### Feedback Control System Analysis#### System Diagram OverviewThe image illustrates a feedback control system with the following components:- **Input**: \( X(s) \)- **Feedback Path**: Includes a summing junction, which subtracts feedback \( Y(s) \) from input \( X(s) \).- **Controller/Subsystem**: \( G(s) \)- **Feedback Element**: \( H(s) \)- **Output**: \( Y(s) \)#### System Equations- **Transfer Function G(s)**: \[ G(s) = \frac{1}{(s+1)(s+k)} \]- **Feedback Transfer Function H(s)**: \[ H(s) = s + 2 \]### Problem Statements1. **Determine the Transform Function \( H_1(s) = \frac{Y(s)}{X(s)} \)** - This involves calculating the overall transfer function of the feedback system.2. **Discuss the BIBO Stability of the System** - Analyze the stability of the system based on the values of \( k \).### Steps to Solve#### (a) Determining the Transform Function \( H_1(s) \)To find \( H_1(s) \), use the formula for the closed-loop transfer function of a feedback system:\[H_1(s) = \frac{G(s)}{1 + G(s)H(s)}\]Substitute the given \( G(s) \) and \( H(s) \) into the equation and simplify.#### (b) BIBO Stability AnalysisFor Bounded Input, Bounded Output (BIBO) stability, assess the poles of the closed-loop transfer function:- Identify the poles of \( H_1(s) \) to determine stability.- The system is stable if all poles have negative real parts.Discuss how \( k \) affects the pole locations and thus the stability.### DiscussionThis analysis provides insights into ensuring system stability, which is critical in control systems design. Understanding feedback loops and their influence on system behavior enables better system performance and reliability.

Answered: Image /qna-images/answer/607cad2b-182d-4a97-b3be-3fe1f3096f61.jpg

Consider the following feedback system X(s) E(s) Y(s) Σ G(0) H(s) 1 where G(s)=- H(s)=s+2. (s+1)(s+k)* (a) Determine the transform function H,(s)=SI X(s) (b) Discuss the BIBO stability of the system in terms of the values of k.

Consider the following feedback system X(s) E(s) Y(s) Σ G(0) H(s) 1 where G(s)=- H(s)=s+2. (s+1)(s+k)* (a) Determine the transform function H,(s)=SI X(s) (b) Discuss the BIBO stability of the system in terms of the values of k.

Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

Section: Chapter Questions

Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...

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