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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
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CL_1
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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
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
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