Page 175 - Modern Control of DC-Based Power Systems

Control Approaches for Parallel Source Converter Systems 139

Figure 5.15 Phase plane of the generic second-order system.

the system [29]. The Eq. (5.82) represents the buck converter system are
defined in section. The system is rewritten with a new set of variables
x 1 5 V ref 2 V and x 2 5 I L , adding a new variable x 3 , that does not repre-
sent any physical element but integrates the voltage error. The system in
the new set of variables is described by the following set of equations:
8
x 2 x 1 P eq
_ x 1 5 2 2
>
> 1 x 3
>
C f R L C f C f x 1
>
>
<
x 1 dUE (5.82)
_ x 2 52 1
>
L f L f
>
>
>
>
:
_ x 3 52k g x 1
The nonlinear target system is a two-dimension system defined by the
equations describing its dynamic:
_ ξ 52 ξ 5

1
_ ξ 52 ξ 1 2 (5.83)
2
3
2
The mapping π : R -R is applied to the system (5.82) such that
π 1 ξðÞ 5 ξ 5 x 1 and π 2 ξðÞ 5 ξ 5 x 3 , resulting in the following nonlinear
2
1
system:
ξ P eq
8
2ξ 5 2 2
> 5 π 3 ðξÞ 1 1 ξ
> 1 2
> C f ξ
C f R L C f
>
> 1
<
ξ c πξðÞÞUE (5.84)
5
2 @π 3 ðξÞ ξ 2 @π 3 ðξÞ ξ 52 1 1 ð
> 1 2
> @ξ @ξ
> L f L f
>
2ξ 52k g ξ 1
>
:
2