Post on 18-Jan-2016
FEEDBACK CONTROL SYSTEMS
Dr. Basil Hamed
Key Words:
• Feedback Systems• Automatic Control• Estimation &
Identification • Mathematical Modeling• Process Optimization• Decision Making
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
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
An Open-Loop Control System
• The controlled ‘output’ is the resulting toast
• System does not reject changes in component characteristics
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
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
Actuation
• A device for acting on the environment
Sensing
• A device for measuring some aspect of the environment
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
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
Examples of Control Applications
Aerospace Applications: Aircraft or missile guidance and control Space vehicles and structures
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
For Example II
2
2 22s s
( )X s( )U s
( )x t
( )u tM
C
K
Physical Model
Block Diagram model
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
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
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
Control Criteria
• Speed of Response• Robustness to unknown
plant and load• Stability
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:
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
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
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
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
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
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
sensorssensors
signal condsignal cond. . & & amplificationamplification
AA//DD computercomputerhardwarehardware
controlcontrolsoftwaresoftware
DD//AA
actuatorsactuators
DYNAMIC SYSTEMDYNAMIC SYSTEMDYNAMIC SYSTEMDYNAMIC SYSTEM
Integrated Product DesignIntegrated Product Design
DESIGNDESIGN
PROTOTYPEPROTOTYPE
TEST &TEST &MEASUREMENTMEASUREMENTSIMULATIONSIMULATION
ANALYSISANALYSIS
DYNAMICDYNAMICMODELMODEL
+-
- +DESIRED PERFORMANCE
PHYSICAL SYSTEM
COMPUTERMODEL
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
Thank you and good luck in your Final
Exams
Thank you and good luck in your Final
Exams