Difference between revisions of "Figure 5.6: Illustration of Lyapunov’s concept of a stable solution"

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(Created page with "{{Figure |Chapter=Dynamic Behavior |Figure number=5.6 |Sort key=506 |Figure title=Illustration of Lyapunov’s concept of a stable solution |GitHub URL=https://github.com/murr...")
 
 
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'''Figure 5.6:''' Illustration of Lyapunov’s concept of a stable solution. The solution represented by the solid line is stable if we can guarantee that all solutions remain within a tube of diameter ǫ by choosing initial conditions sufficiently close the solution.
 
'''Figure 5.6:''' Illustration of Lyapunov’s concept of a stable solution. The solution represented by the solid line is stable if we can guarantee that all solutions remain within a tube of diameter ǫ by choosing initial conditions sufficiently close the solution.
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<nowiki>
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# lyapunov_stability.py - illustration of Lyapunov stability
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# RMM, 6 Apr 2024
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import matplotlib.pyplot as plt
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import numpy as np
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import control as ct
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import fbs                      # FBS plotting customizations
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t = np.linspace(0, 6)
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x0 = np.sin(t) + 0.8 * np.cos(2.2 * t) + 2.5
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# Plot the centerline and bounds
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fbs.figure('211')
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plt.plot(t, x0, 'k')
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plt.plot(t, x0 + 0.5, 'r', t, x0 - 0.5, 'r')
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# Plot the signal
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x = x0 - 0.4 * np.sin(t - 0.3)
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plt.plot(t, x, 'b--')
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# Label the axes
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plt.xlabel("Time $t$")
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plt.ylabel("State $x$")
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# Add some arrows and label the range of stability
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plt.arrow(1.5, 1.3, 0, 0.8, width=0.01, head_width=0.05)
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plt.arrow(1.5, 4.15, 0, -0.8, width=0.01, head_width=0.05)
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plt.text(1.6, 1.4, "$\\epsilon$")
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fbs.savefig('figure-5.6-lyapunov_stability.png')
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</nowiki>

Latest revision as of 05:06, 7 April 2024

Chapter Dynamic Behavior
Figure number 5.6
Figure title Illustration of Lyapunov’s concept of a stable solution
GitHub URL https://github.com/murrayrm/fbs2e-python/blob/main/figure-5-6-lyapunov stability.py
Requires python-control

Figure-5-6-lyapunov stability.png

Figure 5.6: Illustration of Lyapunov’s concept of a stable solution. The solution represented by the solid line is stable if we can guarantee that all solutions remain within a tube of diameter ǫ by choosing initial conditions sufficiently close the solution. 

# lyapunov_stability.py - illustration of Lyapunov stability
# RMM, 6 Apr 2024

import matplotlib.pyplot as plt
import numpy as np
import control as ct
import fbs                      # FBS plotting customizations

t = np.linspace(0, 6)
x0 = np.sin(t) + 0.8 * np.cos(2.2 * t) + 2.5

# Plot the centerline and bounds
fbs.figure('211')
plt.plot(t, x0, 'k')
plt.plot(t, x0 + 0.5, 'r', t, x0 - 0.5, 'r')

# Plot the signal
x = x0 - 0.4 * np.sin(t - 0.3)
plt.plot(t, x, 'b--')

# Label the axes
plt.xlabel("Time $t$")
plt.ylabel("State $x$")

# Add some arrows and label the range of stability
plt.arrow(1.5, 1.3, 0, 0.8, width=0.01, head_width=0.05)
plt.arrow(1.5, 4.15, 0, -0.8, width=0.01, head_width=0.05)
plt.text(1.6, 1.4, "$\\epsilon$")

fbs.savefig('figure-5.6-lyapunov_stability.png')
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