Answer :
### Distinguishing Between Feedback and Feedforward Controllers
#### Feedback Controller:
1. Definition:
A feedback controller adjusts its actions based on the difference between the desired and actual system output. This difference is known as the error signal.
2. Mechanism:
- The error signal is calculated by comparing the process output with the setpoint.
- The controller uses this error signal to compute control actions that will minimize the error over time.
- Common forms of feedback controllers include Proportional (P), Integral (I), and Derivative (D) controllers, often combined as PID controllers.
3. Usage:
- Feedback controllers are typically used to correct disturbances that affect the system after they have occurred.
- They provide robustness against system changes and external disturbances since they constantly adjust the control input to achieve the desired output.
#### Feedforward Controller:
1. Definition:
A feedforward controller predicts the impact of disturbances on the system and compensates for them before they influence the process output.
2. Mechanism:
- The controller measures disturbances and calculates the necessary control action to counteract these disturbances directly.
- It focuses on maintaining the desired output by anticipating changes rather than reacting to them post-occurrence.
3. Usage:
- Feedforward controllers are effective in dealing with predictable and measurable disturbances before they affect the process.
- However, they do not inherently provide correction for unpredicted disturbances or model discrepancies.
#### Combined Usage:
- Together:
- Feedback and feedforward control strategies can be used together to combine the advantages of both systems.
- Feedforward control can handle predictable disturbances ahead of time, while feedback control can correct any residual errors or unexpected disturbances.
- The combination provides a more robust and reliable control solution, optimizing system performance by addressing both anticipated and unforeseen discrepancies.
### Analysis of the Given Process with Unity Feedback and Proportional Controller
Given:
[tex]\[ G(s) = \frac{0.2}{s + 0.4} \][/tex]
is controlled with a proportional controller with gain [tex]\( K \)[/tex], in a negative unity feedback configuration.
1. Open-Loop Transfer Function:
- The open-loop transfer function of the system can be represented as the product of the process transfer function [tex]\( G(s) \)[/tex] and the proportional controller [tex]\( K \)[/tex]:
[tex]\[ L(s) = K \cdot G(s) = K \cdot \frac{0.2}{s + 0.4} \][/tex]
2. Feedback Configuration:
- In a negative unity feedback system, the feedback loop has a transfer function [tex]\( H(s) = 1 \)[/tex].
3. Closed-Loop Transfer Function:
- The closed-loop transfer function [tex]\( T(s) \)[/tex] is given by:
[tex]\[ T(s) = \frac{G(s) \cdot K}{1 + G(s) \cdot K \cdot H(s)} \][/tex]
4. Substituting [tex]\( G(s) \)[/tex] and [tex]\( H(s) = 1 \)[/tex]:
- Substitute [tex]\( G(s) = \frac{0.2}{s + 0.4} \)[/tex] and [tex]\( H(s) = 1 \)[/tex]:
[tex]\[ T(s) = \frac{\frac{0.2K}{s + 0.4}}{1 + \frac{0.2K}{s + 0.4}} \][/tex]
5. Simplify the Transfer Function:
- To simplify, find the common denominator and combine terms:
[tex]\[ T(s) = \frac{0.2K}{s + 0.4 + 0.2K} \][/tex]
So, the closed-loop transfer function [tex]\( T(s) \)[/tex] of the system is:
[tex]\[ T(s) = \frac{0.2K}{s + 0.4 + 0.2K} \][/tex]
#### Feedback Controller:
1. Definition:
A feedback controller adjusts its actions based on the difference between the desired and actual system output. This difference is known as the error signal.
2. Mechanism:
- The error signal is calculated by comparing the process output with the setpoint.
- The controller uses this error signal to compute control actions that will minimize the error over time.
- Common forms of feedback controllers include Proportional (P), Integral (I), and Derivative (D) controllers, often combined as PID controllers.
3. Usage:
- Feedback controllers are typically used to correct disturbances that affect the system after they have occurred.
- They provide robustness against system changes and external disturbances since they constantly adjust the control input to achieve the desired output.
#### Feedforward Controller:
1. Definition:
A feedforward controller predicts the impact of disturbances on the system and compensates for them before they influence the process output.
2. Mechanism:
- The controller measures disturbances and calculates the necessary control action to counteract these disturbances directly.
- It focuses on maintaining the desired output by anticipating changes rather than reacting to them post-occurrence.
3. Usage:
- Feedforward controllers are effective in dealing with predictable and measurable disturbances before they affect the process.
- However, they do not inherently provide correction for unpredicted disturbances or model discrepancies.
#### Combined Usage:
- Together:
- Feedback and feedforward control strategies can be used together to combine the advantages of both systems.
- Feedforward control can handle predictable disturbances ahead of time, while feedback control can correct any residual errors or unexpected disturbances.
- The combination provides a more robust and reliable control solution, optimizing system performance by addressing both anticipated and unforeseen discrepancies.
### Analysis of the Given Process with Unity Feedback and Proportional Controller
Given:
[tex]\[ G(s) = \frac{0.2}{s + 0.4} \][/tex]
is controlled with a proportional controller with gain [tex]\( K \)[/tex], in a negative unity feedback configuration.
1. Open-Loop Transfer Function:
- The open-loop transfer function of the system can be represented as the product of the process transfer function [tex]\( G(s) \)[/tex] and the proportional controller [tex]\( K \)[/tex]:
[tex]\[ L(s) = K \cdot G(s) = K \cdot \frac{0.2}{s + 0.4} \][/tex]
2. Feedback Configuration:
- In a negative unity feedback system, the feedback loop has a transfer function [tex]\( H(s) = 1 \)[/tex].
3. Closed-Loop Transfer Function:
- The closed-loop transfer function [tex]\( T(s) \)[/tex] is given by:
[tex]\[ T(s) = \frac{G(s) \cdot K}{1 + G(s) \cdot K \cdot H(s)} \][/tex]
4. Substituting [tex]\( G(s) \)[/tex] and [tex]\( H(s) = 1 \)[/tex]:
- Substitute [tex]\( G(s) = \frac{0.2}{s + 0.4} \)[/tex] and [tex]\( H(s) = 1 \)[/tex]:
[tex]\[ T(s) = \frac{\frac{0.2K}{s + 0.4}}{1 + \frac{0.2K}{s + 0.4}} \][/tex]
5. Simplify the Transfer Function:
- To simplify, find the common denominator and combine terms:
[tex]\[ T(s) = \frac{0.2K}{s + 0.4 + 0.2K} \][/tex]
So, the closed-loop transfer function [tex]\( T(s) \)[/tex] of the system is:
[tex]\[ T(s) = \frac{0.2K}{s + 0.4 + 0.2K} \][/tex]
