Download Slung Load Control and User Interfaces for a Quadrotor Micro

4.2
Sensors
where
p
ζ ω0
b1 = 1 − α
γ +β
ω = ω0 1 − ζ 2
ω
ζ ω0
b2 = α 2 + α
α = e−ζ ω0 h
γ −β
ω
β = cos(ω h)
a1 = −2αβ
2
a2 = α
γ = sin(ω h)
and ω0 is given in radian per seconds (rad/s). The pulse-transfer function can be
written on backward shift form as
H(z) =
b1 z + b2
b1 + b2 z−1
−1
−1
⇐⇒
H(z
)
=
z
z2 + a1 z + a2
1 + a1 z−1 + a2 z−2
This gives the filter’s equation
y f (kh) = H(z)u(kh)
=⇒
⇐⇒
⇐⇒
y f (kh)z(1 + a1 z−1 + a2 z−2 ) = u(kh)(b1 + b2 z−1 )
y f (kh + 1) + a1 y f (kh) + a2 y f (kh − 1) = b1 u(kh) + b2 u(kh − 1)
y f (kh + 1) = −a1 y f (kh) − a2 y f (kh − 1) + b1 u(kh) + b2 u(kh − 1)
where y f (kh) is the filtered signal, u(kh) is the input sample of the signal to be
filtered and kh is the time instant t = kh when the computer samples the values. This
is advantageous since the output can be calculated a sample in advance and there
will be minimal computational delay when the output is to used.
1
With ω0 = ωc = 2 · 6.2831853 rad/s and ζ = √ , the final filter can be seen (4.12)
2
y f (kh+1) = 1.929y f (kh)−0.9314y f (kh−1)+0.001234u(kh)+0.001205u(kh−1)
(4.12)
The Bode plot of the filter can be seen in Figure 4.6.
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