# eigenvector example

import numpy as np
import matplotlib.pyplot as plt
from scipy import signal

a = np.array([[-1,-10,-10],[1,0,0],[0,1,0]],dtype=float)
b = np.array([[0],[0],[0]],dtype=float)
c = np.eye(3)
d = np.zeros((3,1))
sys = signal.StateSpace(a, b, c, d)

t, yout, xout = signal.lsim(sys, U=0, T=np.linspace(0,10,100), X0=[0,-1,1])

plt.figure(figsize=(8,4), dpi=100)
plt.plot(t,xout)
plt.grid()
plt.show()


# (right) eigenvectors
D,V = np.linalg.eig(a)
print(f'eval_r:\n{D}')
print(f'evec_r:\n{V}')

# left eigenvectors
D,W = np.linalg.eig(a.T)
print(f'eval_l:\n{D}')
print(f'evec_l:\n{W}')

# w3: left eigenvector associated with lambda=-1
w3 = W[:,2]
print(f'w3:\n{w3}')

# (w_3^T)x history

plt.figure(figsize=(8,4), dpi=100)
plt.plot(t,xout.dot(w3))
plt.grid()
plt.show()


# time history for x_0 = V[:,0].real

t, yout, xout = signal.lsim(sys, U=0, T=np.linspace(0,10,100), X0=V[:,0].real)

plt.figure(figsize=(8,4), dpi=100)
plt.plot(t,xout)
plt.grid()
plt.show()