3. Proportional Controller

\[c(t) = c_s + K_c\epsilon(t)\] where

\(c\) = output signal from controller, psig or mA
\(c_s\) = steady state output from controller, the bias value
\(K_c\) = proportional gain
\(\epsilon\) = error = setpoint \(-\) measured variable



In terms of deviation variable \(C=c-c_s\), we have, \[C(t) = K_c\epsilon(t)\] Taking Laplace transform, \[C(s) = K_c\epsilon(s) \qquad \Longrightarrow \quad \frac{C(s)}{\epsilon(s)}=K_c = G_c(s)\]

The controller output will saturate (level out) at \(c_{\text{max}}\) = 15 psig or 20 mA at the upper end and at \(c_{\text{min}}\) = 3 psig or 4 mA at the lower end of the output.