) An ideal PD controller has the following transfer function: \mathrm{G}_{\mathrm{c}}(\mathrm{s})=\frac{\mathrm{P}(\mathrm{s})}{\mathrm{E}(\mathrm{s})}=\mathrm{K}_{c}\left[\mathrm{~T}_{\mathrm{D}} \mathrm{s}+1\right] However, an actual PD controller has the following transfer function: \mathrm{G}_{\mathrm{c}}(\mathrm{s})=\frac{\mathrm{P}(\mathrm{s})}{\mathrm{E}(\mathrm{s})}=\mathrm{K}_{\mathrm{c}}\left[\frac{\mathrm{T}_{\mathrm{D}} \mathrm{s}+1}{\left(\mathrm{~T}_{\mathrm{D}} / \beta\right) \mathrm{s}+1}\right] Where B is

a constant that is characteristic of an industrial PD controller system. ) If a unit step change in the error e(t) is introduced to the controller, determine the idealand actual output responses. i) Determine which of the 2 scenarios (ideal or actual) has faster response kinetics. ) The preheater furnace below is used to increase the temperature of a crude oil from T¡ to T,where T is the target value. The crude oil enters and leaves the furnace at the same flow rate(F= Fi). Fuel and air are mixed and burned in the furnace to heat the crude oil. Determine the manipulated variable(s), controlled variable(s), and disturbancevariable(s) of the system. ) Construct a feedforward (FF) configuration to control the temperature of the hot crudeoil leaving the furnace, and discuss the control logic.