$($d$)$ Plot the time variation of all four state variables

$3.(30$ points$)$ This problem is best done using the aid of computer programming such as Python or MATLAB. The lateral aircraft equations of motion, linear state space form, of a particular aircraft at a particular flight condition, are given by

$\left[\begin{array}{c}{}^{}\\ p^{}\\ r^{}\\ {}^{}\end{array}\right]=\left[\begin{array}{cccc}-0\mathrm{.}34& 0& -1\mathrm{.}0& 0\mathrm{.}22\\ -6\mathrm{.}42& -10\mathrm{.}4& 1\mathrm{.}5& 0\\ 0\mathrm{.}9& -0\mathrm{.}75& -1\mathrm{.}46& 0\\ 0& 1& 0& 0\end{array}\right]\left[\begin{array}{c}\\ p\\ r\\ \end{array}\right]$

$($a$)$ Determine the eigenvalues of this system $($which are same as the eigenvalues of the system matrix$)$

$($b$)$ Determine the damping ratio and natural frequency of each of the modes that demonstrate oscillatory motion.

$($c$)$ Compute the dynamics of this system for the following initial conditions. Note that the dphi$\_$dt equation we did in class has a small typo. dphi$\_$dt should be set equal to $p$ and not $r$

$=5$

$p=10ra\frac{d}{s}$

$r=20ra\frac{d}{s}$

$=0$

$($d$)$ Plot the time variation of all four state variables.

$4.(30$ bonus points$)$ Assume that the dynamics of the aircraft you dealt with in Problem $2$ including control by elevator deflection is given by

$\left[\begin{array}{c}{u}^{}\\ w^{}\\ q^{}\\ {}^{}\end{array}\right]=\left[\begin{array}{cccc}-0\mathrm{.}006& 0\mathrm{.}014& 0\mathrm{.}008& -0\mathrm{.}261\\ -0\mathrm{.}003& -0\mathrm{.}054& 1& 0\\ 0\mathrm{.}16& -4\mathrm{.}08& -0\mathrm{.}054& 0\\ 0& 0& 1& 0\end{array}\right]\left[\begin{array}{c}u\\ w\\ q\\ \end{array}\right]+\left[\begin{array}{c}-0\mathrm{.}003\\ -0\mathrm{.}08\\ -0\mathrm{.}52\\ 0\end{array}\right]{}_e$

$($a$)$ For this system, determine the feedback control gain that would lead to the eigenvalues given by ${}_1=-0\mathrm{.}2,{}_2=-0\mathrm{.}5,{}_{3,4}=-0\mathrm{.}1+-0\mathrm{.}8j$

$($b$)$ For this system, use the LQR approach discussed in class to determine the feedback control gain that would optimize the control effort. Use

$Q=\left[\begin{array}{cccc}0\mathrm{.}1& 0& 0& 0\\ 0& 0\mathrm{.}1& 0& 0\\ 0& 0& 0\mathrm{.}1& 0\\ 0& 0& 0& 0\mathrm{.}1\end{array}\right]$

and

$R=0\mathrm{.}9$

$($c$)$ For both cases, plot $($preferably on the same plot$)$ the system response for an initial condition given by $q=30ra\frac{d}{s}$ with all other variables' initial condition set to $0.$