Page 217 - Modern Control of DC-Based Power Systems

Control Approaches for Parallel Source Converter Systems 181

Variable x 1 corresponds to V and x 2 to I L . Next, the error variable
z 1 is introduced, for shifting the system’s coordinates are shifted by V ref .
This is done in the same way as for the previous Backstepping approach.

(5.192)
z 1 5 x 1 2 V ref
Since the reference voltage is supposed to be constant, _ V ref is zero.
Deriving (5.192) leads to
P
x 2 x 1
_ z 1 5 2 2 : (5.193)
C f R L C f x 1 C f
The goal is to set _z 1 52 c 1 z 1 , where c 1 . 0 is a design parameter. In
order to remove the unwanted terms in blue, the virtual control function
was selected in the following way:
^
x 2 x 1 P 1
α x 1 5 52 c 1 z 1 1 1 (5.194)
ðÞ
C f R L C f x 1 C f
In this case it is not possible to insert the parameter P in the virtual
control law to drive the error z 1 to zero as it is unknown as it was
described in Section 5.6.1. For this reason, it is replaced with the estimate
^
P 1 instead.
The typical Lyapunov function is selected again:

1
2
V 1 x 1 5 z : (5.195)
1
ðÞ
2
Subsequently, an estimation error is induced in the system. That is
why z 1 cannot be rendered to _z 1 52 c 1 z 1 1 z 2 . Inserting (5.194) into
(5.193) leads to the term which is used instead:
1
^

_ z 1 52 c 1 z 1 1 z 2 2 P 2 P 1 U (5.196)
x 1 C f
To guarantee Lyapunov stability, the error between the estimate ^ P 1
and the real value of P has to be considered in the Lyapunov function.
This allows the possibility to select an update law for the estimate which
stabilizes the system. Considering the estimation error in the Lyapunov
function
1 1 2
^
2
V 1 5 z 1 P2P 1 (5.197)
1
2 2Γ 1