Design Project CRaBP ECG $7702024$: Spacecraft Control in the Gravitational

Field of Earth and Moon $40$ Points. Due Date Nov. $30.$

Consider the motion of a spacecraft in is the growitational field of the Earth $($:M$\_(1)$:$)-$ Mooe L$\_(2)$ point $($callod Lignange Point or

Lbration point$).$ Note that L$\_(2)$ point remains fixed on the rotating rod. That is$,$ L$\_(2)$ point, M$\_(2),$

abd M$\_(1)$ remain collineser and are rotating about the baryecenter. Thus, a satellite stationed at L$\_(2)$

point can make a contimuoss obowrvation of a region of the Moon. This satellite transmits the

data to anther satellite which transmits the data to the Earth. Note that the satellite stationod

at the L$\_(2)$ cannot directly transmit data to the Earth bevase the Moon lies betwen the L$\_(2)$ point

abd the Earth.

The planar non$-$dimersionaliand linear second$-$order equations of motion of the spacocraft

$($ m $)$ in the X$-$Y plase: with respent to the L$\_(2)$ point are described by

x$-2$hat$($y$)-($U$\_($xx$))$x$=$u$\_(1)+$d$\_(1)($t$)$

y$+2$x$^()-($U$\_($YY$))$g$=$u$\_(2)+2$d$\_(2)($t$)$

where U$\_($xx$)=7\mathrm{.}3219,$U$\_($V$\backslash$gamma $)=-2\mathrm{.}1904,$ and u$\_(1)$ abd tilde$($i$)\_(2)$ ate applied control acoderation inputs,

and d$\_(1),$d$\_(2)$ are disturbance imputs

d$\_(1)=0\mathrm{.}6$sin$(-\backslash$omega $\_(1)$t$)$;$,$d$\_(2)=$cos$(\backslash$omega $\_(1)$t$+(\backslash$pi $)/(4))$

Chocee $\backslash$omega $\_(1)=1\backslash$omega $\_(1)$ if you like, but plot $($d$\_($i$))\backslash$omega $\_(1).-[$x$,$y$,$y$^(),$y$]^(7)$inR$^(4)$v$-$Aw$+$Bw

z$=[$x$,$y$]^($T$)$inR$^(2)$

where: $2$ is a mescured output.

Simulate the ubontrolled disturbance input$-$free system $(2)$ with instial condition

an Show the plots of x$,$y$.$ Note what you oboerve. $[$USE MATHLAB eatensively$].$

II$.$

$($a$)$ Find the transfer matrix G$($s$)=$Y$($s$)/($U$)($s$).$

$($b$)$ Compute the controllability and obowrvability matricess, abd tramomisoion zerces.

$($c$)$ Is the system observable and coutrollable.

$($d$)$ Is the open$-$loop disturbance frese system systatm stable?

$($e$)$ Doss the frequency of disturbance coincide with tralsmisson zerce? If it coincides, choose

different valoe of $\backslash$omega $\_(1)$w$=0$ d$\_(1)$ and d$\_(2)$x$,$y$,$i$\_(1),$u$\_(3),$d$\_(3),$d$\_(2)(1)/(2)$ page