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

2

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




4

Geometry of Conventional Aircrafts


5

Basic Force Balance

Weight

Lift

Drag

Thrust


6

Basic Moment Balance

X

Y

Z

Rolling

Pitching

Yawing


7

Aileron Roll


8

Elevator Pitch


9

Rudder Yaw


10ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

10

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


11

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∂ ∂⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠


12

Short Period ModeReference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989.

• Heavily damped• Short time period• Constant velocity


13

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 + +


14

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


15

Long Period (Phugoid) Dynamics

11 11 12

21 21 220E

T

a g b bu ua b b

δδθ θ

− ΔΔ Δ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ΔΔ Δ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦



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 =


16

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⎣ ⎦


17

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


18

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


19

Dutch Roll Dynamics

11 12 11 12

21 22 21 22

A

R

a a b ba a b brr

δββδ

ΔΔ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤Δ ⎡ ⎤= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ΔΔΔ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦



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




21


22

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


23

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


24

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


25

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




27

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.


28

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


29

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.


30


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= − =


31

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=



32


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

=+

==


33


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


34


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

=+

=+ +


35

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.



36

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.


37

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.


38

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.


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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.


40

Alignment Control Through Electronic Aid and Pilot InputReference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989.


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Glide Slope: Pitch ControlReference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989.


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Glide Slope: Speed ControlReference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989.


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Flare: Sink Rate ControlReference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989.


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Applications Linear Control Design Techniques in Aircraft Control – I

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