Section 10–2 / Pole Placement 727
(10–9)
[See Problems A–10–2andA–10–3for the derivation of Equations (10–8) and (10–9).]
Equation (10–7) is in the controllable canonical form. Thus, given a state equation, Equa-
tion (10–1), it can be transformed into the controllable canonical form if the system is
completely state controllable and if we transform the state vector xinto state vector
by use of the transformation matrix Tgiven by Equation (10–4).
Let us choose a set of the desired eigenvalues as m 1 ,m 2 ,p,mn.Then the desired
characteristic equation becomes
(10–10)
Let us write
(10–11)
When is used to control the system given by Equation (10–7), the system
equation becomes
The characteristic equation is
This characteristic equation is the same as the characteristic equation for the system,
defined by Equation (10–1), when is used as the control signal. This can be
seen as follows: Since
the characteristic equation for this system is
∑s I-A+BK∑= @T-^1 (s I-A+BK) T@= @s I-T-^1 AT+T-^1 BKT@ = 0
x
=Ax+Bu=(A-BK) x
u=-Kx
@s I-T-^1 AT+T-^1 BKT@ = 0
xˆ
=T-^1 ATxˆ -T-^1 BKTxˆ
u=-KTxˆ
KT=Cdn dn- 1 p d 1 D
As-m 1 BAs-m 2 B p As-mnB=sn+a 1 sn-^1 +p+an- 1 s+an= 0
xˆ
T-^1 B= G
0 0 0 1
W