Hardware Reference
In-Depth Information
Figure 3.40: IVC via injecting a signal.
and X
c
(0) = 0. Thus
∙
¸
N
p
D
+
N
a
n
d
y =
X
p
(0).
(3.69)
D
The objective of the initial value compensator is to find I
v
=
d
,asopposed
to changing the closed-loop system characteristic equation D, such that
N
D
+
N
D
d
has a more desirable dynamics than
N
D
.
According to Yamaguchi et al [219], any N
a
can be represented as
N
a
= N
a
N
n
a
a
contains only the stable zeros and N
a
contains only unstable zeros.
Let the desired poles of the transfer function between the initial states X
p
(0)
and y described by equation 3.69 be located at ζ
i
(i =1, 2, ...l), and let
where N
d
m
=(z −ζ
1
)(z −ζ
2
)...(z −ζ
l
),
and
d = d
m
d
= d
m
N
a
,
(3.70)
where d
contains only stable roots which we choose to be N
a
.Then
y =
N
p
d
d
m
+ N
a
n
Dd
1
d
m
X
p
(0)
]
=
N
p
d
m
+ N
a
n
D
1
d
m
X
p
(0).
]
(3.71)
Now we select n such that the roots of N
p
d
m
+N
a
n in equation 3.71 include
all the roots of D, which are λ
i
,i=1, 2, ...m+n
d
,andλ
i+1
= 0 for continuous
time or λ
i+1
= 1 for discretize time model. Then the transient response of
y(t) is dominated by the desired poles ζ
i
(i =1, 2, ...l). Hence it is necessary
to find n =[n
1
,n
2
, ...n
m
]intheformof
n
i
= a
i,q
z
q
+ a
i,q−1
z
q−1
+ ... + a
i,1
z + a
i,0
,
(3.72)
so that the following equation is satisfied:
N
p
(λ
j
)d
m
(λ
j
)+N
a
(λ
j
)n
i
(λ
j
)=0,i=1, ..., m; j =1, ..., m+ n
d
+1. (3.73)
