ASE6029: Covariance propagation¶

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ASE6029: Linear Optimal Control, Inha University.

Jong-Han Kim (jonghank@inha.ac.kr)

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import numpy as np
import matplotlib.pyplot as plt

A = np.array([[0.6, -0.8],[ 0.7, 0.6]])

eval, evec = np.linalg.eig(A)
print(eval)
print(abs(eval))

[0.6+0.74833148j 0.6-0.74833148j]
[0.9591663 0.9591663]

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Sx = np.eye(2)*1600
Sw = np.eye(2)
Sx11 = []
for i in range(100):
  Sx = A@Sx@A.T + Sw
  Sx11.append(Sx[0,0])
print(Sx)

plt.figure(figsize=(8,4), dpi=100)
plt.plot(Sx11)
plt.show()

[[13.7888224  -0.02868337]
 [-0.02868337 12.08430949]]

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x = 40*np.random.randn(2)
x_hist = []
for t in range(400):
  x = A.dot(x) + np.random.randn(2)
  x_hist.append(x)

plt.figure(figsize=(8,4), dpi=100)
plt.plot(x_hist)
plt.show()

x_ss = np.array(x_hist).T[:,200:]
print(np.cov(x_ss))

[[12.43797211 -0.18757096]
 [-0.18757096 10.46040716]]

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