Question

The z-transform corresponding to the Laplace transform G(s) =10/s(s+5) is

The question was posed to me in homework.

I want to ask this question from The inverse z-transform and Response of Linear Discrete Systrems in division Digital Control Systems of Control Systems

Answer 1

To find the z-transform corresponding to the Laplace transform G(s)=10s(s+5)G(s) = \frac{10}{s(s+5)}G(s)=s(s+5)10​, we need to express the Laplace transform in terms of the z-domain. We will use the bilateral z-transform and standard methods for converting Laplace expressions to z-domain expressions.

Steps:

1. Partial Fraction Decomposition:

   Decompose the Laplace function G(s)=10s(s+5)G(s) = \frac{10}{s(s+5)}G(s)=s(s+5)10​ into simpler terms:

   10s(s+5)=As+Bs+5\frac{10}{s(s+5)} = \frac{A}{s} + \frac{B}{s+5}s(s+5)10​=sA​+s+5B​

   Multiply both sides by s(s+5)s(s+5)s(s+5) and solve for AAA and BBB:

   10=A(s+5)+B(s)10 = A(s+5) + B(s)10=A(s+5)+B(s)

   Setting s=0s = 0s=0, we get:

   10=A(0+5)⇒A=210 = A(0 + 5) \quad \Rightarrow \quad A = 210=A(0+5)⇒A=2

   Setting s=−5s = -5s=−5, we get:

   10=B(−5)⇒B=−210 = B(-5) \quad \Rightarrow \quad B = -210=B(−5)⇒B=−2

   So the partial fraction decomposition is:

   10s(s+5)=2s−2s+5\frac{10}{s(s+5)} = \frac{2}{s} - \frac{2}{s+5}s(s+5)10​=s2​−s+52​

2. Z-Transform of Each Term:

   For each term, we can use standard z-transform pairs to convert them to the z-domain.

   • The z-transform of 1s\frac{1}{s}s1​ is 11−z−1\frac{1}{1-z^{-1}}1−z−11​.
   • The z-transform of 1s+5\frac{1}{s+5}s+51​ is z−11−5z−1\frac{z^{-1}}{1 - 5z^{-1}}1−5z−1z−1​.

   Using these, we get:

   Z{2s}=21−z−1\mathcal{Z}\left\{ \frac{2}{s} \right\} = \frac{2}{1-z^{-1}}Z{s2​}=1−z−12​ Z{2s+5}=2z−11−5z−1\mathcal{Z}\left\{ \frac{2}{s+5} \right\} = \frac{2z^{-1}}{1 - 5z^{-1}}Z{s+52​}=1−5z−12z−1​

3. Combine the Results:

   Combining these two terms:

   G(z)=21−z−1−2z−11−5z−1G(z) = \frac{2}{1 - z^{-1}} - \frac{2z^{-1}}{1 - 5z^{-1}}G(z)=1−z−12​−1−5z−12z−1​

This is the z-transform corresponding to the given Laplace transfer function G(s)G(s)G(s).

Final Answer:

The z-transform corresponding to G(s)=10s(s+5)G(s) = \frac{10}{s(s+5)}G(s)=s(s+5)10​ is:

G(z)=21−z−1−2z−11−5z−1G(z) = \frac{2}{1 - z^{-1}} - \frac{2z^{-1}}{1 - 5z^{-1}}G(z)=1−z−12​−1−5z−12z−1​