DC Gain

The steady-state value of the step response is called the DC gain of the system:

$$\text{DC gain}=y(\infty )=\underset{t\to \infty }{\lim }y(t)$$

where $u(t)=1(t)$.

By the Final Value Theorem, we can compute DC gain with ease (when FVT holds). For example, for the unit step response where we have

$$\begin{array}{rl}u(t)& =1(t)\\ U(s)& =\frac{1}{s}\\ Y(s)& =U(s)H(s)=\frac{H(s)}{s}\end{array}$$

If all poles of $sY(s)=H(s)$ are strictly stable, then by FVT

$$y(\infty )=\underset{s\to 0}{\lim }H(s)$$

Example

Given the system transfer function

$$H(s)=\frac{s^2+5s+3}{s^3+4s^2+2s+5}$$

compute the DC gain of the system.

All the poles of $H(s)$ are strictly stable poles (Routh-Hurwitz Criterion), so

$$y(\infty )=H(s){|}_{s=0}=\frac{3}{5}$$