Page 241 - Modern Control of DC-Based Power Systems

Control Approaches for Parallel Source Converter Systems 205

Figure 5.66 The trajectories of the two areas converges to the switching line.

x 1 5 V bus 2 V ref 5 0 (5.254)

The system (5.252) has a stable equilibrium point x e for z 5 1 where
the following applies:

x e1 5 V bus 2 V ref (5.255)
0

In the case of z 5 0 the following applies:

x e2 5 2V ref (5.256)
0
The switching line, with r 1 . 1, separates the state-space in two areas
(Fig. 5.66).

s xðÞ 5 r 1 x 1 1 x 2 5 0

0; for s xðÞ . 0 (5.257)
z 5
1; for s xðÞ . 0
Now it is necessary to determine the parameter r 1 of the switching
line (5.257) from the necessary condition for the existence of a sliding
state, _ss , 0. It follows from Eq. (5.258) in combination with the system
Eqs. (5.252) and (5.253):

1 1 zV in 2 V ref
r 1 2 x 2 2 x 1 1 , 0 (5.258)
_ ss 5 s xðÞ
R L C LC LC