Page 218 - Modern Control of DC-Based Power Systems

182 Modern Control of DC-Based Power Systems

Differentiation with respect to time leads to:
_
^ P 1

_ V 1 5 z 1 _z 1 1 P 2 ^ P 1
Γ 1
(5.198)
!
2 ^ z 1 ^ P 1
52 c 1 z 1 z 1 z 2 1 P 2 P 1 2 2
1
x 1 C f Γ 1
In line with the method applied in the previous chapter the update
law is chosen in such a way that the last term is set to zero, therefore the
update function of the CPL power ^ P 1 which ensures stability is obtained:
_ z 1
^ P 1 52 Γ 1 (5.199)
x 1 C f
The second step requires another error variable which incorporates a
second estimate as well. The second error variable can defined as:

x 2 x 2 x 1 ^ P 1
z 2 5 2 α 5 1 c 1 z 1 2 2 (5.200)
C f C f R L C f x 1 C f

with the derivative:
_ x 2 2 c 1 x 2
_ z 2 5 2 1
2 c z 1 1 c 1 z 1 2 P 2 ^ P 1
1 2
C f x 1 C f R L C
f
_ !
x 1 P ^ P 1 ^ P 1
1 2 2 (5.201)
2
2
R C 2 f x 1 R L C f 2 x 1 C f x C f
L
1
Again, the Lyapunov function is not just extended by the second error
variable but by the second estimate as well:
1 1 2
2
V 2 5 _ V 1 1 z 1 P2 ^ P 2 (5.202)
2 2 2Γ 2
Now the derivative of the Lyapunov function for the whole system
needs to be calculated while inserting (5.201) into (5.202):
_
^ P 2

_ V 2 5 _ V 1 1 z 2 _z 2 2 P 2 ^ P 2
Γ 2

2 _ x 2 2 c 1
52 c 1 z 1 z 1 z 2 1 z 2 2 c z 1 1 c 1 z 2 2 P 2 ^ P 1
1 1
C f x 1 C f