Question 1
A stable high performance aircraft is fitted with a stability augmentation system using a
proportional gain on pitch rate feedback to elevator. The aircraft short period approximation
expression has the following values:
K q = −1
T 2 = 0.25 s
 sp = 0.2
 sp = 2 rad/s
where the symbols have their usual meanings.
(a) Sketch a block diagram illustration for such a control system, observing the correct sign
conventions for elevator inputs and write an algebraic expression for the closed loop pitch
rate transfer function in terms of the parameters given above. Assume the use of a feedback
gain Gq.
(4 marks)
(b) By collecting the algebraic terms of the pitch rate transfer function in the appropriate
standard form, derive expressions relating the closed loop short period frequency and
damping ratio to the open loop (no control system) aircraft values.
(4 marks)
(c) Calculate the gain necessary to give a short period damping ratio of 0.7, with an appropriate
closed loop short period frequency.
(4 marks)
(d) Sketch the Bode plots for the pitch rate and pitch angle responses for an aircraft and its
controller with the characteristics given in (c). Using the approximate gain and phase for
pitch angle from the Bode plots, sketch the Nichols plot.
(6 marks)
(e) Comment on the features of the Nichols plot in (d) relative to that expected from a physical
implementation of such a control system; comment on the differences and likely impact on
handling qualities.
(2 marks)
turn over...
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Question 2
The linearized lateral-directional equations of motion may be written:
𝑚𝑠 − 𝑌𝑣
[ −𝐿𝑣
−𝑁𝑣
−𝑚𝑔 − 𝑌𝑝 𝑠
𝐼𝑥𝑥 𝑠 2 − 𝐿𝑝 𝑠
−𝐼𝑧𝑥 𝑠 2 − 𝑁𝑝 𝑠
𝑚𝑈 − 𝑌𝑟 𝑣
𝑌𝜉
𝐼𝑧𝑥 𝑠 − 𝐿𝑟 ] [𝜑] = [ 𝐿𝜉
𝑁𝜉
𝐼𝑧𝑧 𝑠 − 𝑁𝑟 𝑟
𝑌𝜁
𝜉
𝐿𝜁 ] [ ]
𝜁
𝑁𝜁
where 𝜉 is aileron deflection, 𝜁 is the rudder deflection and the other symbols have their usual
meanings.
An aircraft has a proportional feedback control system with 𝜉 = −𝜉𝑝𝑖𝑙𝑜𝑡 + 𝐾𝛽 and 𝜁 =
−𝜁𝑝𝑖𝑙𝑜𝑡 + 𝑘𝑟. Here, 𝜉𝑝𝑖𝑙𝑜𝑡 is the pilot’s lateral stick input, 𝜁𝑝𝑖𝑙𝑜𝑡 is pilot rudder pedal input,  is
sideslip angle, r is yaw rate and 𝐾 and 𝑘 are positive gains. Standard sign conventions apply.
(i)
Rewrite the equation showing the pilot’s inputs only on the right-hand side; then reduce
the above equation to a lower-order system that represents an approximation to the
Dutch roll motions; explain your reasoning. By observing the terms in the equations,
identify which stability derivatives the controller augments and in which sense each is
augmented; give a brief explanation of your answers. Hence explain the likely purpose
of both parts of the controller, i.e. what aspects of the airframe dynamics might they be
intended to augment.
(6 marks)
(ii)
Derive an expression for the approximate yaw rate response to rudder input transfer
function. For this and the rest of this question, you may assume 𝐾 = 0. Comment on
the nature of this transfer function.
(4 marks)
(iii)
Write down expressions for the frequency, 𝜔𝑑𝑟 , and damping, 𝜁𝑑𝑟 , of the Dutch roll in
terms of the derivatives (it is not necessary to find the expression for 𝜁𝑑𝑟 itself: it is
acceptable to write down the expression for the whole of 2𝜁𝑑𝑟 𝜔𝑑𝑟 ).
(2 marks)
(iv)
Using the Final Value Theorem, determine the algebraic expression for the steady state
yaw rate per unit step pilot input. How does the gain 𝑘 affect the steady state yaw rate
if it is also set to damp Dutch roll motions and is this desirable? Explain your answer.
(6 marks)
(v)
If the simple gain 𝑘 were replaced with the transfer function
𝑘𝑠
(1+𝑇𝑠)
what effect would
𝑘 have on the steady yaw rate?
(2 marks)
turn over...
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Question 3
a)
Define the state-transition matrix, and provide its formulation for a general
state-space system {A, B, C, D}. Why is the state-transition matrix important?
(2 marks)
b1)
Given the following Navion longitudinal motion state-space model:
𝛥𝑢̇
−0.0450
𝛥𝑤̇
−0.3693
[ ]=[
𝛥𝑞̇
0.0019
0
𝛥𝜃̇
0.0360
−2.0221
−0.0395
0
0
176
−2.9858
1
0
−32.2 𝛥𝑢
𝛥𝑤
−28.1376
0
][ ] + [
] 𝛥𝛿𝑒𝑙𝑒𝑣
𝛥𝑞
−11.7391
0
0
0
𝛥𝜃
Find the eigenvalues for the short-period approximation.
(2 marks)
b2)
For the longitudinal aircraft system in Q3.b1, assess its controllability and
observability. Assume all the states are measurable.
(3 marks)
b3)
For the longitudinal aircraft system in Q3.b1, assume the following closed-loop
where NavionSP is the state-space short-period approximation of Navion but
expressed in terms of  using a speed u0=176 m/s:
Determine the controller gain-matrix K so that the closed-loop poles of the
above system are located at p1= −4  3i.
(3 marks)
c)
Explain the concepts of trimming and linearization.
(2 marks)
nd
d)
Given the following 2 order ODE, transform it into:
𝑑2 𝑥(𝑡)
𝑑𝑥(𝑡)
+
2
+ 3𝑦(𝑡) = 0.5𝑢(𝑡)
𝑑𝑡 2
𝑑𝑡
1. A system of two 1st order ODEs
2. A state-space model {A, B, C, D}.
(3 marks)
e)
For the transfer function given below, derive the two main state-space
canonical representations:
2𝑠 2 + 3𝑠 + 1
𝐺(𝑠) =
𝑠 3 + 0.5𝑠 2 + √6𝑠 + 2⁄3
(5 marks)
turn over...
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Question 4
(a) Explain why additional compensation must be provided in a controller that incorporates
pitch rate command path when carrying out turn manoeuvres. Describe, with sketches, the
ways in which this is accomplished in a typical controller of this type.
(4 marks)
(b) When aircraft fly at high angles of attack, various nonlinear flight dynamics phenomena
may arise due to separated flows, aerodynamic interference and time dependence. Explain
how these might manifest themselves in a commercial airliner that is subjected to an upset
from normal flying conditions.
(12 marks)
(c) The C* handling qualities criterion can be formulated as:
l p q Vco q 

C  = k n +
+

g
g 

where n is the normal acceleration (in “g”) at the centre of gravity and lp is the distance of
the pilot’s seat to the centre of gravity. Explain the meaning of and justification for this
type of criterion.
(4 marks)
END OF PAPER
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