Consider the discrete$-$time linear system

xk$+1=$ Axk $+$ Buk, k $=0,1,2,...$

where A in Rn$$n$,$ B in Rn$.$ The integer k just denotes the $$sample index,$$ so in other

words, if the initial condition is given at k $=0$ to be x$0$ and the input sequence is known

$\{$u$0,$ u$1,$ u$2,...\},$ you can generate $\{$x$1,$ x$2,$ x$3,...\}$ by recursively applying the discrete$-$time

equation. For example, x$1$ is computed to be

x$1=$ Ax$0+$ Bu$0,($ x$0$ and u$0$ are known$)$

so now you can compute x$2,$ etc. Answer the following questions:

$1$

$($a$)$ Starting from zero initial conditions, i$.$e$.$ x$0=0,$ and given some positive integer p$,$

derive a relationship in matrix$-$vector form between between xp and the input sequence

$\{$u$0,$ u$1,$ u$2,...,$ up$1\}.$ In other words, find the matrix M in Rn$$p such that

xp $=$ M u$,$ where u $=$

$$

$$

$$

$$

$$

u$0$

u$1$

$...$

up$1$

$$

$$

$$

$$

$.$

$($b$)$ Now suppose p $>$ n and assume M has full rank $($rank$($M $)=$ n so R$($M $)=$ Rn$).$

Consider the situation in which xp is specified and the objective is to determine u$.$ Show

that this is a least$-$norm problem and write a solution u in terms of the appropriate

pseudo$-$inverse for M $.$ This will yield the least norm solution, denoted uln.

$($c$)$ The solution uln is an input sequence which transfers the system state from the origin to

xp in p time steps. Define the $$energy$$ required to make this transfer to be the square of

the Euclidean norm of uln, i$.$e$.$ energy$=$uln$2$

$2.$ Compute the input energy and express

it in terms of symbols M and xp$.$

$($d$)$ Carry out steps $($a$)$ through $($c$)$ for the following two$-$state system:

xk$+1=$

$[11$

$01$

$]$

xk $+$

$[0$

$1$

$]$

uk$,$

where n $=2,$ p $=4,$ and xp $=$

$[1$

$1$

$]$

$.$

$($e$)$ Using the same system and parameters from $($d$),$ calculate another input that achieves

the goal of xp $=$

$[1$

$1$

$]$

$.$ What is the $$energy$$ associated with this new input? Note: if

$$u $6=0$ such that M $$u $=0,$ in other words, $$u is in the null space of M $,$ then M $($uln $+$u$)=$

xp$.$