Image Processing Reference
In-Depth Information
5.4.3 R EDUCED -O RDER O BSERVER D ESIGN
A full-order observer estimates all the states of the system regardless of whether they
are measured or not. Consider the case when there are many states where most of
them are measured except for a few. In this case, there is no need to estimate the
available states. A reduced-order observer is used to estimate the states of a dynamic
system, when partial measurement of some states is available.
Assume that the states of a dynamic system can be portioned into two parts,
those that are measured directly and those that are not measured directly. That is
"
#
"
#
Measured
Not measured
x 1 ( k )
x 2 ( k )
x ( k ) ¼
¼
(
5
:
86
)
The state and output equations are portioned accordingly as
x 1 ( k þ
) ¼ A 11 x 1 ( k ) þ A 12 x 2 ( k ) þ B 1 u ( k )
1
x 2 ( k þ
) ¼ A 21 x 1 ( k ) þ A 22 x 2 ( k ) þ B 2 u ( k )
y ( k ) ¼ Cx 1 ( k )
1
(
5
:
87
)
where
A 11 , A 12 , A 21 , A 22 , B 1 , B 2 , and C are matrices of appropriate size
C is a nonsingular square matrix
Since x 1 ( k )
is measured, then its estimate is
x 1 ( k ) ¼ x 1 ( k ) ¼ C 1 y ( k )
(
5
:
88
)
The second set of states x 2 ( k )
are estimated as
x 2 ( k ) ¼ Ly ( k ) þ z ( k )
(
5
:
89
)
where
z ( k þ
) ¼ Fz ( k ) þ Gy ( k ) þ Hu ( k )
(
:
)
1
5
90
The estimation error e ( k )
is de
ned as
e ( k ) ¼ x 2 ( k ) x 2 ( k )
(
5
:
91
)
Dynamics of the estimation error e ( k )
is governed by
e ( k þ
1
) ¼ x 2 ( k þ
1
) x 2 ( k þ
1
) ¼ A 21 x 1 ( k A 22 x 2 ( k B 2 u ( k ) Ly ( k þ
1
) z ( k þ
1
)
¼ A 21 x 1
) Fz ( k ) Gy ( k ) Hu ( k )
¼ A 21 x 1 ( k A 22 x 2 ( k B 2 u ( k ) LC ( A 11 x 1 ( k A 12 x 2 ( k B 1 u ( k ))
Fz ( k ) GCx 1 ( k ) Hu ( k )
¼ ( A 21 LCA 11 GC ) x 1 ( k )þ( A 22 LCA 12 ) x 2 ( k )þ( B 2 LCB 1 H ) u ( k ) Fz ( k )
¼ ( A 21
( k A 22 x 2
( k B 2 u ( k ) LCx 1
( k þ
1
LCA 11
GC ) x 1
( k )þ( A 22
LCA 12
) x 2
( k )þ( B 2
LCB 1
H ) u ( k )
(
5
:
92
)
F ( x 2 ( k ) Ly ( k ))
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