Solve the question from a Systems Analysis worksheetFigure $9$a shows a simple device for using ocean waves to compress air. The device comprises two rigid bodies $-$ a cylinder and a piston. Waves cause the cylinder to bob up and down and the piston moves internally as a result. Fig. $9$a shows five valves: four are one$-$way valves and the other valve is controllable.

$($a$)$ With all valves closed, the device is effectively as shown in Fig. $9$b$.$ Downward displacements of the cylinder and the piston $($relative to their mean absolute positions$)$ are denoted, $q_1$ and $q_2$ respectively and the downward displacement of the local water surface is denoted $h.$ At one particular instant, $q_1=0,q_2=-1,q_1^{}=0=q_2^{}$ and the equations of motion are:

$1600q_1^{}+100(q_1^{}-q_2^{})+100q_1^{}+5000(q_1-q_2)+11000q_1-Z=11000h$

$1400q_2^{}+100(q_2^{}-q_1^{})+5000(q_2-q_1)+Z=0$

where $Z$ is the nonlinear term $Z=73258(\frac{1\mathrm{.}7+(q_1-q_2)}{1\mathrm{.}7}{)}^{-1\mathrm{.}4}-9624\mathrm{.}7(\frac{1\mathrm{.}7-(q_1-q_2)}{1\mathrm{.}7}{)}^{-1\mathrm{.}4}$ Verify that this is an equilibrium position if $h=0.$

$($b$)$ Linearise this model and present the matrices $\{S_L,M,D,K,S_R\}$ of the Natural Second Order Form $($NSOF$),($see formula sheet$)$ noting that the only input is $h$ and the required output is $(q_1-q_2).$

$($c$)$ Convert the model from part $($b$)$ into state$-$space form. Describe how to obtain the system eigenvalues from this and explain what these mean but do not attempt to calculate them.

$($d$)$ Sketch a SIMULINK model for this system. Assume that a MATLAB Function has been provided for calculating $Z$ given $(q_1-q_2).$ Make clear what each block represents. The precise icons $($pictures$)$ for each block are not essential.