ABSTRACT
FIGURE 8.1 First canonical form realization.
Dividing both sides by αn and letting ai = αi/αn and bi = βi/αn we have
dny
dtn + an−1
dn−1y dtn−1
+ . . .+ a0y = bn dnu
dtn + bn−1
dn−1u dtn
+ . . .+ b0u. (8.7)
Laplace transforming both sides assuming zero initial conditions( sn + an−1sn−1 + . . .+ a0
) Y (s) =
( bns
n + bn−1sn−1 + . . .+ b0 ) U (s) . (8.8)
The system transfer function is given by
H (s) = Y (s)
=
bns n + bn−1sn−1 + . . .+ b0
. (8.9)
sn
Y (s) = − an−1Y (s) s
− an−2Y (s) s2
− . . .− a0Y (s) sn
+ bnU (s) + bn−1 U (s)
s + . . .+ b0
U (s)
sn = bnU (s) + {bn−1U (s)− an−1Y (s)} (1/s)+ {bn−2U (s) − an−2Y (s)} (1/s2) + . . .+ {b0U (s)− a0Y (s)} (1/sn).
