$\left[\begin{array}{c}{x}_1^{}\end{array}\right]$

$x_2^{}$

$x_3^{}=\left[\begin{array}{ccccccc}0& 1& 0& 0& 1& 0& 0\end{array}\right]$

$0$

$0\left[\begin{array}{c}{x}_1\end{array}\right]$

$x_2$

$x_3+\left[\begin{array}{c}0\end{array}\right]$

$0$

$1u$

$y=\left[\begin{array}{ccc}1& 0& 0\end{array}\right]\left[\begin{array}{c}{x}_1\end{array}\right]$

$x_2$

$x_3$

a$)$ Discuss whether the system should be controllable or not. Check your intuition by

forming the controllability matrix and checking its rank.

b$)$ Assume that all of the state variables can be measured directly; that is$,x$ is measured

Find a state feedback law that places the dominant closed$-$loop poles of the system so

farther to the left so that the design specification

farther to the left so that the design specifications are more likely to hold.

polynomial and closed$-$loop characteristic equation. Then check your result by using

the Matlab function "acker.m$"$ or "place.m$".$

d$)$ Simulate the closed loop system with a Simullink model.

Model the plant with Simulink integrators and Simulink gains

Model the the state feedback controller with a Simulink gain

Add "time" and $"$to workspace" blocks.

Simulate the system and plot $x_1,x_2,x_3,y$ and $u$ versus time.

Check if this system is results.

Check if this system is observable or not. $($Form the observability matrix and check if it is invertable or not.$)$

it is invertable or not.$)$

Suppose now that the state variables are not all available for measurement. Design an

observer which has poles with magnitudes about $4$ times larger than the real parts of

the closed$-$loop system. Determine the observer poles.

g$)$ Do coefficient matching to find observer gains.

i$)$ Simulate the closed loop system with the observer.

Model the observer with a Simulink "State$-$Space" block

Simulate the system and plot $x_1,x_2,x_3,$hat$(x{)}_1,$hat$(x{)}_2,$hat$(x{)}_3,y$ and $u$ versus time.

Discuss simulation results.

Use a step ot $/$ units.

Simulate the system and plot $x_1,x_2,x_3,$hat$(x{)}_1,$hat$(x{)}_2,$hat$(x{)}_3,y$ and $u$ versus time.

Discuss simulation results. Is there a steady state error?

Now we want to avoid the steady state error. For this purpose, use the compensator in

the feedback path structure.

$N_u$: Feedforward gain to avoid steady state error

$N_x$ : State reference gain to convert the reference for $y$ into a reference for $x$

$\left[\begin{array}{c}{N}_x\end{array}\right]$

$N_u=\left[\begin{array}{ccc}F& G& J\end{array}\right]$

$H^{-1}\left[\begin{array}{c}0\end{array}\right]$

$1\frac{?}{b}$

$ar(N)=N_u+K{N}_x$

Use a step of $7$ units

Simulate the system and plot $x_1,x_2,x_3,$hat$(x{)}_1,$hat$(x{)}_2,$hat$(x{)}_3,y$ and $u$ versus time.

Discuss simulation results. Is there a steady state error?