02 - First Order Systems

\[G_p(s) = \frac{Y(s)}{X(s)} = \frac{K_p}{\tau_p s + 1}\] For step input of magnitude \(A\), \(X(t)=A\); and \(X(s)=A/s\). \[\begin{aligned} Y(s) &= \frac{A}{s}\frac{K_p}{\tau_p s +1} \end{aligned}\] Upon partial fraction expansion, we get \[Y(s) = AK_p\left(\frac{1}{s}-\frac{1}{s+1/\tau_p} \right)\] Taking \(\mathcal{L}^{-1}\), \[Y(t) = AK_p(1-e^{-t/\tau_p})\]