An electronic PID temperature controller is at steady state with an output of 12 mA. The set point equals the nominal process temperature initially. At t = 0, the error signal is increased at the rate of 0.5 mA/min (equivalent to a rate of 2^∘F/min). If the current settings are

K_c = 2 (dimensionless)
τ_I = 1.5 min
τ_D = 0.5 min

(a) Derive an expression for the controller output p(t).
(b) Repeat (a) for τ_D = 0 (i.e., a PI controller).
(c) Plot the two controller outputs and qualitatively discuss their differences.