In [2]:

    

import numpy as np
import sympy as sy
sy.init_printing()


    

In [3]:

    

s = sy.symbols('s', real=False)
T,t = sy.symbols('T,t', real=True, positive=True)
T1,T2 = sy.symbols('T1,T2', real=True, positive=True)


    

In [5]:

    

G = 1/(s*T+1)**2
G


    

    
Out[5]:





$$\frac{1}{\left(T s + 1\right)^{2}}$$

In [6]:

    

Y = sy.apart(G*1/s, s)
Y


    

    
Out[6]:





$$- \frac{T}{T s + 1} - \frac{T}{\left(T s + 1\right)^{2}} + \frac{1}{s}$$

In [7]:

    

y = sy.inverse_laplace_transform(Y,s,t)
y


    

    
Out[7]:





$$\frac{1}{T e^{\frac{t}{T}}} \left(T e^{\frac{t}{T}} - T - t\right)$$

In [8]:

    

G2 = 1/((s*T1+1)*(s*T2+1))
G2


    

    
Out[8]:





$$\frac{1}{\left(T_{1} s + 1\right) \left(T_{2} s + 1\right)}$$

In [9]:

    

Y2 = sy.apart(G2*1/s, s)
Y2


    

    
Out[9]:





$$- \frac{T_{1}^{2}}{\left(T_{1} - T_{2}\right) \left(T_{1} s + 1\right)} + \frac{T_{2}^{2}}{\left(T_{1} - T_{2}\right) \left(T_{2} s + 1\right)} + \frac{1}{s}$$

In [10]:

    

y2 = sy.inverse_laplace_transform(Y2,s,t)
y2


    

    
Out[10]:





$$\frac{1}{\left(T_{1} - T_{2}\right) e^{\frac{t}{T_{2}} + \frac{t}{T_{1}}}} \left(- T_{1} e^{\frac{t}{T_{2}}} + T_{1} e^{\frac{t}{T_{2}} + \frac{t}{T_{1}}} + T_{2} e^{\frac{t}{T_{1}}} - T_{2} e^{\frac{t}{T_{2}} + \frac{t}{T_{1}}}\right)$$

In [11]:

    

sy.simplify(sy.expand(y2))


    

    
Out[11]:





$$\frac{1}{\left(T_{1} - T_{2}\right) e^{\frac{t}{T_{2}} + \frac{t}{T_{1}}}} \left(- T_{1} e^{\frac{t}{T_{2}}} + T_{1} e^{\frac{t \left(T_{1} + T_{2}\right)}{T_{1} T_{2}}} + T_{2} e^{\frac{t}{T_{1}}} - T_{2} e^{\frac{t \left(T_{1} + T_{2}\right)}{T_{1} T_{2}}}\right)$$