The D.C. gain and steady state error for step input for \(G(s)=\frac{s+1}{s^2+s+1}\) are:

Concept:

DC gain:

The DC gain is the ratio of the magnitude of the steady-state step response to the magnitude of step input.

DC Gain of a system is the gain at the steady-state which is at t tending to infinity i.e., s tending to zero.

DC gain is nothing but the error coefficients.

For type 0 system: \({K_P} = \mathop {\lim }\limits_{s \to 0} G\left( s \right)\)

For type 1 system: \({K_v} = \mathop {\lim }\limits_{s \to 0} sG\left( s \right)\)

For type 2 system: \({K_a} = \mathop {\lim }\limits_{s \to 0} {s^2}G\left( s \right)\)

Steady state error for different inputs is given by

Calculation:

Given:

\(G(s)=\frac{s+1}{s^2+s+1}\)

It is a type 0 system, so:

\({K_P} = \mathop {\lim }\limits_{s \to 0} G\left( s \right)\)

K_p = 1 = DC Gain

Steady-state error for unit step input is given as:

\(e_{ss}=\frac{1}{1+K_p}\)

e_ss = 0.5