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APPENDIX
Approximate System Linearization
One common method for controlling nonlinear dy-
namical systems is based on approximate feedback
linearization (Isidori, 1995), which depends on the
relative degree of each controlled variable. For new-
tonian systems like the quad-rotor in a simplified ap-
proach, the regulated variables of interest, here repre-
sented as the vector q, have relative degree two
¨q = f(q, ˙q,u) (50)
The control variables are represented by the vector u.
A pseudo control ν is defined such that the dynamic
relation between it and the system state is linear ¨q = ν
where ν = f(q, ˙q,u). Since the function f(q, ˙q,u) is
not exactly known, an approximation ν =
ˆ
f(q, ˙q,u) is
used which is invertible regarding u, resulting in
¨q = ν+ ∆(q, ˙q,u) (51)
where the modeling error is represented by
∆(q, ˙q,u) = f(q, ˙q,u) −
ˆ
f(q, ˙q,u) (52)
So the effective actuator can be calculated as
ˆu =
ˆ
f
−1
(q, ˙q, ν) (53)
Supposing in (51) that ∆(q, ˙q, u) = 0 we can proceed
in the stabilization problem, choosing a linear con-
troller, a PD for instance, that will locally solve the
regulation problem. A single hidden layer (SHL) neu-
ral network with convenientlyadapted weights will be
responsible for modeling error cancelation. Including
a command path generator C, the former linear con-
troller can be augmented through the architecture de-
picted in figure 7.
+
-
+
+
-
C
PD
ˆ
f
−1
(q, ˙q, ν)
Plant
SHL-NN
¨q
r
ν
P D
ν
ν
a
q
r
q
ˆu
e
Figure 7: NN augmented adaptive control architecture.
The pseudo control signal in (51) is the sum of
three components
ν = ¨q
r
+ ν
PD
−ν
a
(54)
where ¨q
r
is generated by C, ν
PD
is generated by
the PD controller and ν
a
is generated by the adap-
tive element introduced to compensate for the model
inversion error. The tracking error is computed as
e = [q
r
−q, ˙q
r
− ˙q]
T
and the PD controller can be re-
presented by
ν
PD
=
K
p
K
d
e (55)
so the tracking error dynamics is given by
˙e = Ae+ B(ν
a
−∆) (56)
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