A method for rapidly driving the state to zero. We consider the discrete$-$time linear

dynamical system

x$($t $+1)=$ Ax$($t$)+$ Bu$($t$),$

where A in Rn$$n and B in Rn$$k$,$ k $<$ n$,$ is full rank.

The goal is to choose an input u that causes x$($t$)$ to converge to zero as t $$ $.$ An

engineer proposes the following simple method: at time t$,$ choose u$($t$)$ that minimizes

$$x$($t $+1).$ The engineer argues that this scheme will work well, since the norm of the

state is made as small as possible at every step. In this problem you will analyze this

scheme.

$($a$)$ Find an explicit expression for the proposed input u$($t$)$ in terms of x$($t$),$ A$,$ and

B $($hint: use least squares$).$

$($b$)$ Now consider the linear dynamical system x$($t $+1)=$ Ax$($t$)+$ Bu$($t$)$ with u$($t$)$

given by the proposed scheme $($i$.$e$.,$ as found in $(1$a$)).$ Show that x satisfies an

autonomous linear dynamical system equation x$($t $+1)=$ F x$($t$).$ Express the

matrix F explicitly in terms of A and B$.$

$($c$)$ Now consider a specific case:

A $=$

$[03$

$00$

$]$

$,$ B $=$

$[1$

$1$

$]$

$.$

Compare the behavior of x$($t $+1)=$ Ax$($t$)($i$.$e$.,$ the orginal system with u$($t$)=0)$

and x$($t $+1)=$ F x$($t$)($i$.$e$.,$ the original system with u$($t$)$ chosen by the scheme

described above$)$ for a few initial conditions. Determine whether each of these

systems is stable.