5.11 Given the SSR x = -20 1 x + [1³] u น y = [1 0]x B a. Obtain the I/O equation for this system where y is the output and u is the input_ b. Obtain the transfer function.
5.11 Given the SSR x = -20 1 x + [1³] u น y = [1 0]x B a. Obtain the I/O equation for this system where y is the output and u is the input_ b. Obtain the transfer function.
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![**5.11 Given the SSR**
\[ \dot{x} = \begin{bmatrix} 0 & 1 \\ -20 & -4 \end{bmatrix} x + \begin{bmatrix} 0.3 \\ 1 \end{bmatrix} u \]
\[ y = \begin{bmatrix} 1 & 0 \end{bmatrix} x \]
a. Obtain the I/O equation for this system where \( y \) is the output and \( u \) is the input.
b. Obtain the transfer function.
---
**Explanation:**
This problem involves a state space representation (SSR) of a dynamic system. The SSR is described by the equations above, where:
- \( \dot{x} \) is the derivative of the state vector \( x \).
- The matrix \( A = \begin{bmatrix} 0 & 1 \\ -20 & -4 \end{bmatrix} \) defines the system dynamics.
- The matrix \( B = \begin{bmatrix} 0.3 \\ 1 \end{bmatrix} \) defines how the input \( u \) affects the state.
- The matrix \( C = \begin{bmatrix} 1 & 0 \end{bmatrix} \) defines how the state \( x \) is observed as the output \( y \).
The tasks are:
- **Task a:** Derive the Input/Output (I/O) equation using the state space representation.
- **Task b:** Determine the transfer function, which relates the input \( u \) to the output \( y \) in the frequency domain.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffe082a62-4024-470b-9494-09a6e9a82c01%2F54d34831-aeac-4b13-a1a5-ae12105a69f0%2F3zu0kb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**5.11 Given the SSR**
\[ \dot{x} = \begin{bmatrix} 0 & 1 \\ -20 & -4 \end{bmatrix} x + \begin{bmatrix} 0.3 \\ 1 \end{bmatrix} u \]
\[ y = \begin{bmatrix} 1 & 0 \end{bmatrix} x \]
a. Obtain the I/O equation for this system where \( y \) is the output and \( u \) is the input.
b. Obtain the transfer function.
---
**Explanation:**
This problem involves a state space representation (SSR) of a dynamic system. The SSR is described by the equations above, where:
- \( \dot{x} \) is the derivative of the state vector \( x \).
- The matrix \( A = \begin{bmatrix} 0 & 1 \\ -20 & -4 \end{bmatrix} \) defines the system dynamics.
- The matrix \( B = \begin{bmatrix} 0.3 \\ 1 \end{bmatrix} \) defines how the input \( u \) affects the state.
- The matrix \( C = \begin{bmatrix} 1 & 0 \end{bmatrix} \) defines how the state \( x \) is observed as the output \( y \).
The tasks are:
- **Task a:** Derive the Input/Output (I/O) equation using the state space representation.
- **Task b:** Determine the transfer function, which relates the input \( u \) to the output \( y \) in the frequency domain.
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