650 Chapter 9 / Control Systems Analysis in State Space(9–4)
The controllable canonical form is important in discussing the pole-placement approach
to control systems design.
Observable Canonical Form. The following state-space representation is called
an observable canonical form:
(9–5)
(9–6)
Note that the n*nstate matrix of the state equation given by Equation (9–5) is the
transpose of that of the state equation defined by Equation (9–3).
Diagonal Canonical Form. Consider the transfer-function system defined by Equa-
tion (9–2). Here we consider the case where the denominator polynomial involves only
distinct roots. For the distinct-roots case, Equation (9–2) can be written as
(9–7)
The diagonal canonical form of the state-space representation of this system is given by
=b 0 +
c 1
s+p 1
+
c 2
s+p 2
+p+
cn
s+pn
Y(s)
U(s)
=
b 0 sn+b 1 sn-^1 +p+bn- 1 s+bn
As+p 1 BAs+p 2 B p As+pnB
y =[0 0 p 0 1]G
x 1
x 2
xn- 1
xn
W +b 0 u
F
x
1x
# 2 x
nV = F
0 1 0
0 0 0
p
p
p
0 0 1
- an
- an- 1
- a 1
VF
x 1
x 2
xn
V + F
bn-an b 0
bn- 1 - an- 1 b 0
b 1 - a 1 b 0
Vu
y= Cbn-an b 0 bn- 1 - an- 1 b 0 pb 1 - a 1 b 0 DF
x 1
x 2
xn
V +b 0 u
Openmirrors.com