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tn
'x
<
o
(o
E
Fig. 6.33     Root-Iocii of the inner loop of the pitch stabilhation system of the general
aviation air )lane.
AIRPLANE RESPONSE AND CLOSED-LOOP CONTROL          599
Using Eqs. (6.295) and (6.296) and the relation q(s) = sO(s), we get
q(s)               s(4.4642 S2 + 9.3994 s + 0.4775)
a~s~ = 0 0602 S3 +4 4326 S2 +0 2008 s +0.3103              (6.299)
     The root-locus of the outer loop is shown in Fig. 6.34. We pick a point for <  =
0.9, which gives krg = 0.1194. We have the closed-loop poles at -9.9710,  - 1.7853
+- j0.8773 (short-period), and -0.0197 +- j0.1502 (phugoid). For these closed-
loop pole locations, we get < = 0.9, COn : 1.9882 rad/s for the short-period mode
and < = 0.13, COn = 0.1515 rad/s for the phugoid mode. Thus, with aipvha and pitch
rate feedback, the relaxed static stability version of the general aviation airplane
has the conventional short-period and phugoid modes with level I flying qualities.
6.5.2  Fufj~State Feedback Design for Longitudinal Stab.7ity
   , ' Augmentation System
       To illustrate the design procedure of a full-state feedback stability augmentation
system, let us consider the general aviation system once again. Here, we assume
that all four longitudinal states u, Aa, AO, and q are accurately measured and
are available for feedback. The air speed is usually measured by a pitot-static
sensor, the pitch angle and pitch rates are measured by positional and rate gy-
ros, and the angle of attack is measured by a vane-type sensor. Out of these four
state variables, angle of attack is the difficult one to measure because the local
fiow around the airplane is altered due to wing-body upwash in front and down-
wash behind the wing. The alpha sensor is usually mounted on the wingtips or
at the nose of the fuselage. Thus, it is subjected to the upwash field. Hence, it is
necessar)r to correct the measured'values of the angle of attack for the induced
upwash/downwash effects. For our purpose, we will assume that such corrections
are done and that all four states are accurately measured and are available for
feedback.
       Let us assume that the desired poles or the roots of the longitudinal characteristic
equations that give the specified handling qualities are as fo,llows.
    For the short-period mode,let
A1.2 - -4.8 :1: j2.16
(6.300)
which corresponds to a damping ratio of 0.8 and a natural frequency of 5.26 rad/s
for the short-period mode. As we have seen earlier, these values of darnping ratio
and natural frequency give us level I short-period flying qualities.
     For the phugoid mode, let
A3.4 - -1.5 :t j0.0191
(6.301)
which corresponds to a damping ratio of 0.15 and a naturaj frequency of 0.121
rad/s, giving level I phugoid flying qualities.
     For the values of the roots given by Eqs. (6.300) and (6.301), the desired char-
acteristic equation is given by
S4 + 9.90 S3 + 30.6085 S2 + 8.5312 s + 0.6335                 (6.302)
600            PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
~
x
<
-
(.,
E
Fig. 6.34    Root-Iocii of the outerlaop of the pitch stabilization system of the general
aviation airplane.
AIRPLANE RESPONSE AND CLOSED-LOOP CONTROL          601
  The objective is to design a full-state feedback law so that the closed-loop
characteristic equation is identical to the desired characteristic equation.
     We have the given system in the state-space form
where
X - AX + BU                          (6.303)
X = ~/]
and the matrices A and B given by Eqs. (6.29) and (6.30) are
(6.304)
(6.305)
       0   '
      (6.306)
-11.8674
         0
   The design procedure for the state feedback was explained in Chapter 5. The
first step is to express the given plant in the phase-variable form* Because the
given plant is not in the phase-variable form, we have to do a transformation. The
　
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