Information Technology Reference
In-Depth Information
The dynamical system described in Eq. (
6.31
) is a differentially flat one with flat
output defined as the vector y D Œy
1;1
;y
1;2
; ;y
1;N
. Indeed all state variables can
be written as functions of the flat output and its derivatives.
Moreover, by defining the new control inputs
K
x
2
y
1;2
2K
x
2
y
1;1
C
K
x
2
0
C f.y
1;1
/
v
1
D
K
2K
K
v
2
D
x
2
y
1;3
x
2
y
1;2
C
x
2
y
1;1
C f.y
1;2
/
K
2K
K
v
3
D
x
2
y
1;4
x
2
y
1;3
C
x
2
y
1;2
C f.y
1;3
/
(6.32)
K
2K
K
v
N1
D
x
2
y
1;N
x
2
y
1;N1
C
x
2
y
1;N2
C f.y
1;N1
/
K
2K
K
v
N
D
x
2
NC1
x
2
y
1;N
C
x
2
y
1;N1
C f.y
1;N
/
the following state-space description is obtained
0
@
1
A
0
@
1
A
0100 0000
0000 0000
0001 0000
0000 0000
0000 0000
0000 0000
0000 0100
0000 0000
0000 0001
0000 0000
000 00
100 00
000 00
010 00
000 00
001 00
000 00
000 10
000 00
000 01
0
@
1
A
0
@
1
A
y
1;1
y
2;1
y
1;2
y
2;2
y
1;N1
y
2;N1
y
1;N
y
2;N
y
1;1
y
2;1
y
1;2
y
2;2
y
1;N1
y
2;N1
y
1;N
y
2;N
0
1
v
1
v
2
v
3
v
N1
v
N
@
A
D
C
(6.33)
By selecting measurements from a subset of points x
j
j2Œ1;2; ;m,the
associated observation (measurement) equation becomes
0
@
1
A
y
1;1
y
2;1
y
1;2
y
2;2
y
1;N
y
2;N
0
1
0
1
100 00
000 00
000 10
000 00
z
1
z
2
z
m
@
A
@
A
D
(6.34)
Thus, in matrix form one has the following state-space description of the system
y D A y C
Bv
z
D C y
(6.35)
