MEC 3716 Present#9&10

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MEC 3716 FLIGHT DYNAMICS and CONTROL

Automatic Control TheoryAutomatic Control Theory

The Classical ApproachThe Classical Approach

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Introduction

deals with the analysis and synthesis of logic for the control of system

Control theory

Classical approach Modern approachAnalytical approach based on:

frequency response method

the root locus technique transfer function

Laplace transform

Analytical approach based on:

time response method

the state-space formulation

The System

Systemnput signal Output signal

Eqq

22

222222

2222 = .... ..

Short-Period model q

E

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Introduction

Control System

SystemCommand Output signalControllerInputsignal

referenceinput Actuationsignal

Open-loop control

SystemCommand Output signalControllerInputsignal

referenceinput+

-

errorsignalClosed-loop control

Feedback signal

Transfer Function (T.F.) = Laplace transform of the OutputLaplace transform of the Input

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Introduction

The symbols used in the block diagram of Control System

R(s)

SystemCommand Output signal

ControllerInputsignal

referenceinput+

-

errorsignal

Feedback signal

C(s)(s)

B(s) Feedback Elements

G(s) = C(s)/E(s) is open-loop transfer functionH(s) is feedback transfer function

error signalE(s) = R(s) -B(s)

Feedback signalB(s) = H(s) C(s)

Output transfer functionC(s) = G(s)E(s)C(s) = G(s) R(s) - G(s) H(s) C(s)

R(s)C(s) = G(s)

1 + G(s) H(s)

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Control System (example)

given model ofp i t c h ing mo t i on of an aircraft as represented by secondorder differential equation below:

E =222 .if in the closed-loop system, pitching rate (q) is used as the feedbackelement, and a gain of .4 is used as a controller, determine the outputtransfer function

G(s)Command Output signal+-

pitching rate

qH(s)

E

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system E =222 . has input and output E

{ } { }ELL =222 .Laplace transform)()()(.)( ssssss E 222

2

222

2

2 += ssss

E .)(

)(

Transfer function of system

222

22

2 += sssG ..

)(

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Transfer function of feedback

ss

sqsH =

)(

)()(

Feedback pitching rate (q) )()( sssq q

the output transfer function

)()(

)(

sHsG

sG

+2sss

ss

222

222

222

22

2

2

+++=

.

..

.

222

22

2 += ss ..

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Consider the characteristic equation

ROUTHs CRITERION

2... 22

2

2

2 =++++

aaaan

nn

nn

n

Rouths Table:

2

2

n

n

n

2

2

2

c

b

aan

n

2

2

2

2

c

b

aan

n

2

222

2

=

n

nnnn

a

aaaab

2

222

2

=

n

nnnn

a

aaaab

2

2

2

2

c

b

aan

n

The Rouths stabil ity criterion states:If all the numbers of the first columnhave the same sign, the roots of thecharacteristic polynomial have negativereal parts.The system is stable.

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ROUTHs CRITERION

Check a stability of this system :

2222

22

22)(+++

=sss

sG1.

222222

22

)( +++=

ssssG.

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Consider the transfer function of a second order system

TIME DOMAIN and FREQUENCY DOMAIN SPECIFICATIONS

)()(2

)(

)(

)(

sHsG

sG

sR

sC

+= 22

2

2 nn

n

ss

++=

0.1

0.5

0.9

1.05

0.95

tr

tdtmax

ts

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TIME DOMAIN and FREQUENCY DOMAIN SPECIFICATIONSDelay time

n

dt

2

22.22.22 ++=

Rise time

n

rt

2

2.22.22 ++=

Time to peak amplitude

22

=

n

pt

Settling time

nst

2.2

=

Percent maximum overshoot

22/

.222

= e

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(2)(2)

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if the closed-loop system has damping ratio of 0.707, determine the forward-pathcompensator (kc), and predict the time to half amplitude of the closed loop system of step

response.

Forward-path compensation

kc

+

- 222

22 +ss .

ssH =)(

E

q

forward-pathcompensator

Transfer function of open-loop system

2222 += ss ksG c.)(

Transfer function of closed-loop system

)()(

)(

sHsG

sGTF +2

s

ss

kss

k

c

c

222

2

222

2

2

++

+=.

.

2222 += sks

k

c

c

).(

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The characteristic equation of CL system

22222 =sks c ).(

22222 .n

).( cn k222

).().)(.( ck22222222222 22.ck

The time to half amplitude

stn

222222222222

22222222

2

2 .).)(.(

.. =

PID C ll

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:controlalDifferenti

:controlIntegral

:controlalProportion

2

sKsE

sUte

dt

dKtu

s

K

sE

sUdtteKtu

KsEsUteKtu

dd

it

i

pp

)(

)()()(

)(

)()()(

)()()()(

PID Controller

PID Controller

R(s) PlanOutput signal

PID+- C(s)

E(s) U(s)

PID C t ll

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PID Controller

PID Controller )()()()( teKdtteKteKtu DIP sK

s

KK

sE

sUD

IP =

)(

)(

Find the gains KP, KI, andKD of system has second order of equation of motion of Plan

TK nnP / 22+TK nI /

2

TK nD /22 + nT /22whereis the time constant of integral control

Note: for Plan has order greater than two, the PID controller is determined by

using the Ziegler-Nichols method

PID C t ll

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PID Controller

Ziegler-Nichols Method

P

PI

PID

T/L

0.9*T/L

1.*T/L

L/0.3

*L 0.5*L

KP TI TD

L T

L delay time

T time constant

Find the gains KP, KI, andKD using the Ziegler-Nichols method (1)

sKs

KK

sE

sUD

IP =

)(

)(

I

pI

T

KK =

DpD TKK =

PID C t ll

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PID Controller

Find the gains KP, KI, andKD using the Ziegler-Nichols method ()Step 1

determine a critical gain (Kcr) of the closed loop system by using Rouths stability criterion

crP KK 2.2=cr

PI

P

KK

2.2

= crPD PKK 222.2=

crP KK 22.2= crPI PKK 22.2=

crP KK 2.2=

PID Controller

PI Controller

P Controller

Ziegler-Nichols Method

Step 2

determine a critical frequency (cr), where the frequency is with corresponding to the

critical gain (Kcr) of the closed loop system.

cr

crP

2=

Cont ol S stem (PID Cont olle )

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Control System (PID Controller)

determine the PID controller.

PID+

- sss 22

222++

The closed-loop transfer function

p

p

Ksss

KsG

+++=

2222

)(

22=cr

K 222)(2)(2)(22

=+++ iii crcrcr

22=PK 2.22=IK 22.2=DK

2

2=crP

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(Application to Aircraft Autopilot(Application to Aircraft Autopilot

Design)Design)

Aircraft Autopilot

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Aircraft Autopilot

AnAircraft Autopilot is a mechanical, electrical, or hydraulic system used toguide an aircraft without assistance from pilot.

Advantages: improves the flying qualities of the aircraft lessen the pilot workloads

The autopilot connectsAttitude indicator(pitch angle), a gyroscopicHeading indicator(yaw angle) and a bankindicator(roll angle) to hydraulicallyoperated elevators , rudder and ailerons.

Autopilot for aircraft to fly straight and level on a compass course

Aircraft Autopilot Design
http://en.wikipedia.org/wiki/Attitude_indicatorhttp://en.wikipedia.org/wiki/Attitude_indicatorhttp://en.wikipedia.org/wiki/Gyroscopehttp://en.wikipedia.org/wiki/Heading_indicatorhttp://en.wikipedia.org/wiki/Heading_indicatorhttp://en.wikipedia.org/wiki/Heading_indicatorhttp://en.wikipedia.org/wiki/Heading_indicatorhttp://en.wikipedia.org/wiki/Elevator_%28aircraft%29http://en.wikipedia.org/wiki/Rudderhttp://en.wikipedia.org/wiki/Aileronhttp://en.wikipedia.org/wiki/Aileronhttp://en.wikipedia.org/wiki/Rudderhttp://en.wikipedia.org/wiki/Elevator_%28aircraft%29http://en.wikipedia.org/wiki/Heading_indicatorhttp://en.wikipedia.org/wiki/Heading_indicatorhttp://en.wikipedia.org/wiki/Gyroscopehttp://en.wikipedia.org/wiki/Attitude_indicator

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Aim: determine compensator and feedback elements such that the closed-loop systemhas characteristics as desired specifications.

kc+

-

input outputcompensatorA/C

dynamicstransfer

function

feedback

Values/parameters to be controlled: output of the dynamics model of an aircraftexample: altitude, velocity, pitch angle, roll/yaw

angle, aoa, etc.

Controller: measurable parameters that has contribution in the dynamics modelexample: engine power, velocity, pitch rate,roll/yaw rate, aoa, etc.


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kc+

-

input outputcompensatorA/C

dynamicstransfer

function

feedback

Phugoid dynamics

2

2

=

u

u

Z

gXu

u

u

22

T

E

u

Z

u

Z

XX

EE

TE

The Laplace transformation

=)()() sgsu(s-Xu )()( sXsX TE TE +=)()( sssu

u

Zu 2

)()( s

u

Zs

u

ZTE

TE

22

Aircraft dynamics transfer function(an example problem)


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Aircraft dynamics transfer functionPhugoid dynamics

EXs

s

gs

su

(s-X EEu

=)()(

)(

)(

)

22 u

Z

s

ss

s

su

u

Z E

EE

u

)(

)(

)(

)(

uu

gu

u

g

E ZsXs

ZsX

ssu

EE

2

22

+

= )( )(

uu

gu

u

XZZX

u

Z

E ZsXs

s

ss

EuEuE

2

22

2

+=

)()(

PID ControllerPID+-

)(tE )(t

)(

)(

s

s

E

Pitch attitude autopilot with a PID controller


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Control Surface Actuator

Model of control surface servo actuator

amplifier

ka-

+cv flap

Position

feedback

kf

Servo motor

sBm

2

Model of elevator servo actuator

Ev E(Elevator)

Servo

2sk

fkk 2=

af

m

kk

B=

Pitch attitude autopilot with a PID controller

PID Controller

PID+- )(tE

)(t

)(

)(

s

s

E

2sk

Elevator Servo

)(tvE


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Ro l l a t t i tude c on t ro l

Roll dynamics

Ap ALpLp =

sensor

+

-)(tA

)(t

)()(ssA

2sk

Aileron Servo

)(tvA

sk


)()()( sLspLsAp A

=

pA Ls

L

s

sp A

)(

)(

=p)()( sssp =

ss

sp =)(

)(

)()(

)(

pA Lss

L

s

s A


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Head i ng a t t i t ude c on t ro l

Pure yawing motion

Rr RNNNN = )(

if the closed-loop system has damping ratio of 0.50, and the settling time of CLsystem of step response is 3 s, determine gains , andsK rK

sN

sN

r /.

/.

222

2222

=

sN

sNR

/.

/.

222

222

=

the stability derivatives

sensor

+

-)(tR

)(t

)()(ss

R

Rudder Servo

)(tvR

sk

rk


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Pure yawing motion

Rr RNNNN = )(


)()(])([ sNsNsNNsRr R

=2

NsNNs

N

s

s

rR

R

+= )()()(

2

by given the stability derivatives

222222

22

2 ..

.

)(

)(

+= ssss

R

sensor

+

-)(tR

)(t

)()(ss

R

Rudder Servo

)(tvR

sk

rk


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sensor

+

-)(tR

)(t

)()(ss

R

rk

Rudder Servo

)(tvR

sk

The closed-loop system T.F.

sss

kss

k

R ksv

s

r

r

222222

22

222222

22

2

2

2..

...

.

)(

)(

++

+=

)..(.

.

sr

r

kkss

k

22222222

22

2 +=

MEC 3716 Present#9&10

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