We consider the scalar system

$($d$)/($dt$)$x$=-$x$+$u

We do not observe the system, but we would like to estimate the state x$.$ For that

purpose, we run the state estimator

$($d$)/($dt$)$hat$($x$)=-$hat$($x$)+$u

$($a$)$ Show that regardless of u$,$hat$($x$)$ converges towards x$.$ Thus the estimator works, even

though it reads no information from the system.

$($b$)$ Would this estimator approach work if the system was

$($d$)/($dt$)$x$=$x$+$u$?$

Explain your reasoning.

$($c$)$ We go back to the system

$($d$)/($dt$)$x$=-$ax$+$u

where a$=1.$ Unfortunately, the control engineer has the wrong estimate of a$,$

and she thinks a$=-0\mathrm{.}5.$ Considering the state estimator

$($d$)/($dt$)$hat$($x$)=-0\mathrm{.}5$hat$($x$)+$u

compute x and hat$($x$)$ when x$(0)=0,$hat$($x$)(0)=0,$u$=\backslash$sqrt$($t$)$sin$($t$),$ and t Would

you consider your estimator to be good?

$($d$)$ Using the same information as in the previous question, we now assume y$=$x is

available to make an estimator

$($d$)/($dt$)$hat$($x$)=-0\mathrm{.}5$hat$($x$)+$u$+$l$($y$-$hat$($x$))$

Simulate x and hat$($x$)$ when x$(0)=0,$hat$($x$)(0)=0,$ and u$=\backslash$sqrt$($t$)$sin$($t$)$ for l$=1,10,100$

and t Share your observations.