Consider the nth order linear differential equation:
qsyt=psut
q
s
y
t
p
s
u
t
(1)
where
s=ⅆⅆt
s
ⅆ
ⅆ
t
and where
qs=sn+
α
n

1
sn−1+…+α1s+α0
q
s
s
n
α
n

1
s
n1
…
α1
s
α0
(2)
ps=
β
n

1
sn−1+…+β1s+β0
p
s
β
n

1
s
n1
…
β1
s
β0
(3)
One way to define state variables is by introducing
the auxiliary variable ww
which satisfies the differential
equation:
qswt=ut
q
s
w
t
u
t
(4)
The state variables can then be chosen as
derivatives of ww . Furthermore the output is related to this auxiliary
variable as follows:
yt=pswt
y
t
p
s
w
t
(5)
The proof in the next three equations shows that the
introduction of this variable ww
does not change the system in any way. The first equation uses a simple substition based on the differential equation. Then the order of
psps
and
qsqs
are interchanged. Lastly, yy
is substituted in place of
pswt
p
s
w
t
(using output equation). The result is the original equation describing our system.
psqswt=psut
p
s
q
s
w
t
p
s
u
t
(6)
qspswt=psut
q
s
p
s
w
t
p
s
u
t
(7)
qsyt=psut
q
s
y
t
p
s
u
t
(8)
Using this auxillary variable, we can directly write
the AA, BB and CC matrices. AA is the companionform matrix; its last row (except for a 00 in
the first position) contains the alpha coefficients from the
qsqs:
A=(
0100…0
0010…0
000⋱…0
⋮⋮⋮⋮⋱⋮
0000…1
−
α
0
−
α
1
−
α
2
−
α
3
…−
α
n

1
)
A
0100…0
0010…0
000⋱…0
⋮⋮⋮⋮⋱
⋮
0000…1
α
0
α
1
α
2
α
3
…
α
n

1
(9)
The BB
vector has zeros except for the nnth row which is a 11.
CC can be expressed as
C=β0β1β2⋮
β
n

1
C
β0
β1
β2
⋮
β
n

1
(11)
When all of these conditions are met, the state is
x=wsws2w⋮sn−1w
x
w
s
w
s
2
w
⋮
s
n
1
w
(12)
In conclusion, if the degree of pp is less than that of qq,
we can obtain a statespace representation by inserting
the coefficcients of pp and qq
in the matrices AA, BB and CC
as shown above.