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For the system shown below do the following: R Determine the range of K for the system to be stable Determine the steady state error for a unit step input and a unit ramp input K s(s+1)(s+2) C

For the system shown below do the following: R Determine the range of K for the system to be stable Determine the steady state error for a unit step input and a unit ramp input K s(s+1)(s+2) C

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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### Control System Analysis #### Problem Statement: For the system shown in the diagram below, complete the following tasks: - Determine the range of \( K \) for the system to be stable. - Determine the steady-state error for a unit step input and a unit ramp input. #### System Diagram: The diagram represents a feedback control system with the following components: - **Block Diagram Components:** - **Input (\(\overline{R}\))**: The reference input to the system. - **Summing Junction**: The input \(\overline{R}\) and the output \(\overline{C}\) are combined. The output is the error signal fed into the next block. - **Transfer Function Block**: The system's transfer function is \(\frac{K}{s(s+1)(s+2)}\), where \( K \) is the gain. - **Feedback Path**: The output \(\overline{C}\) is fed back and subtracted from the input \(\overline{R}\). #### Objectives: 1. **Stability Analysis**: - Determine the range of the gain \( K \) that ensures system stability. Use techniques such as the Routh-Hurwitz criterion or root locus analysis. 2. **Steady-State Error Calculation**: - Use the final value theorem to calculate the steady-state error for: - A unit step input. - A unit ramp input. The analysis involves examining the poles and zeros of the transfer function and assessing the impact of various inputs on system performance. Stability and error analysis are key to ensuring the desired behavior of the control system.
Transcribed Image Text:### Control System Analysis #### Problem Statement: For the system shown in the diagram below, complete the following tasks: - Determine the range of \( K \) for the system to be stable. - Determine the steady-state error for a unit step input and a unit ramp input. #### System Diagram: The diagram represents a feedback control system with the following components: - **Block Diagram Components:** - **Input (\(\overline{R}\))**: The reference input to the system. - **Summing Junction**: The input \(\overline{R}\) and the output \(\overline{C}\) are combined. The output is the error signal fed into the next block. - **Transfer Function Block**: The system's transfer function is \(\frac{K}{s(s+1)(s+2)}\), where \( K \) is the gain. - **Feedback Path**: The output \(\overline{C}\) is fed back and subtracted from the input \(\overline{R}\). #### Objectives: 1. **Stability Analysis**: - Determine the range of the gain \( K \) that ensures system stability. Use techniques such as the Routh-Hurwitz criterion or root locus analysis. 2. **Steady-State Error Calculation**: - Use the final value theorem to calculate the steady-state error for: - A unit step input. - A unit ramp input. The analysis involves examining the poles and zeros of the transfer function and assessing the impact of various inputs on system performance. Stability and error analysis are key to ensuring the desired behavior of the control system.
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