Control of a Small Airplane
For the Piper Dakota shown in Fig. 5.36, the transfer function between the elevator input and the pitch attitude is
G(s) = \frac{θ (s)}{δ_e(s)} = \frac{160(s + 2.5)(s + 0.7)}{(s^2 + 5s + 40)(s^2 + 0.03s + 0.06)} , (5.80)
where
θ = pitch attitude, degrees (see Fig. 10.30),
δ_e = elevator angle, degrees.
(For a more detailed discussion of longitudinal aircraft motion, refer to Section 10.3.)
1. Design an autopilot so that the response to a step elevator input has a rise time of 1 \text{ sec} or less and an overshoot less than 10%.
2. When there is a constant disturbing moment acting on the aircraft so that the pilot must supply a constant force on the controls for steady flight, it is said to be out of trim. The transfer function between the disturbing moment and the attitude is the same as that due to the elevator; that is,
\frac{θ (s)}{M_d(s)} = \frac{160(s + 2.5)(s + 0.7)}{(s^2 + 5s + 40)(s^2 + 0.03s + 0.06)}, (5.81)
where M_d is the moment acting on the aircraft. There is a separate aerodynamic surface for trimming, δ_t, that can be actuated and will change the moment on the aircraft. It is shown in the close-up of the tail in Fig. 5.36. Its influence is depicted in the block diagram shown in Fig. 5.37(a). For both manual and autopilot flight, it is desirable to adjust the trim so that there is no steady-state control effort required from the elevator (that is, so δ_e = 0 ). In manual flight, this means that no force is required by the pilot to keep the aircraft at a constant altitude, whereas in autopilot control it means reducing the amount of electrical power required and saving wear and tear on the servomotor that drives the elevator. Design an autopilot that will command the trim δ_t so as to drive the steady-state value of δ_e to zero for an arbitrary constant moment M_d as well as meet the specifications in part (a).
