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Control objective ; To keep the tank temperature at the desired value by adjusting the rate of heat input from the heater. 8. Feedback Controllers. 8.1 Stirred-Tank Heater Example. Basic components in the feedback control loop Process being controlled(stirred tank).

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8 feedback controllers

  • Control objective ; To keep the tank temperature at the desired value by adjusting the rate of heat input from the heater.

8. Feedback Controllers

8.1 Stirred-Tank Heater Example

  • Basic components in the feedback control loop

  • Process being controlled(stirred tank).

  • Sensor and transmitter.

  • Controller.

  • SCR and final control element(electrical heater)  Actuator.

  • Transmission lines(electrical cables) between the various instruments.

Figure 8.1. Schematic diagram for a stirred-tank control system.


Transmission

line

Controller

Actuator

Process

Sensor

8.2 Controller Implementation Using PC

Figure 8.2. Typical equipment for process control using computer.


8.3 Controllers

8.3.1 Historical Perspectives

250 B.C ; Greeks, water level controller

Their mode of operation was very similar to that of the level regulator in the modern flush toilet.

1788 ; James Watt, fly-ball governor

It played a key role in the development of stream power.

1930s ; PID controller became commercially available

The first theoretical papers on process control were published.

1940s ; Pneumatic PID controller

1950s ; Electronic PID controller

Late 1950 ~ 1960s ; The first computer control applications


8.3.2 PID Controllers

  • PID controller is the controller that has the three basic control modes of Proportional(P), Integral(I) , and Derivative(D).

  • PID controllers are still used widely in industry because of their simplicity, robustness, and successful practical applications.

  • In spite of the development of many advanced control algorithms, nearly 80% of the controllers in the industrial field are PID controller.

Figure 8.3. Flow control system.

Figure 8.4. Schematic diagram of a feedback controller.


Proportional(P) part:

Integral(I) part:

Derivative(D) part:

Where and denote the set point(the desired process output) and process out put. Constants are called proportional gain, integral time and derivative time, respectively.

8.3.2.1 Structure of PID Controllers.

  • Three parts of the PID controller :

  • PID controller is sum of the above three part as follows.


8.3.2.2 Roles of Three Parts

  • Proportional part : Since the control output( ) is proportional to the error , it plays role in pushing the process output to the set point as much as the error.

1. Transfer function.

2. Advantage : immediate corrective action.

3. Disadvantage : steady-state error(offset).

4. Usage : when the steady-state error is tolerable( ex. level control which wants to prevent the system from overflowing or drying), proportional-only controller is attractive because of its simplicity  seldom used only.

To remove the steady-state error(offset), the integral control action should be included in the feedback controller.


  • Steady-state error.

    For usual process(i.e., open-loop stable processes), the control output should be nonzero to keep the process output in a nonzero set point.

  • Consider the following PD controller( the following derivation is applicable to P controller case).

PD(or P) controller output is always zero at steady-state if the error is zero(i.e., ).

  • PD(or P) controller cannot be nonzero constant when the error is zero at steady-state. So, the PD(or P) controller cannot keep the process output in a nonzero set point for open-loop stable processes.

  • Offset can be calculated as follows. Here denotes steady state and is the static gain(or DC gain) of the process.


1. Transfer function.

  • Integral part(= reset or floating control part) : Since the integral part is not necessarily zero even though the error at steady-state is zero, it plays an important role in rejecting the offset.

2. Disadvantages

 Not immediate corrective action.

 Practically PI controller is used.

 Oscillatory response.

 Reduce the stability of the system.

 Solution ; proper tuning of the controller or including derivative control action which tends to counteract the destabilizing effects.

 Reset windup( or integral windup).


  • Reset windup( or Integral windup).

  • Sustained error  Large integral term  Saturation of controller output  Further buildup of the integral term while the controller is saturated is referred to as reset windup or integral windup.

  • Reset windup occurs when a PI or PID controller encounters a sustained error, for example, during the start-up of a batch process or after a large set-point change.

  • The large overshoot occurs because the integral term continues to increase until the error signal changes sign at .

  • Antireset windup ; halting the integral action whenever the controller output saturates. Most of commercial controllers provides antireset windup.

Figure 8.5. Reset windup during a set-point change.


1. Transfer function.

2. Advantage : This part enhance the robustness of the PID controller by considering abrupt change of the error.

Figure 8.6. Extrapolation using the derivative of the error

  • Derivative part(= rate action,pre-act or anticipatory control part) : Since this part represents approximately the increment of the error after time from the present time , it plays a role in rejecting the future error in advance by increasing the control output proportional to the future incremental error.

3. Disadvantage : If the process measurement is noisy, this term will change widely and amplify the noise unless the measurement is filtered.


where is a small number, typically between 0.05 and 0.2.

8.3.2.3 Ideal PID Controller.

1. Transfer function.

2. Disadvantages.

 Derivative kick

 Proportional kick


8.3.3 On-Off Controllers derivative action cannot be built(is

  • The controller output of ideal on-off controller.

where and denote the on and off values, respectively.

  • On-off controller can be considered to be a special case of P controller with a very high controller gain

  • Advantage : Simple and inexpensive controllers.

  • Disadvantage

     Not versatile and ineffective.

     Continuous cycling of the controlled variable and excess

    wear on the final control element.

  • Usage : Thermostats in heating system.

    Domestic refrigerator.

    Noncritical industrial applications.


8.3.4 Typical Response of Feedback Control Systems derivative action cannot be built(is

  • No feedback control make the process slowly reach a new steady-state.

  • Proportional control speeds up the process response and reduces the offset.

  • Integral control eliminates offset but tends to make the response oscillatory.

  • Derivative control reduces both the degree of oscillation and response time.

Figure 8.7. Typical process response with feedback control.

C is the deviation from the initial steady-state.


8.3.4.1 Effect of controller gain . derivative action cannot be built(is

  • Increasing the controller gain.

     less sluggish process response.

  • Too large controller gain.

     undesirable degree of oscillation or even unstable response.

  • An intermediate value of the controller gain

     best control result.

Figure 8.8. Process response with proportional control.


8.3.4.2 Effect of integral time . derivative action cannot be built(is

  • Increasing the integral time.

     more conservative(sluggish) process response.

  • Too large integral time.

     too long time to reach to the set point after load upset or set-point change occurs.

  • Theoretically, offset will be eliminated for all values of .

Figure 8.9. PI control: (a) effect of integral time (b) effect of controller gain.


8.3.4.3 Effect of derivative time . derivative action cannot be built(is

Figure 8.10. PID control: effect of derivative time.

  • Increasing the derivative time.

     improved response by reducing the maximum deviation, response time and the degree of oscillation.

  • Too large derivative time.

     measurement noise tends to be amplified and the response may be oscillatory.

  • Intermediate value of is desirable.


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