Page 215 - Modern Control of DC-Based Power Systems

Control Approaches for Parallel Source Converter Systems 179

At this point it has to be mentioned that (5.175) is not necessarily
negative definite, but the whole system with z 2 is asymptotically stable.
Therefore now an update law for α 2 has to be defined. The virtual con-
trol and intermediate update laws are selected as:
1 T
^
α 2 5 2 c 2 z 2 2 h 1 z 1 2 θ f 1 μ ;c . 0 (5.184)
f 2 2 1
h 2

@α 1 @α 1
f f z 1 2 (5.185)
τ f 12 5 τ f 11 2 Γ f 1 1 z 2 5 Γ f 1 1 z 2
@x 1 @x 1
f z
5 Γ f 2 2 2 (5.186)
τ f 22
_ ^
θ f 2 5 τ f 12 1 τ f 22 (5.187)
Taking the time derivative of (5.182) and inserting (5.197) (5.199)
results in:

_ 2 2 ~ T 21 _ ^
1 2 f 1 θ f 1 2 τ f 12
V 2 52cz 2 c 2 z 1 h 2 z 2 z 3 2 θ Γ f 1

~ T 21 _ ^ (5.188)
2θ Γ f 2 θ f 2 τ f
f 2 2 22
Those steps are identically repeated until step n, where the real control
input u enters the system, which for a second-order corresponds to the
second step so that (5.197) changes to:
1 T
^
u 5 α 2 5 2 c 2 z 2 2 h 1 z 1 2 θ f 1 μ ;c . 0 (5.189)
1
f 2 2
h 2
This means that by setting z 3 5 0 and by substituting (5.198), (5.199)
and (5.202) into (5.201), its blue part disappears and the following
Lyapunov derivate is negative definite:
2
_ V 2 52 c 1 z 2 c 2 z 2 (5.190)
1 2
At this point it should be mentioned explicitly that convergence of
^
the parameter estimate θ is guaranteed, but not necessarily convergence
to the real value of θ.
One problem that arises when using Adaptive Backstepping is that
more than one update law is obtained for each unknown constant. An
example for this effect is given in the following section, where the
unknown power is estimated by two update laws. It is possible to remove
the overparameterization by choosing appropriate tuning functions, but it
was shown that overparametrized controllers can reduce cost [65].