06 - Closed Loop Response

Let us consider a first order system, for which the uncontrolled response is given by \[\tau_p\frac{dY}{dt} + Y = K_p m + K_d d\] Taking Laplace transforms, \[\boxed{Y(s) = \frac{K_p}{\tau_p s+1}m(s) + \frac{K_d}{\tau_p s+1}d(s)}\] i.e., \[Y(s) = G_pm(s) + G_d d(s)\] where the transfer functions \(G_p\) and \(G_d\) are: \[G_p = \frac{K_p}{\tau_p s+1} \qquad \text{and} \qquad G_d =\frac{K_d}{\tau_p s+1}\]