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# Import needed modules
import sympy as sp
import numpy as numpy
from sympy import init_printing
init_printing()


    

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# Define the symbols
# Needed variables for LaPlace Transformation and the Frequency
s = sp.symbols('s', positive = True)
omega = sp.symbols('omega', positive = True, real = True)
# Needed for the plant
K, T, L = sp.symbols('K T L', positive = True, real = True)
# Define the error
R = sp.symbols('R', positive = True, real = True)

# Define a FOTD
def FOTD(x,y,z):
    return x/(y*s+1)*sp.exp(-z*s)


    

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# Define the plant
G = FOTD(1,T,L)
# Get the series expansion for first order
GA1 = sp.series(G,s,0,2).removeO().subs(s,sp.I*omega)
# Second order
GA2 = sp.series(G,s,0,3).removeO().subs(s,sp.I*omega)
dG = sp.simplify(sp.Abs(GA1)-sp.Abs(GA2))
sp.solve(dG,omega)


    

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GA1p = sp.Abs(GA1.removeO().subs(T,10).subs(L,10).subs(s,sp.I*omega))
GA2p = sp.Abs(GA2.removeO().subs(T,10).subs(L,10).subs(s,sp.I*omega))
GA1p, GA2p


    

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sp.plot((GA1p,(omega,0,.1)),(GA2p,(omega,0,.1)))


    

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2./((20)**2+10**2)**(0.5)


    

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