AE4301 AUTOMATIC FLIGHT CONTROL
SYSTEM DESIGN EXAMINATION
JANUARY 21th, 2011
13:00 - 16:00
SOLUTION
Delft University of Technology
Faculty of Aerospace Engineering
Control and Simulation Division
This exam contains 5 questions.
PLEASE NOTE
Write clearly and neatly.
Make sure that you provide calculus procedure as well as your final result.
Clearly indicate your final result.
Make sure that you provide all requested information.
Include clear drawings where necessary.
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January 21, 2011
January 21, 2011
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Question 1: State space representation (2 points)
The general representation of a state space system is:
ẋ = Ax + Bu
y = Cx + Du
The longitudinal dynamics of an aircraft, trimmed at airspeed V0 , can be described by the following
state space system:
 
u̇
a11
 α̇   a21
 

 θ̇  =  a31
a41
q̇

a12
a22
a32
a42
a13
a23
a33
a43

u
a14
 α
a24 

a34   θ
a44
q


b1
  b2 

+
  b3  δ e
b4

1. Matrix A is called the state matrix. Give the names of the other three matrices B, C, and
D.
• B: Input matrix
• C: Output matrix
• D: Feedthrough matrix
2. State space systems can be extended to obtain new states. If we want the output vector to
contain only the flight path angle γ and the altitude h:
y=
γ
h
then give the state space matrices A, B, C, D, together with the state and input vectors x
and u that produce exactly this output. Note: use only variables in your answer that have
already been presented in the question.
•



A=


C=



x=




a11 a12 a13 a14 0

a21 a22 a23 a24 0 



a31 a32 a33 a34 0 
;
B
=



a41 a42 a43 a44 0 
0 −V0 V0 0 0
0 −1 1 0 0
0
; D=
0 
0
0 0 1
0
u
α 

θ 
 ; u = (δe )
q 
h
b1
b2
b3
b4
0






3. Reduce the state space system to a two-state system: α and q. Then convert this reduced
state space system into a single second order differential equation for the output α.
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Y (s)
E (s)
R (s)
Kp
+-
H (s)
Figure 1: Block diagram of the roll rate loop.
•
α̇
a22 a24
α
b2
=
+
δe
a42 a44
q̇
b4
q
2 δe
α̇ = a22 α + a24 q + b2 δe → q = α̇−a22aα−b
24
q̇ = a42 α + a44 q + b4 δe = a42 α + a44
α̇−a22 α−b2 δe
a24
α̈ = a22 α̇ + a24 q̇ + b2 δ̇e = a22 α̇ + a24 a42 α + a44
+ b4 δ e
α̇−a22 α−b2 δe
a24
+ b4 δe + b2 δ̇e
α̈ + (−a22 − a44 ) α̇ + (a22 a44 − a24 a42 ) α = b2 δ̇e + (a24 b4 − a44 b2 ) δe
4. Convert the second order differential equation from the previous question into a transfer
function:
H (s) =
α (s)
δe (s)
•
s2 α (s)
+ (−a22 − a44 ) sα (s) + (a22 a44 − a24 a42 ) α (s) = b2 sδe (s) + (a24 b4 − a44 b2 ) δe (s)
α (s) s2 + (−a22 − a44 ) s + (a22 a44 − a24 a42 ) = δe (s) {b2 s + (a24 b4 − a44 b2 )}
b2 s+(a24 b4 −a44 b2 )
α(s)
δe (s) = s2 +(−a22 −a44 )s+(a22 a44 −a24 a42 )
Question 2: Roll rate control (1.5 points)
The simplified roll dynamics of an aircraft are described by two parameters, Lp and Lδa :
H (s) =
Lδa
p (s)
=
δa (s)
s − Lp
1. Give the definition of the initial and final value theorems.
•
lim f (t) = lim {sF (s)}
t→∞
s→0
lim f (t) = lim {sF (s)}
t→0+
s→∞
2. An autopilot is created for the roll rate control, see figure 1. Compute the steady state error
of this system to a step input when Lp = −0.25, Lδa = −0.8, and Kp = 0.2.
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•
E(s)
R(s)
=
E (s) =
1
1+Kp H(s)
=
s−Lp
s−Lp +Kp Lδa
1
Lδ
a
1+Kp s−L
p
R (s) =
=
s−Lp
s−Lp +Kp Lδa
s−Lp
1
s−Lp +Kp Lδa s
s−L
ess = lim e (t) = lim {sE (s)} = lim s s−Lp +Kpp Lδ
=
t→∞
s→0
s→0
−L
s−L
lim s−Lp +Kpp Lδ = −Lp +Kpp Lδ ≈ 2.78
a
a
s→0
a
1
s
=
3. is the system stable?
•
L
Hcl (s) =
Kp H(s)
1+Kp H(s)
=
δa
Kp s−L
p
L
δa
1+Kp s−L
p
=
Pole :s − Lp + Kp Lδa = 0 → Kp =
5
Unstable :Kp > 16
K p L δa
s−Lp +Kp Lδa
Lp
L δa
=
5
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Question 3: Bode, Nyquist, Root Locus plots (2 points)
1. Describe in your own words what a Bode plot is, what it is used for, what properties make
it especially useful, and what kind of information you can obtain from a Bode plot. Also
explain the concepts of gain and phase margin and indicate these in a sketch of a Bode plot.
• Frequency response.
• Used in control design.
• Multiplication in Laplace domain is addition in Bode plot.
• Magnitude and phase information of the frequency response.
• See figure 2 for gain and phase margin.
2. Draw the Nyquist plot of the system:
1
1 + jωT
• See figure 3. Only half the plot (ω from 0 to ∞ also correct). Any value of T gives the
same shape (only different gradation in ω).
3. Where does the Nyquist plot of a Type 1 system start (i.e. for small ω)?
• A Type 1 system has an extra jω term in the denominator. For ω to zero, this results
in an infinite magnitude and a -90 degree phase. So the Nyquist plot starts along the
negative imaginary axis at infinity.
4. Describe in your own words what a root locus plot is and what it is used for. Explain how the
damping ratio and natural frequency of a second order system can be obtained from a root
locus plot. Finally sketch an example of a root locus (does not have to be a real system) and
indicate in the drawing the following regions: Fast response, slow response, many oscillations,
few oscillations, stable, unstable.
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Bode Diagram
Phase (deg)
Magnitude (dB)
20
0
-20
-40
-60
-80
-100
0
gain margin
-90
phase margin
-180
-270
10−2
10−1
100
101
102
Figure 2: Definition of positive gain and phase margin
Nyquist Diagram
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
-1
-0.8 -0.6 -0.4 -0.2
0
0.2
0.4
0.6
Figure 3: Nyquist plot from question 3
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0.8
1
Figure 4: Question 3: 2nd order damping and natural frequency and requested regions in a root locus
plot
• RL shows the location of the closed loop poles for varying feedback gain.
• Used for closed loop control design.
• See figure 4 for the indicated regions and also the derivation of damping ratio and
natural frequency.
Question 4: Stability Augmentation Systems (1 point)
The pitch rate reference signal for a pitch damper is defined as follows:
q=
g
(n − 1)
V0
where q is the pitch rate in [rad/s], g is the graviational acceleration in [m/s2 ], V0 is the forward
velocity in [m/s], and n is the load factor.
Derive this expression by considering vertical force equilibrium of the aircraft in the bottom of a
steady symmetrical pull up manoeuvre. Include a clear drawing which supports your derivation.
Compute desired pitch rate:
• See figure 5
•
L = nW = W + Fc = W + mq 2 R = W + mV q
nmg = mg + mV q, (n − 1) g = V q
q = Vg (n − 1)
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January 21, 2011
R
L = nW = W + F c
V = qR
W
Figure 5: Solution question 4: balance of forces in pull up manoeuvre.
Question 5: Autopilot navigational mode (2.5 points)
Describe the localizer hold mode of a general autopilot in detail. Your description should include
at least the following:
• A clear picture of the situation in which the to-be-controlled parameter can be found.
• Assumptions which are made in this figure.
• A procedure which shows clearly how the control law is determined.
• A block diagram which represents the control law determined earlier.
• An explanation of the nature of each block in the diagram, and definitions of models or
transfer functions of all these blocks.
Explain why there is an influence of the slant range R on the closed loop performance. How can
this influence be compensated for?
For solution see slides 39,45,46 and 71 of lecture 14
January 21, 2011
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