Environmental Engineering Reference
In-Depth Information
, and
1
sL
+
K
i
is stable for any positive
L
and
K
i
,
H
(
s
) is stable for any passive
Z
(
s
). The stability of the
system can then be checked by plotting the Nyquist diagram of
H
(
s
) and the range of the
parameters can be determined by plotting the root locus of
H
(
s
).
A sufficient condition is given below. The transfer function
G
(
s
) or the system is stable if
1
+
K
R
(
s
)
sL
+
K
i
where
H
(
s
)
=
Z
c
(
s
). Since 1
+
K
R
(
s
) is stable for any positive
ξ
and
ω
Z
c
(
s
)
∞
<
1
+
K
R
(
s
)
1
(8.5)
sL
+
K
i
1
+
K
R
(
s
)
sL
+
K
i
according to the well-known small gain theorem (Zhou
et al.
1996). The norm
∞
is the maximum amplitude obtained from the Bode plot of
H
(
s
). It can also be calculated
iteratively, e.g. via using theMATLAB
R
routine
hinfnorm
. It is worth noting that this condition
is sufficient only and it can be very conservative in some cases. Amore conservative condition,
which can be derived from (8.5), is
Z
c
(
s
)
K
i
Z
c
(
s
)
∞
<
K
hm
,
1
+
where
K
hm
=
max
{
K
h
}
. For a resistive load
Z
(
s
)
=
R
, there is
Z
c
(
s
)
∞
=
R
and, hence, the system is stable if
K
i
K
hm
>
R
1
+
or, equivalently, if
K
i
R
−
K
hm
<
1
.
This condition is likely to be very conservative because of the approximations applied to obtain
it and also because of the inherent conservativeness of the small-gain theorem. What is clear
from this is that there is an upper bound on
K
h
and, hence,
K
h
cannot be too big. The system
is more stable if the load is heavier (with a smaller
R
). A large current feedback gain
K
i
also
helps stabilise the system. However,
K
i
should not be too large because, as discussed before,
a large
K
i
leads to high THD. It will be seen from the next subsection that a large
K
i
also
destabilises the current loop when the sampling effect is taken into consideration.
8.3.2 With Consideration of the Sampling Effect
If the controller is implemented with analogue electronic circuits, then the current feedback
gain
K
i
can be very large. However, if it is implemented with a digital controller, then it is
limited, as will be illustrated below.
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