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Control Approaches for Parallel Source Converter Systems 163

Backstepping is a concept that allows the user to follow a scheme
when trying to find a suitable Lyapunov function candidate for the given
system. Hence, the challenge of finding a Lyapunov function candidate
while using Lyapunovs’s direct method is tackled systematically.

5.5.1.1 Lyapunov
Lyapunov proposed an indirect method and a direct method to prove sys-
tem stability. The indirect method of Lyapunov, also called first method,
uses the linearization of a system to determine the local stability of the
original system. Lyapunov’s direct method, also called the second method
of Lyapunov, allows us to determine the stability of a system without
explicitly integrating the differential Eq. (5.135).
(5.135)
_ xtðÞ 5 fx tðÞÞ
ð
This method is a generalization of the idea that if there is a “measure
of energy” in a system; then, the rate of change of the system’s energy
can be studied to ascertain stability [57]. The second method will be used
further on as it is the foundation of the Backstepping theory.
The following definition from [6] describes the properties of a valid
Lyapunov function:

Direct method of Lyapunov: Let the differential equation _x 5 f ðxÞ
with equilibrium point x E 5 0 have a continuous and unique solution for
every initial condition in a neighborhood D 1 ð0Þ of the equilibrium point.
If a function VðxÞ exists which is positive definite and therefore fulfills:
V 0ðÞ 5 0
(5.136)
VxðÞ , 0; x 6¼ 0
and is continuous with continuous derivatives:
_
VxðÞ # 0 (5.137)

Then, the equilibrium point x E 5 0 is stable in the sense of Lyapunov
in the neighborhood. In addition, the equilibrium point is asymptotically
stable if the inequality of eq. (5.137) holds true for all x.
The derivative of a Lyapunov function is calculated with gradients as
follows
n
X @V
T
_ V xðÞ 5 _x gradV xðÞ 5 (5.138)
_ x i
@x i
i51
where _x 5 f ðxÞ. Finally one can check if Eq. (5.137) is fulfilled.