(9–8)
(9–9)
Jordan Canonical Form. Next we shall consider the case where the denominator
polynomial of Equation (9–2) involves multiple roots. For this case, the preceding
diagonal canonical form must be modified into the Jordan canonical form. Suppose, for
example, that the pi’s are different from one another, except that the first three pi’s are
equal, or p 1 =p 2 =p 3 .Then the factored form of Y(s)/U(s)becomes
The partial-fraction expansion of this last equation becomes
A state-space representation of this system in the Jordan canonical form is given by
(9–10)
y= Cc 1 c 2 p cnDF (9–11)
x 1
x 2
xn
V +b 0 u
H
x
1x
2x
3x
# 4 x
nX = H
- p 1
0 0 0 0
1
- p 1
0
p
p
0
1
- p 1
0
0
0
:
0
- p 4
0
p
p
0
:
0
0
- pn
XH
x 1
x 2
x 3
x 4
xn
X + H
0 0 1 1 1
Xu
Y(s)
U(s)
=b 0 +
c 1
As+p 1 B^3
+
c 2
As+p 1 B^2
+
c 3
s+p 1
+
c 4
s+p 4
+p+
cn
s+pn
Y(s)
U(s)
=
b 0 sn+b 1 sn-^1 +p+bn- 1 s+bn
As+p 1 B
3As+p 4 BAs+p 5 BpAs+pnB
y =Cc 1 c 2 p cnDF
x 1
x 2
xn
V +b 0 u
F
x
1x
# 2 x
nV = F
- p 1
0
- p 2
0
- pn
VF
x 1
x 2
xn
V + F
1 1 1
Vu
Section 9–2 / State-Space Representations of Transfer-Function Systems 651