Page 149 - Modern Control of DC-Based Power Systems

Control Approaches for Parallel Source Converter Systems 113

The focus here lies on the input output Linearization. For the envis-
aged controller design, the form of the above system description in (5.1)
is not favorable. A suitable transformation is therefore needed. For this
purpose the Lie derivative is employed, which is defined as the gradient
of a scalar function h(x) multiplied with a vector field f(x), i.e.:

T
L f hxðÞ 5 @hxðÞ fxðÞ 5 grad hxðÞf ðxÞ (5.4)
@x
Considering the system presented in (5.1), and applying the Lie
derivative yields:

L a cxðÞ 5 @cxðÞ (5.5)
@x axðÞ

Now forming the time derivative of the output y one obtains:

_ ytðÞ 5 dc x ðÞ 5 @cx ðÞ _ x 1 1 .. . 1 @cx ðÞ _ x n x 5 @cx ðÞ _ x (5.6)
dt @x @x n @x
Replacing now:

(5.7)
_ xtðÞ 5 axðÞ 1 bx uðÞ
Results in:

_ ytðÞ 5 @cxðÞ axðÞ 1 @cxðÞ bxðÞu (5.8)
@x @x
Which can also be described as the Lie derivative:
_ ytðÞ 5 L a cxðÞ 1 L b cxðÞu (5.9)

In most technical systems according to [6] is L b 5 0, such that:
(5.10)
_ ytðÞ 5 L a cxðÞ
̈
The next step would be to calculate y while continuing from (5.10):
ÿ 5 dL a cx ðÞ 5 @L a cx ðÞ _ x 5 @L a cx ðÞ axðÞ 1 @L a cx ðÞ bxðÞ
dt @x @x @x (5.11)
2
a
5 L a L a cxðÞ 1 L b L a cxðÞ 5 L cðxÞ