Fuzzy Systems. Fuzzy Control

Transcription

1 Fuzzy Systems Fuzzy Control Prof. Dr. Rudolf Kruse Christoph Doell Otto-von-Guericke University of Magdeburg Faculty of Computer Science Institute for Intelligent Cooperating Systems R. Kruse, C. Doell FS Fuzzy Control Part 3 1 / 59

2 Mamdani Control

3 Architecture of a Fuzzy Controller knowledge base fuzzification interface fuzzy decision logic fuzzy defuzzification interface not fuzzy measured values controlled system controller output not fuzzy R. Kruse, C. Doell FS Fuzzy Control Part 3 2 / 59

4 Example: Cartpole Problem (cont.) X 1 is partitioned into 7 fuzzy sets. Support of fuzzy sets: intervals with length 1 4 of whole range X 1. Similar fuzzy partitions for X 2 and Y. Next step: specify rules if ξ 1 is A (1) and... and ξ n is A (n) then η is B, A (1),..., A (n) and B represent linguistic terms corresponding to µ (1),..., µ (n) and µ according to X 1,..., X n and Y. Let the rule base consist of k rules. R. Kruse, C. Doell FS Fuzzy Control Part 3 3 / 59

5 Example: Cartpole Problem (cont.) θ nb nm ns az ps pm pb nb ps pb nm pm ns nm ns ps θ az nb nm ns az ps pm pb ps ns ps pm pm nm pb nb ns 19 rules for cartpole problem, e.g. If θ is approximately zero and θ is negative medium then F is positive medium. R. Kruse, C. Doell FS Fuzzy Control Part 3 4 / 59

6 Definition of Table-based Control Function Measurement (x 1,..., x n ) X 1... X n is forwarded to decision logic. Consider rule if ξ 1 is A (1) and... and ξ n is A (n) then η is B. Decision logic computes degree to ξ 1,..., ξ n fulfills premise of rule. For 1 ν n, the value µ (ν) (x ν ) is calculated. { Combine values conjunctively by α = min µ (1),..., µ (n)}. For each rule R r with 1 r k, compute { α r = min µ (1) i 1,r (x 1 ),..., µ (n) } i n,r (x n ). R. Kruse, C. Doell FS Fuzzy Control Part 3 5 / 59

7 Definition of Table-based Control Function II Output of R r = fuzzy set of output values. Thus cutting off fuzzy set µ ir associated with conclusion of R r at α r. So for input (x 1,..., x n ), R r implies fuzzy set µ output(rr ) x 1,...,x n : Y [0, 1], { y min µ (1) i 1,r (x 1 ),..., µ (n) } i n,r (x n ), µ ir (y). If µ (1) i 1,r (x 1 ) =... = µ (n) i n,r (x n ) = 1, then µ output(rr ) x 1,...,x n = µ ir. If for all ν {1,..., n}, µ (ν) i 1,r (x ν ) = 0, then µ output(rr ) x 1,...,x n = 0. R. Kruse, C. Doell FS Fuzzy Control Part 3 6 / 59

8 Combination of Rules The decision logic combines the fuzzy sets from all rules. The maximum leads to the output fuzzy set µ output x 1,...,x n : Y [0, 1], y max 1 r k { { min µ (1) i 1,r (x 1 ),..., µ (n) }} i n,r (x n ), µ ir (y). Then µ output x 1,...,x n is passed to defuzzification interface. R. Kruse, C. Doell FS Fuzzy Control Part 3 7 / 59

9 Rule Evaluation θ positive small positive medium θ approx. zero approx. zero Rule evaluation for Mamdani-Assilian controller. Input tuple (25, 4) leads to fuzzy output. Crisp output is determined by defuzzification. θ θ min min positive small 1 F positive medium F max F R. Kruse, C. Doell FS Fuzzy Control Part 3 8 / 59

10 Defuzzification So far: mapping between each (n 1,..., n n ) and µ output x 1,...,x n. Output = description of output value as fuzzy set. Defuzzification interface derives crisp value from µ output x 1,...,x n. This step is called defuzzification. Most common methods: max criterion, mean of maxima, center of gravity. R. Kruse, C. Doell FS Fuzzy Control Part 3 9 / 59

11 The Max Criterion Method Choose an arbitrary y Y for which µ output x 1,...,x n reaches the maximum membership value. Advantages: Applicable for arbitrary fuzzy sets. Applicable for arbitrary domain Y (even for Y IR). Disadvantages: Rather class of defuzzification strategies than single method. Which value of maximum membership? Random values and thus non-deterministic controller. Leads to discontinuous control actions. R. Kruse, C. Doell FS Fuzzy Control Part 3 10 / 59

12 The Mean of Maxima (MOM) Method Preconditions: (i) Y is interval (ii) Y Max = {y Y y Y : µ output x 1,...,x n (y ) µ output x 1,...,x n (y)} is non-empty and measurable (iii) Y Max is set of all y Y such that µ output x 1,...,x n is maximal Crisp output value = mean value of Y Max. if Y Max is finite: η = 1 Y Max y i Y Max y i if Y Max is infinite: y Y η = Max y dy y Y Max dy MOM can lead to discontinuous control actions. R. Kruse, C. Doell FS Fuzzy Control Part 3 11 / 59

13 Center of Gravity (COG) Method Same preconditions as MOM method. η = center of gravity/area of µ output x 1,...,x n If Y is finite, then η = y i Y y i µ output x 1,...,x n (y i ). y i Y µoutput x 1,...,x n (y i ) If Y is infinite, then η = y Y y µoutput x 1,...,x n (y) dy y Y µoutput x 1,...,x n (y) dy. R. Kruse, C. Doell FS Fuzzy Control Part 3 12 / 59

14 Center of Gravity (COG) Method Advantages: Nearly always smooth behavior, If certain rule dominates once, not necessarily dominating again. Disadvantage: No semantic justification, Long computation, Counterintuitive results possible. Also called center of area (COA) method: take value that splits µ output x 1,...,x n into 2 equal parts. R. Kruse, C. Doell FS Fuzzy Control Part 3 13 / 59

15 Example Task: compute η COG and η MOM of fuzzy set shown below. Based on finite set Y = 0, 1,..., 10 and infinite set Y = [0, 10] µ output x 1,...,x n (y) η COG η MOM y R. Kruse, C. Doell FS Fuzzy Control Part 3 14 / 59

16 Example for COG Continuous and Discrete Output Space η COG = = η COG = 10 0 y µ output x 1,...,x n (y) dy 10 0 µ output x 1,...,x n (y) dy y dy (0.2y 0.6)y dy y dy ( ) ( ) = R. Kruse, C. Doell FS Fuzzy Control Part 3 15 / 59

17 Example for MOM Continuous and Discrete Output Space η MOM = = 10 7 y dy 10 7 dy = 8.5 = η MOM = = 34 4 = 8.5 R. Kruse, C. Doell FS Fuzzy Control Part 3 16 / 59

18 Problem Case for MOM and COG 1 µ output x 1,...,x n What would be the output of MOM or COG? Is this desirable or not? R. Kruse, C. Doell FS Fuzzy Control Part 3 17 / 59

19 Example: Engine Idle Speed Control VW 2000cc 116hp Motor (Golf GTI) R. Kruse, C. Doell FS Fuzzy Control Part 3 18 / 59

20 Structure of the Fuzzy Controller R. Kruse, C. Doell FS Fuzzy Control Part 3 19 / 59

21 Deviation of the Number of Revolutions drev nb nm ns zr ps pm pb R. Kruse, C. Doell FS Fuzzy Control Part 3 20 / 59

22 Gradient of the Number of Revolutions grev nb nm ns zr ps pm pb R. Kruse, C. Doell FS Fuzzy Control Part 3 21 / 59

23 Change of Current for Auxiliary Air Regulator daarcur nh nb nm ns zr ps pm pb ph R. Kruse, C. Doell FS Fuzzy Control Part 3 22 / 59

24 Rule Base If the deviation from the desired number of revolutions is negative small and the gradient is negative medium, then the change of the current for the auxiliary air regulation should be positive medium. grev nb nm ns az ps pm pb nb ph pb pb pm pm ps ps nm ph pb pm pm ps ps az ns pb pm ps ps az az az drev az ps ps az az az nm ns ps az az az ns ns nm nb pm az ns ns ns nb nb nh pb ns ns nm nb nb nb nh R. Kruse, C. Doell FS Fuzzy Control Part 3 23 / 59

25 Performance Characteristics R. Kruse, C. Doell FS Fuzzy Control Part 3 24 / 59

26 Example: Automatic Gear Box I VW gear box with 2 modes (eco, sport) in series line until Research issue since 1991: individual adaption of set points and no additional sensors. Idea: car watches driver and classifies him/her into calm, normal, sportive (assign sport factor [0, 1]), or nervous (calm down driver). Test car: different drivers, classification by expert (passenger). Simultaneous measurement of 14 attributes, e.g., speed, position of accelerator pedal, speed of accelerator pedal, kick down, steering wheel angle. R. Kruse, C. Doell FS Fuzzy Control Part 3 25 / 59

27 Example: Automatic Gear Box II Continuously Adapting Gear Shift Schedule in VW New Beetle R. Kruse, C. Doell FS Fuzzy Control Part 3 26 / 59

28 Example: Automatic Gear Box III Technical Details Optimized program on Digimat: 24 byte RAM 702 byte ROM uses 7 Mamdani fuzzy rules Runtime: 80 ms 12 times per second new sport factor is assigned. Research topics: When fuzzy control? How to find fuzzy rules? R. Kruse, C. Doell FS Fuzzy Control Part 3 27 / 59

29 Takagi Sugeno Control

30 Takagi-Sugeno Controller Proposed by Tomohiro Takagi and Michio Sugeno. Modification/extension of Mamdani controller. Both in common: fuzzy partitions of input domain X 1,..., X n. Difference to Mamdani controller: no fuzzy partition of output domain Y, controller rules R 1,..., R k are given by R r : if ξ 1 is A (1) i 1,r and... and ξ n is A (n) i n,r then η r = f r (ξ 1,..., ξ n ), f r : X 1... X n Y. Generally, f r is linear, i.e. f r (x 1,..., x n ) = a (r) 0 + n i=1 a (r) i x i. R. Kruse, C. Doell FS Fuzzy Control Part 3 28 / 59

31 Takagi-Sugeno Controller: Conclusion For given input (x 1,..., x n ) and for each R r, decision logic computes truth value α r of each premise, and then f r (x 1,..., x n ). Analogously to Mamdani controller: { α r = min µ (1) i 1,r (x 1 ),..., µ (n) } i n,r (x n ). Output equals crisp control value η = kr=1 α r f r (x 1,..., x n ) kr=1 α r. Thus no defuzzification method necessary. R. Kruse, C. Doell FS Fuzzy Control Part 3 29 / 59

32 Example R 1 : if ξ 1 is then η 1 = 1 ξ ξ R 2 : if ξ 1 is and ξ 2 is then η 2 = 0.1 ξ ξ R 3 : if ξ 1 is and ξ 2 is then η 3 = 0.9 ξ ξ R 4 : if ξ 1 is and ξ 2 is then η 4 = 0.2 ξ ξ If a certain clause x j is A (j) i j,r in rule R r is missing, then µ ij,r (x j ) 1 for all linguistic values i j,r. For instance, here x 2 in R 1, so µ i2,1 (x 2 ) 1 for all i 2,1. R. Kruse, C. Doell FS Fuzzy Control Part 3 30 / 59

33 Example: Output Computation input: (ξ 1, ξ 2 ) = (6, 7) α 1 = 1 /2 1 = 1 /2 η 1 = /2 + 1 = 10.5 α 2 = 1 /2 2 /3 = 1 /2 η 2 = = 28.6 α 3 = 1 /2 1 /3 = 1 /3 η 3 = = 19.3 α 4 = 0 1 /3 = 0 η 4 = /2 + 1 = 10.5 output: η = f (6, 7) = 1 / / / /2 + 1 /2 + 1 /3 = 19.5 R. Kruse, C. Doell FS Fuzzy Control Part 3 31 / 59

34 Example: Passing a Bend Pass a bend with a car at constant speed. Measured inputs: ξ 1 : distance of car to beginning of bend ξ 2 : distance of car to inner barrier ξ 3 : direction (angle) of car ξ 4 : distance of car to outer barrier ξ 4 ξ 3 ξ 2 ξ 1 η = rotation speed of steering wheel X 1 = [0 cm, 150 cm], X 2 = [0 cm, 150 cm] X 3 = [ 90, 90 ], X 4 = [0 cm, 150 cm] R. Kruse, C. Doell FS Fuzzy Control Part 3 32 / 59

35 Fuzzy Partitions of X 1 and X 2 1 small medium big 0 1 small big ξ ξ 2 R. Kruse, C. Doell FS Fuzzy Control Part 3 33 / 59

36 Fuzzy Partitions of X 3 and X 4 outwards 1 forward inwards ξ 3 1 small ξ 4 R. Kruse, C. Doell FS Fuzzy Control Part 3 34 / 59

37 Form of Rules of Car R r : if ξ 1 is A and ξ 2 is B and ξ 3 is C and ξ 4 is D then η = p (A,B,C,D) 0 + p (A,B,C,D) 1 ξ 1 + p (A,B,C,D) 2 ξ 2 + p (A,B,C,D) 3 ξ 3 + p (A,B,C,D) 4 ξ 4 A {small, medium, big} B {small, big} C {outwards, forward, inwards} D {small} p (A,B,C,D) 0,..., p (A,B,C,D) 4 IR R. Kruse, C. Doell FS Fuzzy Control Part 3 35 / 59

38 Control Rules for the Car rule ξ 1 ξ 2 ξ 3 ξ 4 p 0 p 1 p 2 p 3 p 4 R outwards small R forward small R 3 small small outwards R 4 small small forward R 5 small small inwards R 6 small big outwards R 7 small big forward R 8 small big inwards R 9 medium small outwards R 10 medium small forward R 11 medium small inwards R 12 medium big outwards R 13 medium big forward R 14 medium big inwards R 15 big small outwards R 16 big small forward R 17 big small inwards R 18 big big outwards R 19 big big forward R 20 big big inwards R. Kruse, C. Doell FS Fuzzy Control Part 3 36 / 59

39 Sample Calculation Assume that the car is 10 cm away from beginning of bend (ξ 1 = 10). The distance of the car to the inner barrier be 30 cm (ξ 2 = 30). The distance of the car to the outer barrier be 50 cm (ξ 4 = 50). The direction of the car be forward (ξ 3 = 0). Then according to all rules R 1,..., R 20, only premises of R 4 and R 7 have a value 0. R. Kruse, C. Doell FS Fuzzy Control Part 3 37 / 59

40 Membership Degrees to Control Car small medium big ξ 1 = small big ξ 2 = outwards forward inwards ξ 3 = small ξ 4 = 50 0 R. Kruse, C. Doell FS Fuzzy Control Part 3 38 / 59

41 Sample Calculation (cont.) For the premise of R 4 and R 7, α 4 = 1 /4 and α 7 = 1 /6, resp. The rules weights α 4 = 1/4 1/4+1/6 = 3 /5 for R 4 and α 5 = 2 /5 for R 7. R 4 yields R 7 yields η 4 = = η 7 = = The final value for control variable is thus η = 3 / / = R. Kruse, C. Doell FS Fuzzy Control Part 3 39 / 59

42 Fuzzy Control as Similarity-Based reasoning

43 Interpolation in the Presence of Fuzziness Both Takagi-Sugeno and Mamdani are based on heuristics. They are used without a concrete interpretation. Fuzzy control is interpreted as a method to specify a non-linear transition function by knowledge-based interpolation. A fuzzy controller can be interpreted as fuzzy interpolation. Now recall the concept of fuzzy equivalence relations (also called similarity relations). R. Kruse, C. Doell FS Fuzzy Control Part 3 40 / 59

44 Similarity: An Example Specification of a partial control mapping ( good control actions ): gradient deviation R. Kruse, C. Doell FS Fuzzy Control Part 3 41 / 59

45 Interpolation of Control Table There might be additional knowledge available: Some values are indistinguishable, similar or approximately equal. Or they should be treated in a similar way. Two problems: a) How to model information about similarity? b) How to interpolate in case of an existing similarity information? R. Kruse, C. Doell FS Fuzzy Control Part 3 42 / 59

46 How to Model Similarity? Proposal 1: Equivalence Relation Definition Let A be a set and be a binary relation on A. is called an equivalence relation if and only if a, b, c A, (i) a a (reflexivity) (ii) a b b a (symmetry) (iii) a b b c a c (transitivity). Let us try a b a b < ε where ε is fixed. is not transitive, is no equivalence relation. Recall the Poincaré paradox: a b, b c, a c. This is counterintuitive. R. Kruse, C. Doell FS Fuzzy Control Part 3 43 / 59

47 How to Model Similarity? Proposal 2: Fuzzy Equivalence Relation Definition A function E : X 2 [0, 1] is called a fuzzy equivalence relation with respect to the t-norm if it satisfies the following conditions (i) E(x, x) = 1 (reflexivity) x, y, z X (ii) E(x, y) = E(y, x) (symmetry) (iii) (E(x, y), E(y, z)) E(x, z) (t-transitivity). E(x, y) is the degree to which x y holds. E is also called similarity relation, t-equivalence relation, indistinguishability operator, or tolerance relation. Note that property (iii) corresponds to the vague statement if (x y) (y z) then x z. R. Kruse, C. Doell FS Fuzzy Control Part 3 44 / 59

48 Fuzzy Equivalence Relations: An Example Let δ be a pseudo metric on X. Furthermore (a, b) = max{a + b 1, 0} Łukasiewicz t-norm. Then E δ (x, y) = 1 min{δ(x, y), 1} is a fuzzy equivalence relation. δ(x, y) = 1 E δ (x, y) is the induced pseudo metric. Here, fuzzy equivalence and distance are dual notions in this case. Definition A function E : X 2 [0, 1] is called a fuzzy equivalence relation if x, y, z X (i) E(x, x) = 1 (reflexivity) (ii) E(x, y) = E(y, x) (symmetry) (iii) max{e(x, y) + E(y, z) 1, 0} E(x, z) (Łukasiewicz transitivity). R. Kruse, C. Doell FS Fuzzy Control Part 3 45 / 59

49 Fuzzy Sets as Derived Concept δ(x, y) = x y metric E δ (x, y) = 1 min{ x y, 1} fuzzy equivalence relation 1 µ x0 0 x 0 1 x x 0 x µ x0 : X [0, 1] x E δ (x, x 0 ) fuzzy singleton µ x0 describes local similarities. R. Kruse, C. Doell FS Fuzzy Control Part 3 46 / 59

50 Extensional Hull E : IR IR [0, 1], (x, y) 1 min{ x y, 1} is fuzzy equivalence relation w.r.t. Łuka. Definition Let E be a fuzzy equivalence relation on X w.r.t.. µ F(X) is extensional if and only if x, y X : (µ(x), E(x, y)) µ(y). Definition Let E be a fuzzy equivalence relation on a set X. Then the extensional hull of a set M X is the fuzzy set µ M : X [0, 1], x sup{e(x, y) y M}. The extensional hull of {x 0 } is called a singleton. R. Kruse, C. Doell FS Fuzzy Control Part 3 47 / 59

51 Specification of Fuzzy Equivalence Relation Given a family of fuzzy sets that describes local similarities. 1 µ 1 µ 2 µ 3 µ 4 µ 5 µ 6 µ 7 µ 8 µ X There exists a fuzzy equivalence relation on X with induced singletons µ i if and only if i, j : sup{µ i (x) + µ j (x) 1} inf {1 µ i(y) µ j (y) }. x X y X If µ i (x) + µ j (x) 1 for i j, then there is a fuzzy equivalence relation E on X E(x, y) = inf i I {1 µ i(x) µ i (y) }. R. Kruse, C. Doell FS Fuzzy Control Part 3 48 / 59

52 Necessity of Scaling I Are there other fuzzy equivalence relations on IR than E(x, y) = 1 min{ x y, 1}? Integration of scaling. A fuzzy equivalence relation depends on the measurement unit, e.g. Celsius: E(20 C, 20.5 C) = 0.5, Fahrenheit: E(68 F, 68.9 F) = 0.9, scaling factor for Celsius/Fahrenheit = 1.8 (F = 9/5C + 32). E(x, y) = 1 min{ c x c y, 1} is a fuzzy equivalence relation! R. Kruse, C. Doell FS Fuzzy Control Part 3 49 / 59

53 Necessity of Scaling II How to generalize scaling concept? X = [a, b]. Scaling c : X [0, ). Transformation f : X [0, ), x x a c(t)dt. Fuzzy equivalence relation E : X X [0, 1], (x, y) 1 min{ f (x) f (y), 1}. R. Kruse, C. Doell FS Fuzzy Control Part 3 50 / 59

54 Fuzzy Equivalence Relations: Fuzzy Control The imprecision of measurements is modeled by a fuzzy equivalence relations E 1,..., E n and F on X 1,..., X n and Y, resp. The information provided by control expert are k input-output tuples (x (r) 1,..., x n (r), y (r) ) and the description of the fuzzy equivalence relations for input and output spaces, resp. The goal is to derive a control function ϕ : X 1... X n Y from this information. R. Kruse, C. Doell FS Fuzzy Control Part 3 51 / 59

55 Determine Fuzzy-valued Control Functions I The extensional hull of graph of ϕ must be determined. Then the equivalence relation on X 1... X n Y is E((x 1,..., x n, y), (x 1,..., x n, y )) = min{e 1 (x 1, x 1),..., E n (x n, x n), F(y, y )}. R. Kruse, C. Doell FS Fuzzy Control Part 3 52 / 59

56 Determine Fuzzy-valued Control Functions II For X i and Y, define the sets X (0) { i = x X i r {1,..., k} : x = x (r) } i and Y (0) = { y Y r {1,..., k} : y = y (r)}. X (0) i and Y (0) contain all values of the r input-output tuples (x (r) 1,..., x (r) n, y (r) ). For each x 0 X (0) i, singleton µ x0 is obtained by µ x0 (x) = E i (x, x 0 ). R. Kruse, C. Doell FS Fuzzy Control Part 3 53 / 59

57 Determine Fuzzy-valued Control Functions III If ϕ is only partly given, then use E 1,..., E n, F to fill the gaps of ϕ 0. The extensional hull of ϕ 0 is a fuzzy set µ ϕ0 (x 1,..., x n, y ) { = max min{e 1 (x (r) } 1, x 1),..., E n (x n (r), x n), F(y (r), y )}. r {1,...,k} µ ϕ0 is the smallest fuzzy set containing the graph of ϕ 0. Obviously, µ ϕ0 µ ϕ µ (x 1,...,x n) ϕ 0 : Y [0, 1], y µ ϕ0 (x 1,..., x n, y). R. Kruse, C. Doell FS Fuzzy Control Part 3 54 / 59

58 Reinterpretation of Mamdani Controller For input (x 1,..., x n ), the projection of the extensional hull of graph of ϕ 0 leads to a fuzzy set as output. This is identical to the Mamdani controller output. It identifies the input-output tuples of ϕ 0 by linguistic rules: R r : if X 1 is approximately x (r) 1 and... and X n is approximately x (r) n then Y is y (r). A fuzzy controller based on equivalence relations behaves like a Mamdani controller. R. Kruse, C. Doell FS Fuzzy Control Part 3 55 / 59

59 Reinterpretation of Mamdani Controller 3 fuzzy rules (specified by 3 input-output tuples). The extensional hull is the maximum of all fuzzy rules. R. Kruse, C. Doell FS Fuzzy Control Part 3 56 / 59

60 References I R. Kruse, C. Doell FS Fuzzy Control Part 3 57 / 59

61 References II Jantzen, J. (2013). Foundations of fuzzy control: a practical approach. John Wiley & Sons. Klawonn, F., Gebhardt, J., and Kruse, R. (1995). Fuzzy control on the basis of equality relations with an example from idle speed control. IEEE Transactions on Fuzzy Systems, 3(3): Kruse, R., Borgelt, C., Braune, C., Mostaghim, S., and Steinbrecher, M. (2016). Computational intelligence: a methodological introduction. Springer. Mamdani, E. H. and Assilian, S. (1975). An experiment in linguistic synthesis with a fuzzy logic controller. International journal of man-machine studies, 7(1):1 13. Michels, K., Klawonn, F., Kruse, R., and Nürnberger, A. (2006). Fuzzy Control: Fundamentals, Stability and Design of Fuzzy Controllers, volume 200 of Studies in Fuzziness and Soft Computing. Springer, Berlin / Heidelberg, Germany. R. Kruse, C. Doell FS Fuzzy Control Part 3 58 / 59

62 References III Sugeno, M. (1985). An introductory survey of fuzzy control. Information sciences, 36(1-2): R. Kruse, C. Doell FS Fuzzy Control Part 3 59 / 59