Question

If \( R(s) \) is input signal, \( C(s) \) is output signal and \( G(s)H(s) \) is feedback signal, then the steady-state error is given by

• A. \( \lim_{s \to 0} \frac{SR(s)}{1 + sG(s)H(s)} \)
• B. \( \lim_{s \to 0} \frac{SR(s)}{1 + G(s)H(s)} \)
• C. \( \lim_{s \to 0} \frac{R(s)}{1 + G(s)H(s)} \)
• D. \( \lim_{s \to 0} \frac{sR(s)}{1 + G(s)H(s)} \)

Hint:

Use the final value theorem to calculate steady-state error in control systems.

Answer 1

The Correct Option is D

The steady-state error in a unity feedback system is calculated using the final value theorem: \[ e_{ss} = \lim_{t \to \infty} e(t) = \lim_{s \to 0} sE(s) \] Where \( E(s) = R(s) - C(s) \). For unity feedback: \[ E(s) = \frac{R(s)}{1 + G(s)H(s)} \Rightarrow e_{ss} = \lim_{s \to 0} \frac{sR(s)}{1 + G(s)H(s)} \]