LINEAR CONTROL SYSTEMS

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LINEAR CONTROL SYSTEMS

• Ali Karimpour
• Associate Professor
• Ferdowsi University of Mashhad

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Lecture 16
Time domain design of control systems

• Topics to be covered include
• Design of controller in time domain.
• Various controller configurations.
• Different kind of controllers.
• Controller realization.
• Time domain design of the PID controllers.
• Design of PID controllers.
• Design of PD controllers.
• Design of PI controllers.

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Various controller configurations.
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Various controller configurations.
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Series Compensation Structure
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In this Course we consider the series or cascade
compensation.
PID controllers.
In This course
Lead lag controllers
Different kind of controllers
In Graduate courses
H2 Controllers
H8 Controllers
..
Intelligent Controllers
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PID Controllers
?????? PID

• PID has become almost universally used in
  industrial control.
• These controllers have proven to be robust and
  extremely beneficial in the control of many
  important applications.
• PID stands for P (Proportional)
• I (Integral)
• D (Derivative)
• The standard form PID are
• An Alternative form for PID

Proportional only
Proportional plus Integral
Proportional plus derivative
Proportional, integral and derivative
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Lead-lag Compensators
?????? ??? ??? ?? ???
Closely related to PID control is the idea of
lead-lag compensation. The transfer function of
these compensators is of the form If alt1 ,
then this is a lag network. Or (zgtp) in other
form. If agt1 this is a lead network. Or (zltp)
in other form.
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PID and Operational Amplifiers
?????? PID ? ????? ????? ???????
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PID and Operational Amplifiers
?????? PID ? ????? ????? ???????
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PID and Operational Amplifiers
?????? PID ? ????? ????? ???????
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Effects of the PD control on the time response.
????? ?????? PD ?? ???? ?????
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Effects of the PI control on the time response.
????? ?????? PI ?? ???? ?????
Loop transfer function with controller
Loop transfer function without controller
PI controller can improve error by increases the
type of system by one
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Tuning of PID Controllers
????? ????????? PID

• Because of their widespread use in practice, we
  present below several methods for tuning PID
  controllers. Actually these methods are quite
  old and date back to the 1950s. Nonetheless,
  they remain in widespread use today.
• In particular, we will study.
• Ziegler-Nichols Oscillation Method
• Ziegler-Nichols Reaction Curve Method
• Cohen-Coon Reaction Curve Method
• Time domain design
• Frequency domain design

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Ziegler-Nichols Design
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• This procedure is only valid for open loop stable
  plants.
• Open-Loop Tuning
• Closed-Loop Tuning
• According to Ziegler and Nichols, the open-loop
  transfer function of a system can be approximated
  with time delay and single-order system, i.e.
• where TD is the system time delay and T1 is the
  time constant.

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Ziegler-Nichols Reaction Curve Method(Open-Loop
Case)
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• For open-loop tuning, we first find the plant
  parameters by applying a step input to the
  open-loop system.
• The plant parameters K, TD and T1 are then found
  from the result of the step test as shown in
  Figure.

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Ziegler-Nichols Reaction Curve Method(Open-Loop
Case)
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Numerical Example
???? ????

• Consider step response of an open-loop system as

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Numerical Example
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Ziegler-Nichols Oscillation Method(Closed-loop)
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• This procedure is only valid for open loop stable
  plants and it is carried out through the
  following steps
• Set the true plant under proportional control,
  with a very small gain.
• Increase the gain until the loop starts
  oscillating. Note that linear oscillation is
  required and that it should be detected at the
  controller output.
• Record the controller critical gain Kc and the
  oscillation period of the controller output, T.
• Adjust the controller parameters according to
  Table

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Ziegler-Nichols Oscillation Method(Closed-loop)
????? ????? ?????? ???? ??????(???? ????)
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Numerical Example
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• Consider a plant with a model given by
• Find the parameters of a PID controller using the
  Z-N oscillation method. Obtain a graph of the
  response to a unit step input reference.

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Solution
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• Applying the procedure we find
• Kc 8 and ?c ?3. T3.62
• Hence, from Table, we have
• The closed loop response to a unit step in the
  reference at t 0 is shown in the next figure.

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Response to step reference
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Time domain design
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Time domain design
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Time domain design
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Clearly the design is not possible
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Other controllers
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Tuning PD controller
????? ?????? PD
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Tuning PD controller
????? ?????? PD
Tuning kD by graphical method.
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Tuning PD controller
????? ?????? PD
Tuning kD by mathematical method
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Tuning PD controller
????? ?????? PD
Why P.O. gt 4.3
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Tuning PI controller
????? ?????? PI
Clearly type of system is 2 so
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Tuning PI controller (Continue)
????? ?????? PI (?????)
Root loci with proportional controller
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Tuning PI controller (Continue)
????? ?????? PI (?????)
Root loci with PI controller
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Tuning PI controller
????? ?????? PI
Why P.O. gt 4.3
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Compare PI and PD controllers
?????? ????????? PI ? PD
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Exercises
???????
1- In the following system design a PID
controller with Ziegler-Nichols Oscillation Method
2- In the following system design a PID
controller with Ziegler-Nichols Oscillation Method
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Exercises
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3 In the following system design a PD controller
such that that the damping ratio of complex poles
be 0.6 and ramp error constant be 80.
4 In the following system design a PD controller
such that that the damping ratio of complex poles
be 0.6 and ramp error constant be 80.
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Exercises
???????
5 In the following system design a PI controller
such that that the damping ratio of complex poles
be 0.6 and ramp error constant be 80.
6 In the following system design a PI controller
such that that the damping ratio of complex poles
be 0.6 and ramp error constant be 80.
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Exercises
???????
Let the input impedance be generated by a
resistor R2 be in series with a resistor R1 and a
capacitor C1) that are in parallel, and let the
feedback impedance be generated by a resistor Rf
and a capacitor C f . a) Show that this choices
lead to form a PID controller with high frequency
gain limit as
b) Derive the parameters in the controller.