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from the preselected condition, the gyro senses the change and generates elecffrical
signals, which are amplified and are used to drive the servo devices that move the
appropriate control surfaces in the desired manner.
    Pitch displacement autop17ot.   The function of the pitch displacement au-
topilot is to maintain the given pitch attitude of the aircraft  A schematic diagram
of a typical pitch displacement autopilot is shown in Fig. 6.41a, and the block
diagram implementation is shown intFig. 6.41b. The vertical gyro is replaced by
a summer,"~nd the elevator servo is represented by a first-order lag with a time
constant r - l/a. The parameter a characterizes the response time of the servo
and is usually in the range of 10 to 20.1n this design, we woSlassume a -. 10 so that
r - 0.1. As said before, we choose a negative sign for the elevator servo transfer
function.
   Let us consider the general aviation airplane and design a pitch displacement
autopilot for it. We will use the short-period approximation because it is found
Reter
Input
AIRPLANE RESPONSE AND CLOSED-LOOP CONTROL           613
a) Schematic diagram
b) Block diagram
Fig.6.41   Pitch displacement autopilot.
to be adequate for predicting the free and forced responses. The short-period
transfer function of the general aviation airplane for the pitch angle is given by
Eq. (6.164)
AO(s)               -2.0112 s - 3.9018
        (6.369)
  A8e   = s~0~695 S2+. O 8494 s+2~85 )
614           PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Reat Nds
Fig.6.42      Root-Iocus ofpitch displacement autopilot for the general aviation airplane.
a
8
:,
O
Fig. 6.43    Unit-step response of pitch displacement autopilot of the general aviation
airplane.
)
AIRPLANE RESPONSE AND CLOSED-LOOP CONTROL          615
Fig. 6.44    Pitch displacement autopilot with pitch rate feedback.
a business jet4 (see Exercise 6.1) whose short-period transfer function is given by
AO(s)             dis + d2
A8e~S~ = s~a~s2 + a2s + a3~
  q(s)        s AO(s)
Abe~S~ = A8e~S)
                     dis + d2
           = a~s2 + a2s + 23
(6.370)
(6.371)
(6.372)
 where di  -  - 6.6214, d2 - -- 3.8069, ai - 3.1536, a2 -.4.1604, and a3 - 7.5630.
    The first step is to draw the root-locus of the inner loop as shown in Fig. 6.45
and select a suitable value for the rate gyro gain krg. We select a point on the
root-locus for C  = 0.9 so that the system will have an adequate damping ratio
when the outerloop is closed. For 4-  = 0.9, we get krg  - 0.8322, and the locations
of t.he closed- loop poles for the inner loop are -7.5275, -1.8959 _+  j0.9623.
   The block diagram of the outer loop is shown in Fig. 6.46. Here, Geq is the
equivalent transfer function of the inner loop, which is given by
Geq
-lO(dis + d2)
ais3 + (a2 + lOai)s2 +s(a3 + lOa2 - lOkrgdi) + lO(a3 - krgdZ)
          (6.373)
We draw the outer-loop root-locus using MATLABI as shown in Fig. 6.47, We
select a point on the root-locus to have a damping ratio of < = 0.7 as shown,
which is satisfactory to have levell handling qualities. For this value of <  - 0.70,
we get ki - 0.753 and closed-loop poles at -7.9213, -1.5942 :t_: j1.7139, and
-0.2095. The pole at -7.9213 is due to
is the system pole due to the integrator
the design, we perform a simulation to
the elevator servo and that at -0.2095
(s - O) in the forward path. To check
unit-step (elevator) input as shown in
fn
x
<
a
(u
E
616           PERFORMANCE, STABILITY, DYNAMiCS, AND CONTROL
{t
i
<
o:
(I
E
Fig. 6.45   Inner-Ioop root-locus of the pitch displacement autopilot for the business
jeL
Fig. 6.48. We have zero steady-state error, and the system response is found to be
satisfactory.
    A/titude-ho/d autop17ot.    Altitude-hold autopilots are generally used by com-
　
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