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Fig. 6.46 Block diagram of the outerloop of the pitch displacement autopiIoL
AIRPLANE RESPONSE AND CLOSED-LOOP CONTROL 617
Real A,os
Fig. 6.47 Root-Iocus of the outer loop of the pitch displacement autopilot for the
business jeL
Ilg. 6.48 . Unit-step response of the pitch displacement autopilot for the business jet.
618 PERFORMANCE, STABiLITY, DYNAMICS, AND CONTROL
a) Scbematic diagram
Jng.6.49 Altitude-holdautopiIoL
h
automatically levels the aircraft and, afterwards, it will maintain that altitude until
told to do otherwise.
The schematic diagram ofa typical altitude-hold autopilotis shown in Fig. 6.49a.
The corresponding block diagram is shown in Fig. 6.49b. Wc have a lag compen-
sator in the forward path, which is required to retain stability on closing the outer
loop. The transfer fuPnction of the lag compensator is assumed as
Gc = k(s : ~).8)
(6.374)
The transfer function for the altitude-to-elevator input can be obtained as fol-
lows:
h = Uo siny
y -.O -a
(6.375)
(6.376)
AIRPLANE RESPONSE AND CLOSED-LOOP CONTROL 619
where Uo is the fiight velocity and y is the fiight path angle. Then,
sh(s) = Uoy(s)
h(s) Uoy(s)
Z 8e~S~ = A8e~S~
(6.377)
(6.378)
Uo[O(s) - a(s)]
= ABe(S) (6.379)
To illustrate the design procedure, let us consider the general aviation airplane
once again and design an altitude-hold autopilot for it. Using Eqs. (6.163) and
(6.164), we get
h(s) = Uo (-biS2 +(di - b2)S + (d2 - b3)) (6.380)
A8.(s~ = . a~s3 + a -S2 + a3S
where bi - -0.0273, b2 - -2.0366, b3 - O, ai -0.1695, a2 -.0.8494, a3 -
2.1851, di - -2.0112, and d2 - -3.9018.
At first, let us assume that there is no pitch rate feedback. The corresponding
root-locus of the altitude-hold autopilot with only altitude feedback is shown in
Fig. 6.50. We observe that, as soon as the outer loop is closed, the system becomes
unstable for any positive value of the gain k3. Therefore, we need to add the inner
loop with pitch rate feedback as shown in Fig. 6.51. The transfer function Gq is
Fig. 6.50 Root-Iocus of altitude-hold autopilot for the general aviation airplane.
.:l!
620 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
given by
Fig. 6.51 Inner loop ofthe altitude hold autopiIoL
dis + d2 (6.381)
Gq = Aq8( () ) = a~s2 + a2S + a3
Now we plot the root-locus of the inner loop as shown in Fig. 6.52 and pick a
point on the root~locus corresponding to < = 0.8, which gives the gain ofrate gyro
ki - 0.2474. This is a smallimprovement in the damping ratio compared to the
basic system (open loop) damping of 0.7 but may be sufficient to make the system
stable on closing the outer loop.
The next step is to design the outer loop (see Fig. 6.53). The transfer functions
Geq and h(s)]q(s) are given by the following expressions:
-lO(d} s + d2)
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