Page 208 - Modern Control of DC-Based Power Systems

172 Modern Control of DC-Based Power Systems

Consequently, Eq. (5.159) can be written as follows:
(5.162)
_ z 1 52 c 1 z 1 1 z 2
Differentiating the frequently used Lyapunov function equation with
1 2
respect to time, and inserting V 1 x 1 5 z results in:
2 1
ðÞ
_ 2
V 1 5 z 1 _z 1 52 c 1 z 1 z 1 z 2 (5.163)
1
The second error variable is now defined as:
x 2 x 2 x 1 I d
z 2 5 2 αðx 1 Þ 5 1 c 1 z 1 2 2 (5.164)
C f C f R L C f C f
For which the derivative corresponds to:

_ x 2 2 x 2 x 1 I d _ I d
_ z 2 5 2 c z 1 1 c 1 z 1 2 1 1 2 (5.165)
1 2 2 2 2
C f R L C R C R L C C f
f L f f
Inserting now (5.164) and (5.165) in the Lyapunov function:
1 2
_
V 2 x 1 ; x 2 Þ 5 V 1 1 z 2 (5.166)
ð
2
Enables us now to calculate the derivative in the following way:

_ V 2 5 _ V 1 1 z 2 _z 2

2 _ x 2 2 x 2
52 c 1 z 1 z 1 z 2 1 z 2 2 c z 1 1 c 1 z 2 2
1 1 2
C f R L C
f
_ _
x 1 I d i d
1 1 2 Þ
2
R C f 2 R L C f 2 C f
L

2 2 _ x 2 2
52 c 1 z 2 c 2 z 1 z 2 2 ðc 2 1Þz 1 1 ðc 1 1 c 2 Þz 2
1 2 1
C f
_
x 2 x 1 I d _ I d
2 1 1 2 Þ (5.167)
2
R L C 2 R C 2 R L C 2 C f
f L f f
After assuring the stability of V 1 , it has to be ensured that:
(5.168)
_ z 2 52 c 2 z 2 2 z 1