a. Provide and expression for the transfer function G(s) = Y(s)/U(s) of the system. b. Determine the poles and zeros of this system.
a. Provide and expression for the transfer function G(s) = Y(s)/U(s) of the system. b. Determine the poles and zeros of this system.
Introductory Circuit Analysis (13th Edition)
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ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
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![**State Space Representation of a Single-Input, Single-Output (SISO) System**
A single-input, single-output (SISO) system is defined by the following state equations:
\[
\dot{x}(t) = Ax(t) + Bu(t)
\]
\[
y(t) = Cx(t) + Du(t)
\]
For this system, the matrices are given as:
\[
A = \begin{pmatrix}
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
-5726 & -3765 & -477 & -15
\end{pmatrix}
\]
\[
B = \begin{pmatrix}
0 \\
0 \\
0 \\
1
\end{pmatrix}
\]
\[
C = \begin{pmatrix}
75 & 55 & 13 & 1
\end{pmatrix}
\]
\[
D = 0
\]
**Tasks:**
a. **Derive the Transfer Function \( G(s) = \frac{Y(s)}{U(s)} \) of the System**
To find the transfer function, perform the following steps:
1. Compute the Laplace transform of the state equations assuming zero initial conditions.
2. Obtain \( G(s) = C(sI - A)^{-1}B + D \).
b. **Determine the Poles and Zeros of the System**
- **Poles**: The poles of the system are the eigenvalues of matrix \( A \) or the roots of the characteristic equation \( \det(sI - A) = 0 \).
- **Zeros**: The zeros of the system are the values of \( s \) that make \( G(s) = 0 \).
By solving these, you determine the dynamic characteristics of the system's response.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F11657d80-0b9a-4a9b-9635-8d2c2856a255%2Fa41d3aee-9500-4a71-9781-d2ae7693b83c%2Fj1z83n_processed.png&w=3840&q=75)
Transcribed Image Text:**State Space Representation of a Single-Input, Single-Output (SISO) System**
A single-input, single-output (SISO) system is defined by the following state equations:
\[
\dot{x}(t) = Ax(t) + Bu(t)
\]
\[
y(t) = Cx(t) + Du(t)
\]
For this system, the matrices are given as:
\[
A = \begin{pmatrix}
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
-5726 & -3765 & -477 & -15
\end{pmatrix}
\]
\[
B = \begin{pmatrix}
0 \\
0 \\
0 \\
1
\end{pmatrix}
\]
\[
C = \begin{pmatrix}
75 & 55 & 13 & 1
\end{pmatrix}
\]
\[
D = 0
\]
**Tasks:**
a. **Derive the Transfer Function \( G(s) = \frac{Y(s)}{U(s)} \) of the System**
To find the transfer function, perform the following steps:
1. Compute the Laplace transform of the state equations assuming zero initial conditions.
2. Obtain \( G(s) = C(sI - A)^{-1}B + D \).
b. **Determine the Poles and Zeros of the System**
- **Poles**: The poles of the system are the eigenvalues of matrix \( A \) or the roots of the characteristic equation \( \det(sI - A) = 0 \).
- **Zeros**: The zeros of the system are the values of \( s \) that make \( G(s) = 0 \).
By solving these, you determine the dynamic characteristics of the system's response.
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