In [1]:
from sympy import *
init_printing()
In [2]:
W_P, W_I, G, I, K = var('W_P, W_I, G, I, K', commutative=False)
In [3]:
P = Matrix([[0, 0, W_I],
[W_P*G, W_P, W_P*G],
[-G, -I, -G]])
In [4]:
P
Out[4]:
$$\left[\begin{matrix}0 & 0 & W_{I}\\W_{P} G & W_{P} & W_{P} G\\- G & - I & - G\end{matrix}\right]$$
In [5]:
P11 = P[:2, :2]
P11
Out[5]:
$$\left[\begin{matrix}0 & 0\\W_{P} G & W_{P}\end{matrix}\right]$$
In [6]:
P12 = P[:2,2:]
P12
Out[6]:
$$\left[\begin{matrix}W_{I}\\W_{P} G\end{matrix}\right]$$
In [7]:
P21 = P[2:, :2]
P21
Out[7]:
$$\left[\begin{matrix}- G & - I\end{matrix}\right]$$
In [8]:
P22 = P[2:,2:]
P22
Out[8]:
$$\left[\begin{matrix}- G\end{matrix}\right]$$
In [9]:
Im = Matrix([[I]])
In [10]:
(Im + P22*K).inv()
Out[10]:
$$\left[\begin{matrix}\left(- G K + I\right)^{-1}\end{matrix}\right]$$
In [11]:
N = P11 + P12*K*(Im - P22*K).inv()*P21
N
Out[11]:
$$\left[\begin{matrix}- W_{I} K \left(G K + I\right)^{-1} G & - W_{I} K \left(G K + I\right)^{-1} I\\W_{P} G - W_{P} G K \left(G K + I\right)^{-1} G & W_{P} - W_{P} G K \left(G K + I\right)^{-1} I\end{matrix}\right]$$