Edit Figure: Figure 5.9: Phase portrait and time domain simulation for a system with a single unstable equilibrium point

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{| |- align=top | rowspan=3 | [[Image:figure-5.9-unstable_eqpt-pp.png]] | align=center | <math> \begin{aligned} \dot x_1 &= x_1 - 3 x_2 \\ \dot x_2 &= -3 x_1 + x_2 \end{aligned} </math> |- align=bottom | rowspan=2 | [[Image:figure-5.9-untable_eqpt-time.png]] |- |} '''Figure 5.9:''' Phase portrait and time domain simulation for a system with a single unstable equilibrium point. The equilibrium point $x_\text{e}$ at the origin is unstable since not all trajectories that start near $x_\text{e}$ stay near $x_\text{e}$. The sample trajectory on the right shows that the trajectories very quickly depart from zero. <nowiki> # unstable_eqpt.py - plots for stable equlibrium point # RMM, 7 Apr 2024 import matplotlib.pyplot as plt import numpy as np import control as ct import fbs # FBS plotting customizations saddle = ct.ss([[1, -3], [-3, 1]], [[0], [1]], np.eye(2), 0) # Draw the phase portrait fbs.figure() ct.phase_plane_plot( saddle, [-1, 1, -1, 1], 0.4, gridtype='meshgrid', gridspec=[6, 6]) plt.gca().set_aspect('equal') plt.suptitle("") fbs.savefig('figure-5.9-unystable_eqpt-pp.png') fbs.figure('321') plt.axis([0, 3, -100, 100]) timepts = np.linspace(0, 3) response = ct.input_output_response(saddle, timepts, 0, [1, 0]) plt.plot(response.time, response.outputs[0], 'b', label="$x_1$") plt.plot(response.time, response.outputs[1], 'r--', label="$x_2$") plt.xlabel("Time $t$") plt.ylabel("$x_1, x_2$") plt.legend(loc='upper right', ncols=1, frameon=False) fbs.savefig('figure-5.9-unstable_eqpt-time.png') </nowiki>

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