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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)
Geq = a~S3 +~a2 +~Oa~s2 +~a3 +~Oa2~Okldl s +~O~a3 k d2~
           (6.382)
;((;=(U) -bls2+(dl-b2)s+d2-b)  (6.383)
    ls+d2
The lag-compensator pole (s = -8) is sufficiently away from the origin so that
it has very little effect on the short-period dynamics. The root-locus of the outer
loop is shown in Fig. 6.54. We obse:rve that we have to confine ourselves to ver)/
Iow values of the compensator gain k3 to keep the system stable. A higher value
of k3 will make the system unstable. The effect of the compensator is obvious. It
has pulled to the left the branches of the root-locus that originate from the two
poles at the orig:in and migrate to the right half of the plane. This provides a small
range of stable values for the complex conjugate poles. With k3 = 0.1425, we get
the closed- loop poles at -8.5822,  -6.3151 -i:  j5.0691,  -0.2118 :+: 1.1378,  -1.94,
and -1.6175.  .
     To verify the design, we simulate the response of the system to a unit-step ele-
vator input as shown in Fig. 6.55. The large overshoot is due to the low value
of the damping ratio. Except for this, the system performance is satisfactory.
u
x
<
o.
ai
E
AIRPLANE RESPONSE AND CLOSED-LOOP CONTROL          621
Fig. 6.52    Root-Iocus of the inner loop of the altitude-hold autopilot for the general
aviation airplane.
Fig.6.53    Outerloop of the aMtude-hold autopilot
622         PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
ReaIAxis
Fig. 6.54   Root-Iocus of the outer loop of the pitch displacement autopilot for the
general aviation airplane.
     2
 1.8
 1.6
 1.4
 1.2
s
91
o
 0.8
 0.6
 0.4
 0.2
    o
Stap R8sponse, A~Utude Hold  Auto-Pltot, GA Nrplane
                                                                                                                               
                                                                                                                               
                                                                                                                               
　
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