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s11_ps10_spring04

# s11_ps10_spring04 - Unied Engineering II Spring 2004...

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Unified Engineering II Spring 2004 Problem S11 (Signals and Systems) Consider an aircraft ﬂying in cruise at 250 knots, so that v 0 = 129 m/s Assume that the aircraft has lift-to-drag ratio L 0 = 15 D 0 Then the transfer function from changes in thrust to changes in altitude is 2 g 1 G ( s ) = (1) mv 0 s ( s 2 + 2 ζω n s + ω 2 n ) where the natural frequency of the phugoid mode is g ω n = 2 (2) v 0 the damping ratio is 1 ζ = (3) 2( L 0 /D 0 ) and g = 9 . 82 m/s is the acceleration due to gravity. The transfer function can be 2 g normalized by the constant factor mv 0 , so that 1 ¯ G ( s ) = (4) s ( s 2 + 2 ζω n s + ω 2 n ) is the normalized transfer function, corresponding to normalized input 2 g u ( t ) = δT mv 0 ¯ 1. Find and plot the impulse response corresponding to the transfer function G ( s ), using partial fraction expansion and inverse Laplace techniques. Hint: The poles of the system are complex, so you will have to do complex arithmetic. 2. Suppose we try to control the altitude through a feedback loop, as shown below G(s) + u(t) e(t) r(t) h(t) k

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That is, the control input u ( t ) (normalized throttle) is a gain k times the error, e ( t ), which is the difference between the altitude
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