$[$Controllability indices$]$

Consider the controllability matrix C $=[$B AB A$2$B $$ An$1$B$]$ for a linear system model,

where A is n$$n and B is n$$m for positive integers m$,$ n$.$ Let b$1,...,$ bm denote the columns

of B$.$ Let $$ C be obtained by reordering the columns of C as follows $($such reordering does not

change the column span$)$:

C $=$

b$1$ Ab$1$ A$2$b$1$ An$1$b$1$ b$2$ Ab$2$ Am$1$b$2$ b$3$ An$1$bm

$.$

A basis for the column span of C$,$ or equivalently, the range space of C$,$ can be found by the

following algorithm. Consider the columns of C one by one from left to right, and add any

column to the basis that is not in the span of the columns before it$.$

$($a$)$ Show that whenever a column of the form Ajbi is not included in the basis then any

column of the form Aj$$

bi with j$>$ j will not be included. $($Hint: Start by considering

the columns with i $=1.)$

$($b$)$ Let i denote the number of columns of the form Ajbi that were added to the basis by

the algorithm. The numbers $1,...,$ m are called the controllability indices of $($A$,$B$).$

Under what condition on the controllability indices is $($A$,$B$)$ controllable? $($Controllability

indices are related to the so$-$called Luenberger controllable canonical forms that

generalize the CCF we$$ve seen for SISO systems and which can be found by elementary

row and column operations operating on the A and B matrices.$)$

$($c$)$ According to the theory of Luenberger controllable canonical forms, any state space

model with n $=4,$m $=2$ and controllability indices $1=$ $2=2$ can be put into the

following form by a state space transformation for some values of the constants indicated:

A $=$

$$

$$

$0100$

a b c d

$0001$

e f g h

$$

$$

B $=$

$$

$$

$00$

$1$ x

$00$

$01$

$$

$$

For what values of the constants are the controllablity indices for the above $($A$,$B$)$ given

by $1=$ $2=2?($This shows that not all values of the constants work.$)$