Q.1 Two systems, G1 and G2 are coupled according to Figure 1 G₁ u G₂ Figure 1 Block diagram of the system in Question 1. a. Determine the transfer function between u and y. b. b. Determine the poles and the zeros of the system when the transfer functions are 2 G₁(s) = = and G₂ (s) S+4 s+5 c. Determine and draw the step response of the system. =
Q.1 Two systems, G1 and G2 are coupled according to Figure 1 G₁ u G₂ Figure 1 Block diagram of the system in Question 1. a. Determine the transfer function between u and y. b. b. Determine the poles and the zeros of the system when the transfer functions are 2 G₁(s) = = and G₂ (s) S+4 s+5 c. Determine and draw the step response of the system. =
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#### Figure 1: Block diagram of the system in Question 1.
##### Questions:
1. **Determine the transfer function between u and y.**
2. **Determine the poles and the zeros of the system when the transfer functions are given by:**
\[
G_1(s) = \frac{1}{s+4}
\]
\[
G_2(s) = \frac{2}{s+5}
\]
3. **Determine and draw the step response of the system.**
### Detailed Explanation:
**Block Diagram Description:**
- The diagram shows two systems, \(G_1\) and \(G_2\), connected in a parallel configuration.
- The input to the system is denoted as \(u\) and the output is \(y\).
- Both \(G_1\) and \(G_2\) receive the same input \(u\).
- The outputs of \(G_1\) and \(G_2\) are summed to produce the output \(y\).
### Answer Breakdown:
**(a) Transfer Function Determination:**
Calculate the overall transfer function \( G(s) \) from input \( u \) to output \( y \).
Given:
\[ G_1(s) = \frac{1}{s+4} \]
\[ G_2(s) = \frac{2}{s+5} \]
The overall transfer function for the parallel configuration:
\[ G(s) = G_1(s) + G_2(s) \]
\[ G(s) = \frac{1}{s+4} + \frac{2}{s+5} \]
**(b) Poles and Zeros Determination:**
Identify and calculate the poles and zeros of the combined transfer function \( G(s) \).
**(c) Step Response Determination:**
Compute and plot the step response of the system \( G(s) \) when subjected to a unit step input.
### Solutions:
Work through each problem step-by-step to provide the required information for educational purposes. Graphs and plots will be detailed for clarity.
#### Note:
Ensure to conclude by summarizing findings and emphasizing key learning](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F69a887a2-135a-4778-a06d-e94a31f4008c%2F066ea690-5d23-4a94-8425-de4e5a0d9993%2Ffa89ls6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Control Systems Problem: Analysis of Coupled Systems
**Question 1:**
Two systems, G1 and G2, are coupled according to Figure 1.
#### Figure 1: Block diagram of the system in Question 1.
##### Questions:
1. **Determine the transfer function between u and y.**
2. **Determine the poles and the zeros of the system when the transfer functions are given by:**
\[
G_1(s) = \frac{1}{s+4}
\]
\[
G_2(s) = \frac{2}{s+5}
\]
3. **Determine and draw the step response of the system.**
### Detailed Explanation:
**Block Diagram Description:**
- The diagram shows two systems, \(G_1\) and \(G_2\), connected in a parallel configuration.
- The input to the system is denoted as \(u\) and the output is \(y\).
- Both \(G_1\) and \(G_2\) receive the same input \(u\).
- The outputs of \(G_1\) and \(G_2\) are summed to produce the output \(y\).
### Answer Breakdown:
**(a) Transfer Function Determination:**
Calculate the overall transfer function \( G(s) \) from input \( u \) to output \( y \).
Given:
\[ G_1(s) = \frac{1}{s+4} \]
\[ G_2(s) = \frac{2}{s+5} \]
The overall transfer function for the parallel configuration:
\[ G(s) = G_1(s) + G_2(s) \]
\[ G(s) = \frac{1}{s+4} + \frac{2}{s+5} \]
**(b) Poles and Zeros Determination:**
Identify and calculate the poles and zeros of the combined transfer function \( G(s) \).
**(c) Step Response Determination:**
Compute and plot the step response of the system \( G(s) \) when subjected to a unit step input.
### Solutions:
Work through each problem step-by-step to provide the required information for educational purposes. Graphs and plots will be detailed for clarity.
#### Note:
Ensure to conclude by summarizing findings and emphasizing key learning
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