Hardware Reference
In-Depth Information
of equation 3.5, if available; otherwise one can add real pole, or well damped
complex poles (e.g., complex poles with damping ratio of ζ n2 =0.7) at fre-
quencies that is at or well above the resonance frequency so that the effects
of the changes in phase and amplitude contributed by the denominator on the
overall gain and phase margins become negligible.
With the filter F(s) inlcuded in the loop, the open-loop transfer function
L(s), error rejection (or sensitivity) transfer function S(s), complementary
sensitivity transfer function T (s), and shock transfer function S h (s)are:
L(s)=G c (s)F(s)G p (s),
1
1+G c (s)F(s)G p (s) ,
S(s)=
(3.18)
T (s)= G c (s)F(s)G p (s)
1+G c (s)F(s)G p (s) ,
G p (s)
1+G c (s)F(s)G p (s) .
S h (s)=
For the example plant model shown in Figure 3.2, we have
µ
2.6 × 10 7 × 2.3
s 2 + 131.2s +1.328 × 10 5 +
−2.6 × 10 7
s 2 + 2312s +1.305 × 10 9
G p (s)=
,
8.7880 × 10 14
(s 2 + 131.2s +1.328 × 10 5 )
(s 2 + 3990s +2.309 × 10 9 )
(s 2 + 2312s +1.305 × 10 9 ) . (3.19)
=
The numerator of the filter F(s) are chosen such that its zeros cancel the poles
of the flexible mode, i.e., the roots of (s 2 + 2312s +1.305 × 10 9 ). For this
actuator model there is a pair of stable zeros (s 2 + 3990s+2.309 ×10 9 ). They
areusedasthepolesofF(s) to balance its zeros. Thus,
F(s)= (s 2 + 2312s +1.305 × 10 9 )
(s 2 + 3990s +2.309 × 10 9 ) .
(3.20)
The transfer function of the compensated model, P comp ,is
8.7880 × 10 14
(s 2 + 131.2s +1.328 × 10 5 ) .
P comp (s)=F(s)G p (s)=
(3.21)
The Bode plot of the compensated actuator with controller of equation 3.11
in cascade with notch filter of equation 3.17 is shown in Figure 3.8. The
crossover frequency f v is chosen to be 1500 Hz or 9425 rad/s. It is clearly
visible from this plot that the open loop transfer function looks identical to
the case of rigid body model controlled by the lag-lead compensator (shown by
solid line in the figure). Such a response is achieved because the notch filter of
equation 3.20 is exactly the inverse of the model of resonant mode and cancels
the resonance completely.
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