7. Consider the control system given below, in which a PID controller is used to control the system. R(S) Gc (S) PID Controller 1 s(s+3)(s+7) G(S) Plant c(s) Where G(s) and Gc(s) = k₁ (1 ++ Tas) Apply Zigler Nichols second method to determine the values of Kp, Ti and Ta and obtain the transfer function of the PID controller.
7. Consider the control system given below, in which a PID controller is used to control the system. R(S) Gc (S) PID Controller 1 s(s+3)(s+7) G(S) Plant c(s) Where G(s) and Gc(s) = k₁ (1 ++ Tas) Apply Zigler Nichols second method to determine the values of Kp, Ti and Ta and obtain the transfer function of the PID controller.
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Please assist with exercise 7 with details on how to do it. Thank you.
![**Control System with PID Controller**
This example illustrates a control system in which a PID (Proportional-Integral-Derivative) controller is employed. The diagram represents the system components and their interactions.
**Diagram Explanation:**
- **Blocks and Signals:**
- There are three main blocks: \( G_C(s) \) (PID Controller), \( G(s) \) (Plant), and a summing junction that receives input \( R(s) \) and output feedback.
- The summing junction subtracts the feedback from the input to form the error signal, which is then processed by the PID controller.
- The output of the PID controller is fed into the plant to produce the final output \( C(s) \).
**Mathematical Representation:**
- **Plant Transfer Function:**
- \[ G(s) = \frac{1}{s(s+3)(s+7)} \]
- This defines the mathematical model of the plant in terms of its poles.
- **PID Controller Transfer Function:**
- \[ G_C(s) = k_p \left( 1 + \frac{1}{T_i s} + T_d s \right) \]
- This represents the PID controller with proportional gain \( k_p \), integral time \( T_i \), and derivative time \( T_d \).
**Task:**
Apply Zigler-Nichols second method to determine the values of \( k_p \), \( T_i \), and \( T_d \) to derive the transfer function of the PID controller. This method involves using specific rules and criteria for tuning PID parameters to achieve optimal control.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1296c196-158c-4a2c-9587-82b5996e9fed%2Fee1457a9-98b6-4307-972e-ac428aece3b6%2Fmwtvwzl_processed.png&w=3840&q=75)
Transcribed Image Text:**Control System with PID Controller**
This example illustrates a control system in which a PID (Proportional-Integral-Derivative) controller is employed. The diagram represents the system components and their interactions.
**Diagram Explanation:**
- **Blocks and Signals:**
- There are three main blocks: \( G_C(s) \) (PID Controller), \( G(s) \) (Plant), and a summing junction that receives input \( R(s) \) and output feedback.
- The summing junction subtracts the feedback from the input to form the error signal, which is then processed by the PID controller.
- The output of the PID controller is fed into the plant to produce the final output \( C(s) \).
**Mathematical Representation:**
- **Plant Transfer Function:**
- \[ G(s) = \frac{1}{s(s+3)(s+7)} \]
- This defines the mathematical model of the plant in terms of its poles.
- **PID Controller Transfer Function:**
- \[ G_C(s) = k_p \left( 1 + \frac{1}{T_i s} + T_d s \right) \]
- This represents the PID controller with proportional gain \( k_p \), integral time \( T_i \), and derivative time \( T_d \).
**Task:**
Apply Zigler-Nichols second method to determine the values of \( k_p \), \( T_i \), and \( T_d \) to derive the transfer function of the PID controller. This method involves using specific rules and criteria for tuning PID parameters to achieve optimal control.
Expert Solution
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Given:
To find:
we have to find the values of Kp , Ti and Td and obtain the transfer function Gc(s) of the PID controller.
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