Longitudinal Flight Control Systems
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《Basic Longitudinal Autopilots (I) – Attitude Control System》
◎First idea: A Displacement Autopilot◎
○Typical block diagram:
θref
Vertical eg
Gyro
Amplifier
eδ
Elevator δe
A/C
Servo
Dynamics
θ(t)
○Basic features:
£ A direct command following design ==> The simplest from of autopilot
£ Designed to hold the A/C in straight and level flight by maintaining θ ( t ) = θref .
£ The A/C has to be trimmed for straight and level flight before this autopilot is engaged
○Working principle:
£ Whenever θ ( t ) ≠ θref ⇒ eg ≠ 0 ⇒ eδ ≠ 0 ⇒ A restoring δe appears
£ Mathematically, if we represent the A/C dynamics as θ ( s ) = H (δ ,θ )δe ( s ) , the elevator servo as
e
δe ( s) = H voe g ( s) , and the amplifier as a gain K a , then we will have
θ (s ) =
For θref
K a H (δ e ,θ ) H vo
θref ( s) ⇒ eg ( s) =
1
θref ( s)
1 + K a H (δ e ,θ ) H vo
1 + K a H (δ e ,θ ) H vo
( s) = θ* s ⇒ lim eg ( t ) = seg ( s )
<< 1 , if K a H (δ e ,θ ) H vo
s
=
0
t →∞
s =0
>> 1 .
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○Design considerations:
£Selection of K a must provide adequate stability margin for errors in H (δ ,θ ) and H vo .
e
£Root locus analysis is often used to examine the closed-loop design.
【例一】Design example (a) – Conventional transport:
¢ Simplified longitudinal dynamics: H (δ ,θ ) =
e
− ( s + 3.1)
s( s 2 + 2.8 s + 3.24 )
− 12.5
s + 12.5
Line of 0.6 damping ratio
¢ Simplified servo dynamics: H vo =
jω
4
Dominant mode damping ratio
decreases when Ka increases
2
CL poles at Ka = 0.64 when the dominant
mode damping ratio still > 0.6
Ka max
= 5.96
σ
-14
θref
eg
Ka
-12
-10
− 12.5 δe
s + 12.5
-8
-6
-4
− ( s + 3.1)
s( s 2 + 2.8s + 3.24)
-2
0
θ
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○Difficulties of the system:
(a) Little maneuvering capability is provided with θref being the command.
==> Pitching maneuver is often conducted by commanding the pitching rate.
(b) The design works only if the open-loop system is adequately stable. ==> 【例二】
【例二】Example design for problem (b) – Modern jet aircraft:
¢ Simplified longitudinal dynamics: H (δ ,θ ) =
e
==>
− 1.39( s + 0.306)
s( s 2 + 0.805s + 1.325)
Dominant pole damping ratio of the A/C: 0.35 ( For the A/C in【例一】: 0.77& )
jω
Line of 0.6 damping ratio
1) Dominant mode damping ratio
decreases rapidly as Ka increases
2) No acceptable design can be found.
θref
eg
-14
-12
Ka
− 10 δe
s + 10
==> System goes unstable at
-10
-8
-6
3
2
1
-4
− 1.39( s + 0.306)
s( s + 0.805s + 1.325)
2
Ka max =
4.565
0
-2
σ
θ
K a max = 4 .565 (In【例一】, K a max = 5.96 )
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【Circumvention for system stability】
◎Attitude Autopilot with a rate feedback inner loop◎
○Typical block diagram:
θref
Vertical
Gyro eg Amplifier ea
eδ Elevator
A/C
Servo δe Dynamics
.
erg
θ
Rate Gyro
θ
○Basic features:
£ A rate gyro which senses the pitching rate is used to feedback θ& .
£ A multi-loop feedback structure is employed
==> The differentiator inner loop will increase the damping of the A/C (short period) dynamics
【例三】Example design for the jet aircraft in【例二】
¢ Block diagram representation of the system:
θref
eg
Ka
ea
inner-loop
eδ
erg
− 10
s + 10
Ka: outer-loop feedback gain
δe
-1.39(s+0.306)
s2+0.805s+1.325
.
θ
1
s
θ
Kb
Kb: inner-loop feedback gain
==> Note that the rate gyro senses the pitching rate θ& , and the system output is pitch angle θ .
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Inner-loop root locus
eδ
ea
Stability increases
with increasing Kb
δe
-10
s+10
-1.39(s+0.306)
s2+0.805s+1.325
Kb
3
jω
Solutions with Kb = 1
Solutions with Kb = 2
2
CL poles of the inner-loop
locus become the OL poles
of the outer-loop locus
1
σ
-12
-10
-8
-6
-4
0
-2
Outer-loop root locus
jω
The outer-loop locus will retain the open-loop zero.
The pole at s = 0 reappears in the outer-loop locus.
3
2
with Kb = 2, K a max > 24.9
with Kb = 1, Ka max < 14.4
θref
eg
Ka
.
θ
ea
1
-12
inner-loop .
θ
-10
1
s
-8
-6
-4
-2
0
σ
θ
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○Discussions:
£ A differentiator feedback inner-loop greatly enhances the system stability.
==> In 例二, the dominant pole damping ratio (ζ ) of the CL system is < 0.35 for any K a . In
this design, ζ of the CL system will be > 0.6 for reasonable values of K a and K b .
£ The larger the value of K b , the larger the value of ζ will be for the same K a .
Kb = 0
Kb = 1
Kb = 2
K a = 0.5
ζ = 0.225
ζ = 0.716
ζ = 0.940
==> If we fix the desired
K a = 0.95
ζ = 0.162
ζ = 0.604
ζ = 0.963
K a max
4.56
14.4
24.9
備注
No inner-loop(例二)
With inner-loop(例三)
With inner-loop(例三)
ζ for the CL system, then the larger the value of K b will mean a larger
K a max , hence a larger gain margin for the design.
£ However, it may not always be desirable to use a large K b .
==> Overly increase in ζ will drag down the response speed of the CL system.
==> Often, rate data measurement is noisy. Consequently, increase the value of K b will mean
amplification of the noise signal
○Remaining difficulties of the system:
£ With θref being the command, the problem of little maneuvering capability remains.
==> Pitching maneuver is often conducted by commanding the pitching rate.
£ Remedy of the problem: Replace θref with a pitch rate command, i.e. θ&ref .
==> The resulting design:
A Pitch Orientation Control System
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【An attitude autopilot with maneuverability】
◎Pitch Orientation Control System◎
○Typical block diagram:
.
θref
Integrating
Amplifier
Gyro
eg
ea
Elevator
A/C
eδ Servo δ Dynamics
e
t
eg = ∫ (θ&ref − θ& )dτ
.
θ
Rate Gyro
0
○Basic features and the design considerations:
£ Provide the A/C with a stable longitudinal dynamics and a responsive pitch rate maneuver.
==> The responsive pitch rate maneuver is achieved by replacing θref with θ&ref .
£ A rate gyro feedback inner-loop remains, thereby maintaining the two-loop feedback structure.
£ An integrator (through the integrating gyro) is used to nullify the steady state output error. .
==> The current feedback structure is the same as that of the「Attitude autopilot」.
.
θref
1
s
eg
Ka
ea
inner-loop
eδ
−p
s+ p
δe
Hδe,θ. =
-b(s+z)
s2+a1s+a2
.
θ
Kb
£ Like previous designs, we will decide the inner-loop gain K b and the outer-loop gain K a .
==> The system can be analyzed through the same procedure as that performed previously.
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【Special stability issue on attitude control designs】
○Both the「Attitude Autopilot」and the「Pitch Orientation Control System」are basic to all longitudinal
flight controllers.
○Being the basic flight control systems, they must maintain A/C stability in all working ranges.
£ In general, this working range will cover both low A.O.A and high A.O.A conditions, where
widely different longitudinal dynamics may appear at different A.O.A ranges.
==> For instance, high A.O.A. pitch-up phenomena for high tail fighter aircraft
0.1
CM 0
4
-0.1
5
3
-0.2
-0.3
0
dCM / dCL > 0
2
0.3
0.6 CL 0.9
1
1.2
High AOA pitch up
Due to loss of tail lift U
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3
2
1
High AOA
seperation
£ A single feedback design must ensure CL stability in both working ranges.
==> Control design must proceed with two paths: (a) Low AOA case & (b) High AOA case
【例四】Aircraft with pitch up at high AOA ranges:
− 15( s + 0.4)
£ At low AOA: H (δ ,θ& ,low ) =
e
s2 + 0.9 s + 8
==> Stable, without pitch up
£ At high AOA: H (δ ,θ& ,high ) =
e
==> Unstable, with pitch up.
− 9( s + 0.3)
( s + 3)( s − 2.9)
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jω
9
Inner-loop root locus
The same.Kb is applied
to the both cases
Solutions with Kb = 0.3
Solutions with Kb = 0.6
ea
eδ
-10 δe
s+10
6
The low AOA case.
3
σ
-10
-8
-6
-4
-2
-3
.
θ
-b(s+z)
s2+a1s+a2
-6
-9
jω
Kb
9
6
The high AOA case.
3
σ
Increase in Kb will move -10
the unstable pole leftward,
but will harm the others
Often, we seek a tradeoff design.
-8
-6
-4
-2
2
-3
-6
-9
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The high AOA case.
Outer-loop root locus
jω
9
6
The selected values
of Kb and Ka must be
suitable for both cases.
3
σ
-10
-8
-6
-4
-2
2
-3
Locus with Kb = 0.3
Locus with Kb = 0.6
Sloution with Kb = 0.6 and Ka = 2
.
θref
1 eg
s
Ka
ea
-6
-9
jω
9
.
θ
inner-loop
6
3
Tune up Kb will boost Kamax ,
hence the gain margin.
-10
σ
-8
-6
-4
-2
-3
-6
-9
The low AOA case.
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£ A simulation of the closed-loop system response:
.
θ ref (t)
.
θ (t)
12
8
4
0
deg
α(t)
deg
4
0
-4
A/C is neutrally stable at this AOA
30
25
20
15
10
5
deg
δe(t)
0
5
10
15
t, sec
20
¡ The closed-loop stability is maintained for both the low AOA and the high AOA regimes.
¡ The pitch rate follows the command alright, though with a time delay.
==> Indicating maneuverability with control stick steeling.
【Further comments on attitude control designs】
○ Though a pitch rate maneuver is emphasized in the design of the Pitch orientation control system, it
is true that a pitch angle command may be more desirable in certain instances. In these cases, an
Attitude autopilot will be the basic attitude control design used.
○ Often, an attitude autopilot does not work along without some sort of velocity control. ==> Next
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