Solve this question from a Systems Analysis worksheet

Figure $10$ shows a diesel$-$generator set $($bounded by the dashed curve$)$ with two closed$-$loop controllers on it$.$ One controller adjusts the strength of the field, $k_T,$ on the generator to regulate the voltage output, $V_L$ so that it does not differ greatly from $V_{Cmmd.}.$ The second controller adjusts the throttle setting, $,$ for the diesel$-$engine in order to regulate the speed, $,$ so that it does not differ greatly from ${}_{Cmmd}.$

Both controllers are proportional controllers and the equations describing them are:

$k_T=p(V_{\text{C}\text{i}\text{m}\text{d}}-V_L)$

$=q({}_{Cmmd}-)$

The diesel$-$generator set itself is characterised

$1\mathrm{.}25{}^{}=250*(1\mathrm{.}0-0\mathrm{.}8(1-\frac{}{60(1+2)}{)}^2)-k_{T}I$

$0\mathrm{.}06I^{}=(k_T-(R_L+0\mathrm{.}01)I)$

$V_L=R_{L}I$

$($Part a$)$ There is an equilibrium operating condition for the generator set at throttle position $=0\mathrm{.}9$ for which $=148ra\frac{d}{s}$ and $I=165\mathrm{.}0A.$ Calculate the generator field strength, $k_T$ and the load resistance, $R_L.$

$($Part b$)$ If the command voltage, $V_{Cmmd},$ is $200V$ and if the command speed, ${}_{Cmmd},$ is $150ra\frac{d}{s},$ deduce the controller gains, $p$ and $q.$

$($Part c$)$ This model is to be linearised to produce a state$-$space representation $($see formula sheet$).$ State what entries you would place in the vectors $u,x$ and $y.$ Do not calculate matrices, A and $B$ but do report $C$ and $D.$

$($Part d$)$ Sketch a SIMULINK model which represents the dynamics of the subsystem enclosed within the dashed line. Indicate each of the three inputs, $\{,R_L,k_T\},$ with an input tab, Label every line in the SIMULINK model to indicate what quantity it is carrying.

$($Note: It does not matter if you cannot recall the exact icons for particular blocks so long as your diagram is clear. Use a single block for the evaluation of $250.(1\mathrm{.}0-0\mathrm{.}8(1-\frac{}{60(1+2)}{)}^2).$