Page 176 - Modern Control of DC-Based Power Systems

140 Modern Control of DC-Based Power Systems

The solution of the first equation of the system (5.84) results in:
ξ P eq
5
π 3 ξðÞ 52 C f ξ 1 1 1 (5.85)
2
1 ξ 2 ξ C f
R L 1
An implicit definition of the manifold can be obtained by defining:
5
5 x 2 1 C f x 2 x 1 2 P eq 1 x 3 C f 5 0
Φ xðÞ 5 x 2 2 π 3 ξðÞ 1
jξ 1 5x 1 ;ξ 2 5x 3
R L x 1
(5.86)
As shown before, the dynamic of the off-the-manifold trajectories is
defined by:

_ z 52 K 2 z (5.87)
The result of the Eq. (5.87) is the control output that stabilizes the
system:
!
L f x 1 x 2 x 1 P eq
d 5cðπÞ5 1 2 2 1x 3
E L f C f R L C f C f x 1

! !!
1 P eq x 1 P eq
4
5
25x C f 1 2 1k g C f x 1 2K 2 x 2 1C f x 2 2
1 2 1 1x 3 C f
R L x R L x 1
1
(5.88)
Since the term P eq =x 1 represents the current absorbed by the CPL
load, the measurement of the current I CPL can be used for calculating the
control law, resulting in:
!
L f x 1 x 2 x 1 I CPL
d 5 1 2 2 1x 3
E L f C f R L C f C f
!!
!
1 I CPL 5 x 1
4
25x C f 1 2 1k g C f x 1 2K 2 x 2 1C f x 2 2I CPL 1x 3 C f
1 1
R L x 1 R L
(5.89)
The overall structure of the I&I control applied to the buck converter
is described in Fig. 5.16.
The application of the I&I control on a system characterized by
parallel-connected buck converters consists of replicating the procedure