P6.4 A feedback control system is shown in Figure P6.4. The controller and process transfer functions are given by s + 40 s(s + 10) G(s) = K and G(s) and the feedback transfer ftinction is H(s) = 1/(s + 20). (a) Determine the limiting value of gain K for a stable system. (b) For the gain that results in marginal stability, determine the magnitude of the imaginary roots.
P6.4 A feedback control system is shown in Figure P6.4. The controller and process transfer functions are given by s + 40 s(s + 10) G(s) = K and G(s) and the feedback transfer ftinction is H(s) = 1/(s + 20). (a) Determine the limiting value of gain K for a stable system. (b) For the gain that results in marginal stability, determine the magnitude of the imaginary roots.
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![**P6.4 Feedback Control System Analysis**
A feedback control system is illustrated in Figure P6.4. The system's behavior is described through its controller and process transfer functions, which are given by:
\[ G_c(s) = K \]
\[ G(s) = \frac{s + 40}{s(s + 10)} \]
The feedback transfer function is:
\[ H(s) = \frac{1}{s + 20} \]
**Questions for Analysis:**
(a) **Determine the Limiting Value of Gain \( K \):** Establish the limiting value for the gain \( K \) that ensures the system remains stable.
(b) **Magnitude of Imaginary Roots for Marginal Stability:** Calculate the magnitude of the imaginary roots when the system is on the verge of instability, i.e., when it achieves marginal stability.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdb6e49c2-d01d-4438-a99a-92f670cec770%2F1d7cd3ee-de68-46a8-b49d-f75db7d8e890%2Fzubb8r4_processed.png&w=3840&q=75)
Transcribed Image Text:**P6.4 Feedback Control System Analysis**
A feedback control system is illustrated in Figure P6.4. The system's behavior is described through its controller and process transfer functions, which are given by:
\[ G_c(s) = K \]
\[ G(s) = \frac{s + 40}{s(s + 10)} \]
The feedback transfer function is:
\[ H(s) = \frac{1}{s + 20} \]
**Questions for Analysis:**
(a) **Determine the Limiting Value of Gain \( K \):** Establish the limiting value for the gain \( K \) that ensures the system remains stable.
(b) **Magnitude of Imaginary Roots for Marginal Stability:** Calculate the magnitude of the imaginary roots when the system is on the verge of instability, i.e., when it achieves marginal stability.
Transcribed Image Text:### Diagram Explanation: Control System Block Diagram
This block diagram, labeled "Figure P6.4," represents a feedback control system commonly used in engineering for regulating dynamic systems.
- **R(s):** This is the reference input signal, representing the desired output of the system.
- **Σ (Summing Junction):** The input reference R(s) is added with feedback, yielding the error signal \( E_a(s) \). The summing junction combines or subtracts these signals as required.
- **Controller \( G_c(s) \):** The error signal \( E_a(s) \) is processed by the controller. The controller modifies the signal based on the control law designed to minimize the error.
- **Process \( G(s) \):** The output of the controller is fed into the process or plant, represented by the transfer function \( G(s) \). This block represents the system under control.
- **Y(s):** The output of the process is denoted as Y(s), which is the actual output being controlled.
- **Sensor \( H(s) \):** The output Y(s) is fed back through a sensor with a transfer function \( H(s) \). The sensor measures the output and provides feedback to the summing junction to compare with the reference input R(s).
- **Feedback Loop:** The system uses feedback to reduce the error by adjusting the system input, enhancing stability and accuracy.
This diagram exemplifies how various components of a feedback control system interact to achieve the desired system performance.
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