Page 171 - Modern Control of DC-Based Power Systems

Control Approaches for Parallel Source Converter Systems 135

The calculation of the control law starts with the definition of the tar-
get system to map into the nonlinear system to be controlled. The target
system has typically a low-order dimension, and it is described by the fol-
lowing expression:
_
ξ 5 αξðÞ (5.66)
ξ 5 0 (5.67)

As described in (5.67), the target system has an equilibrium point at
the origin that must be asymptotically stable.
The target system is usually defined a priori by the designer, by speci-
fying the desired dynamic. This allows the target system to comply with
the theorem in [28], because it can be defined such as (5.67) is satisfied.
In the I&I approach, the target system defines also the dynamic of the
closed-loop system, since the system controlled by the feedback loop con-
_
tains the copy of the dynamic of ξ 5 αξðÞ.
p
n
Supposing that there exists a mapping π:R -R such as x 5 πξ ðÞ,
p
with ξAR , the following equations must hold:
πξ ðÞ 5 x (5.68)

πð0Þ 5 0 (5.69)
The application of the mapping x 5 πξðÞ and of the control law to the
nonlinear system (5.65) results in the partial differential Eq. (5.70).
@π
f πξðÞÞ 1 g πξðÞÞc πξðÞÞ 5 (5.70)
@ξ αðξÞ
ð
ð
ð
Once the conditions (5.68) (5.70) are satisfied, the manifold M,
which is the geometric interpretation of the mapping procedure, can be
defined implicitly by:

n n jx 5 πξðÞfor some ξAR p (5.71)
f xAR jΦ xðÞ 5 0 5 xARf
The Eq. (5.71) shows that the aim of the controller is to drive to zero
all the off-the-manifold trajectories the manifold, described by z 5 Φ xðÞ.
The implicit definition of the manifold demonstrates that the only trajec-
tories permitted are the ones that satisfy Eq. (5.71), which are strictly
connected to the target system.
Moreover, the equation demonstrates an equivalence between the map
p
n
π:R -R and the manifold, defining a correlation between the mathe-
matical and geometrical interpretation of the stabilization procedure.