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ki -(0.4253 + k2) -(48.8614 +k3) -(14.1354 + k4) -(9.4685 +ks)
(6.364)
The characteristic equation of this transformed phase-variable form is
S[S4 + (9.4685 + ks)S3 + (14.1354 + k4)S2 + (48.8614 + k3)S
+ (0.4253 + k2) + ki] = O (6,365)
Comparing the coefficient of Eqs. (6.340) and (6.365), we obtain ki - 0, k2 -
0.1869, k3 = 27.8950, k4 - 15.3543, and ks = 1.4395.
Then next step is to transform back to the original state-space. Substituting for
Z, Ap, and Bp in Eq. (6.363), we obtain
X = (A - BK P)X + Br(t)
- AX + B[r(t) - K P(X)]
This gives us the desired state feedback control law as
(6.366)
(6.367)
U - r (t) - K PX (6.368)
where the feedback gain matrix K is given by Eq. (6.362) with Ki-Ks obtained
as above and the transformation matrix P is given by Eq. (6.358).
To verifiy the design,we obtain the free response [r (t) = 0] using MATLAB1 for
the initial conditions AB = 5 deg, A4 = p = Al~/ - r - O as shown in Fig. 6.40.
We observe that the system with full-state feedback law performs as expected.
Autopilots come in various t)rpes and degrees of sophistication and perform
different functions. Autopilots are commonly used for maintaining pitch attitude,
612 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
~o
CL_1
.Znl
-o
cn
q
J
D
o
g "-:~/2~/
h _101_L___1_L
0246
8 10 12 14 16 18 20
llme, s
Fig. 6.40 Lateral-directional full-state feedback controller for the general aviation
airplane.
heading, altitude, Mach number, prescribed glide slope during approach, and
landing.
The principal element of an autopilot is a sensing element called gyro (gyro-
scope). It may be a positional gyro or a rate gyro. Whenever there is a deviation
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