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gyros and the roll and yaw rates by the rate gyros. The sideslip angle is perhaps the
most difficult one to measure. A vane-type instrument is usually used to measure
the sideslip but, like angle of attack sensor, itis also subject to positional errors due
to fuselage sidewash. Another method used to obtain sideslip is the integration of
~
o,
:d
E
608 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
accelerometer outputs. However, this is also subject to measurement noise. In any
case, it is essential that the sideslip be accurately estimated and be available for
feedback. For our purpose, we will assume that all five state variables are measured
accurately and are available for feedback.
Let us consider the oeneral aviation airplane once again and assume that the
desired poles of the lateral-directional closed-loop characteristic equations are as
follows.
For Dutch-roll mode, let
Adr - -1.21- j2.75 (6.337)
so that we have a damping ratio of 0.4 and a natural frequency Crh of 3.0 rad/s,
corresponding to level I Dutch-roll flying qualities.
For roll subsidence and spiral mode, let
A,r - -8.0 (6.338)
\sp - -0.05 (6.339)
The other root is the zero root, A - O.
We have a roll-time constant of 0.125, which gives level I fiying qualities for
the roll-subsidence mode. The spiral mode is damped and, therefore, it also has
level I fiying qualities. Thus, overall, the assumed pole locations give us level I
lateral-directional fiying qualities.
For these values of five roots, the desired lateral- directional characteristic equa-
tion is given by
S(S4 + 10.9080 S3 + 29.4897 S2 + 76.7565 s + 0.6122) : O (6.340)
The objective is to design a full-state feedbacklaw so that the closed-loop lateral-
directional characteristic equation of the general aviation airplane is identical to
the desired characteristic equation.
From Eqs. (6.196) and (6.197), we have
-0.2557 0.1820 0 0 -1.0000
0 O 1.0 0 0
A- -16.1572 O ~8.4481 O 2.2048 '(6.341)
O O O 0 1.0
4.5440 O , -0.3517 0 -0.7647
0 0.0712
0 0
B -. 29.3013 2.5764
O 0
-0.2243 -4.6477
(6.342)
AIRPLANE RESPONSE AND CLOSED-LOOP CONTROL 609
Because the given plant is not in phase-variable form, we have to do a transfor-
mation and express it in the phase-variable form. Then, do the full-state feedback
design in this new transformed space, Finally, do a reverse transformation to obtain
the full-state feedback law in the original state-space.
The phase~variable form is given by
Z = ApZ +'BpU
(6.343)
where Z is the new state vector and matrices Ap and Bp are in the phase-variable
form
Z-. PX
Ap = PAP-I
Bp = PB
(6.344)
(6.345)
(6.346)
The state vector x was defined in Eq. (6.183). The transformation matrix P is
constructed as follows.
1) Obtain the lateral directional controllability matrix Qc given by
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