Solve this question from a Systems Analysis worksheet

Fig. $11$ shows an alignment system being considered for directing a beam of microwave power onto a

satellite in orbit. A rigid parabolic reflector unit has mass $30kg$ and its center of gravity is at point $B.$ Its

moment of inertia about $B$ is $4kg{m}^2.$ Beneath this, there is a rigid platform having mass $70kg.$ Its c$\mathrm{.}0.$g$.$ is at

point $E$ and its moment of inertia about $E$ is $8kg{m}^2.$ Points $D$ and $F$ on the rigid platform are each supported

from ground by a spring$-$damper unit as shown where $k=4000\frac{N}{m}$ and $c=0\mathrm{.}5N\frac{s}{m}.$ There are also springs

of stiffness $k$ separating points $A,D$ and points $C,F.$

$($a$)$ Prepare an approximate STATE$-$SPACE model for the dynamics of this system. The inputs are to be the

forces $\{f_1,f_2\}($acting to separate $A,D$ and $C,F$ respectively$).$ The outputs from this model are to be the

vertical deflection $($positive upwards$),u,$ and angle $($anticlockwise positive$),,$ of the reflector about its

centre of gravity. Use the symbols $v$ and $$ to represent the vertical deflection and angle of the rigid

platform.

$($b$)$ A controller proposed for this system is intended to keep both $u$ and $({}_{\text{r}\text{e}\text{f}}-)$ close to zero where ${}_{\text{r}\text{e}\text{f}}$ is

a reference angle for the reflector determined by satellite position. The controller is described by:

$f_1=-200u-10000({}_{rff}-)$

$f_2=-200u+10000({}_{rff}-)$

Prepare a new STATE$-$SPACE model for the closed$-$loop system where ${}_{\text{r}\text{e}\text{f}}$ is the only input and the

output is $({}_{\text{r}\text{e}\text{f}}-).$

$($c$)$ Describe how you could determine whether the closed$-$loop system is stable and comment on what

other information you could obtain from the same calculation. $($Do not attempt to do the actual

calculation$)$