EEE-8005 Industrial automation SDL

Module leader: Dr. Damian Giaouris

Email: Damian.Giaouris@ncl.ac.uk

Room: E3.16

Phone: 0191 222 -7327

Module Leader of: Digital Control (EEE 8007)Degree Program Director of MSc: Automation and Control

Scopes / Objectives

Lecture Scope: • To give a mathematical background on set theory

Lecture Outcomes: • Syllabus outline• Explain the SDL part of the course• Boolean set theory – definition, intersection, union…• Need for fuzzy logic• Fuzzy logic set theory – membership functions: form, domain,

image• Logical operators OR AND Min Max…• Linguistic variables

Module Structure

Student Directed LearningStudent Directed Learning

Some lectures => Trigger further individualindividual study

Normal Lectures: 2hs/week

1h session/week: SDL

Provisional syllabus

Artificial IntelligenceFuzzy Logic

Theory Matlab

Neural NetworksGenetic algorithms

Provisional syllabus

Week 1: Intro – Basic set theoryWeek 2: Design of fuzzy logic controllersWeek 3: Design of fuzzy logic controllers IIWeek 4: TS Fuzzy Logic Weeks 5 - 7: Matlab programming Week 8: ANN – Matlab Week 9: ANN – Matlab IIWeek 10: Genetic AlgorithmsWeek 11: RevisionWeek 12: ???

Control strategy

Conventionalcontrol Model of the actual plant

Deterministic Stochastic

Inaccurate

Complex methods

Human reasoning and experience

Complicated processes Controlled by experiencedpractical engineers

Have no ideaabout the model

Use their knowledge &experience

Human reasoningNo model neededSatisfactory performance

Artificial Intelligence

Artificial Intelligence

•Expert Systems (ES)•Fuzzy Logic (FL)•Artificial Neural Networks (ANN)•Genetic Algorithms (GAs)•A combination of all these

Set theory I

Shape A Shape B Shape C Shape D

Shape E Shape F Shape G Shape H

H Shape F, Shape D, Shape C, ShapeA

G Shape E, Shape B, Shape A, ShapeB

H Shape G, Shape F, Shape E, Shape D, Shape C, Shape B, Shape A, ShapeC

Set theory II

H Shape F, Shape D, Shape C, ShapeA

G Shape E, Shape B, Shape A, ShapeB

H Shape G, Shape F, Shape E, Shape D, Shape C, Shape B, Shape A, ShapeC

Subset: A set that has some elements from another set

G Shape B, Shape A, ShapeD BD Union: A set that has all the elements of two other sets

G Shape B, Shape A, Shape H, Shape F, Shape D, Shape C, Shape AD

DAAD

Intersection: A set that has all the common elements of two other sets

G ShapeEB H Shape G, ShapeEwhere

BEEB

Boolean Logic

esTemperaturA HotesTemperaturBAB ,

25 esTemperaturesTemperaturB

25 Temperature

MembershipFunction

100%

0%

Boolean and Fuzzy Logic (FL)

Temperature=24.99 ??? Not so HotNot so Hot

Temperature=25 100 %

Temperature=24 90 %Temperature=15 0 %

Element Membership function, I.e. How much an element belongs to a set

HotMuchHowesTemperaturC ,

Fuzzy Logic

25 Temperature

MembershipFunction

15

100%

0%

25 Temperature

MembershipFunction

100%

0%

Fuzzy Sets I

Triangular

Trapezoidal

Fuzzy Sets II

Gaussian

Sigmoidal

Polynomial

Fuzzy Sets III

Logical Operators

6 4, 2, 1,A

1 2 3 4 5 6

Set A

Union•For element 1: Is 1 a member of set A OR set BIntersection•For element 1: Is 1 a member of set A AND set B

6 5, 2,3, B

1 2 3 4 5 6

Set B

Logical Operators Discrete Sets

6 3,4,5, 2, 1,BA

1 2 3 4 5 6

UnionA B AND OR

1 0 0 1

0 1 0 1

1 1 1 1

0 0 0 0

6 2,BA

1 2 3 4 5 6

Intersection

Logical Operators Continuous Sets

25 esTemperaturesTemperaturA

30 eTemperatureTemperaturInter

25 etemperaturetemperaturUnion

25 Temperature30 25 Temperature30

03 esTemperaturesTemperaturB

Fuzzy sets & Logical Operators I

OR=MAXAND=MIN

A B Min(A, B)and

Max(A, B)or

1 0 0 1

0 1 0 1

1 1 1 1

0 0 0 0

Fuzzy sets & Logical Operators II

Example – Matlab Exercise

Two fuzzy sets have the following membership functions

35,30,75

1

30,25,55

1

3525,0

xxx

xxx

xorxx

A

40,35,85

1

35,30,65

1

4030,0

xxx

xxx

xorxx

B

Plot the two setsFind the union and the intersection of them, and explain the results through the min, max operator

Linguistic variables

The room is cold lets switch on the heaterNot The temperature is 17.5 degrees

)(x

esTemperatur

1

1510 20

cold

Lecture 1

Lecture scope

Lecture Scope: • To define advanced concepts on FL set theory• Connection between classical and FS theory

Lecture Outcomes: • Notation• Definitions like support, height…• Union, intersection, max and min• Negation, bounded sums• Cartesian products on crisp and FS• Extension principle• Fuzziness

Lecture Outcomes

Lecture Scope: • Basic steps in the design of a Fuzzy Logic Controller

Lecture Outcomes:

• Basic Control strategy• Fuzzification• Fuzzy Inference System• Multiple Inputs – And/Or operators• Overlapping Fuzzy Sets• Defuzzyfication

Design of a FLC - Basic Concept

FL mimics Human Reasoning:

If … Then…

IF THEN RULES

R1: If the room is very cold then switch on the heater to fullR2: If the room is cold then switch on the heater to mediumR3: If the room is normal then switch off the heater

If part: premise - Then part: conclusion

Design of a FLC - Fuzzification I

)(x

Temperatures

1

1510 20

VeryCold

Cold Warm Hot

1. Cover I/O the universe of discourse with FS2. Assign to every real input a membership function at each set This process is called Fuzzyfication

Design of a FLC - Fuzzification II

)(x

Temperatures

1

1510 20

VeryCold

Cold Warm Hot

11

0.7

0.5

With this way every real input is mapped to a fuzzy setThe value of the membership function that will be assigned dependson the shape of the membership function

Design of a FLC – If… Then…

1. If … Then … Rules2. Input Fuzzy Sets (Fuzzification)3. Output Fuzzy Sets

Associate

If Then Rules

Input Linguistic Variable

Output Linguistic Variable

If … Then … Rules associate the input fuzzy sets to the output fuzzy setsIf … Then … Rules associate the input fuzzy sets to the output fuzzy sets

Design of a FLC – If… Then…

)(x

eTemperatur0

Very Cold Cold Normal

35 10050 80

)(x

%

Heater0

MaxMedOff

35 10050 80

R1: If temp is Very Cold Then Heater is MaxR2: If temp is Cold Then Heater is MedR3: If temp is Normal Then Heater is Off

Design of a FLC - Degree of Support Boolean sets

Assume an IF THEN rule with Boolean sets:R1: IF student fails THEN his/her parents are Sad

Hence if a student x fails 100% then his/her parentswill be 100% sad.

Therefore how much truth is the premise defines how much truth is the conclusion

The value of 100% or 0% is called degree of support of R1

Design of a FLC - Degree of Support Fuzzy sets I

Exactly the same stands for fuzzy setsR1: If temp is Cold Then Heater is Med

)(x

1

5030 80

Cold)(x

1

50%35% 80%

Med

Assume temp=35oC

Design of a FLC - Degree of Support Fuzzy sets II

So the degree of support is 0.7So the output “Med” is true 0.7

)(x

1

5030 80

Cold

35

0.7

???

R1: If temp is Cold Then Heater is Med

Design of a FLC - Degree of Support Fuzzy sets III

I have to take 70% of the output

)(x

1

5030 80

Cold

35

0.7

)(x

1

50%35% 80%

Med

0.7

Design of a FLC - Degree of Support Fuzzy sets IV

)(x

1

50%35% 80%

Med

0.7

)(x

1

50%35% 80%

Med

0.7

Min method Product method

Design of a FLC - 2nd example

)(x

hour

milesSpeed0

Slow Normal Fast

35 10050 80

)(x

%

LevelBrake0

Min Med Max

35 10050 80

R1: If speed is Slow Then Brake is MinR2: If speed is Normal Then Brake is MedR3: If speed is Fast Then Brake is Max

Design of a FLC - Degree of Support Fuzzy sets II

85 miles/hour -> Input: Max 0.5Hence Output: 0.5

)(x

hourmilesSpeed

0.5

9080 100

Fast

85

)(x

0.5

90%80% 100%

Maximum

Brake level

Design of a FLC - Degree of Support Fuzzy sets III

85 miles/hour -> Input: High 0.5Hence Output: 0.5

)(x

0.5

90%80% 100%

High

Brake level

)(x

hourmilesSpeed

0.5

9080 100

Fast

85

Design of a FLC – Number of Inputs

Has the previous controller a satisfactory performance?

No, what about if the speed is medium and there is a car in 5m

We need another input, the distance from the front car.

Hence the rules will have the following form:

R1: If Speed is High OR/AND the Distance is Small Then Brake is Max

Hence we have to use logical operators: Max & Min

Design of a FLC – Or / AND I

The problem now is the degree of support of this rulesince there are two fuzzy sets that are activated

High Speed and Small Distance

)(x

hkmSpeed /,

1

9080 100

High)(x

m,distance

1

2010 30

Close

Design of a FLC – Or / AND II

Assume that the actual speed is 85 and the actual distance is 18 meters:

)(x

hkmSpeed /,

1

9080 100

High

85

0.5

)(x

m,distance

1

2010 30

Close

0.6

18

Degree from input 1=0.5Degree from input 2=0.6

Design of a FLC – Or / AND III

Since the OR operator was used then the overall degree of support is found by the max operation:Degree of Support for rule 1: max(0.5,0.6)=0.6

If the operator was the AND then we would use min:Degree of Support for rule 1: min(0.5,0.6)=0.5

)(x

0.6

90%80% 100%

Maximum

Brake level

)(x

0.6

90%80% 100%

High

Brake level

Maximum

Design of a FLC – Multiple Input FS I

The universe of discourse must be fully covered by FSHence now the controller could be:

)(x

hkmSpeed /,

9080 100

High

60504030

MedLow

Input Output

)(x

scaleBrake

9080 100

Full

60504030

SomeLittle

If Speed==Low Then Brake==Little

If Speed==Some Then Brake==Some

If Speed==High Then Brake==Full

Design of a FLC – Multiple Input FS II

Hence if input=35km/h:

Input Output

)(x

hkmSpeed /,

9080 100

High

60504030

MedLow

0.5

)(x

scaleBrake

9080 10060504030

Little

Design of a FLC – Overlapping Input FS I

What about if speed is 50km/h?The controller will do nothing!!!

For this reason we overlap the FS:

)(x

hkmSpeed /,

9080 1007050403020100 60

VeryLow

Low HighVeryHigh

)(x

hkmSpeed /,

9080 1007050403020100 60

Nothing Little Some Full

Brake scale %

Design of a FLC – Overlapping Input FS II

1. If Speed==Very Low Then Brake==Nothing2. If Speed==Low Then Brake==Little3. If Speed==High Then Brake==Some 4. If Speed==Very High Then Brake==Full

Speed=25 km/h Very Low 0.8 Low 0.2

Hence degree of support for R1 is 0.8 and for R2 is 0.2

Design of a FLC – Overlapping Input FS III

)(x

hkmSpeed /,

9080 1007050403020100 60

Nothing

Brake scale%

)(x

hkmSpeed /,

9080 1007050403020100 60

Little

Brake scale%

Aggregation MethodAggregation Method

1. Max (Maximum) 2. Prodor (Probabilistic Or) 3. Sum

)(x

hkmSpeed /,

9080 1007050403020100 60

Nothing

Brake scale %

Design of a FLC – Overlapping Input FS V

Brake scale %

)(x

hkmSpeed /,

9080 1007050403020100 60

Nothing

Brake scale %

)(x

hkmSpeed /,

9080 1007050403020100 60

Nothing)(x

hkmSpeed /,

9080 1007050403020100 60

Nothing

Design of a FLC – Defuzzification

)( x

0.5

90%80% 100%

Maximum

)( x

0.5

90%80% 100%

Mean Of Maxima

Max Of Maxima

Least Of Maxima

)( x

0.5

90%80% 100%

Centre of area (COA)

Design of a FLC – Defuzzification

max: yyout Maximum

Mean Of Maxima (MOM) max:1

1

j

m

jj yy

mout

Centre of area (COA)

m

ij

m

ijj

y

yyout

1

1

Largest of maximumSmallest of maximum

Design of a FLC – Summary

The first step is to Fuzzify the real inputs:Appropriate cover the universe of discourse with FS

The second step is to create the FIS:Create the IF THEN rules using AND/OR operatorAggregate all the FLR to get the final output FS

Initially choose the number of inputs/outputs and their universe of discourse

The last step is to defuzzify the output fuzzy sets to a real value

Lecture 3

Artificial Neural Networks (ANNs)

Human Brain:

Memory Processor

Small “computing” element: Neuron

NucleusCell bodyAxon/Nuerous dendritic links Synapses

1010 to 1012

Adaptive connections

Structure of ANNs

Σ f(net)net y

w1

w2

w3

w n

x1

x2

x3

xn

Activationfunction

Inputs

1

b

Inputs: x1 ,x2,x3,…,xn Weights w1 ,w2,w3,…,wn

bxwxwxwxwbxwnet nn

n

iii

.......332211

1

bxwfnetfy

n

iii

1

Activation function

bxwfnetfy

n

iii

1

Linear activation function

y

net

Threshold activation function

y

net

+1

-1

Activation function…cont

bxwfnetfy

n

iii

1

net

+1

0.5

y

Sigmoid function

Tansigmoid function

y

net

+1

-1

Architecture of ANNs

Combinations of ANNs

y1

x2

x3

x1

y2

y3

+1 +1

Threshold Threshold

b11

b12

w111

w112

w431 w34

2

InputLayer

OutputLayer

HiddenLayer

o1

o2

o3

o4

Multi-layer feedforward

Σ f(net)net y

w1

w2

w3

w n

x1

x2

x3

xn

Activationfunction

Inputs

1

b

3 inputs x 4 outputs from o 3 outputs y

T

T

T

yyy

oooo

xxx

321

4321

321

y

o

x

Multi-layer feedforward

T

T

T

yyy

oooo

xxx

321

4321

321

y

o

x

14

13

12

11

143

142

141

133

132

131

123

122

121

113

112

111

,

b

b

b

b

www

www

www

www

11 bw

y1

x2

x3

x1

y2

y3

+1 +1

Threshold Threshold

b11

b12

w111

w112

w431 w34

2

InputLayer

OutputLayer

HiddenLayer

o1

o2

o3

o4

1st Layer

Hidden Layer

113

1132

1121

111

11 bxwxwxwfo

214

2143

2132

2121

211

21 bowowowowfy

Recurrent neural networks

y1

x2

x1

y2

y3

w111

w112

w431 w34

2

Delay

ExternalInputs

Delay

Classification of ANN

Supervised Learning:

Unsupervised Learning

Teacher Input/ Target data

Network weight correction Learning algorithm

Minimize an error function

Mean-squared error (MSE)

Learning algorithm

Back propagationNon-LinearFunction

NeuralNetwork

Learning Algorithm

x

Input

y

y

+

-

error )(1 kwkwkw ijijij

ijij w

Ew

n: Learning Rate

)1(

)(1

kw

kwkwkw

ij

ijijij

ANNs Strategy

1.Assemble the suitable training data2.Create the network object

3.Train the network 4.Simulate the network response to new inputs

Application of ANNs

1. Classification and diagnostic2. Pattern recognition 3. Modelling 4. Forecasting and prediction5. Estimation and Control

Revision

Σ f(net)net y

w1

w2

w3

w n

x1

x2

x3

xn

Activationfunction

Inputs

1

b

y1

x2

x3

x1

y2

y3

+1 +1

Threshold Threshold

b11

b12

w111

w112

w431 w34

2

InputLayer

OutputLayer

HiddenLayer

o1

o2

o3

o4Non-Linear

Function

NeuralNetwork

Learning Algorithm

x

Input

y

y

+

-

error

Matlab

net= newff ([-4 3; -5 5], [4,1], {‘tansig’,’purelin’},’traingda’ )

net.trainParam.lr

net.trainParam.epochs

net.trainParam.goal

562.003.0 23 xxxxfy x=0-20 input=x

target=f(x)

>> net=newff([0,20],[10,1],{'tansig','purelin'},'trainlm');>> net.trainParam.goal=1e-5;>> net.trainParam.epochs=500;

>> [net,tr]=train(net,p,t);

>> a=sim(net,x)