
Recall the parameters for a PID compensator:
Gc(s) = K1+ K21
s + K3s
PID P
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PID Parameters
Recall the parameters for a PID compensator:
Gc(s) = K1+ K21
s + K3s
We will use the constants Kp, Ki, and Kdwhich stand for
Proportional, Integral, and Derivative.
PID P t
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PID Parameters
Recall the parameters for a PID compensator:
Gc(s) = K1+ K21
s + K3s
We will use the constants Kp, Ki, and Kdwhich stand for
Proportional, Integral, and Derivative.
Gc(s) = Kp+ Ki1
s + Kds
PID P t
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PID Parameters
Recall the parameters for a PID compensator:
Gc(s) = K1+ K21
s + K3s
We will use the constants Kp, Ki, and Kdwhich stand for
Proportional, Integral, and Derivative.
Gc(s) = Kp+ Ki1
s + Kds
IfKi and Kdare zero we have a simple P controller.
PID Parameters
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PID Parameters
Recall the parameters for a PID compensator:
Gc(s) = K1+ K21
s + K3s
We will use the constants Kp, Ki, and Kdwhich stand for
Proportional, Integral, and Derivative.
Gc(s) = Kp+ Ki1
s + Kds
IfKi and Kdare zero we have a simple P controller.
If only Kd is zero we have a PI controller.
PID Parameters
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PID Parameters
Recall the parameters for a PID compensator:
Gc(s) = K1+ K21
s + K3s
We will use the constants Kp, Ki, and Kdwhich stand for
Proportional, Integral, and Derivative.
Gc(s) = Kp+ Ki1
s + Kds
IfKi and Kdare zero we have a simple P controller.
If only Kd is zero we have a PI controller.
If only Ki is zero we have a PD controller.
The Ziegler Nichols Rules
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The Ziegler-Nichols Rules
Ziegler and Nichols came up with two methods for setting theparameters of PID controllers.
The Ziegler Nichols Rules
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The Ziegler-Nichols Rules
Ziegler and Nichols came up with two methods for setting theparameters of PID controllers. These are rules of thumb andthere is no guarantee that the resulting system behaves optimally.
The Ziegler-Nichols Rules
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The Ziegler-Nichols Rules
Ziegler and Nichols came up with two methods for setting theparameters of PID controllers. These are rules of thumb andthere is no guarantee that the resulting system behaves optimally.
Ziegler-Nichols provides only a starting point for further tuning.
The Ziegler-Nichols Rules
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The Ziegler-Nichols Rules
Ziegler and Nichols came up with two methods for setting theparameters of PID controllers. These are rules of thumb andthere is no guarantee that the resulting system behaves optimally.
Ziegler-Nichols provides only a starting point for further tuning.
Method 1: Applies if the systems response to a unit-step isS-shaped, indicating that the plant involves no pureintegration and the system response is not dominated by a
pair of complex-conjugate poles:
The Ziegler-Nichols Rules
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The Ziegler Nichols Rules
Ziegler and Nichols came up with two methods for setting theparameters of PID controllers. These are rules of thumb andthere is no guarantee that the resulting system behaves optimally.
Ziegler-Nichols provides only a starting point for further tuning.
Method 1: Applies if the systems response to a unit-step isS-shaped, indicating that the plant involves no pureintegration and the system response is not dominated by a
pair of complex-conjugate poles:
The Ziegler-Nichols Rules
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The Ziegler Nichols Rules
Ziegler and Nichols came up with two methods for setting theparameters of PID controllers. These are rules of thumb andthere is no guarantee that the resulting system behaves optimally.
Ziegler-Nichols provides only a starting point for further tuning.
Method 1: Applies if the systems response to a unit-step isS-shaped, indicating that the plant involves no pureintegration and the system response is not dominated by a
pair of complex-conjugate poles:
Notice that this method is applied on the plant itself, without
feedback.
The Ziegler-Nichols Rules
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The Ziegler Nichols Rules
Ziegler and Nichols came up with two methods for setting theparameters of PID controllers. These are rules of thumb andthere is no guarantee that the resulting system behaves optimally.
Ziegler-Nichols provides only a starting point for further tuning.
Method 1: Applies if the systems response to a unit-step isS-shaped, indicating that the plant involves no pureintegration and the system response is not dominated by a
pair of complex-conjugate poles:
Notice that this method is applied on the plant itself, without
feedback.Method 2: The system appears to involve some pureintegration and/or dominant complex-conjugate poles (i.e.the response is similar to an underdamped 2nd orderresponse).
The Ziegler-Nichols Rules
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The Ziegler Nichols Rules
Ziegler and Nichols came up with two methods for setting theparameters of PID controllers. These are rules of thumb andthere is no guarantee that the resulting system behaves optimally.
Ziegler-Nichols provides only a starting point for further tuning.
Method 1: Applies if the systems response to a unit-step isS-shaped, indicating that the plant involves no pureintegration and the system response is not dominated by a
pair of complex-conjugate poles:
Notice that this method is applied on the plant itself, without
feedback.Method 2: The system appears to involve some pureintegration and/or dominant complex-conjugate poles (i.e.the response is similar to an underdamped 2nd orderresponse).
The Ziegler-Nichols Rules
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g
Ziegler and Nichols came up with two methods for setting theparameters of PID controllers. These are rules of thumb andthere is no guarantee that the resulting system behaves optimally.
Ziegler-Nichols provides only a starting point for further tuning.
Method 1: Applies if the systems response to a unit-step isS-shaped, indicating that the plant involves no pureintegration and the system response is not dominated by a
pair of complex-conjugate poles:
Notice that this method is applied on the plant itself, without
feedback.Method 2: The system appears to involve some pureintegration and/or dominant complex-conjugate poles (i.e.the response is similar to an underdamped 2nd orderresponse). This method is applied on the closed-loop system,with feedback.
Ziegler-Nichols: Method 1
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g
The systems S-shaped response can be characterized by twoconstants, the delay time L and time constant T.
Ziegler-Nichols: Method 1
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g
The systems S-shaped response can be characterized by twoconstants, the delay time L and time constant T. These
parameters can be obtained by drawing a tangent line at theinflection point of the curve:
Ziegler-Nichols: Method 1
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g
The systems S-shaped response can be characterized by twoconstants, the delay time L and time constant T. These
parameters can be obtained by drawing a tangent line at theinflection point of the curve:
L is the intersection of the tangent line with the time axis.
Ziegler-Nichols: Method 1
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g
The systems S-shaped response can be characterized by twoconstants, the delay time L and time constant T. These
parameters can be obtained by drawing a tangent line at theinflection point of the curve:
L is the intersection of the tangent line with the time axis. L + Tis the time at which the tangent line intersects the steady-state
value.
The following table gives the gains for Method 1
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The following table gives the gains for Method 1.
The following table gives the gains for Method 1
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The following table gives the gains for Method 1.
Controller Kp Ki Kd
P T
/L
0 0PI 0.9T/L 0.27T/L2 0
PID 1.2T/L 0.6T/L2 0.6T
The following table gives the gains for Method 1
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The following table gives the gains for Method 1.
Controller Kp Ki Kd
P T
/L
0 0PI 0.9T/L 0.27T/L2 0
PID 1.2T/L 0.6T/L2 0.6T
A PID controller tuned by this method has a pole at the origin anddouble zeros at s= 1/L:
The following table gives the gains for Method 1
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The following table gives the gains for Method 1.
Controller Kp Ki Kd
P T
/L
0 0PI 0.9T/L 0.27T/L2 0
PID 1.2T/L 0.6T/L2 0.6T
A PID controller tuned by this method has a pole at the origin anddouble zeros at s= 1/L:
Gc(s) = Kp+ Ki1
s + Kds
= 1.2T/L + 0.6T/L2 1
s + 0.6Ts
= 0.6T
s+ 1
L
22
s
Ziegler-Nichols: Method 2
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To apply the second method we do a test on the system that variesKpwhile keeping Kd= 0 and Ki= 0.
Ziegler-Nichols: Method 2
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To apply the second method we do a test on the system that variesKpwhile keeping Kd= 0 and Ki= 0. The system being tested isas follows:
Ziegler-Nichols: Method 2
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To apply the second method we do a test on the system that variesKpwhile keeping Kd= 0 and Ki= 0. The system being tested isas follows:
Ziegler-Nichols: Method 2
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To apply the second method we do a test on the system that variesKpwhile keeping Kd= 0 and Ki= 0. The system being tested isas follows:
Kpis increased from 0 until it reaches a critical value Kcrat which
the output exhibits sustained oscillations...
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At Kp= Kcrthe systems output will oscillate with period Pcr.
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At Kp= Kcrthe systems output will oscillate with period Pcr.These two values are used to determine the PID gains:
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At Kp= Kcrthe systems output will oscillate with period Pcr.These two values are used to determine the PID gains:
Controller Kp Ki KdP 0.5Kcr 0 0
PI 0.45Kcr 0.54Kcr/Pcr 0PID 0.6Kcr 1.2Kcr/Pcr 0.075KcrPcr
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At Kp= Kcrthe systems output will oscillate with period Pcr.These two values are used to determine the PID gains:
Controller Kp Ki KdP 0.5Kcr 0 0
PI 0.45Kcr 0.54Kcr/Pcr 0PID 0.6Kcr 1.2Kcr/Pcr 0.075KcrPcr
Computational SearchIf we have a model of the system or if the system can somehow be
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If we have a model of the system orif the system can somehow besimulated, we can simply search for the PID parameters that bestsatisfy our design criteria.
Computational SearchIf we have a model of the system or if the system can somehow be
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If we have a model of the system orif the system can somehow besimulated, we can simply search for the PID parameters that bestsatisfy our design criteria.
e.g. Assume we have the following system:
Computational SearchIf we have a model of the system or if the system can somehow be
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If we have a model of the system orif the system can somehow besimulated, we can simply search for the PID parameters that bestsatisfy our design criteria.
e.g. Assume we have the following system:
Computational SearchIf we have a model of the system or if the system can somehow be
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If we have a model of the system orif the system can somehow besimulated, we can simply search for the PID parameters that bestsatisfy our design criteria.
e.g. Assume we have the following system:
Like the controllers produced by Ziegler-Nichols, this PID controllerhas a pole at the origin and a pair of double zeros at a.
Computational SearchIf we have a model of the system or if the system can somehow be
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If we have a model of the system orif the system can somehow besimulated, we can simply search for the PID parameters that bestsatisfy our design criteria.
e.g. Assume we have the following system:
Like the controllers produced by Ziegler-Nichols, this PID controllerhas a pole at the origin and a pair of double zeros at a. Thisgives us a 2-D parameter space which is relatively easy to search.
Computational SearchIf we have a model of the system or if the system can somehow be
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If we have a model of the system orif the system can somehow besimulated, we can simply search for the PID parameters that bestsatisfy our design criteria.
e.g. Assume we have the following system:
Like the controllers produced by Ziegler-Nichols, this PID controllerhas a pole at the origin and a pair of double zeros at a. Thisgives us a 2-D parameter space which is relatively easy to search.
The goal is to satisfy the following requirements:%OS
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If we have a model of the system orif the system can somehow besimulated, we can simply search for the PID parameters that bestsatisfy our design criteria.
e.g. Assume we have the following system:
Like the controllers produced by Ziegler-Nichols, this PID controllerhas a pole at the origin and a pair of double zeros at a. Thisgives us a 2-D parameter space which is relatively easy to search.
The goal is to satisfy the following requirements:%OS
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y ysimulated, we can simply search for the PID parameters that bestsatisfy our design criteria.
e.g. Assume we have the following system:
Like the controllers produced by Ziegler-Nichols, this PID controllerhas a pole at the origin and a pair of double zeros at a. Thisgives us a 2-D parameter space which is relatively easy to search.
The goal is to satisfy the following requirements:%OS