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Obtain the transfer function of the system (steps required) -5 -25 -5] 1 x2 +0 u 1 I3 y = 0 25 5] r2

Obtain the transfer function of the system (steps required) -5 -25 -5] 1 x2 +0 u 1 I3 y = 0 25 5] r2

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
Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
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To obtain the transfer function of the system, follow these steps: **State-Space Representation:** Consider the system given in the state-space form: \[ \begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \\ \dot{x}_3 \end{bmatrix} = \begin{bmatrix} -5 & -25 & -5 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} + \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} u \] The output equation is: \[ y = \begin{bmatrix} 0 & 25 & 5 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \] **Explanation:** 1. **State Vector:** - \(\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}\) represents the state variables of the system. 2. **State Matrix (A):** - \(\begin{bmatrix} -5 & -25 & -5 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}\) is the system matrix. 3. **Input Matrix (B):** - \(\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}\) represents how the input \( u \) affects the states. 4. **Output Matrix (C):** - \(\begin{bmatrix} 0 & 25 & 5 \end{bmatrix}\) is used to compute the output \( y \) from the state vector. **Objective:** The aim is to derive the transfer function by transforming the state-space model into the frequency (Laplace) domain and expressing \( Y(s) \) in terms of \( U(s) \). **Steps:** 1. Using Laplace transforms, convert
Transcribed Image Text:To obtain the transfer function of the system, follow these steps: **State-Space Representation:** Consider the system given in the state-space form: \[ \begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \\ \dot{x}_3 \end{bmatrix} = \begin{bmatrix} -5 & -25 & -5 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} + \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} u \] The output equation is: \[ y = \begin{bmatrix} 0 & 25 & 5 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \] **Explanation:** 1. **State Vector:** - \(\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}\) represents the state variables of the system. 2. **State Matrix (A):** - \(\begin{bmatrix} -5 & -25 & -5 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}\) is the system matrix. 3. **Input Matrix (B):** - \(\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}\) represents how the input \( u \) affects the states. 4. **Output Matrix (C):** - \(\begin{bmatrix} 0 & 25 & 5 \end{bmatrix}\) is used to compute the output \( y \) from the state vector. **Objective:** The aim is to derive the transfer function by transforming the state-space model into the frequency (Laplace) domain and expressing \( Y(s) \) in terms of \( U(s) \). **Steps:** 1. Using Laplace transforms, convert
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