5. Lecture Fuzzy Systems

Transcription

1 Soft Control (AT 3, RMA) 5. Lecture Fuzzy Systems Fuzzy Control

2 5. Structure of the lecture. 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. 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 2

3 Contents of the Lecture 5.. Fuzzy Systems. Fuzzification 2. Defuzzyfying 3. Operation of the overall system 2. Fuzzy Control. Rules 2. Control 3. Fuzzy Control 4. Design Process 3. Summary 2

4 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 355) engl.: Fuzzy system Fuzzification Inference Defuzzyfication 22

5 Fuzzification Conversion of a numeric size in a degree of membership to linguistic expressions of a linguistic size (VDI/VDE 355) engl.: fuzzification Fuzzification Inference Defuzzyfication 23

6 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 very low low medium high very high T/ C T = 58 C T * = (.5.5 ) 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 24

7 Example for Fuzzification.8.5 very low low medium high very high.5 T = 28 C 5 T 2 = 58 C T 3 = 95 C T/ C T = 28 C T *= (,8 ) The temperature T = 28 C is low T 2 = 58 C T 2 *= (,5,5 ) The temperature T 2 = 58 C is between medium and high, more medium T 3 = 95 C T 3 *= ( ) The temperature T 3 = 95 C is very high 25

8 Defuzzyfication Conversion of a fuzzy-quantity in a numeric output value (e.g. in a control variable). (VDI/VDE 355) Engl.: defuzzyfication Fuzzification Inference Defuzzyfication 26

9 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 27

10 Example: linguistic variables very low low medium high very high 5 T/ C very low low medium high very high 5 W/% 28

11 Example: rule base and factum Rule base R: 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 = 5 C.75 very low low medium high very high very low low medium high very high W/% 5 T/ C very low low medium high very high 5 Fuzzification: T * = ( ).25 W/% 5 29

12 Example: Accumulation (MAX) Very Low Low Medium High Very High 5 W/%.75 Very Low Low Medium High Very High.25 Very Low Low Medium High Very High 5 W/% 5 W/%.75 High Very High.25 5 W/% 3

13 Maximum-Defuzzyfication High Very High 5 High Very High W/% Where is the maximum?.75 Mean-of-Maxima (mean value of the Maxima) Smallest-of-Maxima (first Maximum) Largest of maxima (last peak) High Very High.75 High Very High.25 5 W/%.25 5 W/%.25 5 W/% MOM: Y D = SOM: Y D = 87.5 LOM: Y D = 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 3

14 Gravity method General = Center of gravity (COG) Simplified or for Singletons = Center of singletons (COS), centroide y D y D y n i n y y High Very High 5 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) y i i dy dy y y i i COG: Y D = High Very High 5 COS: Y D = 85 W/% W/% 32

15 Area median method = Center of area (COA) y D mit y D y dy y D y dy High Very High 5 COA: Y D = W/% 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 33

16 Operation of a Fuzzy-System. Fuzzification Determination of the degrees of membership of the sharp input variables 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. Activation Determination of the single Output-Fuzzy-Quantities (for each rule) 4. Accumulation Overlap of the single Output-Fuzzy-Quantities to an overall Output- Fuzzy-Quantity (function of attractiveness) 5. Defuzzyfication Determination of the sharp output values from the function of attractiveness 34

17 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 35

18 Control Block diagram of a control Comparing element Control output Process variable reference variable w - Algorithm Actuators route Control element Disturbances (incl. EMC, environment,... ) Feedback variable Sensors 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 36

19 Control Block Diagram Actuator feedback Input Variables Control Signals Control Part Algorithms Actuators route Output variables Disturbances (incl. EMF, environment,...) Feedback variables Sensors Characteristics Variables in the loop do NOTcontinously retroact themselves Binary values No standardized task Example: Positioning of an inverted pendulum 37

20 Comparison of Automation and Control Variables Mathematics Feedback system Feedback variables Disturbances Amplifier loop Number of signals Specification continous Automation Differential equations Permanently synchronised closed loop Variables in loop retroact themselves unknown disturbances can be corrected Amplification loop is defined Stability problem >95% of control loops are oneloop/einschleifig ( Sensor, Actuator) Always same, standardized: Controlled variable adjust the reference input discrete Boolean Algebra, Automata, Petri Nets Variables in loop effects other variables No amplifier loop Control Asynchronous binary feedback variables( Events) only known in advance and trackable disturbances can be corrected Always several loops/mehrschleifig, i.e. several hundred sensors and actuators Complexity Always restart, "not standardized bar: usually extensive Rules can be applied 38

21 Fuzzy-Control Application of a fuzzy system for the control and automation 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 (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. 39

22 Fuzzy Control in the example of inverted pendulum Regel : IF Pendulum angle positiv-down AND Angular acceleration negative AND Wagon position middle THEN acceleration should be negative m negative-up negative-down negativeup negativeup middle-up middleup positive-up positive-down positiveup positivedown

23 Swing up with Fuzzy Controller 4 E

24 Static characteristics of fuzzy controllers Control base: R: 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 42

25 Control and Variables characteristics Control variale y Variable u 43

26 Example with two input variables 44

27 Design parameters of a fuzzy controller Problem oriented design parameters Method oriented design parameters Inferencemethods (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) Control base Defuzzification methods x Fuzzification Inference Defuzzification y ZGF Input variables ZGF Output variables 45

28 Design process of a fuzzy controller Design process = method for determining the method and parameters of the problem Design process. Defining the parameters method 2. Defining the parameters problem. 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 ) Note: Even method parameters usually have not much influence on the behaviour method parameters will be partially used by the design tool set 46

29 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 47

30 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 48