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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Zero order hold sampling of a first order system
\begin{equation}
G(s) = \frac{1}{s-\lambda}
\end{equation}
Step1:  Sampling and taking the z-transform of the step-response
\begin{equation}
Y(z) = \frac{1}{\lambda} \left( \frac{z}{z-\mathrm{e}^{\lambda h}} - \frac{z}{z-1} \right).
\end{equation}
Dividing by the z-transform of the input signal
\begin{equation}
H(z) = \frac{z-1}{z}Y(z) = \frac{1}{\lambda} \left( \frac{ \mathrm{e}^{\lambda h} - 1 }{ z - \mathrm{e}^{\lambda h} } \right)
\end{equation}
Verifying for specific value of lambda
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Python Code:
h, lam = sy.symbols('h, lambda', real=True, positive=True)
s, z = sy.symbols('s, z', real=False)
G = 1/(s-lam)
Y = G/s
Yp = sy.apart(Y, s)
Yp
from sympy.integrals.transforms import inverse_laplace_transform
from sympy.abc import t
inverse_laplace_transform(Yp, s, t)
Explanation: Zero order hold sampling of a first order system
\begin{equation}
G(s) = \frac{1}{s-\lambda}
\end{equation}
End of explanation
lam = -0.5
h = 0.1
G = cm.tf([1], [1, -lam])
Gd = cm.c2d(G, h)
Hd = 1/lam * cm.tf([np.exp(lam*h)-1],[1, np.exp(lam*h)])
print(Gd)
print(Hd)
Explanation: Sampling and taking the z-transform of the step-response
\begin{equation}
Y(z) = \frac{1}{\lambda} \left( \frac{z}{z-\mathrm{e}^{\lambda h}} - \frac{z}{z-1} \right).
\end{equation}
Dividing by the z-transform of the input signal
\begin{equation}
H(z) = \frac{z-1}{z}Y(z) = \frac{1}{\lambda} \left( \frac{ \mathrm{e}^{\lambda h} - 1 }{ z - \mathrm{e}^{\lambda h} } \right)
\end{equation}
Verifying for specific value of lambda
End of explanation
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