06 - Closed Loop Response

For the typical closed loop system, the response is given by \[Y(s) = \frac{G_pG_fG_c}{1+G_pG_fG_cG_m}Y_{sp}(s) + \frac{G_d}{1+G_pG_fG_cG_m}d(s)\] For proportional controller, \[G_c = K_c\] For simplicity, let us consider the case with \(G_f=G_m=1\).

Then, \[Y(s) = \frac{G_pK_c}{1+G_pK_c}Y_{sp}(s) + \frac{G_d}{1+G_pK_c}d(s)\]

Substituting for \(G_p\) and \(G_d\), we get, \[\begin{aligned} Y(s) &= \frac{\dfrac{K_p}{\tau_p s+1}K_c}{1+\dfrac{K_p}{\tau_p s+1}K_c}Y_{sp}(s) + \frac{\dfrac{K_d}{\tau_p s+1}}{1+\dfrac{K_p}{\tau_p s+1}K_c}d(s) \\ &= \frac{K_pK_c}{\tau_p s+1 + K_pK_c}Y_{sp}(s) + \frac{K_d}{\tau_p s+1 + K_pK_c}d(s) \end{aligned}\] i.e., \[\boxed{Y(s) = \frac{K_p'}{\tau_p' s+1}Y_{sp}(s) + \frac{K_d'}{\tau_p' s+1}d(s)}\] where \[\tau_p' =\frac{\tau_p}{1+K_pK_c} \qquad K_p'=\frac{K_pK_c}{1+K_pK_c} \qquad K_d'=\frac{K_d}{1+K_pK_c}\] The parameters \(K_p'\) and \(K_d'\) are known as closed-loop static gains.

Closed loop response of first order system due to proportional control has the following characteristics:

It remains first-order with respect to load and setpoint changes.

The time constant has been reduced (i.e., \(\tau_p' <\tau_p\)), which means that the closed loop response become faster, than the open loop response, to changes in setpoint or load.

The static gains have been decreased.