ECE 565 - Digital Control Systems

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15 Νοε 2013 (πριν από 3 χρόνια και 9 μήνες)

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1
ECE 565 - Digital Control Systems
Objective:Studnets should acquire a fundamental understanding of digital dontrol systems and
their design.
Assumptions
1.We Assume that a mathamatical mode of the plant can be derived
plants can be a combination of one or more basic plants
an electrical systemthat consist of resistors,capacitors,inductors and has
has a voltage or current source and some sort of load
a mechanical systemthat consists of springs,dampers,mass and can move linearly
(velocity) or it can rotate (torque)
a hydraulic system:fluid pressure driven
a pnuematic system:air pressure driven
2.we assume that a computer serves as the controller
analog input signals are digitized
output signals may be converted to an analog form
3.we assume systems are linear systems
4.we assume the systemare time invarient
2
CH 1:Introduction
1.1 Overview:Analysis and Design of Closed Loop Control Systems using Digital Computers
plant:physical systemto be controlled
control actuator (A):drives plant (can be included as part of plant)
sensors(S):measures plant response
controller (or compensator,filter):
compare:sensor data to desired response
generate:difference signal that attempts to minimize difference
compensation is function of the plant dynamics
plant dynamics (behavior):generally not selectable
systemdynamics:must be made satisfactory by controller
SystemModel
i.linear:any input x
i
(t) generates output y
i
(t)
then
Σ
i
a
i
x
i
(t) =
Σ
i
ai y
i
(t)
- all systems are inherently non-linerar
- many systems are linear over a limited range,can be modelled as linear systems
ii.time invarient:constant systemparameters (coefficients)
iii.discrete time:signals only change at discrete instants
(as opposed to non-linear,time varient,continuous systems)
Plant
A
controller
S
+
difference
signal
desired
response
plant input response
3
1.2 Digital Control Systems
radar unit:measures approximate lateral & verticle position
control unit:calculates approximate pitch & bank commands
transmitter:sends commands to to autopilot
controllers &commands are independent
lateral controllers:controls aircrafts lateral position
verticle controllers:controls aircrafts altitude
pitch command used to affect altitude
bank command used to affect lateral position
y(t):lateral distance of aircraft fromdesired position
y(kT):sampled value of y(t) at intervals of T (sample period)
φ
φφ
φ(kT):digital control command:make y(t) = 0
φ
φφ
φ(t):data hold clamps bank command constant at last value ( onboard the aircraft)
w(t):disturbances:unwanted inputs:wind,white noise,
- some sensor noise is always present inaccuracy in estimating systemstate
design problem:maintain y(t) small in the presence of disturbances (wind,radar noise)
- must understand mathmatical model:relationship between noise,commands,and position
- specify controller processing as function of aircraft model and noise
Radar
controller
b
ank command
p
itch commasnd
lateral position
verticle position
Transmitter
noise w(t)
Radar
+
desired
position
D
ata
Hold
Lateral
Digital
Controler
Aircraft
Lateral
System
position,y(t)
bank command,
φ
φφ
φ(t)
r
adar noise
Ty(kT)
4
1.3:Control Problem:control physical system
- closed loop:systemresponse is used to generate new commands
- error signal:difference between systemresponse &desired response
Possible Design Criteria

disturbance rejection

steady state errors

transient response

sensitivity to parameters changes in plant
Issues in Solving the Control Problem

selecting sensors & feedback signals

selecting actuators

developing models (equations) for plant,sensors,actuators

designing the controller

design evaluation:analytical,simulation,physical test

iteration until satisfactory response obtained
Conceptual Aspects
M
athmatical
Solution
M
athmatical
M
odel
Physical
System
problem
formulation
solution
translation
5
1.4 Satellite Model:development of mathimatical model
plant description:spherical satellite with thrusters

6 degrees of freedom:roll,pitch,yaw

consider only yawangle,θ(t)

thrusters generate torque,τ(t) to reduce θ(t)

assume no friction

ignore initial conditions when first developing transfer function
i.2
nd
order diffeq plant model:
)(
)(
t
dt
td
J
2
2
τ
θ
=
==
=
J:satellites moment of intertia about yaw axis
laplace transform:
)]([)()( tLssJs
2
τΤΘ =
==
==
==
=
ii.2
nd
order plant transfer function:
2
p
Js
1
s
s
sG =
==
==
==
=
)(
)(
)(
Τ
Θ
iii.state variable model:define x
1
(t) = θ(t) (controlled variable)
)(
)(
)(
)(
)(
)(
)()(
)()()(
)()(
t
J
1
0
tx
tx
00
10
tx
tx
J
t
ttx
ttxtx
ttx
2
1
2
1
2
12
1
τ
τ
θ
θ
θ


















+
++
+




































=
==
=


















=
==
==
==
=
=
==
==
==
=
=
==
=
&
&
&&
&
&
&
θ
θθ
θ
(t)
thrusters
6
1.5 Servo Motor (positioning system)
- tracking system:electric motor used to rotate radar antenna
- radar antenna tracks aircraft position
- error signal = proportional to difference between antenna direction &LOS of aircraft
DC Motor system:
- armature controlled motor with constant field
- assume L
a
can be ignored (ok for many motors)
- input:e(t)
- output:motor position = θ
i.motor back emf:e
m
(t) = K
b
w(t) = K
b
(dθ/dt)
θ(t) = shaft position
w(t) = angular velocity of shaft
K
b
= motor constant
J = moment of inertia connected to shaft
B = t otal viscous friction
ii.torque developed by motor:τ(t)
dt
td
B
dt
td
Jt
2
2
)()(
)(
θθ
τ +
++
+=
==
=
iii.developed torque for motor:τ(t) = K
T
i(t)
K
T
= motor parameter
i(t) = aramature current
iv.voltage equation for Armature:e(t) = R
a
i(t) + e
m
(t)
Solve for θ in terms of e(t)
dt
td
R
K
R
te
R
tete
ti
a
b
aa
m
)()()()(
)(
θ

−−
−=
==
=

−−

=
==
=
dt
td
B
dt
td
J
dt
td
R
KK
R
teK
t
2
2
a
bT
a
T
)()()(
)(
)(
θθθ
τ +
++
+=
==
=−
−−
−=
==
=
a
T
a
bTa
2
2
R
te
K
dt
td
R
KKBR
dt
td
J
)()()(
=
==
=
+
++
+
+
++
+
θθ
J
i
f
= constant
i
e
R
a
L
a
e
m
θ
θθ
θ
7
Laplace Transform 

transfer function
























+
++
+
+
++
+
=
==
=
+
++
+
+
++
+
=
==
=
=
==
=
+
++
+
+
++
+
a
bTa
aT
a
bTa
2
aT
a
T
a
bTa2
R
KKBR
ss
JRK
s
R
KKBR
Js
RK
sE
s
sE
R
K
ss
R
KKBR
sJs
)(
)(
)()()(
Θ
ΘΘ
State Space Representation
)(
)(
)(
)(
)(
)()()()(
)()()(
)()(
te
JR
K
0
tx
tx
JR
KKBR
0
10
tx
tx
te
JR
K
tx
JR
KKBR
ttx
ttxtx
ttx
a
T
2
1
a
bTa
2
1
a
T
2
a
bTa
2
12
1
























+
++
+










































+
++
+

−−

=
==
=


















+
++
+
+
++
+

−−
−=
==
==
==
=
=
==
==
==
=
=
==
=
&
&
&&
&
&
&
θ
θ
θ
8
Antenna Pointing System
- consists of 2 servomechanisms (servo):systemin which mechanical position is controlled
- controlled by electric motor &gear system
1.yaw axis control
v
o
(t):output voltage of shaft directly proportional to angle of rotation
v
i
(t):proprotional to desired yaw angle
e(t):error voltage = v
o
(t) - v
i
(t)
- power amplfier required to amplify error to drive motor
- assume amplifier has max gain = 5 and max output of 24v
- if error > 4.8v non-linear output
- may be important to keep systemin linear region
- some systemdesigned to operate in non-linear region
2.pitch axis control is similar
+
K
1
s(s+a)
K
S
error
s
ervo motor
amplifier
Output
24
-24
-4.8 4.8 Input
pitch axis
φ(t)
θ(t)
Yawaxis