Solve this problem ONLY in MATLAB, please attach the entire script. Please note that for part $A,C,D$ and $E$ the final answers are given.

A continuous ball mill with four classes of particle sizes, constant total mass and perfect mixing throughout the mill, where only class $1$ and class $2$ particles are fed, is described by the mathematical model:

$\frac{d{}_1}{dt}=-S_1{}_1+0({}_{1,0}-{}_1)$

$\frac{d{}_2}{dt}=S_1{b}_{21}{}_1-S_2{}_2+(1-{}_{1,0}-{}_2)$

$\frac{d{}_3}{dt}=S_1{b}_{31}{}_1+S_2{b}_{32}{}_2-S_3{}_3-{}_3$

Where ${}_i(t)$ is the mass fraction of particles of class $i$ in the mill, $S_i$ is the selection rate of breaking of particles of class $i(S_1=19h^{-1},S_2=2\mathrm{.}8h^{-1},S_3=0\mathrm{.}2h^{-1}),$ and $b_{ij}$ is the breakdown distribution of particles of class $j$ in class $i,(b_{21}=0\mathrm{.}8,b_{31}=0\mathrm{.}16,b_{32}=0\mathrm{.}8)$

for $i=1,2,3,j,$ where class $1$ are the largest particles and class $4,$ the smallest. The mill feed has a mass fraction $of$ class $1$ particles, ${}_{1,0}(t)$ whose value $is$ occasionally irregular.

The turnover frequency $(t)of$ the mill $is$ manipulated with the mass flow fed $to$ the mill $by$ conveyor belts $to$ achieve the desired mass fraction $of$ class $3$ particles.

Manipulated input variable: $,$ Disturbance: ${}_{1,0},$ State variables:${}_1,{}_2,{}_3,$ Output variable: ${}_3,$ Model parameters: $S_1,S_2,S_3,b_{21},b_{31},b_{32}.$

$A.$ Find the numerical values $of$ the steady state when ${}_{1,0,s}=1y{}_s=1h^{-1}.$ Answer for this section: ${}_{1s}=0\mathrm{.}05,{}_{2s}=0\mathrm{.}2,{}_{3s}=0\mathrm{.}5.$

$B.$ Linearize the model, write $it$ into deviation variables, and substitute numerical values into the linearization matrices.

$C.$ Determine $if$ the system $is$ controllable and observable. Answer for this section: $Itis$ observable and controllable.

$D.$ Obtain the transfer functions $Gp(s)$ and $Gd(s)$ and determine the zeros and poles $of$ each. Answer for this section:

$Gp(s)=\frac{-0\mathrm{.}5s^2-9\mathrm{.}46s-3\mathrm{.}64}{s^3+25s^2+104\mathrm{.}56s+91\mathrm{.}2},Gd(s)=\frac{0\mathrm{.}8s+0\mathrm{.}8}{s^3+25s^2+104\mathrm{.}56s+91\mathrm{.}2}$

$E.IfGc=Kc,Gf=1yGm=1,$ determine the open loop stability. Answer for this section: $Itis$ stable

$F.$ For the conditions $of$ the previous section, determine the range $ofKc$ values that guarantee that the closed loop $of$ a feedback control scheme $is$ stable.

$G.$ For the conditions $of$ the previous section, determine the ultimate frequencies.