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┃L _ _ _ _ _ ___ _ __ _ ~ _ ~ _ _ _ __ ~ _ _ _ _ __ _ _ _ J ┃
┃ r:: -i_ ┃
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b) Block diagram
Fig.631 Pitch stabilizaOon system.
AIRPLANE RESPONSE AND CLOSED-LOOP CONTROL 597
valid. Using Cm r = 0.07/rad and Cmq and Cma at 50% of the baseline values, we
get
Aa (s) -(0.0602 S3 + 4.4326 S2 + 0.2008 s + 0.3103)
~8e~s~=03~39S4+~3390S3+05~40S2T6-.t :ef.~ (6.295)
Aq(s) -s(4.4642 S2 + 9.3994 s + 0.4775)
A8~s~=03~39s4+~3390S3+05~40S2+00469s 00235 (6.296)
In Fig. 6.31b, Ga, denotes the transfer function given by Eq. (6.295). Because Gct
has a negative sign, we choose a negative sign for the transfer function of the ele-
vator servo so that the loop transfer function is positive, and we can use a negative
feedback as usual.
The first step is to design the inner Ioop. The block diagram of the inner loop is
shown in Fig. 6.32a. We draw the root-locus of the innerloop and select a point on
the root-locus corresponding to r = 0.8 as shownin Fig. 6.33. We get ki -- 0.1848.
Next, we proceed to the design of the outer loop. The block diagram of the outer
Ioop is shown in Fig. 6.32b. Here, 7; represents the closed-loop transfer function
of the inner ,loop and is given by
-kj a G,
Tcr = s+u
1 kaGy
s+a
(6.297)
The transfer function for the pitch rate to angle of attack is obtained as follows:
Abe,e
q(s)
A (.) = (Aq8(: ())) (A8 (,))) (6.298)
a) Inner loop
A8e,d c8) _U.-V~~Lc
-V,-
~
b) Outerloop '
Fig. 632 Inner and outerloops of the pitch stabilization system.
s)
598 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
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 < =
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