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Aircraft Altitude Control System: Transfer Function, Impulse Response, and Stability, Exercises of Signals and Systems Theory

Tamil Nadu Open UniversitySignals and Systems Theory

The transfer function of an aircraft's altitude control system from changes in thrust to changes in altitude. It includes the calculation of the natural frequency, damping ratio, and normalized transfer function. The document also discusses the impulse response using partial fraction expansion and inverse laplace techniques, and the stability analysis through pole placement in the complex plane for positive and negative gains.

Typology: Exercises

2011/2012

Uploaded on 07/20/2012

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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
v0 = 129 m/s
Assume that the aircraft has lift-to-drag ratio
L0 = 15
D0
Then the transfer function from changes in thrust to changes in altitude is
2g 1
G(s) = (1)
mv0 s (s2 + 2ζωns + ω2
n)
where the natural frequency of the phugoid mode is
g
ωn = √2 (2)
v0
the damping ratio is
1
ζ = (3)
√2(L0/D0)
and g = 9.82 m/s is the acceleration due to gravity. The transfer function can be
2g
normalized by the constant factor mv0 , so that
1
¯
G(s) = (4)
s (s2 + 2ζωns + ω2
n)
is the normalized transfer function, corresponding to normalized input
2g
u(t) = δT
mv0
¯
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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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 lifttodrag 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 + 2ζωns + ω^2 n)

where the natural frequency of the phugoid mode is

g ωn =

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 normalized by the constant factor 2 g mv 0 , so that

¯^1 G(s) = (4) s (s^2 + 2ζωns + 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)

r(t) e(t) u(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 h(t) and the altitude reference r(t). The transfer function from r(t) to h(t) can be shown to be

1 H(s) = 1 + kG(s)

For the gain k in the range [0, 0 .1], plot the poles of the system in the complex plane. You should find that for any positive k, the complex poles are made less stable. What gain k makes the complex poles unstable, i.e., for what gain is the damping ratio zero?

  1. For the gain k in the range [− 0. 1 , 0], plot the poles of the system in the complex plane. You should find that for any negative k, the real pole is unstable.

Note that neither positive gain or negative gain makes the system more stable than without feedback control. It is possible to do better with a dynamic gain, but this problem should give you an idea of why the phugoid dynamics are so hard to control with throttle only.

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