Problem pts$]$

Given a homogeneous state$-$space dynamic system defined by:

$x^{}=\left[\begin{array}{ccc}0& 4& 0\end{array}\right]$

$-9x$

$y=\left[\begin{array}{cc}1& 1\end{array}\right]x$

Use the method of diagonalization to derive an analytic expression for the state transition matrix.

You may use the Symbolic Toolbox. You may use the Control System Toolbox. You may use the LiveScript equation editor. You may use basic Matlab functions. You MUST show the steps and display all intermediate and final results. If you have the correct answer, but do not show how you got that answer, points will be taken off.

Solve for the initial value response $y(t)$ associated with initial conditions: $,x(0)=\left[\begin{array}{c}-2\\ 3\end{array}\right].$

Using your analytic result for the initial value response $y(t),($do NOT use LSIM or any other simulation tool!$),$ numerically evaluate the time response of the output signal, and plot the data on a properly annotated figure. Explain why the time response looks the way it does in terms of the eigenvalues.

Two circular plates are located with a distance of $h_0$ between them. At $t=0,$ the plates begin to move towards each other in the velocity $\frac{V}{2}.$ Between the plates there is an incompressible fluid. Assuming the velocity of the fluid velocity is not a function of the gap between the plates find the fluid velocity as a function of the radius $(r)$ and time $(t).$