A unity feedback system is characterized by the open-loop transfer function

\(G(s)=\frac{100}{s(5 s+10)(2 s+10)}\)

The steady-state errors for unit-step and unit-ramp inputs are :

Concept:

For a unity feedback system with open-loop transfer function \( G(s) \), the steady-state error depends on the input type and the system type.

The system type is determined by the number of integrators (poles at origin) in the open-loop transfer function.

The steady-state error \( e_{ss} \) is given by:

• For unit-step input: \( e_{ss} = \frac{1}{1 + K_p} \), where \( K_p = \lim_{s \to 0} G(s) \)
• For unit-ramp input: \( e_{ss} = \frac{1}{K_v} \), where \( K_v = \lim_{s \to 0} sG(s) \)

Given:

\( G(s) = \frac{100}{s(5s+10)(2s+10)} \)

This system has one pole at origin → Type 1 system

Calculation:

Step input:

\( K_p = \lim_{s \to 0} G(s) = \infty \Rightarrow e_{ss} = \frac{1}{1+\infty} = 0 \)

Ramp input:

\( K_v = \lim_{s \to 0} sG(s) = \lim_{s \to 0} \frac{100}{(5s+10)(2s+10)} = \frac{100}{10 \cdot 10} = 1 \Rightarrow e_{ss} = \frac{1}{1} = 1 \)

Correct Answer: 1) 1, 1