A plane is heading to LAX with constant acceleration of 60 km/h². At time t = 0, the plane stays still at position 0 km. The state of the plane is measured every hour. (In other words, there is an hour between t and t + 1). The state vector x has 3 components. x = x1 X2 X3 position velocity [acceleration] For motion with constant acceleration, the velocity v of the object at time t2 is calculated using equation Vt₂ = Vt₁ + a(t2 − t₁). The position a of the object at time tą is calculated using equation æ₁₂ = x₁, +v₁₂(t2 − t₁) + ½ a(t2 − t₁)². 1 (a) Model the problem as a linear dynamical system.
A plane is heading to LAX with constant acceleration of 60 km/h². At time t = 0, the plane stays still at position 0 km. The state of the plane is measured every hour. (In other words, there is an hour between t and t + 1). The state vector x has 3 components. x = x1 X2 X3 position velocity [acceleration] For motion with constant acceleration, the velocity v of the object at time t2 is calculated using equation Vt₂ = Vt₁ + a(t2 − t₁). The position a of the object at time tą is calculated using equation æ₁₂ = x₁, +v₁₂(t2 − t₁) + ½ a(t2 − t₁)². 1 (a) Model the problem as a linear dynamical system.
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Transcribed Image Text:A plane is heading to LAX with constant acceleration of 60 km/h². At time t = 0, the plane stays still at position 0 km. The
state of the plane is measured every hour. (In other words, there is an hour between t and t + 1). The state vector x has 3
components. x =
x1
X2
x3
=
position
velocity
Lacceleration_
=
For motion with constant acceleration, the velocity v of the object at time tê is calculated using equation V2
The position of the object at time t2 is calculated using equation ₂ = xt₁ + vt₁(t2 − t₁) + = a(t2 − t₁)² .
(a) Model the problem as a linear dynamical system.
(b) Use python to simulate the state vector for 10 hours.
vt₁ + a(t2 − t₁).
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I'm still comfusion about Part A. Can you explain it more Clearly?
And also, I got the sample code for part B, Can you modify the code base on it?
![```python
import matplotlib.pyplot as plt
import numpy as np
x0 = ...
A = ...
states = np.zeros((11,3))
states[0] = x0
# Use for loop to calculate x_t
for t in range(1,11):
...
# Plotting
times = np.arange(0, 11, 1)
plt.plot(times, ..., color='r', label='Plane Position')
plt.plot(times, ..., color='g', label='Plane Velocity')
plt.plot(times, ..., color='b', label='Plane Acceleration')
# Naming the x-axis, y-axis and the whole graph
plt.xlabel("Time")
plt.ylabel("State Vector")
plt.title("State Trajectory of the Plane")
# Adding legend
plt.legend()
# To load the display window
plt.show()
```
### Explanation of the Code:
1. **Imports:**
- `matplotlib.pyplot` as `plt`: Used for plotting graphs.
- `numpy` as `np`: Used for numerical operations and handling arrays.
2. **State Initialization:**
- `x0` and `A`: Placeholder variables representing initial conditions and a transformation matrix, respectively.
- `states`: A NumPy array initialized with zeros, having dimensions `11x3`, intended to store state vectors over time.
- `states[0] = x0`: Sets the initial state.
3. **State Calculation Loop:**
- A loop intended to iterate from 1 to 10 to calculate state vectors (details not provided).
4. **Plotting:**
- `times = np.arange(0, 11, 1)`: Generates an array representing discrete time points from 0 to 10.
- `plt.plot(...)`: Commands to plot state trajectories, with:
- `color='r'`: Red for plane position.
- `color='g'`: Green for plane velocity.
- `color='b'`: Blue for plane acceleration.
5. **Graph Labeling:**
- `plt.xlabel("Time")`: Labels the x-axis as "Time".
- `plt.ylabel("State Vector")`: Labels the y-axis as "State Vector".
- `plt.title("State Trajectory of the Plane")`: Sets the graph title.
6. **Legend:**
- `plt.legend()`: Adds a legend to describe the plotted lines.
7. **Display](https://content.bartleby.com/qna-images/question/a98117fc-4537-4efb-9016-9993926a8208/81fad2ad-d008-47a9-842e-a33d477b6b7a/tgybfni_processed.png)
Transcribed Image Text:```python
import matplotlib.pyplot as plt
import numpy as np
x0 = ...
A = ...
states = np.zeros((11,3))
states[0] = x0
# Use for loop to calculate x_t
for t in range(1,11):
...
# Plotting
times = np.arange(0, 11, 1)
plt.plot(times, ..., color='r', label='Plane Position')
plt.plot(times, ..., color='g', label='Plane Velocity')
plt.plot(times, ..., color='b', label='Plane Acceleration')
# Naming the x-axis, y-axis and the whole graph
plt.xlabel("Time")
plt.ylabel("State Vector")
plt.title("State Trajectory of the Plane")
# Adding legend
plt.legend()
# To load the display window
plt.show()
```
### Explanation of the Code:
1. **Imports:**
- `matplotlib.pyplot` as `plt`: Used for plotting graphs.
- `numpy` as `np`: Used for numerical operations and handling arrays.
2. **State Initialization:**
- `x0` and `A`: Placeholder variables representing initial conditions and a transformation matrix, respectively.
- `states`: A NumPy array initialized with zeros, having dimensions `11x3`, intended to store state vectors over time.
- `states[0] = x0`: Sets the initial state.
3. **State Calculation Loop:**
- A loop intended to iterate from 1 to 10 to calculate state vectors (details not provided).
4. **Plotting:**
- `times = np.arange(0, 11, 1)`: Generates an array representing discrete time points from 0 to 10.
- `plt.plot(...)`: Commands to plot state trajectories, with:
- `color='r'`: Red for plane position.
- `color='g'`: Green for plane velocity.
- `color='b'`: Blue for plane acceleration.
5. **Graph Labeling:**
- `plt.xlabel("Time")`: Labels the x-axis as "Time".
- `plt.ylabel("State Vector")`: Labels the y-axis as "State Vector".
- `plt.title("State Trajectory of the Plane")`: Sets the graph title.
6. **Legend:**
- `plt.legend()`: Adds a legend to describe the plotted lines.
7. **Display
Solution
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