678 Chapter 9 / Control Systems Analysis in State SpaceEXAMPLE 9–11 Consider the system given by
For this case,The system is therefore completely state controllable.Alternative Form of the Condition for Complete State Controllability. Consider
the system defined by
(9–56)
where
If the eigenvectors of Aare distinct, then it is possible to find a transformation matrix
Psuch that
Note that if the eigenvalues of Aare distinct, then the eigenvectors of Aare distinct; how-
ever, the converse is not true. For example, an n*nreal symmetric matrix having
multiple eigenvalues has ndistinct eigenvectors. Note also that each column of the P
matrix is an eigenvector of Aassociated with
Let us define
(9–57)
Substituting Equation (9–57) into Equation (9–56), we obtain
(9–58)
By defining
P-^1 B=F=AfijB
z# =P-^1 APz+P-^1 Bu
x=Pz
li(i=1, 2,p,n).
P-^1 AP=D= F
l 1
0
l 2
0
ln
V
B=n*r matrix
A=n*n matrix
u=control vector (r-vector)
x=state vector (n-vector)
x
=Ax+Bu
CBABD= B
0
1
1
- 1
R =nonsingular
B
x# 1
x# 2R= B
1
2
1
- 1
RB
x 1
x 2R + B
0
1
R[u]
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