Solve this question from a Systems Analysis worksheet

State$-$space models are used extensively in modelling dynamic systems. Continuoustime state$-$space models take the form:$-$

$x^{}=Ax+Bu$

$y=Cx+Du$

$($a$)$ Name each of the quantities in the state$-$space model and provide a brief description of the quantity.

$($b$)$ Using the example of a passenger car travelling on a straight flat road, suggest what might be the input$($s$),$ state$-$variable$($s$)$ and output$($s$)$ of the model of its dynamics.

$($c$)$ Explain the concept of "equivalence" between state$-$space models. Prove that two equivalent state$-$space models will return the same output steady$-$state responses from harmonic $($sinusoidal$)$ excitation if the same input is applied to both models.

$($d$)$ If the matrix $A$ of a state$-$space model is diagonal $($i$.$e$.$ contains only ${}_1,{}_2$dots${}_n$ as entries along the leading diagonal$),$ what conclusions can be drawn immediately and directly from inspection of the model matrices $?$

Figure $6$ shows a SIMULINK model of a dynamic system. Note that the square blocks $\frac{1}{s}$ denote integrators, the circular icons denote arithmetic differences, triangular icons represent pure gain blocks and the tabs on the extreme left and extreme right represent input and output signals respectively.

$($e$)$ Using the notation of the diagram, complete these equations $...$

$x_1^{}=$dots.,$x_2^{}=$dots.,$x_3^{}=$dots.,$y=$dots.

$($f$)$ Present the above equations in state$-$space form.