Solve this question from a System Analysis worksheet

Figure $7$ illustrates a proposed tower$-$climbing system for wind$-$turbine

maintenance. In this system, a carriage holds a set of powered wheels pressed

firmly to the tower and these set the height of $A-$ a horizontal pivot line

through the centreline of the tower. Another horizontal pivot line, D$,$ also

passes through the tower centreline and is a fixed distance below $A.$ A flexible

beam ABC determines the height of the vertical platform and control over this

height is achieved by adjusting the force $F$ in the link $BD.$ The following

equations govern the dynamic behaviour of the platform:

$=ta{n}^{-1}(5cos\frac{}{8+5sin()})$

$40,000{}^{}+200{}^{}=5$Fcos$(+)-(250,000+12\mathrm{.}5V)cos()$

$8,000z^{}+500z^{}+100,000z=2$Fcos$(+)-(50,000+0\mathrm{.}2V)cos()$

$h=12\mathrm{.}5sin()-0\mathrm{.}1zcos()$

$($a$)$ Identify the state$-$variables, inputs and outputs from the above

equations.

$($b$)$ Sketch a SIMULINK model of this system. Use appropriate icons if

possible but alternative icons may be used provided that the meaning is clear

in every case. Use icons to represent inputs and outputs, use a single block to

represent the function relating $$ to $$ and label signal lines.

$($c$)$ An equilibrium condition exists where $V=60kN$ and $=30.$ Determine

the equilibrium values of $F,z$ and $h.$ Suggest how the SIMULINK model might

be used to verify this equilibrium.

$($d$)$ Based on the outcomes from $($c$),$ prepare a linearised model of the

dynamic system in standard state$-$space format making clear what are the

input, state and output vectors, $(U,x$ and $y$ respectively$)$ and the matrices $A,$

B$,$ C$,$ D$.$ Note that at the equilibrium state $\frac{d}{d}=-0\mathrm{.}348837.$