Question

Consider a unity feedback system with closed-loop transfer function \[ C(s) = \frac{s + 90}{s^2 + 10s + 90} \] The steady-state error with respect to a unit ramp input is \(\underline{\hspace{2cm}}\). (rounded off to one decimal place)

Hint:

For unit ramp input, the steady-state error is calculated using the velocity error constant \( K_v \).

Answer 1

Correct Answer: 0.1

For a unity feedback system, the steady-state error with respect to a unit ramp input is given by: \[ e_{\text{ss}} = \frac{1}{1 + K_v} \] Where \( K_v \) is the velocity error constant. The closed-loop transfer function is \( T(s) = \frac{C(s)}{1 + C(s)} \), and the velocity error constant is: \[ K_v = \lim_{s \to 0} s \times T(s) \] \[ K_v = \lim_{s \to 0} \frac{s(s + 90)}{s^2 + 10s + 90} = \frac{0 \times 90}{0 + 0 + 90} = \frac{90}{90} = 1 \] Thus: \[ e_{\text{ss}} = \frac{1}{1 + 1} = \frac{1}{2} = 0.1 \] \[ \boxed{0.1} \]