Question $3$: $(1)(10$ points$)$ Controller Design: Is this system controllable?. Explain by creating the controllability matrix

and reporting rank.

$=[50$

$]+[0$

$]$

$1021$

$=[10]$

Controllability Matrix: Rank # of States? Controllable?

$(2)(40$ points$)$ Design a Full State$-$Feedback Controller: A second$-$order plant is modeled by the state space

equation:

$=[016$

$]+[1$

$]$

$100$

$=[10]$

a$.)$ Please describe the system dynamics. Is there anything of particular importance to be aware of for

this system?

b$.)$ Calculate the feedback gains given the desired pole placement of $1,2=22.$ What important

characteristic of the system has changed with this pole placement.

Feedback Gains

What important characteristic has changed?

c$.)$ Calculate the controller effort u if reference r $=0$ at some time t when state values are $=[2$

$]$

$4$ Effort $=$

d$.)$ We would need an observer to perform full state feedback for this system, given the poles of the

controller, provide an estimate for the observer poles. This doesn$$t need to be exact and should

focus on the magnitude of the real part of the poles. $($More generally, how should we choose

observer poles in relation to controller poles?$)$

Observer pole location guess: