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the des;gPn procedure, consider the general aviation airplane and use the Dutch-roll
AIRPLANE RESPONSE AND CLOSED-LOOP CONTROL 605
Refer
input
Referen
input
a) Schematic diagram
b) Block diagram
Fig.6.36 Yawdamper.
transfer function to rudder input. From Eq. (6.285),
AVt(s) -(0.1582 s + 0.0294)
A8r(S) = s~0 0340 S2+0 034~s+0~6~3~ (6.333)
so that
rls) sAp(s) .
A8r~S~ = ~8~s)
~(0.1582 s + 0.0294)
0.0340 S2 + 0.0347 s + 0~6~3
(6.334)
(6.335)
We assume that the rudder servo is a first-order system with a -10 so that
the time constant r = l/ti -. 0.10. As said before, we use the negative sign in
the numerator of the rudder servo transfer function because we have a negative
sign in the numerator of the yaw-rate-to-rudder-input transfer function as given by
Eq. (6.335). With this sign adjustment, we can use the negative feedback as shown
in Fig. 6.36b.
We assume that the transfer function of the wash-out circuit is given by
Gc=- s
: s +0 3333 (6.336)
Now we draw the root-locus as shown in Figs. 6.37 and 6.38. The root-locus of
Fig. 6.38 is a close-up of the root-locus around the origin. We select a point on
root-locus to have < - 0.8 for the Dutch-roll mode. We get krg -- 0.4228 and the
location of the closed-loop poles at ~6.9705, -2.0108 -1: j1.5140, -0.3619. This
gives us a damping ratio of 0.8 and a natural frequency of 2.5170 rad/s, giving us
level I Dutch-roll flying qualities for the general aviation airplane, which is a class
I airplane.
m 0.1
xo
o -0.1
o 0.05
a0
< -0.05
606
u,
x
<
o
d
E
PERFORMANCE, STABILI-fY, DYNAMICS, AND CONTROL
Fig. 637 Root-Iocus of the yaw damper system for the general aviation airplane.
Fig. 638 Root-Iocus of the yaw damper system for the general aviation airplane
(close-up around the origin).
AIRPLANE RESPONSE AND CLOSED-LOOP CONTROL 607
Yaw Damper On
- YawDampcrOff(BasicSystem)
a}
7a
o:
3
d
>-
a) Dutch-roll approximation
b) Complete fifth-order system
Fig. 639 Unit-step response of the yaw damper for the general aviation airplane.
We can now verify the design by simulating the response to a unit-step function
as shown in Fig. 6.39a. We observe that the closed-loop system with yaw damperis
performing satisfactory. To check the design further, we perform a simulation for
the complete fifth-order system as shown in Fig. 6.39b. We observe that the design
is satisfactory. The difference in the steady-state behaviors is due to the approxi-
mations irrvolved in deriving Dutch-roll transfer functions as we have discussed
earlier.
6.5.4 Full-State Feedback Design for Lateral-Directional Stabr7ity
Augmentation System
In this section we will design a full-state feedback control law for a lateral-
directional stability augmentation system. Such a flight control system needs that
all five state variables p, ~, p, p, and r be accurately measured and be available
for feedback. The roll and yaw angles can be accurately measured by positional
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