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