Problem $4.$ Block Diagram and Responses of a Feedforward and Feedforward$-$Feedback System. $(25$ points$)$

Consider the block diagram shown in the text by Fig. $15\mathrm{.}11.$ This is repeated below in Figure $4\mathrm{.}1.$ The arrangement is similar to Fig. $11\mathrm{.}8$ given in the text for feedback control, but an additional signal path has been added using the transfer functions, $G_l$ and $G_l$ to provide feedforward control action.

Referring to the figure, the disturbance transmitter with transfer function $G_t$ sends a measurement of the disturbance variable to the feedforward controller $G.$ The outputs of the feedforward and feedback controllers are then added together, and the resulting sum is then sent to the control valve described by the transfer function Gv$.$ In contrast to the steady$-$state design methods of Section $15\mathrm{.}3,$ the block diagram in Figure $4\mathrm{.}1$ is based on deviation variables.

Answer the questions given below.

Figure $4\mathrm{.}1$ Block diagram of a feedfonward$-$feedback process control system

$($a$)$ Using block diagram algebra, show that the above system can be represented by the following transfer function, which is given as eq$.15-20$ in the text.

$\frac{Y(s)}{D(s)}=\frac{G_s+G_1{G}_f{G}_v{G}_p}{1+G_c{G}_v{G}_p{G}_m}$

$($b$)$ Refer to Example $15\mathrm{.}5$ in the text. Figure $15\mathrm{.}13$ compares the closed$-$loop responses for two cases: $($a$)$ feedfonward controllers with and without dynamic compensation, and $($b$)$ a system that includes both feedback control and feedfonward control. Referring to the text, fill in the missing steps and develop the working equations and Matlab code that reproduces the results shown on these twa figures.