Page 201 - Modern Control of DC-Based Power Systems

Control Approaches for Parallel Source Converter Systems 165

At first, a general dynamic system is considered, where x is the system
state and u is the control input.
_ x 5 fx; uÞ; f 0; 0Þ 5 0 (5.139)
ð
ð
The goal is to achieve global asymptotic stability for the equilibrium
point x 5 0. Therefore, one needs to find a feedback law u 5 axðÞ that
guarantees for all x 6¼ 0.

_ V ðxÞ # 0 (5.140)
It is possible to choose u in such a way to attain the infimum, which
is the greatest lower bound of the term in brackets. This will lead to min-
imal response time see Eq. (5.141).
inf
_ T (5.141)
u fVxðÞ 5 f ð x; uÞUgrad VðxÞg
The Lyapunov function VðxÞ is used to find a stabilizing control law
u 5 kx ðÞ; hence, it is named CLF. The existence of a globally stabilizing
control law is equivalent to the existence of a CLF which is known as
Artstein’s theorem [58]. This approach can be illustrated by considering
the system, which is affine (i.e., linear) in the control input:
_ x 5 fxðÞ 1 hxðÞu (5.142)

Under the assumption that a CLF for the system is known, Sontag
proposes a particular choice of control law, which is commonly referred
to as Sontag’s formula [59]:

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
ða 2 a 1 b 2
u 5 kxðÞ 52
b
(5.143)
a 5 V x xðÞfxðÞ
b 5 V x xðÞhðxÞ
This control law often uses parts of the system to help stabilize it. This
reduces the magnitude of the control input u.
!
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ða 2 a 1 b 2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
_
V 5 V x x ðÞ fx ðÞ 1 hx ðÞuÞ 5 a 1 b 2 52 a 1 b 2
2
ð
b
(5.144)