Applications Linear Control Design Techniques in Aircraft Control – I
Transcript of Applications Linear Control Design Techniques in Aircraft Control – I
Lecture – 29
Applications Linear Control Design Techniques in Aircraft Control – I
Dr. Radhakant PadhiAsst. Professor
Dept. of Aerospace EngineeringIndian Institute of Science - Bangalore
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Topics
Brief Review of Aircraft Flight Dynamics
An Overview of Automatic Flight Control Systems
Automatic Flight Control Systems: Classical (Frequency Domain) Designs
Brief Review of Aircraft Flight Dynamics
Dr. Radhakant PadhiAsst. Professor
Dept. of Aerospace EngineeringIndian Institute of Science - Bangalore
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Geometry of Conventional Aircrafts
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Basic Force Balance
Weight
Lift
Drag
Thrust
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Basic Moment Balance
X
Y
Z
Rolling
Pitching
Yawing
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Aileron Roll
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Elevator Pitch
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Rudder Yaw
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10ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Airplane Dynamics: Six Degree-of-Freedom ModelRef: Roskam J., Airplane Flight Dynamics and Automatic Controls, 1995
( ) ( )( ) ( )
( ) ( )
1 2 3 4
2 25 6 7
8 2 4 9
= c PQ+c L +c
= c
= c
T T
T
T T
P QR c L N N
Q PR c P R c M M
R PQ c QR c L L c N N
+ + +
− − + +
− + + + +
( )
( )
( )
1 sin
1 sin cos
1 cos cos
T
T
T
U VR WQ g X Xm
V WP UR g Y Ym
W UQ VP g Z Zm
= − − Θ+ +
= − + Φ Θ+ +
= − + Φ Θ+ +
( )
= sin tan cos tan = cos sin = sin cos sec
P Q RQ R
Q R
Φ + Φ Θ+ Φ Θ
Θ Φ − Φ
Ψ Φ + Φ Θ
cos sin 0 cos 0 sin 1 0 0sin cos 0 0 1 0 0 cos sin
0 0 1 sin 0 cos 0 sin cos
I
I
I
x Uy Vz W
ψ ψ θ θψ ψ φ φ
θ θ φ φ
−⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥= −⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦
= sin cos sin cos cosIh z U V W− = Θ− Θ Φ − Θ Φ
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Representation of Longitudinal Dynamics in Small Perturbation
State space form:
cX AX BU= +
0
0
000
0 0 1 0
U W
U W
U U W W QW W W
X X gZ Z U
AM M Z M M Z M M U
−⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥+ + +⎢ ⎥⎣ ⎦
0 0
E T
E T
E E TTW W
X X
Z ZB
M M Z M M Z
δ δ
δ δ
δ δ δ δ
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥+ +⎢ ⎥⎢ ⎥⎣ ⎦
UW
XQθ
Δ⎡ ⎤⎢ ⎥Δ⎢ ⎥=⎢ ⎥Δ⎢ ⎥Δ⎣ ⎦
Ec
T
Uδδ
Δ⎡ ⎤= ⎢ ⎥Δ⎣ ⎦
1 1, .U WX XX X etc
m U m W∂ ∂⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Short Period ModeReference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989.
• Heavily damped• Short time period• Constant velocity
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Short Period Dynamics
11 1
21 22 2
1E
a ba a bq q
α αδ
Δ Δ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤= + Δ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥Δ Δ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦
State Space Equation:
Transfer Function Equations:
2
2
( )( )
( )( )
E
q q
E
A s Bss As Bs C
A s Bq ss As Bs C
α ααδ
δ
+Δ=
Δ + +
+Δ=
Δ + +
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Long Period (Phugoid) DynamicsReference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989.
• Lightly damped• Changes in pitch attitude, altitude, velocity• Constant angle of attack
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Long Period (Phugoid) Dynamics
11 11 12
21 21 220E
T
a g b bu ua b b
δδθ θ
− ΔΔ Δ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ΔΔ Δ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦
State Space Equation:
Transfer Function Equations:
2
2
( )( )
( )( )
u u
E
E
A s Bu ss As Bs C
A s Bss As Bs C
θ θ
δ
θδ
+Δ=
Δ + +
+Δ=
Δ + +
Assumption: 0TδΔ =
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Representation of Lateral Dynamics in Small Perturbation
State space form: cX AX BU= +0 0
*
( ) cos
0
0
0 1 0 1
V P R
XZ XZ XZV V P P R R
X X X
XZ XZ XZV V P P R R
Z Z Z
Y Y U Y gI I IL N L N L NI I I
AI I IN L N L N LI I I
θ
∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗
− −⎡ ⎤⎢ ⎥⎢ ⎥+ + +⎢ ⎥
= ⎢ ⎥⎢ ⎥+ + +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
*
0
0 0
R
A A R R
A A R R
XZ XZ
X X
XZ XZ
Z Z
Y
I IL N L NI I
BI IN L N LI I
δ
δ δ δ δ
δ δ δ δ
∗
∗ ∗ ∗ ∗
⎡ ⎤⎢ ⎥⎢ ⎥+ +⎢ ⎥
= ⎢ ⎥⎢ ⎥+ +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
VP
XRφ
Δ⎡ ⎤⎢ ⎥Δ⎢ ⎥=⎢ ⎥Δ⎢ ⎥Δ⎣ ⎦
Ac
R
Uδδ
Δ⎡ ⎤= ⎢ ⎥Δ⎣ ⎦
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Lateral Dynamic InstabilitiesReference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989.
Directional divergence
Spiral divergence
• Do not possess directional stability• Tend towards ever-increasing angle
of sideslip• Largely controlled by rudder
• Spiral divergence tends to gradual spiraling motion & leads to high speed spiral dive
• Non–oscillatory divergent motion• Largely controlled by ailerons
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Lateral Dynamic InstabilitiesReference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989.
Dutch roll oscillation
• Coupled directional-spiral oscillation
• Combination of rolling and yawing oscillation of same frequency but out of phase each other
• Controlled by using both ailerons and rudders
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Dutch Roll Dynamics
11 12 11 12
21 22 21 22
A
R
a a b ba a b brr
δββδ
ΔΔ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤Δ ⎡ ⎤= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ΔΔΔ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦
State Space Equation:
Transfer Function Equations:
2 2
2 2
ˆ ˆ( ) ( )( ) ( )
ˆ ˆ( ) ( )( ) ( )
A R
r r r r
A R
A s B A s Bs ss As Bs C s As Bs C
r s A s B r s A s Bs As Bs C s As Bs C
β β β ββ βδ δ
δ δ
+ +Δ Δ= =
Δ + + Δ + +
Δ + Δ += =
Δ + + Δ + +
Automatic Flight Control Systems: An Overview
Dr. Radhakant PadhiAsst. Professor
Dept. of Aerospace EngineeringIndian Institute of Science - Bangalore
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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SensorsAltimeter: Height above sea levelAir Data System: Airspeed, Angle of Attack, Mach No., Air Temperature etc.Magnetometer: HeadingInertial Navigation System (INS) • Accelerometers: Translational motion of the aircraft
in the three axes• Gyroscopes: Rotational motion of the aircraft in the
three axesGPS: Accurate position, ground speed
The transfer function for most sensors can be approximated by a gain k
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Actuators
Electrical actuators: • Its a second order system in general• It can be approximated to a first order system
with small angle displacements
Hydraulic / Pneumatic actuators• First order system
Combination of the above
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Applications of Automatic Flight Control Systems
Cruise Control Systems• Attitude control (to maintain pitch, roll and heading)• Altitude hold (to maintain a desired altitude)• Speed control (to maintain constant speed or Mach no.)
Stability Augmentation Systems• Stability enhancement • Handling quality enhancement
Landing Aids• Alignment control (to align wrt. runway centre line)• Glideslope control• Flare control
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Techniques for Autopilot Design
Frequency domain techniques:• Root locus• Bode plot • Nyquist plot• PID design etc.
Time domain techniques:• Pole placement design• Lyapunov design• Optimal control design etc.
Automatic Flight Control Systems: Frequency Domain Designs
Dr. Radhakant PadhiAsst. Professor
Dept. of Aerospace EngineeringIndian Institute of Science - Bangalore
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Velocity Hold Control SystemReference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989.
The forward speed of an airplane can be controlled by changing the thrust produced by propulsion system.
The function of the speed control system is to maintain the some desired flight speed.
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Attitude Control SystemReference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989.
Sense the angular position
Compare the angular position with the desired angular position
Generate commands proportional to the error signal
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Roll Attitude AutopilotReference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989.
Components:• Model for roll dynamics
(transfer function) • Attitude gyro• Comparator• Aileron actuator
Reference:R. C. Nelson: Flight Stability and Automatic Control, McGraw-Hill, 1989.
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Roll Attitude AutopilotReference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989.
Problem:
Design a roll attitude control system to maintain a wings levelattitude for a vehicle having the following characteristics.
The system performance is to have damping ratio ζ = 0.707 and an undamped natural frequency ωn = 10 rad/s.
Assume the aileron actuator and the sensor (gyro) can be represented by the gains ka and ks
20.5 / 2.0 /p aL rad s L sδ= − =
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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The system transfer function( )( ) ( )
Forward path transfer function( ) ( )( )
( ) ( ) ( )
( ) 1 (unity feedback assumption)
The loop transfer function
( ) ( ) ( 0.5
a
a p
a aa
a p
s
s Ls s s L
s s LG s ke s s s s L
H s k
kG s H ss s
δ
δ
φδ
δ φδ
Δ=
Δ −
Δ Δ= =
Δ −
= =
=+
, where ) a ak k Lδ=
Roll Attitude AutopilotReference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989.
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Roll Attitude AutopilotReference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989.
0
The desired damping needed is 0.707We know cos . Hence,draw a line of 45 from the origin. Any root intersecting this line will have 0.707.
ζζ θ
ζ
==
=
Gain is determined from:
1,0.5
This leads to 0.0139. However,0.35 / (much lower than desired!)n
ks s
krad sω
=+
==
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Roll Attitude AutopilotReference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989.
1 2Moving pole to is impractical, unless there is an increase in wingspan of the aircraft..!Hence, the cure is to have a stability augmentation system.
P P
achieved
desired
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Roll Attitude AutopilotReference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989.
Compensator is added in form of rate feedback loop to meet desired damping and natural frequency.The inner loop transfer function can be expressed as follows:
Transfer function of inner loop
( )( ) ( )
a
a p
Lp ss s L
δ
δΔ
=Δ −
( ) , ( ) 1 ( ),0.5IL
IL IL rg IL as akG s H s k k k L
s δ= = =+
IL( )M(s)
1 ( ) ( )
0.5
IL
IL IL
IL
IL
G sG s H sk
s k
=+
=+ +
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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n
Inner loop gain is selected to movethe augmented root farther on the negative real axis. If the inner loop is located at
14.14, then the root locuswill shift to the left and desired
and can be
s
ξ ω
= −
achieved.
Roll Attitude AutopilotReference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989.
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Stability Augmentation System (SAS)Reference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989.
Inherent stability of an airplane depends on the aerodynamic stability derivatives.
Magnitude of derivatives affects both damping and frequency of the longitudinal and lateral motion of an airplane.
Derivatives are function of the flying characteristics which change during the entire flight envelope.
Control systems which provide artificial stability to an airplane having undesirable flying characteristics are commonly called as stability augmentation systems.
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Consider an aircraft with poor short period dynamic characteristics. Assume one degree of freedom (only pitching motion about CG) todemonstrate the SAS.
( )
Substituting the numerical values
0.071 5.49 6.71
This leads to 0.015 2.34 /
It is seen that the airplane has poor damping (flying quality).
q
e
sp nsp
M M M M
rad s
α α δθ θ θ δ
θ θ θ δ
ζ ω
− + + =
− + = −
= =
Short perioddynamics:
Example: SASReference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989.
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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e
One way to improve damping is to provide rate feedback.Artificial damping is provided by producing an elevator deflection in proportion to pitch rate and then adding it to the pilot's input; i.e.
δ δ=
( )
e
Rate gyro measures the and creates an electrical signal to provide in addtion to .With this, the modified dynamics becomes:
0.071 6.71 5.49 6.71
2
epArtificialPilot commandinput
e
n
k
k
k
θ
θ
θ δ
θ θ θ δ
ζω
+
− + + = −20.071 6.71 , 5.49
Hence, by varying , the desired damping is achieved.nk
kω= + =
Example: SASReference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989.
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Key Components:• Alignment control (to align wrt. runway
centre line)
• Glideslope control
• Flare control
Landing SystemReference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989.
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Alignment Control Through Electronic Aid and Pilot InputReference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989.
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Glide Slope: Pitch ControlReference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989.
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Glide Slope: Speed ControlReference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989.
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Flare: Sink Rate ControlReference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989.
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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