Page 237 - Modern Control of DC-Based Power Systems

Control Approaches for Parallel Source Converter Systems 201

_ s , 0; for s xðÞ . 0
(5.233)
_ s . 0; for s xðÞ , 0
If you summarize both conditions, you get

s_ , 0; (5.234)
s
as a condition for a sliding state. The following applies

T
_ s 5 grad s xðÞ _x (5.235)
T
T
and for the case of s xðÞ 5 r x; it follows that _s 5 r _x. Unfortunately, the
s
condition s_ , 0, does not secure for every conceivable case that the tra-
jectories reach the switching surface in finite time. The condition is
therefore necessary, but not sufficient for a sliding state control.
There are different approaches to ensure the reachability of the sliding
surface for all trajectories in finite time. A very common approach is that
mentioned in [75]. Here, the decrease of the switching function along
the trajectories xðtÞ is specified. It applies

(5.236)
_ s xðÞ 52 q sgn s xðÞÞ 2 ksðxÞ
ð
with positive constants q and k. Obviously the sliding surface s in combi-
nation with Eq. (5.236) satisfies the necessary condition.
2
s
s_ 52 qs jj 2 ks , 0 (5.237)
Since the Eq. (5.236) has a decrease rate of _s ,2 q or an increase rate
s
of _ . q even for very small values of s jj, the trajectories x(t) reach the
sliding surface also in finite time. If you consider that:
T T
_ s xðÞ 5 grad s xðÞ _x 5 grad sðxÞ Ax 1 buð Þ (5.238)
then one obtains from (5.236) the control law:
T
u xðÞ 52 grad s xðÞ Ax 1 q sgn s xðÞÞ 1 ksðxÞ (5.239)
ð
T
grad sðxÞ b
In the case of a switching hyperplane as switching surface
T

T
T

r Ax 1 q sgn r x 1 kr x
u xðÞ 52 (5.240)
T
r b
The freely selectable positive parameters q and k can be used to influ-
ence the dynamics of the control.