5. Lecture Fuzzy Systems...Genetic algorithms, Simulated annealing, Differential evolution...
Transcript of 5. Lecture Fuzzy Systems...Genetic algorithms, Simulated annealing, Differential evolution...
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5. Structure of the lecture
1. Introduction Soft Control: Definition and delimitation, basic of 'intelligent'
systems
2. Knowledge representation and knowledge engineering (symbolic AI)
Application: Expert Systems
3. FuzzySystems: dealing with fuzzy knowledge
Application: Fuzzy control
1. Fuzzy-Quantity
2. Fuzzy-Relations, Fuzzy-Inference
3. Fuzzy-System, Fuzzy-Control
4. Connective Systems: Neural Networks
Application: Identification and neural control
5. Genetic algorithms, Simulated annealing, Differential evolution
Application: Optimization
6. Summary & References
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Contents of the Lecture 5.
1. Fuzzy Systems
1. Fuzzification
2. Defuzzyfying
3. Operation of the overall system
2. Fuzzy Control
1. Rules
2. Control
3. Fuzzy Control
4. Design Process
3. Summary
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Fuzzy System
• engl.: Fuzzy system
System, that used linguistic rules and with the help of the partial blocks
fuzzification, inference and defuzzyfying, mapped the numeric input variables
to numeric output variables
(VDI/VDE 3550)
Fuzzification Inference Defuzzyfication
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Fuzzification
• engl.: fuzzification
Conversion of a numeric size in a degree of membership to linguistic
expressions of a linguistic size
(VDI/VDE 3550)
Fuzzification Inference Defuzzyfication
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Fuzzification
• Transition from a sharp signal value X to a fuzzy signal value X*
• Assignment of the degrees of membership for all linguistic terms of the
corresponding linguistic variable
• For n linguistic terms, there is a n-tuples of degrees of membership
In the fuzzification, a sharp signal is not transferred in a fuzzy-quantity, but in a
vector of sharp degrees of memberships of fuzzy-quantities
1
0
μ
T/°C
50 1000
very low low very highhighmedium
T = 58°C T * = (0 0 0.5 0.15 0)
0.5
0.15
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Example for Fuzzification
• T1 = 28 °C T1*= (0 0,8 0 0 0) The temperature T1 = 28 °C is low
• T2 = 58°C T2*= (0 0 0,5 0,15 0) The temperature T2 = 58 °C isbetween medium and high, more medium
• T3 = 95°C T3*= (0 0 0 0 1) The temperature T3 = 95 °C is very high
0
μ
T/°C
50 1000
very low low very highhighmedium
T2 = 58°C
0.5
0.15
1
T3 = 95°CT1 = 28°C
0.8
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Defuzzyfication
• Engl.: defuzzyfication
Conversion of a fuzzy-quantity in a numeric output value (e.g. in a control
variable).
(VDI/VDE 3550)
Fuzzification Inference Defuzzyfication
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Thoughts about Defuzzyfication
• The output fuzzy-quantity represents a activation function
• Question: What exact value best describes the result of the inference?
• Basic Ideas:
Maxima of the function:
Value, that is the maximum in the fuzzy quantity
(Problem: Definition by multiple maxima)
"Middle" of the area
Center or median of the area under the curve
(Problem: complex calculation)
• Methods
Maximum-Defuzzyfication
gravity method
Area median method
• First an example
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Example: linguistic variables
1
0
μ
T/°C
50 1000
very low low very highhighmedium
1
0
μ
W/%
50 1000
very low low very highhighmedium
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Example: rule base and factum
Rule base
• R1: IF T = very low THEN W = very high
• R2: IF T = low THEN W = high
• R3: IF T = medium THEN W = medium
• R4: IF T = high THEN W = low
• R5: IF T = very high THEN W = very low
• Input Variable: T = 15 °C
1
0
μ
T/°C
50 1000
very low low very highhighmedium
1
0
μ
W/%
50 1000
very low low very highhighmedium
1
0
μ
W/%
50 1000
very low low very highhighmedium
0.75
0.25
Fuzzification: T * = (0.75 0.25 0 0 0)
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Example: Accumulation (MAX)
1
0
μ
W/%
50 1000
Very Low Low Very HighHighMedium
1
0
μ
W/%
50 1000
Very Low Low Very HighHighMedium
1
0
μ
W/%
50 1000
Very Low Low Very HighHighMedium
0.75
0.25
μ
W/%
50 100
Very HighHigh1
0
0.75
0.25
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Maximum-Defuzzyfication
• Where is the maximum ?
Mean-of-Maxima (mean value of the
Maxima)
Smallest-of-Maxima (first Maximum)
Largest of maxima (last peak)
μ
W/%
50 100
Very HighHigh1
0
0.75
0.25
μ
W/%
50 100
Very HighHigh1
0
0.75
0.25
μ
W/%
50 100
Very HighHigh1
0
0.75
0.25
μ
W/%
50 100
Very HighHigh1
0
0.75
0.25
MOM: YD = 93.75 SOM: YD = 87.5 LOM: YD = 100
Evaluation
Simple Calculation
Only rules with a maximum degree of fulfillment go to the result (usually one)
The degree of fulfillment of the rule is not taken into account (for MOM and
triangular-structured ZGF, others partially).
Range boundaries are not always possible (depends on ZGF)
Discontinuous output values
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Gravity method
• General
= Center of
gravity (COG)
μ
W/%
50 100
Very HighHigh1
0
0.75
0.25
Evaluation
All the rules are taken into account
Continuous output values
Levels of fulfillment are taken into account
Complex calculation
Range boundaries are not possible ( Advanced gravity method)
dyy
dyyy
yD
COG: YD =
• Simplified
or for Singletons
= Center of singletons
(COS), centroide
μ
W/%
50 100
Very HighHigh1
0
0.75
0.25
n
i
i
n
i
ii
D
y
yy
y
1
1
COS: YD = 85
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Area median method
• = Center of
area (COA)
μ
W/%
50 100
Very HighHigh1
0
0.75
0.25
Evaluation (almost like in gravity method)
All the rules are taken into account
Continuous output values
Levels of fulfillment are taken into account
Complex calculation (more complex than in gravity method)
Range boundaries are not possible
For singletons in output Fuzzy-Quantities unsuitable
D
D
y
y
D dyydyymity
COA: YD =
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Operation of a Fuzzy-System
1. FuzzificationDetermination of the degrees of membership of the sharp inputvariables to the Input-Fuzzy-Quantities
2. Aggregation (premise analysis)Determination of the levels of fulfillment of the single rule premises(Determination of active rules)
3. ActivationDetermination of the single Output-Fuzzy-Quantities (for each rule)
4. AccumulationOverlap of the single Output-Fuzzy-Quantities to an overall Output-Fuzzy-Quantity (function of attractiveness)
5. DefuzzyficationDetermination of the sharp output values from the function ofattractiveness
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Application: Fuzzy control
• Basics
Properties of a scheme
Properties of a control
Comparison of control (close loop and open loop)
• Fuzzy control
Application of a Fuzzy-System to control
• Design Methodology
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Block diagram of a control
Process variable
routeActuators
Sensors
Control element
Control output
reference variable w
-
Feedback variable
Comparing
element
Algorithm
Disturbances(incl. EMC, environment, ... )
Control
Characteristics
• Sphere of influence, where variables continuously retroact to themself
• Continuous values
• Standardized task: disturbance correction, tuning the reference variable
Example: Balancing of an inverted pendulum
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Control
Block Diagram
Output variablesControl Part
Control SignalsInput Variables
route
Actuator feedback
Actuators
SensorsFeedback variables
Disturbances(incl. EMF, environment, ...)
Algorithms
Characteristics
• Variables in the loop do NOTcontinously retroact themselves
• Binary values
• No standardized task
Example: Positioning of an inverted pendulum
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„Always restart, "not standardized bar: usually extensive
Rules can be applied“
„Always same“, standardized: „Controlled variable adjust the reference input“
Specification
Always several loops/mehrschleifig, i.e. several hundred sensors and actuators Complexity
>95% of control loops are one-loop/einschleifig (1 Sensor, 1 Actuator)
Number of signals
Variables in loop effects other variables
Variables in loop retroact themselves
Feedback variables
discretecontinousVariables
Boolean Algebra, Automata, Petri Nets
Differential equationsMathematics
Amplifier loop
Disturbances
Feedback system
No amplifier loopAmplification loop is defined Stability problem
only known in advance and trackabledisturbances can be corrected
unknown disturbances can be corrected
Asynchronous binary feedback variables( Events)
Permanently synchronised closed loop
ControlAutomation
Comparison of Automation and Control
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Fuzzy-Control
• Fuzzy controller (fuzzy controller) can be used for regulatory as well as for control tasks. Often combinations of the two are found.
• The resulting controller can be the described link between inputs and outputs
Characterstics curve
In general not-linear
Application of a fuzzy system for the control and automation
(Control)
Fuzzy controllers are not novel controller types.They belong to the class of nonlinear curves or
Characterstics diagram controller.
However, there are new design methods and the interpretation of results.
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m
1
negative-up positive-up
positive-downnegative-down
middle-up
180120900-90-120-180
0
negative-up
middle-
up
negative-
up
positive-
down
positive-
up
-30 30
Fuzzy Control in the example of inverted pendulum
Regel 1:
IF Pendulum angle
positiv-downAND
Angular acceleration negative
ANDWagon position
middle
THENacceleration should be
negative
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Static characteristics of fuzzy controllers
Control base:
R1: IF e = NG THEN u = NG
R2: IF e = NU THEN u = NU
R3: IF e = PG DANN u = PG
• Examples with mixed Degree of overlap Input fuzzy quantities
• Max-Min-Inference
• COS-Defuzzification
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Control and Variables characteristics
Control variale y Variable u
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Design parameters of a fuzzy controller
Fuzzification Inference Defuzzification
Control base
y
ZGFZGF Input variables Output variables
x
Problem orienteddesign parameters
Method oriented design parameters
Defuzzificationmethods
Inference-methods
(see 4. VL)
•Premise evaluation: Operators for AND and OR
(t-Norm und s-Norm)• Activation: Operator for
the closing of the Premise Conclusion (t-Norm)
• Accumulation: Operator for the
summary of single
control output (s-Norm)
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Design process of a fuzzy controller
Design process
1. Defining the parameters method
2. Defining the parameters problem
1. Define the linguistic variables and the number of terms
2. Defining the membership functions
3. Defining the rules (expertise)
3. Simulation using a model (if possible)
4. Implementation
Depending on the result of 3 (or 4): Optimization through interventions in 2 (or 1)
• Note: Even method parameters usually have not much influence on the behaviour
• method parameters will be partially used by the design tool set
Design process = method for determining the method and parameters of the problem
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Dynamic fuzzy controller
• Fuzzy controllers are initially static
• Dynamic behaviour can only be produced by external components are
Post-processing of output variables(integration)
pre-processing of input variables (Derivation)
• Example: Fuzzy-PID-Controller
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Summary and learning for 5th Lecture
To know the concept of fuzzy system
Fuzzification
Apply and describe the methods of De-fuzzification
Functionality of Fuzzy sytems
Concept of fuzzy controller with respect to with control and regulation
Design process of fuzzy controller