feedback control systems dr. basil hamed key words: feedback systems automatic control estimation...
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FEEDBACK CONTROL SYSTEMS
Dr. Basil Hamed
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Key Words:
• Feedback Systems• Automatic Control• Estimation &
Identification • Mathematical Modeling• Process Optimization• Decision Making
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Systems and Control
• A System is a device or process that takes a given input and produces some output:– A DC motor takes as input a voltage and produces
as output rotary motion– A chemical plant takes in raw chemicals and
produces a required chemical product
SystemInput Output
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Closed Loop Control
• Closed-loop control takes account of actual output and compares this to desired output
Measurement
DesiredOutput
+-
ProcessDynamics
Controller/Amplifier
OutputInput
• Open-loop control is ‘blind’ to actual output
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An Open-Loop Control System
• The controlled ‘output’ is the resulting toast
• System does not reject changes in component characteristics
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What is a Control System ?
• A process to be controlled• A measurement of process output• A comparison between desired and actual output• A controller that generates inputs from comparison
Measurement
+ -ProcessController Output
DesiredOutput
Comparison
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Control
• Many control systems can be characterised by these components
Sensor
Actuator ProcessControl
Referencer(t)
Outputy(t)
-+
Errore(t)
ControlSignal
u(t)
Plant
Disturbance
Sensor Noise
Feedback
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Actuation
• A device for acting on the environment
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Sensing
• A device for measuring some aspect of the environment
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Computing
• A combination of electronics and software
Empty
Wash
FillDone
Ready
SpinRinse
Stop
FailFail
Timeout
Overflow
Yes
Check Level
Fill Achieved?
Start
Fail
Stop
Open Valve
No
Close Valve
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Examples of Control Applications
Biological Systems:Central Nervous System is the controller for the body Robotics:
Robots perform automated tasks in assembly lines, where precision is important and dangerous tasks physically impossible for humans
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Examples of Control Applications
Aerospace Applications: Aircraft or missile guidance and control Space vehicles and structures
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Examples : Washing Machine
• System Requirements– Understanding of load sizes– Receptacle to hold clothes– ‘Plumbing’– Agitation of drum– Ease of use, Reliability– Low Cost
• Actuators– AC or DC Motors– Water inlet/drain
• Sensors– Water level– Load speed/balance
• Control– Choice depends on design
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Examples : The CD Player
• A CD player is an example of control system
• Requires– Accurate positioning of
the laser read head– Precise control of media
speed– Conversion of digital data
to analogue signal
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Examples : Hard Drive
• A computer disk drive is another example of a rotary control system
• Requires– Accurate positioning of
the magnetic read head– Precise control of media
speed– Extraction of digital data
from magnetic media
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Examples : Modern Automobiles
• Modern Automobiles are controlled by a number of computer components
• Requires– Control of automobile sub
systems• Brakes and acceleration
• Cruise control
• ABS
• Climate control
• GPS
– Reliability– Low cost– Ease of use
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Example: DC Motor Speed Control
• Desired speed d
• Actual speed • Tachometer measurements plus noise• Control signal is a voltage• Variations in Load Torque
Actual Speed Measurement
+
-Load Torque
PowerAmplifier
ControllerMotor
Tacho
d
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Example: Batch Reactor Temperature Control
• Goal: Keep Temperature at desired value Td• If T is too large, exothermic reaction may cause explosion• If T is too low, poor productivity may result• Feedback is essential because process dynamics are not
well known
ControllerSteam
Water
Measured Temperature
Coolant
ReactantsDesiredTemperature
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Example: Aircraft Autopilot
• Standard components in modern aircraft• Goal: Keep aircraft on desired path• Disturbances due to wind gust, air density, etc.• Feedback used to reject disturbances
GPS/Inertial
Path controller
RudderElevons
Measured pathRoute
SensorsActuators
Disturbances
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Mathematical Modelling
• To understand system performance, a mathematical model of the plant is required
• This will eventually allow us to design control systems to achieve a particular specification
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Block Diagrams
• Formalise control systems as ‘pictures’• Components can be combined to produce
an overall mathematical description of systems
• Interaction between elements is well defined
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Block Diagrams: Summation
• Ideal, no delay or dynamics• Two inputs: ( ) ( ) ( )z t d t y t
• Three or more: ( ) ( ) ( ) ( )z t f t g t y t
( )z t( )z t
( )y t( )y t
( )d t( )f t
( )g t
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Laplace Example I
( ) ( ) ( ) ( ) ( )p p
dymc y t u t sY s mc Y s U s
dt
pm c
( )Q u t
( )T y t
Physical model
( ) ( ) ( )
( ) ( ) ( )
1( ) ( )
p
p
p
sY s mc Y s U s
s mc Y s U s
Y s U ss mc
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For Example I
( ) ( ) ( ) ( ) ( )p p
dymc y t u t sY s mc Y s U s
dt
pm c
( )Q u t
( )T y t
Physical model
1
ps mc( )U s ( )Y s
Block Diagram model
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For Example I
( ) ( ) ( ) ( ) ( )p p
dymc y t u t sY s mc Y s U s
dt
pm c
( )Q u t
( )T y t
Physical model
( )G s( )U s ( )Y s
Transfer Function
1( )
p
G ss mc
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For Example II
( )x t
( )u tM
C
K
22 2
22 ( ) ( )
d x dxx t u t
dt dt
2C
M
2 K
M
K
M
2
C
KM
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For Example II
2 2 22 ( ) ( )s s X s U s
22 2
22 ( ) ( )
d x dxx t u t
dt dt
2 2 2. ( ) 2 . ( ) . ( ) ( )s X s s X s X s U s
Laplace Transform
2
2 2( ) ( )
2X s U s
s s
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For Example II
2
2 22s s
( )X s( )U s
( )x t
( )u tM
C
K
Physical Model
Block Diagram model
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Block Diagrams: Transfer Functions
• Transfer Function G(s) describes system component
• An operator that transfers input to output• Described as a Laplace transform because
( )Y s( )X s ( )G s
( ) ( ) ( )Y s G s U s ( ) ( ) ( )y t g t u t
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Single-Loop Feedback System
DesiredValue
Output
Transducer
+-
FeedbackSignal
error
Controller Plant
ControlSignal
( )C s ( )G s
K
( )d t ( )e t ( )u t ( )y t
( )f t
• Error Signal• The goal of the Controller C(s) is:
To produce a control signal u(t)Which drives the ‘error’ e(t) to zero
( ) ( ) ( ) ( ) ( )e t d t f t d t Ky t
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Controller Objectives
• Controller cannot drive error to zero instantaneously as the plant G(s) has dynamics
• Clearly a ‘large’ control signal will move the plant more quickly
• The gain of the controller should be large so that even small values of e(t) will produce large values of u(t)
• However, large values of gain will cause instability
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Control Criteria
• Speed of Response• Robustness to unknown
plant and load• Stability
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Response of a First-Order System
1( ) ( ) ( ) ( )
dyay t x t Y s X s
dt s a
0 1 2 3 4 5 6 70
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time
Out
put
Response of First Order Lag to Impulse Input
0( ) aty t y e
General Solution:
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Step Response
0 1 2 3 4 5 6 70
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Out
put
Time
Response of First Order Lag to Step Input
( ) (1 )atfy t y e
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Speed of Response
ux ye
K1
s a
, ( )dy
ay u u K x ydt
Equations:
( )dy
ay K x ydt
( )dy
a K y Kxdt
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System Descriptions
( ) ( )( )
KY s X s
s a K
( )
K
s a K ( )X s ( )Y s
( )dy
a K y Kxdt
( )0( ) a k ty t y e
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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (s)
Out
put
Speed of Response
( )0( ) a k ty t y e
0( ) aty t y e
Increasing K increases Speed of Response
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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Out
put
Time (s)
Speed of Response to Step
( )0( ) 1 a k ty t y e
0( ) (1 )aty t y e
Increasing K increases Speed of Response
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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Time (s)
Tracking Error
Steady-State Error
Initial Response
Input
Output
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sensorssensors
signal condsignal cond. . & & amplificationamplification
AA//DD computercomputerhardwarehardware
controlcontrolsoftwaresoftware
DD//AA
actuatorsactuators
DYNAMIC SYSTEMDYNAMIC SYSTEMDYNAMIC SYSTEMDYNAMIC SYSTEM
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Integrated Product DesignIntegrated Product Design
DESIGNDESIGN
PROTOTYPEPROTOTYPE
TEST &TEST &MEASUREMENTMEASUREMENTSIMULATIONSIMULATION
ANALYSISANALYSIS
DYNAMICDYNAMICMODELMODEL
+-
- +DESIRED PERFORMANCE
PHYSICAL SYSTEM
COMPUTERMODEL
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42
A Word About Stability
0 10 20 30 40 50 60-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50 60-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Start hereG
180o phaseinversion
0 10 20 30 40 50 60-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Bigger hereK
Another phase inversion
If G is such that input is phase reversed (180o out of phase) for any frequency, then input will be back in phase
If loop gain >1 then system will be unstable
BANG !
If System is unstable for one input, it will be unstable for all inputs
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Thank you and good luck in your Final
Exams
Thank you and good luck in your Final
Exams