Page 213 - Modern Control of DC-Based Power Systems

Control Approaches for Parallel Source Converter Systems 177

T
ðÞUx 2
_ x 1 5 θ f x 1 1 h 1 x 1
f 1 1 ðÞ
T
_ x 2 5 θ f x 1 ; x 2 Þ 1 h 2 x 1 ; x 2 ÞUx 3
f 2 2 ð ð (5.172)
^
T
_ x k 5 θ f x 1 ; x 2 ; .. . ; x 3 Þ 1 h k x 1 ; x 2 ; .. . ; x k ÞUu
f k k
ð
ð
This time the known functions f return vectors. Also, the unknown
k
parameter θ is considered to be a vector. The output is defined as y 5 x 1
which should track a reference signal y ref ðtÞ. This tracking control prob-
lem can be transformed to a regulation problem by introducing the track-
ing error variable. The error signal is defined as z i 5 x i 2 α i21 (e.g.,
z 1 5 x 1 2 y ref ). Afterwards the first system equation is rewritten:
T

_ z 1 5 _x 1 2 _y 5 θ f 1 z 1 1 y ref 1 h 1 z 1 1 y ref ð z 2 1 α 1 Þ 2 _y ref
ref
f 1
(5.173)

For simplicity f z 1 1 y ref will be referred to simply as f , and simi-
1 1
larly for h 1 ;f ;h 2 .
2
At this stage the Backstepping algorithm is expanded, by introducing
^ ~ ^
_
an estimate θ f 1 of θ f 1 with the estimation error θ f 1 5 θ f 1 2 θ f 1 , (with
~ _ ^
θ f 1 52 θ f 1 ). By applying now the typical Lyapunov function candidate,
that not only penalizes the tracking errors but also the estimation errors:
1 1 T T
21 ~
^ 5 z 1 θ Γ θ
~
2
V 1 z 1 ; θ f 1 1 f 1 f 1 f 1 (5.174)
2 2
T
The newly introduced term Γ 5 Γ . 0 is the adaptation gain matrix,
^
and determines how fast the parameters of θ are adapted. This leads to
the following Lyapunov derivative:
21 _
T
~
_ V 1 5 z 1 _z 1 1 θ Γ θ ~
~
T 21 _ (5.175)

T
5 z 1 θ f 1 h 1 ðz 2 1 α 1 Þ 2 _y 2 θ Γ ^ 2 Γ f 1 1 1
f z
f 1 1 ref f 1 θ f 1
To cancel the indefinite terms, virtual control α 1 , and the intermedi-
are defined as:
ate update laws τ f 11
1 ^ T
α 1 5 2 c 1 z 1 2 θ f 1 _y ;c . 0 (5.176)
f 1 1 ref
h 1
_ ^
f z
τ f 11 5 θ f 1 5 Γ f 1 1 1 (5.177)