# ECE 565 - Digital Control Systems

Ηλεκτρονική - Συσκευές

15 Νοε 2013 (πριν από 4 χρόνια και 5 μήνες)

104 εμφανίσεις

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
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
controller
b
ank command
p
itch commasnd
lateral position
verticle position
Transmitter
noise w(t)
+
desired
position
D
ata
Hold
Lateral
Digital
Controler
Aircraft
Lateral
System
position,y(t)
bank command,
φ
φφ
φ(t)
r
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

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