In one mode of operation, the autopilot of a jet transport is used to control altitude. For the purpose of designing the altitude portion of the autopilot loop, only the long-period airplane dynamics are important. The linearized relationship between altitude and elevator angle for the long-period dynamics is
The autopilot receives from the altimeter an electrical signal proportional to altitude. This signal is compared with a command signal (proportional to the altitude selected by the pilot), and the difference provides an error signal. The error signal is processed through compensation, and the result is used to command the elevator actuators. A block diagram of this system is shown in Fig. 6.103. You have been given the task of designing the compensation. Begin by considering a proportional control law D(s) = K. (a) Use MATLAB to draw a Bode plot of the open-loop system for D(s) = K = 1.
(b) What value of K would provide a crossover frequency (i.e., where |G| = 1) of 0.16 rad/sec?
(c) For this value of K, would the system be stable if the loop were closed?
(d) What is the PM for this value of K?
(e) Sketch the Nyquist plot of the system, and locate carefully any points where the phase angle is 180° or the magnitude is unity.
(f) Use MATLAB to plot the root locus with respect to K, and locate the roots for your value of K from part (b).
(g) What steady-state error would result if the command was a step change in altitude of 1000 ft? For parts (h) and (i), assume a compensator of the form
(h) Choose the parameters K, T, and α so that the crossover frequency is 0.16 rad/sec and the PM is greater than 50°. Verify your design by superimposing a Bode plot of D(s)G(s)/K on top of the Bode plot you obtained for part (a), and measure the PM directly.
(i) Use MATLAB to plot the root locus with respect to K for the system, including the compensator you designed in part (h). Locate the roots for your value of K from part (h).
(j) Altitude autopilots also have a mode in which the rate of climb is sensed directly and commanded by the pilot.
(i) Sketch the block diagram for this mode.
(ii) Define the pertinent G(s).
(iii) Design D(s) so that the system has the same crossover frequency as the altitude hold mode and the PM is greater than 50°.