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13. (s+2) (s+1)²+(1)² Let G[s] = = produce a stable closed-loop system with a closed-loop pole at s = -1. and Ge[s] = K. Select a non-negative gain, K, to (a) K doesn't exist. (b) K = 1.0 (c) K = 0.5 (d) K = 2.0

13. (s+2) (s+1)²+(1)² Let G[s] = = produce a stable closed-loop system with a closed-loop pole at s = -1. and Ge[s] = K. Select a non-negative gain, K, to (a) K doesn't exist. (b) K = 1.0 (c) K = 0.5 (d) K = 2.0

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
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**Problem Statement:** Let \( G[s] = \frac{(s+2)}{(s+1)^2(s+1)^2} \) and \( G_c[s] = K \). Select a non-negative gain, \( K \), to produce a stable closed-loop system with a closed-loop pole at \( s = -1 \). **Options:** (a) \( K \) doesn’t exist. (b) \( K = 1.0 \) (c) \( K = 0.5 \) (d) \( K = 2.0 \) **Explanation:** This problem involves finding the appropriate gain \( K \) for a given transfer function \( G[s] \) so that the closed-loop system is stable and has a closed-loop pole at \( s = -1 \). The transfer function \( G[s] \) has a numerator \( (s+2) \) and a denominator \( (s+1)^2(s+1)^2 \). The gain \( K \) is part of the controller transfer function \( G_c[s] \). To solve the problem, you must analyze the system and determine the conditions under which the pole of the closed-loop system lies at the specified location.
Transcribed Image Text:**Problem Statement:** Let \( G[s] = \frac{(s+2)}{(s+1)^2(s+1)^2} \) and \( G_c[s] = K \). Select a non-negative gain, \( K \), to produce a stable closed-loop system with a closed-loop pole at \( s = -1 \). **Options:** (a) \( K \) doesn’t exist. (b) \( K = 1.0 \) (c) \( K = 0.5 \) (d) \( K = 2.0 \) **Explanation:** This problem involves finding the appropriate gain \( K \) for a given transfer function \( G[s] \) so that the closed-loop system is stable and has a closed-loop pole at \( s = -1 \). The transfer function \( G[s] \) has a numerator \( (s+2) \) and a denominator \( (s+1)^2(s+1)^2 \). The gain \( K \) is part of the controller transfer function \( G_c[s] \). To solve the problem, you must analyze the system and determine the conditions under which the pole of the closed-loop system lies at the specified location.
### Block Diagram Description The image presents a block diagram that is essential for understanding a particular system configuration, which will be referenced throughout the exam. Below is a detailed explanation of the components and flow within the diagram: 1. **Inputs:** - **R(λ):** This represents the input signal or reference input. - The symbol "+", next to R(λ), indicates that this input enters a summation junction. 2. **Summation Junction (Σ):** - This component combines the input signals. In this case, signals R(λ) and another feedback loop signal (discussed later) enter the summation junction. 3. **Error Signal (E(λ)):** - Labeled as E(λ), this signal represents the result of the summation process at the junction. It then enters the next block. 4. **Transfer Function Blocks:** - **Gc(λ):** This block represents a transfer function or component of the system, which processes the error signal E(λ). - **G(λ):** The output from Gc(λ) flows into another block represented by G(λ), which further processes the signal. 5. **Output:** - **Y(λ):** The final output of the system after processing through both Gc(λ) and G(λ). 6. **Feedback Loop:** - A feedback loop is present, where the output Y(λ) is fed back into the summation junction. This loop is essential for control systems to adjust the input based on the output. Overall, this block diagram illustrates a typical feedback control system where R(λ) is the reference input, E(λ) is the error signal, and Y(λ) is the system output. The diagram highlights the signal flow and processing through various system components. Understanding this setup is crucial for addressing exam questions related to control system behavior and analysis.
Transcribed Image Text:### Block Diagram Description The image presents a block diagram that is essential for understanding a particular system configuration, which will be referenced throughout the exam. Below is a detailed explanation of the components and flow within the diagram: 1. **Inputs:** - **R(λ):** This represents the input signal or reference input. - The symbol "+", next to R(λ), indicates that this input enters a summation junction. 2. **Summation Junction (Σ):** - This component combines the input signals. In this case, signals R(λ) and another feedback loop signal (discussed later) enter the summation junction. 3. **Error Signal (E(λ)):** - Labeled as E(λ), this signal represents the result of the summation process at the junction. It then enters the next block. 4. **Transfer Function Blocks:** - **Gc(λ):** This block represents a transfer function or component of the system, which processes the error signal E(λ). - **G(λ):** The output from Gc(λ) flows into another block represented by G(λ), which further processes the signal. 5. **Output:** - **Y(λ):** The final output of the system after processing through both Gc(λ) and G(λ). 6. **Feedback Loop:** - A feedback loop is present, where the output Y(λ) is fed back into the summation junction. This loop is essential for control systems to adjust the input based on the output. Overall, this block diagram illustrates a typical feedback control system where R(λ) is the reference input, E(λ) is the error signal, and Y(λ) is the system output. The diagram highlights the signal flow and processing through various system components. Understanding this setup is crucial for addressing exam questions related to control system behavior and analysis.
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