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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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
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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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
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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MEC 3716 FLIGHT DYNAMICS and CONTROL
Automatic Control TheoryAutomatic Control Theory
The Classical ApproachThe Classical Approach
(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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MEC 3716 FLIGHT DYNAMICS and CONTROL
Automatic Control TheoryAutomatic Control Theory
The Classical ApproachThe Classical Approach
(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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Aircraft Autopilot Design
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.
Aircraft Autopilot Design
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Aircraft Autopilot Design
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)
Aircraft Autopilot Design
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Aircraft Autopilot Design
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
Aircraft Autopilot Design
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Aircraft Autopilot Design
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
Aircraft Autopilot Design
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Aircraft Autopilot Design
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
The Laplace transformation
)()()( sLspLsAp A
=
pA Ls
L
s
sp A
)(
)(
=p)()( sssp =
ss
sp =)(
)(
)()(
)(
pA Lss
L
s
s A
Aircraft Autopilot Design
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Aircraft Autopilot Design
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
Aircraft Autopilot Design
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Aircraft Autopilot Design
Head i ng a t t i t ude c on t ro l
Pure yawing motion
Rr RNNNN = )(
The Laplace transformation
)()(])([ 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
Aircraft Autopilot Design
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Aircraft Autopilot Design
Head i ng a t t i t ude c on t ro l
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 +=
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