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Industrial Automation SEQUENTIAL FLOW CHARTSDL.ppt

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The document outlines the structure and objectives of the EEE-8005 Industrial Automation SDL module, led by Dr. Damian Giaouris, focusing on digital control and fuzzy logic. It covers mathematical foundations of set theory, fuzzy systems, and artificial intelligence applications, along with a provisional syllabus detailing weekly topics and learning outcomes. The course involves student-directed learning with lectures, practical applications involving MATLAB, and the design principles of fuzzy logic controllers.

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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 Learning Student Directed Learning Some lectures => Trigger further individual individual study Normal Lectures: 2hs/week 1h session/week: SDL
Provisional syllabus Artificial Intelligence Fuzzy Logic Theory Matlab Neural Networks Genetic algorithms
Provisional syllabus Week 1: Intro – Basic set theory Week 2: Design of fuzzy logic controllers Week 3: Design of fuzzy logic controllers II Week 4: TS Fuzzy Logic Weeks 5 - 7: Matlab programming Week 8: ANN – Matlab Week 9: ANN – Matlab II Week 10: Genetic Algorithms Week 11: Revision Week 12: ???
Control strategy Conventional control Model of the actual plant Deterministic Stochastic Inaccurate Complex methods
Human reasoning and experience Complicated processes Controlled by experienced practical engineers Have no idea about the model Use their knowledge & experience Human reasoning No model needed Satisfactory 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, Shape  A   G Shape E, Shape B, Shape A, Shape  B   H Shape G, Shape F, Shape E, Shape D, Shape C, Shape B, Shape A, Shape  C
Set theory II   H Shape F, Shape D, Shape C, Shape  A   G Shape E, Shape B, Shape A, Shape  B   H Shape G, Shape F, Shape E, Shape D, Shape C, Shape B, Shape A, Shape  C Subset: A set that has some elements from another set   G Shape B, Shape A, Shape  D B D  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   A D D A A D    Intersection: A set that has all the common elements of two other sets   G Shape   E B   H Shape G, Shape  E where B E E B   
Boolean Logic   es Temperatur A    Hot es Temperatur B A B   ,   25   es Temperatur es Temperatur B 25 Temperature Membership Function  100% 0%
Boolean and Fuzzy Logic (FL) Temperature=24.99 ??? Not so Hot Not so Hot Temperature=25 100 % Temperature=24 90 % Temperature=15 0 % Element Membership function, I.e. How much an element belongs to a set   Hot Much How es Temperatur C , 
Fuzzy Logic 25 Temperature Membership Function  15 100% 0% 25 Temperature Membership Function  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 B Intersection •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,   B A 1 2 3 4 5 6 Union A B AND OR 1 0 0 1 0 1 0 1 1 1 1 1 0 0 0 0   6 2,   B A 1 2 3 4 5 6 Intersection
Logical Operators Continuous Sets   25   es Temperatur es Temperatur A   30   e Temperatur e Temperatur Inter   25   e temperatur e temperatur Union 25 Temperature 30 25 Temperature 30   0 3   es Temperatur es Temperatur B
Fuzzy sets & Logical Operators I OR=MAX AND=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 , 7 5 1 30 , 25 , 5 5 1 35 25 , 0 x x x x x x x or x x A                                  40 , 35 , 8 5 1 35 , 30 , 6 5 1 40 30 , 0 x x x x x x x or x x B    Plot the two sets Find 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 heater Not The temperature is 17.5 degrees ) (x  es Temperatur 1 15 10 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 full R2: If the room is cold then switch on the heater to medium R3: 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 15 10 20 Very Cold Cold Warm Hot 1. Cover I/O the universe of discourse with FS 2. 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 15 10 20 Very Cold Cold Warm Hot 11 0.7 0.5 With this way every real input is mapped to a fuzzy set The value of the membership function that will be assigned depends on the shape of the membership function
Design of a FLC – If… Then… 1. If … Then … Rules 2. 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 sets If … Then … Rules associate the input fuzzy sets to the output fuzzy sets
Design of a FLC – If… Then… ) (x  e Temperatur 0 Very Cold Cold Normal 35 100 50 80 ) (x  % Heater 0 Max Med Off 35 100 50 80 R1: If temp is Very Cold Then Heater is Max R2: If temp is Cold Then Heater is Med R3: 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 parents will 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 sets R1: If temp is Cold Then Heater is Med ) (x  1 50 30 80 Cold ) (x  1 50% 35% 80% Med Assume temp=35o C
Design of a FLC - Degree of Support Fuzzy sets II So the degree of support is 0.7 So the output “Med” is true 0.7 ) (x  1 50 30 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 50 30 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 miles Speed 0 Slow Normal Fast 35 100 50 80 ) (x  % Level Brake 0 Min Med Max 35 100 50 80 R1: If speed is Slow Then Brake is Min R2: If speed is Normal Then Brake is Med R3: If speed is Fast Then Brake is Max
Design of a FLC - Degree of Support Fuzzy sets II 85 miles/hour -> Input: Max 0.5 Hence Output: 0.5 ) (x        hour miles Speed 0.5 90 80 100 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 0.5 Hence Output: 0.5 ) (x  0.5 90% 80% 100% High Brake level ) (x        hour miles Speed 0.5 90 80 100 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 rule since there are two fuzzy sets that are activated High Speed and Small Distance ) (x  h km Speed / , 1 90 80 100 High ) (x  m , distance 1 20 10 30 Close
Design of a FLC – Or / AND II Assume that the actual speed is 85 and the actual distance is 18 meters: ) (x  h km Speed / , 1 90 80 100 High 85 0.5 ) (x  m , distance 1 20 10 30 Close 0.6 18 Degree from input 1=0.5 Degree 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 FS Hence now the controller could be: ) (x  h km Speed / , 90 80 100 High 60 50 40 30 Med Low Input Output ) (x  scale Brake 90 80 100 Full 60 50 40 30 Some Little 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  h km Speed / , 90 80 100 High 60 50 40 30 Med Low 0.5 ) (x  scale Brake 90 80 100 60 50 40 30 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  h km Speed / , 90 80 100 70 50 40 30 20 10 0 60 Very Low Low High Very High ) (x  h km Speed / , 90 80 100 70 50 40 30 20 10 0 60 Nothing Little Some Full Brake scale %
Design of a FLC – Overlapping Input FS II 1. If Speed==Very Low Then Brake==Nothing 2. If Speed==Low Then Brake==Little 3. 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  h km Speed / , 90 80 100 70 50 40 30 20 10 0 60 Nothing Brake scale % ) (x  h km Speed / , 90 80 100 70 50 40 30 20 10 0 60 Little Brake scale %
Aggregation Method Aggregation Method 1. Max (Maximum) 2. Prodor (Probabilistic Or) 3. Sum ) (x  h km Speed / , 90 80 100 70 50 40 30 20 10 0 60 Nothing Brake scale %
Design of a FLC – Overlapping Input FS V Brake scale % ) (x  h km Speed / , 90 80 100 70 50 40 30 20 10 0 60 Nothing Brake scale % ) (x  h km Speed / , 90 80 100 70 50 40 30 20 10 0 60 Nothing ) (x  h km Speed / , 90 80 100 70 50 40 30 20 10 0 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 :   y y out  Maximum Mean Of Maxima (MOM)   max : 1 1     j m j j y y m out  Centre of area (COA)          m i j m i j j y y y out 1 1   Largest of maximum Smallest 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 operator Aggregate 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 Nucleus Cell body Axon/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 Activation function Inputs 1 b Inputs: x1 ,x2 ,x3 ,…,xn Weights w1 ,w2 ,w3 ,…,wn b x w x w x w x w b x w net n n n i i i          ....... 3 3 2 2 1 1 1                b x w f net f y n i i i 1
Activation function                b x w f net f y n i i i 1 Linear activation function y net Threshold activation function y net +1 -1
Activation function…cont                b x w f net f y n i i i 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 b1 1 b1 2 w11 1 w11 2 w43 1 w34 2 Input Layer Output Layer Hidden Layer o1 o2 o3 o4 Multi-layer feedforward Σ f(net) net y w1 w2 w3 w n x1 x2 x3 xn Activation function Inputs 1 b 3 inputs x 4 outputs from o 3 outputs y       T T T y y y o o o o x x x 3 2 1 4 3 2 1 3 2 1    y o x
Multi-layer feedforward       T T T y y y o o o o x x x 3 2 1 4 3 2 1 3 2 1    y o x                           1 4 1 3 1 2 1 1 1 43 1 42 1 41 1 33 1 32 1 31 1 23 1 22 1 21 1 13 1 12 1 11 , b b b b w w w w w w w w w w w w 1 1 b w y1 x2 x3 x1 y2 y3 +1 +1 Threshold Threshold b1 1 b1 2 w11 1 w11 2 w43 1 w34 2 Input Layer Output Layer Hidden Layer o1 o2 o3 o4 1st Layer Hidden Layer   1 1 3 1 13 2 1 12 1 1 11 1 1 b x w x w x w f o       2 1 4 2 14 3 2 13 2 2 12 1 2 11 2 1 b o w o w o w o w f y     
Recurrent neural networks y1 x2 x1 y2 y3 w11 1 w11 2 w43 1 w34 2 Delay External Inputs 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 propagation Non-Linear Function Neural Network Learning Algorithm x Input y y ^ + - error     ) ( 1 k w k w k w ij ij ij     ij ij w E w       n: Learning Rate     ) 1 ( ) ( 1         k w k w k w k w ij ij ij ij 
ANNs Strategy 1.Assemble the suitable training data 2.Create the network object 3.Train the network 4.Simulate the network response to new inputs
Application of ANNs 1. Classification and diagnostic 2. Pattern recognition 3. Modelling 4. Forecasting and prediction 5. Estimation and Control
Revision Σ f(net) net y w1 w2 w3 w n x1 x2 x3 xn Activation function Inputs 1 b y1 x2 x3 x1 y2 y3 +1 +1 Threshold Threshold b1 1 b1 2 w11 1 w11 2 w43 1 w34 2 Input Layer Output Layer Hidden Layer o1 o2 o3 o4 Non-Linear Function Neural Network 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   5 6 2 . 0 03 . 0 2 3      x x x x f y 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)

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Industrial Automation SEQUENTIAL FLOW CHARTSDL.ppt

  • 1. 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
  • 2. 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
  • 3. Module Structure Student Directed Learning Student Directed Learning Some lectures => Trigger further individual individual study Normal Lectures: 2hs/week 1h session/week: SDL
  • 4. Provisional syllabus Artificial Intelligence Fuzzy Logic Theory Matlab Neural Networks Genetic algorithms
  • 5. Provisional syllabus Week 1: Intro – Basic set theory Week 2: Design of fuzzy logic controllers Week 3: Design of fuzzy logic controllers II Week 4: TS Fuzzy Logic Weeks 5 - 7: Matlab programming Week 8: ANN – Matlab Week 9: ANN – Matlab II Week 10: Genetic Algorithms Week 11: Revision Week 12: ???
  • 6. Control strategy Conventional control Model of the actual plant Deterministic Stochastic Inaccurate Complex methods
  • 7. Human reasoning and experience Complicated processes Controlled by experienced practical engineers Have no idea about the model Use their knowledge & experience Human reasoning No model needed Satisfactory performance Artificial Intelligence
  • 8. Artificial Intelligence •Expert Systems (ES) •Fuzzy Logic (FL) •Artificial Neural Networks (ANN) •Genetic Algorithms (GAs) •A combination of all these
  • 9. 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, Shape  A   G Shape E, Shape B, Shape A, Shape  B   H Shape G, Shape F, Shape E, Shape D, Shape C, Shape B, Shape A, Shape  C
  • 10. Set theory II   H Shape F, Shape D, Shape C, Shape  A   G Shape E, Shape B, Shape A, Shape  B   H Shape G, Shape F, Shape E, Shape D, Shape C, Shape B, Shape A, Shape  C Subset: A set that has some elements from another set   G Shape B, Shape A, Shape  D B D  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   A D D A A D    Intersection: A set that has all the common elements of two other sets   G Shape   E B   H Shape G, Shape  E where B E E B   
  • 11. Boolean Logic   es Temperatur A    Hot es Temperatur B A B   ,   25   es Temperatur es Temperatur B 25 Temperature Membership Function  100% 0%
  • 12. Boolean and Fuzzy Logic (FL) Temperature=24.99 ??? Not so Hot Not so Hot Temperature=25 100 % Temperature=24 90 % Temperature=15 0 % Element Membership function, I.e. How much an element belongs to a set   Hot Much How es Temperatur C , 
  • 13. Fuzzy Logic 25 Temperature Membership Function  15 100% 0% 25 Temperature Membership Function  100% 0%
  • 17. 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 B Intersection •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
  • 18. Logical Operators Discrete Sets   6 3,4,5, 2, 1,   B A 1 2 3 4 5 6 Union A B AND OR 1 0 0 1 0 1 0 1 1 1 1 1 0 0 0 0   6 2,   B A 1 2 3 4 5 6 Intersection
  • 19. Logical Operators Continuous Sets   25   es Temperatur es Temperatur A   30   e Temperatur e Temperatur Inter   25   e temperatur e temperatur Union 25 Temperature 30 25 Temperature 30   0 3   es Temperatur es Temperatur B
  • 20. Fuzzy sets & Logical Operators I OR=MAX AND=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
  • 21. Fuzzy sets & Logical Operators II
  • 22. Example – Matlab Exercise Two fuzzy sets have the following membership functions                               35 , 30 , 7 5 1 30 , 25 , 5 5 1 35 25 , 0 x x x x x x x or x x A                                  40 , 35 , 8 5 1 35 , 30 , 6 5 1 40 30 , 0 x x x x x x x or x x B    Plot the two sets Find the union and the intersection of them, and explain the results through the min, max operator
  • 23. Linguistic variables The room is cold lets switch on the heater Not The temperature is 17.5 degrees ) (x  es Temperatur 1 15 10 20 cold Lecture 1
  • 24. 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
  • 25. 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
  • 26. 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 full R2: If the room is cold then switch on the heater to medium R3: If the room is normal then switch off the heater If part: premise - Then part: conclusion
  • 27. Design of a FLC - Fuzzification I ) (x  Temperatures 1 15 10 20 Very Cold Cold Warm Hot 1. Cover I/O the universe of discourse with FS 2. Assign to every real input a membership function at each set This process is called Fuzzyfication
  • 28. Design of a FLC - Fuzzification II ) (x  Temperatures 1 15 10 20 Very Cold Cold Warm Hot 11 0.7 0.5 With this way every real input is mapped to a fuzzy set The value of the membership function that will be assigned depends on the shape of the membership function
  • 29. Design of a FLC – If… Then… 1. If … Then … Rules 2. 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 sets If … Then … Rules associate the input fuzzy sets to the output fuzzy sets
  • 30. Design of a FLC – If… Then… ) (x  e Temperatur 0 Very Cold Cold Normal 35 100 50 80 ) (x  % Heater 0 Max Med Off 35 100 50 80 R1: If temp is Very Cold Then Heater is Max R2: If temp is Cold Then Heater is Med R3: If temp is Normal Then Heater is Off
  • 31. 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 parents will 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
  • 32. Design of a FLC - Degree of Support Fuzzy sets I Exactly the same stands for fuzzy sets R1: If temp is Cold Then Heater is Med ) (x  1 50 30 80 Cold ) (x  1 50% 35% 80% Med Assume temp=35o C
  • 33. Design of a FLC - Degree of Support Fuzzy sets II So the degree of support is 0.7 So the output “Med” is true 0.7 ) (x  1 50 30 80 Cold 35 0.7 ??? R1: If temp is Cold Then Heater is Med
  • 34. Design of a FLC - Degree of Support Fuzzy sets III I have to take 70% of the output ) (x  1 50 30 80 Cold 35 0.7 ) (x  1 50% 35% 80% Med 0.7
  • 35. 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
  • 36. Design of a FLC - 2nd example ) (x        hour miles Speed 0 Slow Normal Fast 35 100 50 80 ) (x  % Level Brake 0 Min Med Max 35 100 50 80 R1: If speed is Slow Then Brake is Min R2: If speed is Normal Then Brake is Med R3: If speed is Fast Then Brake is Max
  • 37. Design of a FLC - Degree of Support Fuzzy sets II 85 miles/hour -> Input: Max 0.5 Hence Output: 0.5 ) (x        hour miles Speed 0.5 90 80 100 Fast 85 ) (x  0.5 90% 80% 100% Maximum Brake level
  • 38. Design of a FLC - Degree of Support Fuzzy sets III 85 miles/hour -> Input: High 0.5 Hence Output: 0.5 ) (x  0.5 90% 80% 100% High Brake level ) (x        hour miles Speed 0.5 90 80 100 Fast 85
  • 39. 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
  • 40. Design of a FLC – Or / AND I The problem now is the degree of support of this rule since there are two fuzzy sets that are activated High Speed and Small Distance ) (x  h km Speed / , 1 90 80 100 High ) (x  m , distance 1 20 10 30 Close
  • 41. Design of a FLC – Or / AND II Assume that the actual speed is 85 and the actual distance is 18 meters: ) (x  h km Speed / , 1 90 80 100 High 85 0.5 ) (x  m , distance 1 20 10 30 Close 0.6 18 Degree from input 1=0.5 Degree from input 2=0.6
  • 42. 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
  • 43. Design of a FLC – Multiple Input FS I The universe of discourse must be fully covered by FS Hence now the controller could be: ) (x  h km Speed / , 90 80 100 High 60 50 40 30 Med Low Input Output ) (x  scale Brake 90 80 100 Full 60 50 40 30 Some Little If Speed==Low Then Brake==Little If Speed==Some Then Brake==Some If Speed==High Then Brake==Full
  • 44. Design of a FLC – Multiple Input FS II Hence if input=35km/h: Input Output ) (x  h km Speed / , 90 80 100 High 60 50 40 30 Med Low 0.5 ) (x  scale Brake 90 80 100 60 50 40 30 Little
  • 45. 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  h km Speed / , 90 80 100 70 50 40 30 20 10 0 60 Very Low Low High Very High ) (x  h km Speed / , 90 80 100 70 50 40 30 20 10 0 60 Nothing Little Some Full Brake scale %
  • 46. Design of a FLC – Overlapping Input FS II 1. If Speed==Very Low Then Brake==Nothing 2. If Speed==Low Then Brake==Little 3. 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
  • 47. Design of a FLC – Overlapping Input FS III ) (x  h km Speed / , 90 80 100 70 50 40 30 20 10 0 60 Nothing Brake scale % ) (x  h km Speed / , 90 80 100 70 50 40 30 20 10 0 60 Little Brake scale %
  • 48. Aggregation Method Aggregation Method 1. Max (Maximum) 2. Prodor (Probabilistic Or) 3. Sum ) (x  h km Speed / , 90 80 100 70 50 40 30 20 10 0 60 Nothing Brake scale %
  • 49. Design of a FLC – Overlapping Input FS V Brake scale % ) (x  h km Speed / , 90 80 100 70 50 40 30 20 10 0 60 Nothing Brake scale % ) (x  h km Speed / , 90 80 100 70 50 40 30 20 10 0 60 Nothing ) (x  h km Speed / , 90 80 100 70 50 40 30 20 10 0 60 Nothing
  • 50. 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)
  • 51. Design of a FLC – Defuzzification   max :   y y out  Maximum Mean Of Maxima (MOM)   max : 1 1     j m j j y y m out  Centre of area (COA)          m i j m i j j y y y out 1 1   Largest of maximum Smallest of maximum
  • 52. 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 operator Aggregate 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
  • 53. Artificial Neural Networks (ANNs) Human Brain: Memory Processor Small “computing” element: Neuron Nucleus Cell body Axon/Nuerous dendritic links Synapses 1010 to 1012 Adaptive connections
  • 54. Structure of ANNs Σ f(net) net y w1 w2 w3 w n x1 x2 x3 xn Activation function Inputs 1 b Inputs: x1 ,x2 ,x3 ,…,xn Weights w1 ,w2 ,w3 ,…,wn b x w x w x w x w b x w net n n n i i i          ....... 3 3 2 2 1 1 1                b x w f net f y n i i i 1
  • 55. Activation function                b x w f net f y n i i i 1 Linear activation function y net Threshold activation function y net +1 -1
  • 56. Activation function…cont                b x w f net f y n i i i 1 net +1 0.5 y Sigmoid function Tansigmoid function y net +1 -1
  • 57. Architecture of ANNs Combinations of ANNs y1 x2 x3 x1 y2 y3 +1 +1 Threshold Threshold b1 1 b1 2 w11 1 w11 2 w43 1 w34 2 Input Layer Output Layer Hidden Layer o1 o2 o3 o4 Multi-layer feedforward Σ f(net) net y w1 w2 w3 w n x1 x2 x3 xn Activation function Inputs 1 b 3 inputs x 4 outputs from o 3 outputs y       T T T y y y o o o o x x x 3 2 1 4 3 2 1 3 2 1    y o x
  • 58. Multi-layer feedforward       T T T y y y o o o o x x x 3 2 1 4 3 2 1 3 2 1    y o x                           1 4 1 3 1 2 1 1 1 43 1 42 1 41 1 33 1 32 1 31 1 23 1 22 1 21 1 13 1 12 1 11 , b b b b w w w w w w w w w w w w 1 1 b w y1 x2 x3 x1 y2 y3 +1 +1 Threshold Threshold b1 1 b1 2 w11 1 w11 2 w43 1 w34 2 Input Layer Output Layer Hidden Layer o1 o2 o3 o4 1st Layer Hidden Layer   1 1 3 1 13 2 1 12 1 1 11 1 1 b x w x w x w f o       2 1 4 2 14 3 2 13 2 2 12 1 2 11 2 1 b o w o w o w o w f y     
  • 60. Classification of ANN Supervised Learning: Unsupervised Learning Teacher Input/ Target data Network weight correction Learning algorithm Minimize an error function Mean-squared error (MSE)
  • 61. Learning algorithm Back propagation Non-Linear Function Neural Network Learning Algorithm x Input y y ^ + - error     ) ( 1 k w k w k w ij ij ij     ij ij w E w       n: Learning Rate     ) 1 ( ) ( 1         k w k w k w k w ij ij ij ij 
  • 62. ANNs Strategy 1.Assemble the suitable training data 2.Create the network object 3.Train the network 4.Simulate the network response to new inputs
  • 63. Application of ANNs 1. Classification and diagnostic 2. Pattern recognition 3. Modelling 4. Forecasting and prediction 5. Estimation and Control
  • 64. Revision Σ f(net) net y w1 w2 w3 w n x1 x2 x3 xn Activation function Inputs 1 b y1 x2 x3 x1 y2 y3 +1 +1 Threshold Threshold b1 1 b1 2 w11 1 w11 2 w43 1 w34 2 Input Layer Output Layer Hidden Layer o1 o2 o3 o4 Non-Linear Function Neural Network Learning Algorithm x Input y y ^ + - error
  • 65. Matlab net= newff ([-4 3; -5 5], [4,1], {‘tansig’,’purelin’},’traingda’ ) net.trainParam.lr net.trainParam.epochs net.trainParam.goal   5 6 2 . 0 03 . 0 2 3      x x x x f y 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)