$(35$ pts$)$ A heating tank is to have a "feedforward $+$ feedback" control

system as illustrated on the side diagram. The goal is to control the

temperature of the tank $T_{\text{o}\text{u}\text{t}}$ by manipulating the flow of hot oil $q_{\text{o}\text{i}\text{l}}$ running

through the heating coil. The main disturbance is $T_{in},$ which is known to have

an irregular behavior. The feedback controller provides "trim" to correct for

inaccuracies in the model used to design the feedforward control and to

account for disturbances other than $T_{in}.$ Your goal is to estimate parameters

for the feedback and feedforward controller. The block diagram for the

control system is illustrated below.

After calibrating the transmitter and the transducer, designing the valve, and deriving the process model to find the

relation between $T_{\text{o}\text{u}\text{t}}$ and $T_{in}$ and $T_{\text{o}\text{u}\text{t}}$ and $q_{\text{o}\text{i}\text{l}}$ you found the following transfer functions.

$K_{TT1}=K_{TT2}=3,K_{\frac{l}{p}}=0\mathrm{.}75,G_v=\frac{4}{2s+1},G_p=\frac{e^{-5s}}{9s+1},G_d=\frac{1}{20s+1}$

a$)(15$ pts$)$ Demonstrate that the ultimate gain and period for the feedback controller are $K_{cu}0\mathrm{.}36$ and $P_{u}20$

b$)(5$ pts$)$ Use Ziegler$-$Nichols tuning relations to estimate parameters for the feedback controller, if such controller is PID

c$)(10$ pts$)$ Find $K_{FF},{}_a$ and ${}_1$ for the feedforward controller, if such controller is a lead$-$lag unit.