Repeat Problem 6.6-1, but with the process transfer function given by

\[
G(s)=\frac{20}{s(3 s+1)}
\]

Solve for the unit-step responses for \(0 \leq t \leq 1.0 \mathrm{~s}\).

Problem 6.6-1

Consider the system of Fig. P6.5-2, with \(D(z)=1\).

(a) With the sampler and zero-order hold removed, write the system differential equation.

(b) Using the rectangular rule for numerical integration with the numerical integration increment equal to \(0.25 \mathrm{~s}\), evaluate the unit-step response for \(0 \leq t \leq 1.5 \mathrm{~s}\).

(c) Repeat part (b) with the sampler and zero-order hold in the system, and with a sample period \(T=0.5 \mathrm{~s}\).

(d) Solve for the exact unit-step responses for parts (b) and (c), and compare results.

Fig. P6.5-2

R(s) + D(z) S T-1s Gp(s) M(s) 1 C(s) s+2