Solve this question from a system analysis worksheetState$-$space models are used extensively in modelling dynamic systems

Continuous$-$time state$-$space models take the form:

$x^{}=Ax+Bu$

$y=Cx+Du$

$($a$)$

Name each of the quantities in the above state$-$space model and provide a

brief description of the quantity. Make clear which quantities depend on

$($b$)$ Using the example of the inverted pendulum shown in $2$a$,$ suggest what

might be the input$($s$),$ state$-$variable$($s$)$ and output$($s$)$ of the dynamic

model.

Figure $2$a Inverted Pendulum

$($c$)$ Explain the concept of a 'coordinate transformation' of a state vector.

Describe how a new state$-$space model obtained from a coordinate

transformation is equivalent to the original model.

$($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$),$ state the conditions for which

the system is stable, controllable and observable.

$($e$)$ Figure $2$b shows a Simulink model of a dynamic system. In this, the

square blocks $\frac{1}{s}$ denote integrators, the circular icons denote arithmetic

differences or sums, triangular icons represent pure gains and the tabs on

the extreme left and extreme right represent input and output signals

respectively.

i$)$ Using the notation of the diagram, complete these equations

$x_1^{}=$dots,$x_2^{}=$dots

ii$)$ Present the above equations in a linearised state$-$space form at the

unique equilibrium position where $u=20(N)$