Page 242 - Modern Control of DC-Based Power Systems

206 Modern Control of DC-Based Power Systems

By choosing r 1 5 1 it is possible to decouple the condition of exis-
R L C
tence (5.258) from the state x 2 5 _ V bus . This selection in combination
with Eq. (5.251) leads to the following condition that needs to be
satisfied:
1
s zV in 2 V bus Þ , 0 (5.259)
LC
s_ 5 s xðÞ ð
From this condition we obtain for the case 1: s xðÞ , 0 and z 5 1.
V in . V bus (5.260)
For the case 2 s x ðÞ . 0 and z 5 0, the condition (5.259) is used to
obtain

V bus . 0 (5.261)
For the existence of a sliding necessary condition (5.259) next to the
Eqs. (5.260) and (5.261) has to be fulfilled. Furthermore from (5.262) it
can be seen that for case 1 has a decrease rate of 2 V bus , 0. For case 2 an
LC
increase rate of V IN 2V bus . 0 is present. The trajectories of the control
LC
converge in finite time towards the sliding state.
1
_ s 5 ð zV in 2 V bus Þ (5.262)
LC
Ideally, a converter will switch at infinite frequency with its phase tra-
jectory moving on the sliding surface when it enters SM operation
However, in the presence of finite switching time and time delay, this
ideal behavior is not possible. The discontinuity in the feedback control
will produce a particular dynamic behavior in the vicinity of the surface
trajectory known as chattering.
If the chattering is uncontrolled, the converter system will become
self-oscillating at a very high switching frequency corresponding to the
chattering dynamics. This undesirable high switching frequency will result
in excessive switching losses, inductor and transformer core losses, and
EMI noise issues. To solve these problems, the control law in (5.257) is
redefined as:
0 for s xðÞ ,2 β

u 5 (5.263)
1 for s xðÞ . β