Golden Nugget Airlines has opened a free bar in the tail of their airplanes in an attempt to lure customers. In order to automatically adjust for the sudden weight shift due to passengers rushing to the bar when it first opens, the airline is mechanizing a pitch-attitude autopilot. Figure 5.68 shows the block diagram of the proposed arrangement. We will model the passenger moment as a step disturbance Mp (s) = M0/s, with a maximum expected value for M0 of 0.6.
(a) What value of K is required to keep the steady-state error in θ to less than 0.02 rad (≅ 1°)? (Assume the system is stable.)
(b) Draw a root locus with respect to K.
(c) Based on your root locus, what is the value of K when the system becomes unstable?
(d) Suppose the value of K required for acceptable steady-state behavior is 600. Show that this value yields an unstable system with roots at
s = -2.9,-13.5,+1.2 ± 6.6j.
(e) You are given a black box with rate gyro written on the side and told that, when installed, it provides a perfect measure of , with output KT . Assume that K = 600 as in part (d) and draw a block diagram indicating how you would incorporate the rate gyro into the autopilot. (Include transfer functions in boxes.)
(f) For the rate gyro in part (e), sketch a root locus with respect to KT.
(g) What is the maximum damping factor of the complex roots obtainable with the configuration in part (e)?
(h) What is the value of KT for part (g)?
(i) Suppose you are not satisfied with the steady-state errors and damping ratio of the system with a rate gyro in parts
(e) through
(h). Discuss the advantages and disadvantages of adding an integral term and extra lead networks in the control law. Support your comments using MATLAB or with rough root-locus sketches.
