Page 57 - Modern Control of DC-Based Power Systems

22 Modern Control of DC-Based Power Systems

shows how the addition of a feedback loop modifies the transfer function
of the system:
1 TðsÞ G vg ^ Z out
v
^ v out 5 ^v ref 1 ^ in 2 i Lo1 (2.9)
HðsÞ 1 1 TðsÞ 1 1 TðsÞ 1 1 TðsÞ
1
(2.10)
TsðÞ 5 HðsÞG c sðÞG vd sðÞ
V m

G vg CL sðÞ 5 ^ v out ðsÞ 5 G vg (2.11)

^ v in ðsÞ ^ v ref ; ^ i Lo1 50 1 1 TðsÞ

Z out CL sðÞ 5 ^ v out ðsÞ 5 Z out (2.12)

2 ^ i Lo1 ðsÞ ^ v ref ; ^ i Lo1 50 1 1 TðsÞ
From the resulting Eq. (2.12) it can be observed that, through the feed-
back, the closed-loop transfer function and the output impedances are
reduced by the factor of 1=ð1 1 TsðÞÞ which is called the sensitivity func-
tion [14]. It describes how the influence of disturbances in the input values
is attenuated by good control design on the output voltage. An optimal
controller design would ensure that the norm of (1 1 T) is very large over
all frequencies. This would mean that the transfer functions would become
insignificant and with them their effects on the output voltage.
Due to the limited bandwidth of a real controller this is not possible
to realize. Consequently, the controller is designed such that the norm of
T is large for frequencies smaller than a specified crossover frequency and
becomes infinite for DC values. For frequencies that are above the cross-
over frequency, the norm of T becomes marginal compared to 1 and the
closed-loop transfer functions by this means retrieve the original charac-
teristics of the open-loop transfer functions [15]. To limit the disturbances
at high frequencies the filter of the converter has to be designed to limit
the bandwidth of the system so that the effects on the output voltage are
limited over the whole frequency range.
The definition of the closed-loop input impedance, Z IN CL ,isthe ratio
of perturbations in the input voltage ^ in to the perturbations in the input cur-
v
^
^
rent i in ðsÞ, while the perturbations in the load current i Lo1 are set to be zero.

Z IN CL sðÞ 5 ^ v in ðsÞ (2.13)
^
^ v ref ; ^ i Lo1 50
i In ðsÞ