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system and the inputs to the system, the only unknown in determining the states
of the system is the initial conditions. Therefore, if we can
find all the initial
conditions from inputs and outputs of the system, then the system is completely
state observable. In this section, we examine the observability of LTI system. Since
the results obtained are identical to continuous and discrete-time systems, we only
consider observability of discrete-time LTI system [6,7].
4.10.1 D EFINITION OF O BSERVABILITY
A discrete-time LTI system is said to be completely state observable if the states
of the system can be estimated from the knowledge of inputs and outputs of the
system.
4.10.2 O BSERVABILITY C ONDITION
The following theorem gives the necessary and suf
cient condition for observability
of a discrete-time linear system.
THEOREM 4.2
A discrete-time LTI system given by the input and output equations
x ( k þ
1
) ¼ Ax ( k ) þ Bu ( k )
(
4
:
125
)
y ( k ) ¼ Cx ( k ) þ Du ( k )
is completely state observable if the observability matrix Q is full rank, where Q is
de
ned as
2
3
C
CA
CA 2
.
CA N 1
4
5
P ¼
(
4
:
126
)
Proof: Let the initial state be x (
)
at time k ¼
0
0 and the input and output for time
k
0beu ( k )
and y ( k )
, respectively, then
) þ C X
k
1
y ( k ) ¼ Cx ( k ) þ Du ( k ) ¼ CA k x (
A k 1 n Bu ( n ) þ Du ( k )
0
(
4
:
127
)
n ¼ 0
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