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Problem 10.12PP
Altitude Control o f a Hot-Air Balloon: American solo balloonist Steve Fossett landed in the 
Australian outback aboard Spirit o f Freedom on July 3rd, 2002, becoming the first solo balloonist 
to circumnavigate the globe (see Fig. 1). The equations of vertical motion for a hot-air balloon 
(Fig. 2), linearized about vertical equilibrium, are
S t + —ST = Sq, 
r i
T2? + Z = o8T -\-W t
6T = deviation of the hot-air temperature from the equilibrium temperature where buoyant force 
equals weight.
z = altitude of the balloon,
e ^ q u ' a ^ ' w S g f i T , ^ .....
z = altitude of the balloon,
6q = deviation in the burner heating rate from the equilibrium rate 
(normalized by the thermal capacity of the hot air), 
w = vertical component of wind velocity, 
r1,72, a = parameters of the equations.
An altitude-hold autopilot Is to be designed for a balloon whose parameters 
are
r1 = 250 sec t2 = 25 sec a = 0.3 m/(sec .”C).
Figure 1 Spirit of Freedom balloon Source: Steve Holland/AP Images
Figure 2 Hot-air balloon
Only altitude is sensed, so a control law of the form 
6q(s) = Dc(s)[zd(s) - z(s)],
will be used, where zd Is the desired (commanded) altitude.
(a) Sketch a root locus of the closed-loop eigenvalues with respect to the gain K for a 
proportional feedback controller, 5q = -K(z-zd). Use Routh’s criterion (or let s= jo) and find the 
roots of the characteristic polynomial) to determine the value of the gain and the associated 
frequency at which the system is marginally stable.
(b) Our intuition and the results of part (a) indicate that a relatively large amount of lead 
compensation is required to produce a satisfactory autopilot. Because Steve Fossett was a 
millionaire, he could afford a more complex controller implementation. Sketch a root locus of the 
closed-loop eigenvalues with respect to the gain K for a double-lead compensator, 5q = Dc(s)(zd 
- z). where
/«+araV
U+0.I2,;
(c) Sketch the magnitude portions of the Bode plots (straight-line asymptotes only) for the open- 
loop transfer functions of the proportional feedback and lead-compensated systems.
(d) Select a gain K for the lead-compensated system to give a crossover frequency of 0.06 
rad/sec.
(e) Select a gain K for the lead-compensated system to give a crossover frequency of 0.06 
rad/sec.
(f) If the emor in part (e) is too large, how would you modify the compensation to give higher low- 
frequency gain? (Give a qualitative answer only.)
S tep -by-s tep s o lu tio n
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