Page 214 - Modern Control of DC-Based Power Systems

178 Modern Control of DC-Based Power Systems

Eq. (5.177) is often referred to in the literature as a so-called tuning
function [64]. Like the control law, these parameter update laws are built
recursively. Substituting into the derivative of V 1 leads to:

T

~
_ V 1 52 c 1 z 1 1 h 1 z 1 z 2 2 θ Γ 21 _ ^ (5.178)
θ f 1 2 τ f 11
f 1 f 1
If the system would be of relative degree 5 1 this would be the final
design step, the update laws would cancel the indefinite term while z 5 0
and the derivate would be reduced to:
_ V 1 52 c 1 z 2 (5.179)
1
which implies that the z 1 subsystem would be stabilized.
Thus, the next logical step lies on assuring that the z 2 subsystem con-
verges; by using α 1 it is possible to step back to the second equation,
which can be written after the introduction of the error variables
z i 5 x i 2 α i21 :
T
_ z 2 5 θ f 1 h 2 ðz 3 1 α 2 Þ 2 _ α 1 (5.180)
f 2 2
The second Lyapunov function can be therefore written as:

1 1 T T
21 ~
~
2
V 2 5 V 1 1 z 1 θ Γ θ (5.181)
2 2 2 f 2 f 2 f 2
The derivative of (5.181) along the solutions of (5.173) and (5.180)
corresponds to:
!
T
~
_ V 2 52 c 1 z 1 h 1 z 2 _z 2 2 θ Γ 21 _ ^ f @α 1
2
1 f 1 θ f 1 2 τ f 11 1 Γ f 1 1 z 2
@x 1
^
h T i
1z 2 θ f 1 h 2 z 3 1 α 2 Þ 1 μ (5.182)
f 2 2 ð 1

~
T
2θ Γ 21 _ ^ T f z
θ 2 Γ f 2 2 2
f 2 f 2
The term μ represents the dynamics of _ α 1 and is calculated by the
1
partial derivatives:
T @α 1 _
@α 1 @α 1 ^ ^ @α 1 @α 1
μ 5 5 θ f 1 h 1 x 2 1 1 _ y 1
1 f 1 1 ^ θ f 1 ref ÿ ref
@t @x 1 @y ref @_y
@θ f 1 ref
(5.183)