In [1]:
import control as ct
import numpy as np
from numpy import linalg as LA
In [2]:
A =[[-7, -12],
     [1,    0]]
B = [1, 0]
C = [1, 2]
D = 0
sys = ct.ss(A, B, C, D)
In [3]:
ct.ss2tf(sys)
Out[3]:
$$\frac{s + 2}{s^2 + 7 s + 12}$$
In [4]:
eigenvalues, eigenvectors = LA.eig(A)
print(eigenvalues)
print(eigenvectors)

[-4. -3.]
[[-0.9701425   0.9486833 ]
 [ 0.24253563 -0.31622777]]
In [5]:
#eigenvector inverse가 제대로 되었는지 확인
inv_eigen = LA.inv(eigenvectors)
np.dot(inv_eigen, eigenvectors)
Out[5]:
array([[1.0000000e+00, 4.4408921e-16],
       [0.0000000e+00, 1.0000000e+00]])
In [6]:
# A * x = x * lamda 확인
A_x = np.dot(A, eigenvectors) # A @ eigenvectors로 써도 같은 결과.
print("A dot eigenvector x is \n{}\n".format(A_x))
#lamda matrix 설정
lamda_matrix = [[-4, 0],[0,-3]]
x_lamda = eigenvectors@lamda_matrix
print("eigenvector dot lamda is\n{}".format(x_lamda))

A dot eigenvector x is 
[[ 3.88057    -2.84604989]
 [-0.9701425   0.9486833 ]]

eigenvector dot lamda is
[[ 3.88057    -2.84604989]
 [-0.9701425   0.9486833 ]]
In [7]:
#modal canonical form
#lamda = inv(eigenvector) @ A @ eigenvector
lamda_calculated = inv_eigen @ A @ eigenvectors
print("modal canonical form  is \n{}".format(lamda_calculated))

modal canonical form  is 
[[-4.00000000e+00  1.77635684e-15]
 [ 0.00000000e+00 -3.00000000e+00]]