Question #$3(25$ points$)$: Reduction of Multiple Subsystems

A hybrid solar cell and diesel power distribution system has been proposed and tested. The system has been shown to have a very good uninterruptible power supply as well as line voltage regulation capabilities. The system's block diagram is shown in the following figure. Take $C_f=$widehat$(C_f)=0\mathrm{.}5,L_f=$widehat$(L_f)=1,R_f=10,$widehat$(R_f)=2,K_{p1}=5,K_{p2}=20,K_{i1}=10.$

a$.$ Find the transfer function, $G(s)=\frac{V_{\text{L}\text{o}\text{a}\text{d}}(s)}{I_{Cf}(s)}$ by using block diagram reduction. $($Show all steps$)[10$ points$]$

b$.$ Convert the block diagram into a signal flow graph and apply Mason's Rule to reduce it into a single transfer function. $[10$ points$]$

c$.$ Compare your answers in parts a and b$.$ Explain why $($or why not$)$ you have the same answers. $[5$ points$]$