Page 150 - Modern Control of DC-Based Power Systems

114 Modern Control of DC-Based Power Systems

For the higher derivates the following is valid:

y 5 cxðÞ
_ y 5 L a cxðÞ
̈ 2
a
y 5 L cxðÞ
^
y ðδ21Þ 5 L δ21
a cxðÞ (5.12)
@L a δ21 cxðÞ
δ
y ðδÞ 5 L cxðÞ 1 bxðÞu
a
@x
i
a
i
L b L cxðÞ 5 @L cxðÞ bxðÞ 5 0;i 5 0; .. . ; δ 2 2
a
@x
i
Only for the index δ 2 1the term L b L cxðÞ is not 0. δ is also called the
a
relative degree of the system. The relative degree is in linear system equal
to the difference between numerator and denominator of the transfer func-
tion. If n 5 dim x ðÞ 5 δ the system has therefore no internal dynamics.
Those internal dynamics are not observable and thus cannot be linearized.
According to [6] the input signal has to be in the following form, to
linearize the system, this corresponds to the nonlinear Ackermann for-
mula, which is only dependent on the original state x:
n
L c xðÞ 1 a n21 L δ21 V
u 52 a a cxðÞ 1 .. . 1 a 1 L a cxðÞ 1 a 0 cðxÞ 1 w
L b L n21 L b L n21
a c xðÞ a c xðÞ
(5.13)
By means of the transformations, the nonlinear system is converted into
a linear representation. For this procedure, the name of exact input output
linearization or exact linearization is used. The choice of the coefficients a i
of the control normal form is unrestricted, so that one can imprint any
desired eigenvalue configuration on the transformed system. Also, the value
V can be freely selected. Comparing Eq. (5.13) with (5.2) it is possible to
say that rðxÞ is the controller, while vxðÞ has the function of a prefilter [6].

5.1.2 Application to MVDC System
The LSF controller is a centralized control architecture, which means that
it collects the measurements of each generator and load. Therefore, the
controller has perfect knowledge as all relevant states can be measured.
This can be a significant advantage in terms of control performance.