Balanced model reduction examples¶
Code¶
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 | #!/usr/bin/env python
import os
import numpy as np
import control.modelsimp as msimp
import control.matlab as mt
from control.statesp import StateSpace
import matplotlib.pyplot as plt
plt.close('all')
# controllable canonical realization computed in MATLAB for the
# transfer function: num = [1 11 45 32], den = [1 15 60 200 60]
A = np.array([
[-15., -7.5, -6.25, -1.875],
[8., 0., 0., 0.],
[0., 4., 0., 0.],
[0., 0., 1., 0.]
])
B = np.array([
[2.],
[0.],
[0.],
[0.]
])
C = np.array([[0.5, 0.6875, 0.7031, 0.5]])
D = np.array([[0.]])
# The full system
fsys = StateSpace(A, B, C, D)
# The reduced system, truncating the order by 1
n = 3
rsys = msimp.balred(fsys, n, method='truncate')
# Comparison of the step responses of the full and reduced systems
plt.figure(1)
y, t = mt.step(fsys)
yr, tr = mt.step(rsys)
plt.plot(t.T, y.T)
plt.plot(tr.T, yr.T)
# Repeat balanced reduction, now with 100-dimensional random state space
sysrand = mt.rss(100, 1, 1)
rsysrand = msimp.balred(sysrand, 10, method='truncate')
# Comparison of the impulse responses of the full and reduced random systems
plt.figure(2)
yrand, trand = mt.impulse(sysrand)
yrandr, trandr = mt.impulse(rsysrand)
plt.plot(trand.T, yrand.T, trandr.T, yrandr.T)
if 'PYCONTROL_TEST_EXAMPLES' not in os.environ:
plt.show()
|
Notes¶
1. The environment variable PYCONTROL_TEST_EXAMPLES is used for testing to turn off plotting of the outputs.