Image Processing Reference
In-Depth Information
4.9.2 C
ONTROLLABILITY
C
ONDITION
The following theorem gives the necessary and suf
cient conditions for controllabil-
ity of a discrete-time LTI system.
THEOREM 4.1
A discrete-time LTI system described by
x
(
k
þ
1
) ¼
Ax
(
k
) þ
Bu
(
k
)
(
4
:
120
)
is completely state controllable if the controllability matrix Q is full rank, where
Q
¼
BABA
2
B
A
N
1
B
(
:
)
4
121
Proof:
We assume that the system has one input. Let the initial state be x
(
k
0
)
at
time k
0
and the input for time k
k
0
be u
(
k
0
)
, u
(
k
0
þ
1
)
,
...
, u
(
k
0
þ
N
1
)
, then the
state of the system at time k
¼
N
þ
k
0
is given by
N
þ
k
0
1
X
x
(
N
þ
k
0
) ¼
A
N
x
(
k
0
) þ
A
N
þ
k
0
1
n
Bu
(
n
)
(
4
:
122
)
n
¼
k
0
To drive the system to origin, we must have
A
N
x
(
k
0
) þ
A
N
1
Bu
(
k
0
) þ
A
N
2
Bu
(
k
0
þ
1
) þþ
Bu
(
N
þ
k
0
1
) ¼
0
(
4
:
123
)
or
2
3
u
(
N
þ
k
0
1
)
4
5
u
(
N
þ
k
0
2
)
.
u
(
k
0
þ
¼
A
N
x
(
k
0
)
BABA
2
B
A
N
1
B
(
4
:
124
)
1
)
u
(
k
0
)
is full
Equation 4.124 has a solution if matrix Q
¼
BABA
2
B
A
N
1
B
rank.
Example 4.14
Consider the following system:
x(k)
u(k)
1
0
:
25
1
1
x(k þ
1)
¼
þ
1
:
5
0
:
25
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