Consider industrial automatic controllers whose control actions are proportional, proportionalplus- integral, proportional-plus-derivative, and proportional-plus- integral-plus-derivative. the transfer functions of these controllers can be given, respectively, by ((*) ,(*) = kp ((*) ,(*) = kp (1+ - ./ * ) ((*) ,(*) = kp (1+ td s ) ((*) ,(*) = kp (1+ - ./ * + td s ) where u(s) is the laplace transform of u(t), the controller output, and e(s) is the laplace transform of e( the actuating error signal. sketch u(t) versus t curves for each of the five types of controllers when the actuating error signal is: (a) e(t) = unit step function (b) e(t) unit ramp function in sketching curves assume that the numerical values of kp, ki, ti, and td are given as: kp = proportional gain = 4 ki= integral gain =2 ti = integral time =2 sec td = derivative time 0.8 sec