Section 10–2 / Pole Placement 725
Kis chosen properly, the matrix A-BKcan be made an asymptotically stable matrix,
and for all x(0)Z 0 , it is possible to make x(t)approach 0 astapproaches infinity. The
eigenvalues of matrix A-BKare called the regulator poles. If these regulator poles are
placed in the left-half splane, then x(t)approaches 0 astapproaches infinity. The prob-
lem of placing the regulator poles (closed-loop poles) at the desired location is called a
pole-placement problem.
In what follows, we shall prove that arbitrary pole placement for a given system is
possible if and only if the system is completely state controllable.
Necessary and Sufficient Condition for Arbitrary Pole Placement We shall now
prove that a necessary and sufficient condition for arbitrary pole placement is that the
system be completely state controllable. We shall first derive the necessary condition. We
begin by proving that if the system is not completely state controllable, then there are
eigenvalues of matrix A-BKthat cannot be controlled by state feedback.
Suppose that the system of Equation (10–1) is not completely state controllable.
Then the rank of the controllability matrix is less than n,or
This means that there are qlinearly independent column vectors in the controllability
matrix. Let us define such qlinearly independent column vectors as f 1 ,f 2 ,p,fq.Also,
let us choose n-qadditionaln-vectorsvq+1,vq+2,p,vnsuch that
is of rank n.Then it can be shown that
(See Problem A–10–1for the derivation of these equations.) Now define
Then we have
whereIqis a q-dimensional identity matrix and In-qis an (n-q)-dimensional identity
matrix.
= @s Iq-A 11 +B 11 k 1 @@s In-q-A 22 @ = 0
=^2
s Iq-A 11 +B 11 k 1
0
- A 12 +B 11 k 2
s In-q-A 22
2
=^2 s I- c
A 11
0
A 12
A 22
d + c
B 11
0
dCk 1 k 2 D^2
= @s I-Aˆ +BˆKˆ@
= @s I-P-^1 AP+P-^1 BKP@
∑s I-A+BK∑= @P-^1 (s I-A+BK)P@
Kˆ =KP= Ck 1 k 2 D
Aˆ =P-^1 AP= c
A 11
0
A 12
A 22
d, Bˆ =P-^1 B=c
B 11
0
d