Tuning Inclass

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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.


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.


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


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


Gc(s) = K1+ K21

s + K3s



Gc(s) = Kp+ Ki1

s + Kds

PID P t


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PID Parameters


Gc(s) = K1+ K21

s + K3s



Gc(s) = Kp+ Ki1

s + Kds

IfKi and Kdare zero we have a simple P controller.

PID Parameters


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PID Parameters


Gc(s) = K1+ K21

s + K3s



Gc(s) = Kp+ Ki1

s + Kds


If only Kd is zero we have a PI controller.

PID Parameters


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PID Parameters


Gc(s) = K1+ K21

s + K3s



Gc(s) = Kp+ Ki1

s + Kds


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.



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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.



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Ziegler-Nichols provides only a starting point for further tuning.



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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:



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Notice that this method is applied on the plant itself, without

feedback.



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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).



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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).



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g






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.



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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:



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g



L is the intersection of the tangent line with the time axis.



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g



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.



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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



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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:



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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



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To apply the second method we do a test on the system that variesKpwhile keeping Kd= 0 and Ki= 0.



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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:



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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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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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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.



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e.g. Assume we have the following system:



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Like the controllers produced by Ziegler-Nichols, this PID controllerhas a pole at the origin and a pair of double zeros at a.



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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.



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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.




Tuning Inclass

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