Page 173 - Modern Control of DC-Based Power Systems

Control Approaches for Parallel Source Converter Systems 137

The aim of the controller is to bring the system to the set-points
values ½x ; 0Š. Therefore, the system variables are modified in order to

1
comply with the Eq. (5.68) as follows:

v 1 5 x 1 2 x
1 (5.74)
v 2 5 x 2
The nonlinear target system is given by:
_ ξ 52 K 1 ξ 3 (5.75)

which has an asymptotically stable equilibrium point in ξ 5 0.

2
The mapping function is π:R-R , meaning that
ðv 1 ; v 2 Þ 5 ðπ 1 ξðÞ; π 2 ξðÞÞ. By setting π 1 ξðÞ 5 ξ 5 v 1 , the equilibrium point
of π 1 ξðÞ 5 ξ 5 0, which is equal to the equilibrium point of v 5 0 and

1
satisfies the hypothesis (5.68).
The application of the map π to the system (5.73) in the new variables
ðv 1 ; v 2 Þ results in a nonlinear system with the partial differential
Eq. (5.70):
2K 1 ξ 5 ξπ 3
8 _
> 3 2
<
@π 2 2 (5.76)
2
2
> @ξ K 1 ξ 5 π 1 ξ 1 c πξðÞð Þ
:
where u 5 c πξðÞÞ is the control output that drives to zero the off-the-
ð
manifold trajectories and π 1 ξðÞ has been already substituted with ξ. The
solution of the first equation of (5.76) results in π 2 52 K 1 ξ 2=3 :
The implicit definition of the manifold, which forces the trajectories
of the off-the-manifold coordinate z 5 Φ xðÞ to zero, can be described as:
Φ xðÞ 5 v 2 2 π 2 ξ 5 v 2 1 K 1 v 1 2=3 5 0 (5.77)

1 jξ 1 5v 1
The dynamic of z must have bounded trajectories to satisfy the theo-
rem in [28], therefore the time derivative of (5.77) is defined by the fol-
lowing equation:

_ z 52 K 2 z (5.78)

The result of (5.78) defines the control output that drives to zero the
coordinate z and renders the manifold invariant is given by:

2 2=3 3 2=3
2
u 5 c πðÞ 52 v 2 v 1 2 K 1 v v 2 K 2 v 2 1 K 1 v (5.79)
2 1 2 1
3